Fourier Clouds: Fast Bias Correction for Imbalanced Semi-Supervised Learning
Authors: Jiawei Gu
Yidi Wang
Qingqiang Sun
Xinming Li
Xiao Luo
Ziyue Qiao
Venue: NeurIPS 2025 poster ()
Keywords: Bias Correction
Semi-Supervised Learning
Random-Phase Images
Long-Tailed Distribution
Primary area: general_machine_learning

Abstract: Pseudo-label-based Semi-Supervised Learning (SSL) often suffers from classifier bias, particularly under class imbalance, as inaccurate pseudo-labels tend to exacerbate existing biases towards majority classes. Existing methods, such as \textit{CDMAD}\cite{cdmad}, utilize simplistic reference inputs—typically uniform or blank-colored images—to estimate and correct this bias. However, such simplistic references fundamentally ignore realistic statistical information inherent to real datasets, specifically typical color distributions, texture details, and frequency characteristics. This lack of \emph{statistical representativeness} can lead the model to inaccurately estimate its inherent bias, limiting the effectiveness of bias correction, particularly under severe class imbalance or substantial distribution mismatches between labeled and unlabeled datasets. To overcome these limitations, we introduce the \textbf{FARAD} (Fourier-Adapted Reference for Accurate Debiasing) System. This system utilizes random-phase images, constructed by preserving the amplitude spectrum of real data while randomizing the phase spectrum. This strategy ensures two critical properties: (1) \textbf{Semantic Irrelevance}, as randomizing phase removes any structural or recognizable semantic cues, and (2) \textbf{Statistical Representativeness}, as preserving the amplitude spectrum maintains realistic textures, color distributions, and frequency characteristics. Grounded theoretically in classical Fourier analysis, the FARAD System provides a robust, accurate estimation of per-class biases. Furthermore, computational efficiency is enhanced through optimized real-to-complex (R2C) batched Fast Fourier Transforms (FFTs). Comprehensive experiments demonstrate that our approach, significantly improves minority-class accuracy and overall SSL performance, particularly under challenging imbalance scenarios, compared with existing reference-based bias correction methods.

Fourier Clouds: Fast Bias Correction for Imbalanced Semi-Supervised Learning
Jiawei Gu', Yidi Wang', Qingqiang Sun?, Xinming Li", Xiao Luo', Ziyue Qiao
School of Computing and Information Technology, Great Bay University, Schooo of Advanced Engineering. Great Bay University, BBejjing University of Civil Engineering and Architecture, *University of Califomia, Los Angeles sjwcseoution com, (ydwang, qqsun @gbu. edu. 1xm1 8118411961111111 com, xiaolucêcs ucla. edu,
com
Abstract
Pseudo-labellbasee Semi-Supervised Learning (SSL) often suffers from classi- fier bias, particularly under class imbalance, as inaccurate pseudo-labels tend to exacerbate existing biases towards majority classes. Existing methods, such CD- MAD[30], utilize simplistic reference inputs-typicalll uniform or blank-colored images to estimate and correct this bias. However, such simplistic references fundamentally ignore realistic statistical information inherent to real datasets, specifically typical color distributions, texture details, and frequency characteris- tics. This lack of statistical representativeenss can lead the model to inaccurately estimate its inherent bias, limiting the effectiveness of bias correction, particularly under severe class imbalance or substantial distribution mismatches between la- beled and unlabeled datasets. To overcome these limitations, we introduce the FARAD (Fourier-Adaie Adapted Reference for Accurate Debiasing) System. This system utilizes random- -haose images, constructed by preserving the amplitude spectrum of real data while randomizing the phase spectrum. This strategy ensures two critical properties: (1) Semantic Irrelevance, as randomizing phase removes any structural or recognizable semantic cues, and (2) Statistical Representativeness, as preserving the amplitude spectrum maintains realistic textures, color distribu- tions, and frequency characteristics. Grounded theoretically in classical Fourier analysis, the FARAD System provides a robust, accurate estimation of per-class biases. Furthermore, computational efficiency is enhanced through optimized coomplxxc (R2C) batched Fast Fourier Transforms (FFTs). Comprehensive experiments demonstrate that our approach, significantly improves minority-clar accuracy and overall SSL performance, particularly under challenging imbalance scenarios, compared with existing reference-based bias correction methods.
Introduction
Long-tailed semi-supervised learning (SSL) has emerged core challenge in deep learning, as must simultaneously contend with two difficulties: severe class imbalance and limited labeled data. Real-world datasets often contain handful of majority classes with abundant labels, alongside a "long
tail" of classes that have very few (or even zero) labeled examples [39 51 52]
57
35 13] In such
scenarios, standard learning algorithms tend to overfit to the majority classes, leaving minority classes underrepresented and poorly modeled[40] [5]] Meanwhile, semi-supervised approaches typically
rely on plentiful unlabeled data compensate for sparse labeled samples 47 23 14). However, this
"oorrespodiing author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

strategy can inadvertently exacerbate imbalance if the unlabeled data distribution skews similarly or even diverges from the labeled set, further amplifying the classifier's inherent bias toward majority across diverse domains from medicine to autonomous systems.
classes 1 59 33
A promising avenue address this issue lies in estimating and subtracting the classifier's intrinsic bias bia. Concretely, suppose we feed the classifier an image devoid of meaningful semantic content; any strong class preferences in the oulput must reflect learned biases rather than genuine discriminative features. By capturing the classifier's 's response (logits) to such a carefully crafted "reference image" and subtracting these reference logits from the classifier's ' predictions on real inputs, we can mitigate the bias that naturally arises from class imbalance. This correctonn step can be applied both during training and inference: at training, helps to produce more balanced pseudo-labes and gradient signals; at inference, it continues to offset the learned bias so that minority- instances are not systematically overlooked [58 55
Core Insight. The key intuition behind reference-based bias subtraction is straight- forward: if a classifier assigns strong class preferences to an image devoid of meaningful semantic content, these preferences must reflect learned biases rather than genuine semantic features. Eliminating semantic cues is thus essential; however, ensuring that the reference images still respect the realistic statistics of the target data distribution equally crucial. Overly simplistic inputs (e.g., uniform-gre -gaa images) can misrepresent important color or texture patterns, potentially leading to suboptimal bias estimation and compromised learning performance.
Compurtson of Reference Methods: Shape vs Staristical Fidelity
190
Daepr fier 10.%%
Coodin chei aunn Plan Phan fAer 10.00
anutetaal it
Our Mutbo
Lan Concri
Shape Prnuecce (Lower Better'
Motivation and Limitations of Existing Refer- ences. Previous methods, such as CDMAD employ minimalistic references (e.g.. images with uniform or blank color) that deliberately remove recognizable shapes guarantee seman- tic irrelevance. While effective at eradicating structured semantic cues, these references fail to capture critical statistical characteristics of real datasets-such as realistic color distributions, texture patterns, and spatial frequency energy. Consequently, these methods often yield suboptimal results, particularly for minority categories in distribution scenarios 31 38
Figure 1: Experimental comparison on CIFAR- 10-LT[8] references, mapping shape presence vs. statistical fidelity. Our random-phase im- ages achieve high fidelity (color/frequency align- ment) with minimal shape cues, while purely blank references (CDMAD(30) may fail to cap- ture realistic data distributions.
highly imbalanced or mismatched
Two Key Properties for Accurate Bias Estimation. To overcome these shortcomings, we pro- pose replacing simplistic references with random-phase images that rigorously fulfill two critical properties:
Semantic Irrelevance
Statistical Representatiieness
Drawing from insights in cognitive and perceptual psychology [I]. we note that classifiers often rely heavily on shape and edge information. Our reference images must therefore be free from any recognizable shapes or edges. Achieving complete semantic irrele- vance ensures that any classifier response to these images cannot stem from genuine semantic clucs, thereby purely reflecting intrinsic biases.
Aligning with foundational concepts in statistical Icarning theory [), bias reference must adequately represent global statistical properties of real data. Un- like blank images, statistically representative refer- ence preserves realistic color distributions, textures, and frequency spectra, thereby letting the classifier's 's response reflect genuine biases learned from real data rather than artifacts introduced by trivial references.
Our Random-Phase Approach. Concretely, we construct reference images by preserving the amplitude spectrum of actual dataset samples, thereby maintaining realistic global statistics such as color distribution and texture patterns. Simultaneously, by randomizing the phase spectrum, eliminate shape- based cues and object structures. This design results in references that exhibit no coherent edges or meaningful forms, yet still align with the target dataset's overall energy distribution. In Figure [] we illustrate a small-scale experiment on CIFAR- 10-LT, mapping each reference type
2

---

onto plane defined by shape presence" (horizontal axis) vs. statistical fidelity" (vertical axis). The specific metrics used to quantify 'shape presence' and 'statistical fidelity' are detailed in Appendix C While uniform-color references appear near minimal shape cues but also have low fidelity, and real images excel at fidelity but inevitably contain structural information, our random- phase images better balance both dimensions across all dataset complexity levels. CDMAD's blank references lie
somewhere in between, indicating that a purely blank or partially structured input may not effectively address the tension between semantic irrelevance and statistical representativennss.
Contributions and Paper Outline. We validate the effectiveness of our method on multiple Long-tailed benchmarks, showing that subtracting the logits on random-phes -paaos images consistently improves minority-chee class recognition more substantially than subtracting uniform-color references. This improvement remains robust under various SSL pipelines (e.g., Fix.Match. ReMixMatch). confirming the method's s broad applicability. We also demonstrate that the added computational cost can be mitigated via real-to-complex (R2C) FFT implementations, allowing our method to scale efficiently larger datasets. Our contributions can be summarized as follows:
We present framework for reference- bias subtraction in long-tailed SSL,, constructing random-phase images that jointly satisfy Semantic Irrelevance and Statistical Representativeness.
By preserving amplitude spectra and randomizing phase spectra, we retain alignment with real data's 's color and frequency energy while thoroughly removing shape cues.
Through extensive experiments, we show that subtracting the classifier's logits on random- references significantly boosts minority-class recognition, especially under severe imbalances.
We demonstrate that the approach's 's computational overhead is manageable using batched R2C FFT, ensuring scalability to larger datasets and higher-resolutie images.
2 Method
In this section, we present our framework for estimating and correcting class bias via random-phase images that preserve the amplitude spectrum of real data. The goal is to address severe class imbalance in both labeled and unlabeled sets, especially when their distributions are mismatched. We begin by giving a high-level motivation and overview of our method in 1 We then detail two key properties: how randomizing the phase spectrum ensures semantic irrelevance ($2.2) and how preserving the amplitude spectrum ensures statistical representativeness ($.3) We next explain how these images serve as a bias estimator for the classifier ($2.4) and how the same principle applies at inference
($2.5) In 2.6. discuss computational considerations for large-scale FFT.
2.1 Overall Framework
Mlass-ibbllanced data, often referred to "long-tailed" data, naturally biases the classifier toward majority classes. In the semi-supervised setting, this bias can be even more pronounced because the unlabeled set may have an unknown distribution that differs substantially from the labeled set. Recent work such as CDMAD proposes subtracting classifier outputs classifier's inherent bias. While conceptually simple, solid-color images (often a uniform gray or RGB constant) have no structure and also do not capture realistic statistics (e.g., texture or color variability) that exist in real images across diverse natural and synthetic domains.
solid-color image to remove the
We propose replacing such simple reference images with random-phass images that preserve the amplitude spectrum of real data. These images: (1) exhibit no recognizable structure (semantic irrelevance), because the phase spectrum is randomized, which destroys spatial correlations (further theoretical validation in Appendix G): (2) preserve statistical properties such as overall energy or color distribution, because the amplitude spectrum is preserved from real samples or a batch-average amplitude (with theoretical support Appendix [H). Hence, the classifier's is output on such a random- phase image better reflects the "intrinsic bias" it has learned (analyzed theoretically in Appendix I).. Formally subtracting that output from the logits of a real input thus removes a large fraction of the bias, improving minority- class recognition significantly across diverse datasets.
3

---

2.2 Semantic Irrelevance via Phase Randomization
Let x E HR be an input image with height H. width W, and 3 color channels. Let F(x) denote its 2D Discrete Fourier Transform (2D-DFT). For each channel, the 2D-DFT at frequency coordinates (u, v) can be expressed in polar form as:
F(x)(u,)) A(u,v) exp()
where A(u, v) (u, is the amplitude spectra, $(u, v) (u, t) is the phase spectra, and j == V is the imaginary unit. (Fundamental concepts of the 2D-DFT are reviewed in Appendix E It well known in image processing that the phase spectrum $(u, v) encodes much of the semantic or structural information (e.g., contours, shapes), whereas the amplitude A(u, v) primarily reflects overall energy or color intensity distributions.
To eliminate semantic cues from æ, we randomize its phase spectrum, replacing Φ(u, v) with a(u, v) drawn from [-x, π. or from a shuffled pool of existing phases. The corresponding random- -paase image is:
Frand =F ((u,,, v) exp(j D(u, v(), }
A(u,
(2)
Because the original spatial structure is corrupted by Ó(u, v), Trand contains no coherent edges or objects. In practice, such mandom-phasee images often appear as turbulent or cloud-like patterns with seemingly random visual characteristics, Consequently, any strong class-specific response from the classifier when presented with zaand is likely attributable to learned internal bias, rather than genuine evidence for that class. Appendix G provides a formal theoretical justification for why randomizing the phase ensures this semantic irrelevance.
2.3 Statistical Representativenes: via Amplitude Preservation
Whereas randomizing the phase spectrum removes meaningful shapes, preserving the amplitude spectrum retains the color intensity and frequency energy similar to real data. In contrast, purely uniform image keeps only the zero-frequency component (DC value), discarding essentially all figher-frequency information. Therefore, such uniform or overly simplistic references might fail to capture how the classifier responds to the normal variability real images.
To obtain a more stable and representative amplitude spectrum, we average amplitudes over mini- batch of real images. Suppose we have a batch (I:(1), _,(2),_... _(B)), }. Let (1, v) and (b() (u, v) be their amplitude and phase spectra, respectively. We compute the batch-averaged amplitude
A(u, v)
Σ 6=1
A (,
B
We then pair A(u, v) with new random phase 6(u, v) to construct Frand This ensures that, while no true object boundaries remain, the overall energy and color distributions match the typical real data in the batch. Empirically, we find that using A reduces variance in the random-phase images and yields more robust bias estimates than simply using amplitude from a single sample. The theoretical basis for why preserving the amplitude spectrum, especially when averaged, reflects real data statistics across diverse natural and synthetic contexts further discussed in Appendix H
2.4 Class Bias Estimation and Correction
Let fe(:) X3 + {1,...,C]} C'} be the classifier, and ge E IRC be its logits. For a random- phase image Trand, we compute
Biias = go go(naa)d)
which reflects the model's 's intrinsic preference over C' classes, in the absence of meaningful semantic inputs (a rigorous explanation of how ge (Irand) reveals this default bias is provided in Appendix I). We then define the bias-corrected logits for any real input æ::
9e (a)) = go (x) Zbiass
By subtracting Ziias: we discount the portion of the logits that arises from the model's prior bias, rather than genuine class evidence. In the context of training on labeled sample (Ix, yn) or

---

unlabeled sample um, we replace (e( with 96 (a) in the loss function (e.g., supervised cross- entsopp or consistency regularization loss). This yields more balanced pseudo-labels for the unlabeled data and reduces overconfidence majority classes while improving minority class recognition.
Notably, this approach is related in spirit logit adjustment methods reference-image subtraction (e.g., CDMAD), but by preserving the real amplitude spectrum, we aim to make the bias estimator more reflective of how the classifier responds to typical color/texture distributions in the dataset, leading to more accurate bias correction across diverse visual domains and learning paradigms.
2.5 Inference Stage Bias Correction
Class imbalance often persists even after training. To further mitigate bias test time, continue apply bias correction. For each incoming test image stest, we create a random-phase image I rand using the same amplitude-pecctrum averaging and phase randomization described above. Then,
test
Z
trast biss
ga (esaed)
(6)
and
g6 (ear))
go (xx (z=est)
Ziias
(7)
We finally predict
fø (stest) = arg max gø (z.tes)
(8)
Although this requires generating a random-phase image (and one extra forward pass) for each test sample mini-batch, it adaptively reduces bias inference time, providing additional gains minority classes, especially if the training distribution was highly skewed or the test distribution differs from the training distribution across diverse real-world testing scenarios.
2.6 FFT Computation and Acceleration
potential concern is the computational overhead of 2D Fourier transforms on large batches. We address this by using batched real-to-complex (R2C) FFT, which exploits the conjugate symmetry of real-valued inputs (as detailed in Appendix specifically E.4). Specifically, for a real-valued image x, its Fourier transform satisfies
F(x)(u,v) F(x)(-u,-o)
where the overline denotes complex conjugation. This property allows us to compute only half the spectrum, reducing both runtime and memory usage. Modern libraries such cuFFT can process entire mini-batches of images in parallel, further accelerating training.
Crucially, we do not discard high-frequency components of during randomization. While randomizing the phase is sufficient to remove recognizable structure, is essential to preserve the full amplitude spectrum to retain realistic color, texture, and energy statistics. Appendix (ppoodidiiidoiiesss theoretical analysis showing that these FFT accelerations and practical computational considerations do not violate the assumptions underpinning our method.
3 Experimental Results
In this section, we aim to address the following four key questions about FARAD in the context of long-tailed semi-supervised learning:
Q1: Does FARAD consistently outperform baselines on standard benchmarks?
Q2: Which FARAD components are critical, and how they compare to other correction strategies?
Q3: Why do random- phase images achieve semantic irrelevance & statistical representativeness??
In addition to these core investigations, extensive further experiments are detailed in the Appendix. These confirm FARAD's robustness across wider array of distribution mismatches (including ex-
treme imbalances and domain shifts, as shown in Appendices D.2 D.6 and (.8) and demonstrate its
versatility with additional SSL. algorithms and network architectures (Appendices) D33 and D.7). Fur- thermore, the Appendix presents in-depth studies optimal reference image design (Appendix D.4
5

---

Table CIFAR-10-LT results with " = Ya. We report bACC/ GM (%) = std. on three imbalance
factors. "Ours" denotes our proposed method. Best entries in bold.
CIFAR-10-LT (bACC / GM)
Algorithm
Y
100
150
Vanilla
65.2= 01.1
08.8:0.13/58.2-000.
(0.-:00.3//440-000000
Re-sampling
64.3=0.48 6.66
55.8::0.47 0..0
52.2++0.05 13..40..5/8..2211.44 38,2
LDAM-DRW
0.9=00..766.010000
62.8:0.17 58.9
57.9=0.20 50.4=0.30
cRT
67.8:0.13/66.311.111
63.2:0.45/ 59.900.40
39.310.10055.6-0772 54.6=00722
59.310.10/
FixMatch
79.2-0.33
11.5+0072266.8-...5
68.440.15/59999045
4+DARP+oRT
85.8=0.43/ 65.6-000/56
82.4+0.26/ 81.---.117
79,6+0.42/78.--00
+CRRSS+LAA
85.6=0.36/81.999 01.
1.2-10.70/74.5-000
1.942.24/64A41...
/ABC
8.26 85.2.10.29
81.1+114// 1.29
77331125/75.6-1
4CCSSL
0.30 8..6
83.2.10.49/ 82.7.0.60
80.310.55 0.57
ASAW+LA
86.2=0.15 15 03.9
80.7 7..5
73.7+0006 / 01.2
A+Adsh
/3.4=0.06/-
76.5=0.35
71.5+0.30/-
4+DebisPL
80.6=0.50/-
4UDAL
86.5
81.4
77.940.33
4L2AC
-/-
82.1
77.6:10.53 7,7
4CDMAD
88.1 00.00/8///7....... 07.9
/5.3:0.48/85.1-000
2..--022/8888-222
/FARAD
0L5=0.38/91133 0.43
0566:0.478885-100
85.4:0.35/85.111 85.1
ReMixMatch
115==0.26/8002220.2.. 0..2
73.8:0.33/66.5510
69.9=0.47 /
/DARR+<RTT
87.3=0.61 / 87.
83.5++0.07 / /3.1
79.7+00.4/ 04.0555/788.90.44
ACREST+LA
84.2+0.11/-
81.3++034/
79.2+0.31 /-
4ABC
87.9=0.47 /7.6600.7
84.5++0.2/ 04.16
80.5+1.18 19.5
/CCoSL
7..200.1187..61100
84.110.56/ 03.7
81.310.83 0.76
4SAW+CRT
87.620.21/87.41..26
85.4.10.22/ 14490.22/3..--0.211 83.9
79.910.15/7999-000
4CDMAD
88.1-0.34/88.01..33.
85.5110.46 85.310.44
82.5 82.0
AFARAD
0.2=00.39991.0210
88.4=0.44/ 88.2=0.46
85.3+0.24/ 84.8=0.27
and effective bias correction application strategies (Appendix D.5
alongside evaluations of its inter- action with data augmentation (Appendix D.9) and scalability to higher resolutions (Appendix D.10) with significant implications for practical deployment scenarios.
3.1 Experimental Settings and Evaluation Metrics
We conduct our experiments four widely-used long-tailed benchmarks: CIFAR-IO-LT CIFAR- 100-LT(8) STL-10-LT (5)) and Small-ImageNet, 127 10). Following previous work, each dataset is artificially made long-tailed by assigning different numbers of labeled samples per class according to an imbalance factor n for labeled data and Th. for unlabeled data. In some cases, ? * Tu. which simulates realistic scenarios where the labeled and unlabeled data distributions are not perfectly aligned, reflecting common challenges in practical machine learning deployments.
All methods are trained under the same hyperparameter settings and data augmentations (weak and strong) as in CDMAD 30 for fair comparisons. For each dataset, we keep the same random seeds and batch sizes to minimize variance. Our implementation adopts standard Adam optimizer, with initial learning rate and weight decay selected via small validation set or the protocol suggested in related SSL studies. The detailed experimental details can be found in Appendix B
Evaluation Metrics. We primarily report the balanced accuracy (bACC) [9]. defined as the average of per-class accuracies, to highlight performance minority classes. For CIFAR- and STL-10 10- LT, we also present the geometric mean (G6 across classes. In the case of hagher-cardinality datasets such CIFAR-100-LT and Small- Amall-ImageNNet-122 reflects performance under significant label imbalance. We additionally provide standard accuracy and class-wise confusion matrices in the appendix for further insights.
we concentrate bACC since it better
Under these settings, we compare our approach with range of baselines, including vanilla al- gorithm (Deep CNN trained with cross- entropy loss), classic long-tailed methods (e.g., RE- SAMPLING 22], L.AM-DRW(E)) (5), cRT(24) I.A (36)), vanilla semi-supervised methods (e.g.. FixMatch 41) ReMixMatch 3) and CISSL approaches such as DARP(25), CReST[50), ABC

---

Table 3: Small-ImaeeNet-22) results with = Ya- We report balanced accuracy (bACC, %). Best
in bold.
Small-ImageNet-1277 (bACC)
Algorithm
32x
64x64
FixMatch
29.7
42.3
DAMR
30.5
42.5
FiXMatch+DARP+cRT
39.7
51.0
FixMatch+CReST
32.5
44.7
FiXMatch+CR<STTLL.
40.9
55.9
FixMatch+ABBC
46.9
56.1
FiXMatch+CoSSL
43.7
53.8
FiXMaD
48.4
59.3
FixMatch+FAAAD
50.6
62.1
CDMAD 30 and CoSSL 10) In the following sections, we present our experimental results and demonstrate how FARAD tackles each of the four key questions posed above through rigorous comparative analysis.
SAW (7), Adsh[[3] DebiasPI. 48 UDAL 28
1.2AC 46).
3.2 Comparative Evaluation on Long- SSL Benchmarks(AAdrresinn Q1)
We begin by examining whether FARAD can surpass prior art-including the recent CDMAD approach-on three standard long-tailed semi-supervised benchmarks where labeled and unlabeled data share the same imbalance factor (?) = 7w). These benchmarks range from a 10-class scenario more demanding 100-class problem (CIFAR-100-L1) and finally larger- scale ImageNet subset (Small- (amall-ImageNNt-127) We measure balanced accuracy (bACC) and geometric mean (GM), using the same splits and baselines as [30] with minor numerical variations. As shown below, FARAD achieves consistent gains of around 3% over the strongest competitor (CDMAD) across these datasets and imbalance levels, largely because our bias references retain realistic statisties while stripping away structural cues, thus yielding more accurate measure of intrinsic bias.
CUFAAR-I-LLTY
CIFAR-10-LI, Table I compares classical rebalancing (e.g. LDAM-DRW, cRT), SSL baselines (Fix- Match, ReMixMatch), and CISSL variants three imbalance levels ((y E 50, 100, 150)). CDMAD previously reached around 88-89% bACC/GM for the least skewed case ((> = 50)), but FARAD consistently improves these scores by about 3This boost arises because random-phase reference images capture more realistic color/frequency statistics, enabling more precise bias subtraction. Even heavier imbalance ((y 150)), our method maintains a solid margin, highlighting its robustness under severe skew across all evaluated metrics.
CIFAR-100-LTT Table evaluates a more challeng- ing dataset with 100 classes, each having a sharp imbal- ance. Although CDMAD already surpasses simpler baselines, FARAD consis- tently adds another z 3% bACC improvement the most extreme setting (7 100), achieving 47.8% vs. CDMAD'S 44.9%. This jump reflects how preserv- ing realistic amplitude spec- tra yields more faithful bias estimates even as the class space expands, thereby bet- ter correcting the model's skew toward heavily sam- pled categories.
Table 2: CIFAR-100-LT with N = = We report balanced accuracy (bACC, %) std. Best entries bold.
CIFAR-100-LT (bACC)
Algorithm
7 20
7 50
Y 100
FixMatch
49.6+-078
42.1+0033
37.6++0.48
FixMatch+DARP
50.8+10.7
43.1-0.54
38.3
FiXMaich+DARP+RRT
51.410.68
44.910.54
40.411..8
FixMatch+CReST
51.810.12
44.910.50
40.1110.5
FiXMaichCCReSTHLA
52.9+0007
47.3+0.17
42.7+0.70
FixMatch+ABC
53.3+0.79
46.7-00.6
41.2+:0.6
FixMatch+CoSSL
53.9
47.6+0.57
43.0+0.61
UiXMatch+UDAL
48.0.10.66
43.7100.1
FixMalch-CDMAD
$4.310.44
48.8.10.55
44.1110.9
ReMixMatch
51.6.10.43
44.2-10.99
39.3110.3
ReMixMatch+DARRP
51,920035
44.7200.6
39.8+0.53
ReMixMaach-DAARP-RRT
54.5+:0.2
48.5+:0911
43.7+00.1
RcMixMatch+CRRaST
51.310.34
45.5-20.6
41.0=0.78
ReMissMaich-CReSTHHAA
51.9++..0
46.6+-1.14
41.7++0.99
RcMixMatch++ABC
55.6++0.35
47.9+0.10
42.2+0012
ReMixManch+CoSSI.
55.8++0.22
48.9+0.61
44.1+0059
Small--mageNet-112,, Fi- nally, Table assesses
ReMixManch-CDMAD
$7.010.32
51.110.46
44,910.42
FixMatch+EARAD
57.310
51.510.41
47.8.0.44

---

Table 4: Ablation study on CIFAR-10-LT with ' 100, Yu
1.
We compare CDMAD and
FARAD under FixMatch and ReMixMatch pipelines, reporting bACC/ GM (%).
CDMAD
FARAD
Ablation Setting
FixMatch
ReMixMatch
FixMatch
ReMixMatch
Full Method
87.5 87.1
89.9
90.5 /
92.2/91.8
No Correction in Training
78.2 75.8
72.3/
82.5 / 82.0
77.2 72.3
No Correction at Test
84.9/ 84.1
88.2/87.7
88.8 / 88.3
90.7 / 99.1
Use Hard Pseudo-labels
86.7 / 86.3
88.9 88.6
89.4/
91.3 / 90.8
High Threshold (r 0.95)
86.8 /6.3
80.4 78.5
89.6 89.0
83.1 / 81.1
method on a scaled-down
ImageNet subset with 127 classes, where each class exhibits high imbalance. Following [30] we adopt FixMatch as the base SSL. algorithm. Even in this more diverse, larger-scale setting, FARAD consistently improves upon CDMAD by roughly 2-3% in bACC for both 32x32 and 64 x4 resolutions, achieving up to 62. vs. 59.3%. Such improvements illustrate that FARAD maintains its bias-correction benefits as the dataset complexity grows.
Summary. Across CIFAR-IO-LT, CIFAR-I00-LT, and 1mall-ImageNett-17 with (> = 7w), FARAD outperforms CDMAD and other strong baselines by roughly 3%. These gains underscore the method's ability to isolate the model's 's inherent preference for majority classes by subtracting reference logits derived from data-like color/frequency distributions but devoid of semantic structure. This principled bias subtraction scales well from small to larger, more complex multi-class scenarios, and consistently delivers notable performance improvements under severe imbalance.
3.3 Ablation Analysis and Computational Performance(Addrresing Q2)
We finally analyze the internal design of FARAD to identify which components contribute most to its overall effectiveness. We also evaluate a practical concern: how to accelerate our bias-correction step via real-to-complex (R2C) FFT to ensure feasible training and inference speeds.
Ablation Studies. We follow the ablation protocol in 33 but replace CDMAD's modules with our 100, 7h. = 1. We compare both FixMatch and ReMixMatch variants of CDMAD vs. FARAD five configurations. Removing bias- correction for pseudo-label refinement severely degrades bACC/GM in both methods, highlighting that adapting the pseudo-labels to correct class imbalance is vital for minority classes. Disabling the bias-correction at test time also yields nontrivial performance drops, implying a final logit subtraction still necessary to offset the learned priors. Switching from soft to hard pseudo-labels or fixing a high confidence threshold likewise reduces accuracy, but the effect is less pronounced than removing bias-correction altogether. These observations underscore that FARAD requires bias subtraction during both training and inference fully climinate majority-clas bias.
own Table 4 presents the results on CIFAR- 10-LT under
FFT Acceleration via R2C. To ensure that our random-phass reference image gener- ation does not become a bottleneck, we leverage a real-to-come compple- (R2C) batched FFT implementation. This achieves at least 50% faster transforms than naive complex- to-complex routines by exploiting conju- gate symmetry for real-valued inputs. Fig- ure compares epoch-wiss training time and per-image inference latency for FARAD with and without R2C acceleration on a sin- gle GPU at various batch sizes. By halving the FFT cost, R2C shortens overall epoch duration by up to 50% and reduces infer- ence latency proportionally. We use pastel color fills and hatching to visually distinguish
RECUPTT No NTFFF Speed Compartesm
31-
2.
12
250
44
128
254
Figure 2: R2C FFT Acceleration provides a 2 50% speedup over naive FFT. We report average epoch time (left) and inference latency (right) on single GPU. Bars show both color and hatch pattern dis- tinctions for clarity. Error bars indicate standard deviations over three runs.
"No FFT" from "R2C FFT" bars.
Summary. These experiments verify that both training-phase and test-phase bias subtraction are essential in FARAD, and that advanced pscudo- label handling (soft labels, data-adaptive thresholds)
8

---

further improves minority-clann class performance. Meanwhile, R2C-based FFT acceleration alleviates the main computational overhead, enabling FARAD to scale efficiently to large semi -speervised datasets.
3.4 Empirical Validation of Random-Phase Image Properties(Addrestee Q3)
Verifying Semantic Irrelevance. We first confirm that random-phan -hasse images effectively remove recognizable semantic cues. Concretely, construct three image sets of size 1,000 each on CIFAR- 10-LT (% = 100, You- = 1): (i) Real images drawn from the training data, (ii) Uniform- gray images (a fixed pixel value), and (ii)) Random-phase images generated by preserving the average amplitude spectrum but randomizing phases. We measure: (a) classifier-outpe entropy, which is higher when the network finds no strong class evidence, and (b) embedding distance to the real-image centroid in pretrained ResNet-50 feature space. Table 5 shows that random- random-phase images yield the highest entropy (2.26 bits), exceeding uniform-gray (1.42 bits) and real samples (1.05 bits). As illustrated in Figure 3 (left), handom-phase images also have the greatest embedding distances, confirming minimal semantic overlap with real data.
Verifying Statistical Representativeness. Next, we verify that random- -hateehhaseeeeee images also preserve realistic color and opatial-frequenee characteristics. From another subset of 500 real samples, generate 500 random-phase references by averaging their amplitude spectra and randomizing phases. T color histograms via KI. divergence, showing that random-phase images (0.015) align far more closely with real data than uniform-gray or random noise. Meanwhile, Figure 3 (right) plots the radial frequency amplitude A(r), revealing that random-phase tracks real images almost exactly across low to high frequencies, unlike uniform-gray gram.. This faithful reproduction of energy spectra confirms that preserving A(u, v) from real data captures the essential color/texture distribution needed for bias estimation.
Table 5: Classifier entropy (bits): higher means fewer recognizable cues. Standard deviations in parentheses.
Table 6: Color distribution KL divergence (lower is more similar to real). Random- closely matches real data's color histogram.
Image Type
Entropy
Reference Type
KL Divergence
Real
1.05 (0.14)
Uniform-Gray
0.476
Unifonn-Gre Gray
1.42 (0.12)
Random Noise
0.235
Random-Phee Phase (Ours)
2.26 (0.09)
Random-Phen (Ours)
0.015
Comparative Analysis Embedding C Characterhrics
Enleddiigg Distribution
Redial
23
Fell
Bunkon Peos
Ludee-Car
10-
Drenkty
Anpetede
0.50
1:0
090
20
14 C.
Eimbulding Ulstnee
Nermaled Erequcery Bodtas
Figure 3: (Left) Embedding distance to real-image centroid (larger distance = fewer semantic features). Random-phase (orange) is farthest, indicating minimal semantic cues. (Right) Ra- dial frequency amplitude. Random-phase (orange) closely matches real (blue), preserving frequency energy while losing spatial structure.
Summary. In these experiments, randomizing phase effectively destroys any semantic structure (high entropy, large embedding distance) while preserving amplitude ensures the color/frequency profiles remain similar to real data (low KL, matching radial spectra). These two semantic irrelevance and statistical representativenesss underlie references than uniform images naive random noise.
underll why our method yields more faithful bias

---

4 Conclusion
In this paper, we introduced a novel bias-correction framework for semi-supervised learning (SSL) under long-tailed distributions. By generating random-pia phanse images that preserve the amplitude spectrum of real data while randomizing the phase spectrum, our method achieves two key prop- erties: (1) semantic irrelevance, which removes all recognizable structural cues, and (2) statistical representativeness, which faithfully reflects the overall color and frequency energy distributions found in the original dataset. Experimental results multiple benchmarks confirm that these proper- ties lead to more accurate bias estimates compared to simpler reference images. Furthermore, we demonstrated how the framework seamlessly integrates with different SSL algorithms and can be efficiently implemented with batched real-to-complex (R2C) FFT, achieving competitive throughput on large-scale datasets. Our approach consistently outperforms prior methods, especially minority classes, thereby highlighting the importance of a complete spectral characterization bias-correction strategies for robust and balanced representation learning across diverse data regimes.
Acknowledge
The work of Ziyue Qiao partially supported by the National Natural Science Foundation of China (Grant No. 62406056), Guangdong Basic and Applied Basic Research Foundation (Grant No.2024A11151401 14), and Guangdong Research Team for Communication and Sensing Integrated with Intelligent Computing (Project No. 2024KCXTD047).
References
[1] Jae Soon Baik, In Young Yoon, Kun Hoon Kim, and Jun Won Choi. Distribution-awe aware robust learning from long-tailed tailed data with noisy labels. European Conference on Computer Vision, pages 160-177. Springer, 2024.
[2] Julius S Bendat and Allan G Piersol. Random data: analysis and measurement procedures. John Wiley & Sons, 2011.
[3] David Berthelot, Nicholas Carlini, Ekin D. Cubuk, Alex Kurakin, Kihyuk Sohn, Han Zhang, and Colin Raffel. Remixmatch: Semi-supervised learning with distribution matching and augmentation anchoring. In International Conference on Learning Representations, 2020.
[4] Christopher M Bishop and Nasser M Nasrabadi. Pattern recognition and machine learning, volume 4. Springer, 2006.
[5] Kaidi Cao. Colin Wei, Adrien Gaidon, Nikos Arechiga, and Tengyu Ma. Learning imbalanced datasets with abbel-distribution-aaaa margin loss. Advances in neural information processing systems, 32, 2019.
[6] Wai-Kai Chen. The Circuits and Filters Handbook (Five Volume Slipcase Ser). CRC Press, 2018.
[7] François Chollet. Xception: Deep learning with depthwise separable convolutions. In Proceed- ings of the IEEE conference on computer vision and paltern recognition, pages 1251-1258, 2017.
[8] Yin Cui, Menglin Jia, Tsung- Yi Lin, Yang Song, and Serge Belongie. on effective number of samples. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 9268-9277 2019.
loss based
[9] Xin Ding, Yongwei Wang, Zuheng Xu, William J Welch, and Z Jane Wang. Cogan: Continuous conditional generative adversarial networks for image generation. In International conference on learning representations, 2021.
[10] Yue Fan, Dengxin Dai, Anna Kukleva, and Bernt Schiele. Cossl: Co-learning of representation and classifier for imbalanced semi-supervised learning. In Proceedings of the IEEF/CVF conference computer vision and pattern recognition, pages 14574-14584, 2022.
10

---

[11] Robert Geirhos, Patricia Rubisch, Claudio Michaelis, Matthias Bethge, Felix A Wichmann, and Wieland Brendel. Imagenet-trained trained enns are biased towards texture; increasing shape bias improves accuracy and robustness. In International conference on learning representations, 2018.
[12] Jiawei Gu, Ziyue Qiao, and Zechao Li, Gradient short-circuit: Efficient out-of-distribution detection via feature intervention. arXiv preprint arXiiv2500.01441, 2025.
[13] Jiawei Gu, Ziyue Qiao, and Zechao Li. Spectralgap: Graph-level disttiibuitiitti detection via laplacian eigenvalue gaps. arXiv preprint arXiv:2505.55777, 2025.
[14] Jiawei Gu, Ziyue Qiao, and Xiao Luo. Neubm: Mitigating model bias in graph neural networks through neutral input calibration. arXiv preprint urXiv:2505. 15180, 2025.
[15] Lan-Zhe Guo and Yu-Feng Li. Class-imbalanced semi-supervised learning with adaptive thresholding. In International Conference on Machine Learning, pages 8082-8094. PMLR, 2022.
[16] Peter E Hart, David G Stork, and RO Duda. Pattern classification. Wiley Hoboken, 2001.
[17] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770-778, 2016.
[18] Nicholas
Higham. Accuracy and stability of numerical algorithms. SIAM, 2002.
[19) Chen Huang, Yining Li, Chen Change Loy, and Xiaoou Tang. Learning deep representation for imbalanced classification. In Proceedings of the IEEE conference computer vision and pattern recognition, pages 5375-5384, 2016.
[20] Max Jaderberg, Andrea Vedaldi, and Andrew Zisserman. Speeding up convolutional neural networks with low rank expansions. arXiv preprint arXiv:1405.386. 2014.
[21] Bernd Jähne. Digital image processing. Springer, 2005.
[22] Nathalie Japkowicz. The class imbalance problem: Significance and strategies. In Proc. the Int'l Conf. on artificial intelligence, volume 56, pages 111117, 2000.
[23] Yuheng Jia, Xiaorui Peng, Ran Wang, and Min-Ling Zhang. Long-tailed partial label learning by head classifier and tail classifier cooperation. Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 12857-12865, 2024.
[24] Bingyi Kang. Saining Xie, Mareus Rohrbach, Zhicheng Yan, Albert Gordo, Jiashi Feng, and Yannis Kalantidis. Decoupling representation and classifier for long-tailed recognition. arXiv preprint arXiv: 19X000991 17, 2019.
[25] Jaehyung Kim, Youngbum Hur, Sejun Park, Eunho Yang. Sung Ju Hwang, and Jinwoo Shin. Dis- tribution aligning refinery of pseudo-labe for imbalanced sumi-super learning. Advances in neural information processing systems, 33:14567-14579, 2020.
[26] Miroslav Kubat. Addressing the curse of imbalanced training sets: one-sided selection. In Proceedings of the 14th international conference on machine learning, pages 179-186. Morgan Kaufmann, 1997.
[27] Zhengfeng Lai, Chao Wang, Henrry Gunawan, Sen-Ching S Cheung, and Chen-Nee Chuah. Smoothed adaptive weighting for imbalanced semi supervise learning: Improve reliability against unknown distribution data. In International Conference on Machine Learning, pages 11828-1 11843, PMLR, 2022.
[28] Justin Lazarow, Kihyuk Sohn, Chen- Lee, Chun-Liang Li, Zizhao Zhang, and Tomas Pfis- ter. Unifying distribution alignment as a loss for imbalanced semi-supervised learning. In Proceedings of the IEFF/CVF Winter Conference on Applications of Computer Vision, pages 5644-5653, 2023.

---

[29] Michel Ledoux and Michel Talagrand. Probability in Banach Spaces: isoperimetry and processes. Springer Science & Business Media, 2013.
[30] Hyuck Lee and Heeyoung Kim. Cdmad: class-imbalanced semi-supervised learning. In Proceedings of the IEEE/CVE Conference on Computer Vision and Pattern Recognition, pages 23891-23900, 2024.
debiasing for
[31] Hyuck Lee, Seungjae Shin, and Heeyoung Kim. Abe: Auxiliary balanced classifier for class- imbalanced semi- suprrvisee learning. Advances Neural Information Processing Systems, 34:7082-7094, 2021.
[32] Ming Li, Qingli Li, and Yan Wang. Class balanced adaptive pseudo labeling for federated semi-supervised learning. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 16292-16301, 2023.
[33] Shuang Li, Kaixiong Gong, Chi Harold Liu, Yulin Wang, Feng Qiao, and Xinjing Cheng. Metasaug: Meta semantic augmentation for long-tailed visual recognition. In Proceedings of the IEEF/CVF conference on computer vision and pattern recognition, pages 5212-5221, 2021.
[34] Ci-Siang Lin, Min-Hung Chen, and Yu-Chiang Frank Wang. Frequency-aware self-supervised long-tailed learning. In Proceedings of the IEEE/CVE International Conference on Computer Vision, pages 963-972, 2023.
[35] Chengcheng Ma, Ismail Elezi, Jiankang Deng, Weiming Dong, and Changsheng Xu. Three heads are better than one: Complementary experts for long-tailed semi semi-supervised learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 2024.
[36] Aditya Krishna Menon, Sadeep Jayasumana, Ankit Singh Rawat, Himanshu Jain, Andreas Veit, and Sanjiv Kumar. Long-tal tail learning via logit adjustment. arXiv preprint WxXiv2200.007314 2020.
[37] Alan Oppenheim. Discrete-time signal processing. Pearson Education India, 1999.
[38] Taemin Park, Hyuck Lee, and Heeyoung Kim. Rebalancing using estimated class distribution for imbalanced semi-supervised learning under class distribution mismatch. In European Conference on Computer Vision, pages 388-404. Springer, 2024.
[39] Hanyu Peng, Weiguo Pian, Mingming Sun, and Ping Li. Dynamic re-weighting for tailed semi-supervised learning. In Proceedings of the IEEE/CVF winter conference on applications of computer vision, pages 6464 6474, 2023.
[40] Emanuel Sanchez Aimar, Nathaniel Helgesen, Yonghao Xu, Marco Kuhlmann, and Michael Felsberg. Flexible distribution alignment: Towards long-tailed semi-supervised learning with proper calibration. In European Conference on Computer Vision, pages 307-327. Springer, 2024.
[41] Kihyuk Sohn, David Berthelot, Nicholas Carlini, Zizhao Zhang, Han Zhang, Colin A Raf- fel, Ekin Dogus Cubuk, Alexey Kurakin, and Chun-Liang Li. Fixmatch: Simplifying semi- supervised learning with consistency and confidence. Advances in neural information processing systems, 33:596-608, 2020.
[42] Mingxing Tan and Quoc Le. Efficientnet: Rethinking model scaling for convolutional neural networks. International conference on machine learning, pages 6105
1105-61144 61. PMLR, 2019.
[43] Antti Tarvainen and Harri Valpola. Mean teachers are better role models: Weight-averaged con- sistency targets improve supervieee deep learning results. Advances neural information processing systems, 30, 2017.
[44] Charles Van Loan. Computational frameworks for the fast Fourier transform. SIAM, 1992.
[45] Roman Vershynin. High-ddieensional probability: An introduction with applications data science, volume 47, Cambridge university press, 2018.
12

---

[46] Renzhen Wang, Xixi Jia, Quanziang Wang, Yichen Wu, and Deyu Meng. Imbalanced semi- supervised learning with bias adaptive classifier, arXiv preprint arXiv:2207.13856, 2022.
[47] Shanshan Wang, Zhen Zeng, Xun Yang, and Xingyi Zhang. Self-supervised graph learning for long-tailed cognitive diagnosis. In Proceedings of the AAAI conference on artificial intelligence, volume 37, pages 110-118. 118, 2023.
[48] Xudong Wang, Zhirong Wu, Long Lian, and Stella X Yu. Debiased learning from naturally imbalanced pseudo-labels. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 14647-14657, 2022.
[49] Yidong Wang, Hao Chen, Qiang Heng. Wenxin Hou, Yue Fan, Zhen Wu, Jindong Wang. Marios Savvides, Takahiro Shinozaki, Bhiksha Raj. et al. Freematch: Self-adaptive thresholding for semi-supervised learning. arXiv preprint arXiv:2205.0772444 2022.
[50] Chen Wei, Kihyuk Sohn, Clayton Mellina, Alan Yuille, and Fan Yang. Crest: A class- rebalancing self-training framework for imbalanced semi aaii-suprrvised learning. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pages 10857-10866, 2021.
[51] Tong Wei and Kai Gan. Towards realistic long-tailes supervipe learning: Consistency is all you need. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 3469-3478, 2023.
[52] Tong Wei, Qian- Yu Liu, Jiang- Xin Shi, Wei- Wei Tu, and Lan-Zhe Guo. Transfer and share: semi-supervised learning from long-tailed data. Machine Learning, 113(4):1725-7442 2024.
[53] Qizhe Xie, Zihang Dai, Eduard Hovy, Thang Luong, and Quoe Le. Unsupervised data augmenta- tion for consistency training. Advances in neural information processing systems, 33:6256-6268, 2020.
[54] Lu Yang, He Jiang, Qing Song, and Jun Guo. A survey on long-tailed visual recognition. International Journal of Computer Vision 130(7):1837-1872, 2022.
[55] Xuanyu Yi, Kaihua Tang, Xian-Sheng Hua, Joo-Hwee Lim, and Hanwang Zhang. Identifying hard noise in long-tailed sample distribution. In European Conference on Computer Vision, pages 739-756. Springer, 2022.
[56] Sergey Zagoruyko and Nikos Komodakis. 1rXiv:1605.07146, 2016.
Wide residual networks. arXiv preprint
[57] Yuhang Zang, Kaiyang Zhou, Chen Huang, and Chen Change Loy. Semi- and long-tailed object detection with cascadematch. International Journal of Computer Vision, 131(4):987-1001, 2023.
[58] Manyi Zhang, Xuyang Zhao, Jun Yao, Chun Yuan, and Weiran Huang. When noisy labels meet long tail dilemmas: A representation calibration method. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 15890-15900, 2023.
[59] Jianggang Zhu, Zheng Wang, Jingjing Chen, Yi-Ping Phoebe Chen, and Yu-Gang Jiang. Bal- anced contrastive learning for long-tailed visual recognition. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 6908-6917, 2022.
NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
13

---

Justification: The abstract and introduction (Section particularly the "Contributions and Paper Outline" subsection) clearly state the main claims regarding the FARAD system: using random-phase images for bias correction by ensuring semantic irrelevance and sta- tistical representativeness, its computational efficiency via R2C FFT, and its effectiveness in improving minority-class accuracy in long-tailed SSL. These claims are supported by theoretical analysis Section 2 (and Appendices G
[ J and extensive experimental
results in Section (Tables 14 Figure and Appendix D.2 D.10
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine to include aspirational goals as motivation as long it is clear that these goals are not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: A dedicated "Limitations" paragraph at Sec. (and further notes in App. D.4 discusses assumptions such as FFT availability, modest extra inference cost, and potential failure when amplitude statistics differ drastically from test data.
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should refleet the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
Answer: [Yes]
14

---

Justification:
All formal results (e.g. Vanishing Semantic Overlap Theorem) list assumptions
explicitly and are proved in App. G sketches are summarised in Sec. 2.2
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Question:
Answer: [Yes]
Justification: Sec. 3.1 and App. provide full hyper-parameters. data splits, and training scripts; code will be released in the supplementary zip for exact replication.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution a specific model and empirical evaluation, it may be necessary either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of large language model), releasing of a model checkpoint, or other means that appropriate to the research performed.
While NeurlPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) the contribution is primarily a new algorithm, the paper should make clear how reproduce that algorithm. (b) If the contribution is primarily new model architecture, the paper should describe the architecture clearly and fully.
(c) the contribution is a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
15

---

Question: Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [Yes]
Justification: All datasets publicly available (CIFAR, STI.-10, ImageNet subsets); training/inference code and configuration files are included in the supplementary materials.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please the NeurIPS code and data submission guidelines (https://nips) cc/
public/ /ttp://iiee///eeeuummmm ss onn ollcy) for more details.
While encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See NeurlPS code and data submission guidelines (https
//nips. httccc/ppbbt wtcc/pbblcc/gee wttp//uwweeeeeee// Sudee//mddm ss onPol icyy for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the proposed method and baselines. only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: Dataset splits, imbalance factors, network backbones, optimisers, learning-rate schedules, and augmentation policies are detailed in Sec. 3.1 and
D.I
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented the core the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, as supplemental material.
7. Experiment statistical significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [Yes]
Justification: Each main table reports mean over three random seeds; Sec. 3.3 explains that std computed across independent runs with identical hyper-pane paree-errameters...
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by emoor bars, confi- dence intervals, or statistical significance tests, least for the experiments that support the main claims of the paper.
16

---

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g., Normally distributed errors).
It should be clear whether the error bar is the standard deviation the standard error of the mean.
is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out range (e.g. negative error rates).
If error bars reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification: App. B discloses GPU type (RTX A6000), memory (48GB), batch sizes, and
average epoch/inference timings (Fig. 2 and Table 25
Guidelines:
The answer NA means that the paper does include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental rums as well as estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't iudn't make it into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics httpp:://eerrpss.co httphh/www.sss...a...../.oooo//a
Answer: [Yes]
Justification: The work was checked against the NeurIPS Code of Ethics; sensitive data or human subjects are involved and all
assets respect their licences.
Guidelines:
The answer NA means that the authors have reviewed the NeurIPS Code of Ethics.
the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make to preserve anonymity (e.g., if there is special consid- eration due to laws or regulations their jurisdiction..
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
"Broader Impact" paragraph discusses positive impacts on fair model training and notes potential misuse for adversarial debiasing with mitigation strategies.
Justification: Sec. 4
Guidelines:
17

---

The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No. they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (c.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate point that improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e-g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of MI.).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: No new high-risk model or large generative model is released; standard image-clasiification networks are used.
Guidelines:
The answer NA means that the paper poses such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification: All datasets cite original papers and list licences (e.g. CIFAR under MIT; ImageNet under non non-commercial research); code dependencies are Apache-2.0 or MIT, noted in README.
Guidelines:
The answer NA means that the paper does not existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset used and, possible, include URL.
18

---

The name of the license (e.g., CC-BY 4.0) should be included for each asset. For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
assets are released, the license, copyright information, and terms of use the package should be provided. For popular datasets, paperssithoeee com/ dat has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that re- re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
If this information is not available online, the authors are encouraged to reach out to the asset's 's creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification: The study does not introduce new datasets or pretrained models beyond weights derived from public data.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
No crowdsourcing or human-subjeds study is involved.
Guidelines:
The answer NA means that the paper does involve crowdsourcing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According to the NeurlPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based on the requirements of your country institution) were obtained?
Answer: [NA]
Justification: Not applicable-no no human subjects personally identifiable information.
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
19

---

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if' applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Does the paper describe the usage of LLMs if is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, originality of the research, declaration is not required.
Question:
Answer: [NA]
Justification: Large language models were not used in the core methodology: they were only employed for minor text polishing after experiments concluded.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, non non-standard components.
Please refer to our LLM policy (https: petttps://neure hat.ar/gss..cc/oe.eeeeneee////0
LLM) for what should or should not be described.
20

---

A Related Work
In recent years, Suni-Supervieed Learning (SSL) under class imbalance conditions (also known as class-imbin masalanead SSL) has received significant attention. Due to the skewed distribution of labeled and unlabeled data, models are prone to bias during training, leading poor performance on minority classes. Research in this area mainly focuses on the following aspects.
A.1 Pseudo-Labeling and Its Performance under Class Imbalance
SSL methods typically leverage the model's ' own predictions generate pseudo-labels for unlabeled data. However, under class imbalance, pscudo-labeling faces the issue of confirmation bias: the model tends to predict the majority class for unlabeled samples, thereby introducing inaccurate pseudo-labels that reinforce the model's bias towards the majority class. Studies have shown that SSL models on long- tailed datasets often fail to generate high-confidence pseudo-labels for minority class samples, resulting in pseudo-label distributions that are more skewed than the true distribution. For instance, with true imbalance ratio of 150:1 in CIFAR-10, the pseudo-label imbalance ratio generated by FixMatch can reach 1046:1. This severely harms the accuracy of the minority class, even leading scenarios where SSL performance the minority class is worse than a baseline model trained only on labeled data [51].
One direct cause of the pseudo-label bias is the fixed confidence threshold strategy. In classic algorithms such as FixMatch, pscudo-labels are only assigned to unlabeled samples whose predicted probabilities exceed predefined threshold. However, in imbalanced data, the majority class is more likely to have high-confidence predictions, making difficult for minority class samples to pass the threshold. As a result, the majority class receives large amount of additional training data, while the minority class receives very few pseudo-labels, exacerbating the imbalance. To alleviate this, Guo et al. proposed an adaptive thresholding method, dynamically adjusting the threshold for each class [32] This approach significantly improves SSL performance in long-tailed settings, especially for the minority classes.
A.2 SSL Methods for Long- Long-Tailed Distributions (FixMatch, ReMixMatch, FrecMatch)
Many SSL algorithms perform well under standard settings, but their performance drops significantly when applied to long-tailed data. FixMatch is simple yet effective SSL method that combines consistency regularization and threshold-based pseudo-labeling. It generates pseudo-labels for weakly augmented unlabeled samples and applies consistency loss to the strongly augmented counterparts. While Fix.aachh performs well on balanced datasets, it lacks specific design considerations for imbalanced data. As mentioned earlier, the fixed threshold strategy leads to the issue of minority classes rarely getting selected for pseudo-labeling [1).
ReMixMatch, built on MixMatth, introduces several key improvements, one of which is distribution alignment. ReMixMatch tracks a moving average distribution q(v) for the unlabeled data and normalizes the predictions using the labeled data's prior distribution g(y). This alignment process forces the model's predictions to follow the overall distribution, improving its ability to handle long-tailed distributions. In addition, ReMixMatch incorporates consistency regularization, entropy minimization, and other augmentation techniques, which help mitigate over-confidence in the majority class and improve overall performance (3) Compared to FixMatch, ReMixMatch focuses more global balance and often performs better on imbalanced datasets.
FreeMatch takes a different approach by introducing adaptive thresholds and a class fairness regular- izer to complement FixMatch. Instead of using a fixed threshold, FreeMatch automatically adjusts the threshold for each class based on the model's 's learning state (4). It also adds class faimess regularization term to encourage the model to make balanced predictions during training, which further improves its performance
long-tailed SSL tasks. For example, in extreme settings where each class has only one labeled sample (CIFAR-10) or four labeled samples (STI.-10), FreeMatch outperforms previous methods by large margin.
21

---

A.3 Bias in SSL Classifiers and Correction (e.g,, CDMAD)
In SSL under class imbalance, the trained model often exhibits classifier bias, meaning it inherently favors the majority class. This bias not only affects pseudo-labels but also manifests in the final model's predictions. Lee et al. demonstrated the presence of bias through a simple experiment: feeding& semantic-less reference image (such as a pure color image) to trained model should idcally result in a uniform prediction across all classes, as the image carries no meaningful structure. However, if the model is trained on imbalanced data, it may still show skewed distribution of predictions, heavily favoring the majority class [30]. This indicates that the model's decision boundaries are biased towards certain classes even in the absence of any content in the input image.
To correct this bias, traditional methods for supervised long-tailed learning often apply logit adjust- ment techniques, where the output logits are adjusted according to the class frequency [5]. In SSL, however, the distribution of the unlabeled data is unknown, making bias correction more complex. Debiasing) method specifically addresses this issuc. The key idea behind CDMAD is to dynamically measure and counteract the bias of the model during training. By inputting blank image into the model, CDMAD measures how biased the model is towards certain classes by observing the output logits for the blank input. These logits represent the model's intrinsic bias, which is then subtracted from the logits of real data samples during training, thus neutralizing the model's bias towards majority classes 33
The CDMAD Liaas-DDstributionn Aestrbutioo-MMsmmtchh
CDMAD's ' approach is both simple and effective: it applies bias correction to the logit outputs by dynamically adjusting for the class preferences that the model has learned during training and inference. Experimental results show that CDMAD, when integrated into SSL algorithms like FixMatch or RcMixMatch, leads to substantial improvements in minority class recognition.
A.4 Spectral Methods in Deep Learning (Fourier Transforms and Phase Randomization)
Another line of research addresses frequency-domain characteristics of deep models, utilizing Fourier analysis to improve model generalization. The phase spectrum of image carries most of the high- level semantic information, while the amplitude spectrum reflects lower-level statistics such as texture and color distributions. has been shown that CNNs tend to more sensitive to amplitude variations (texture) rather than structural features, even when the structural features are more important for human vision. Geirhos et al. demonstrated that CNNs trained on ImageNet often rely on texture rather than object shapes to make predictions, Similarly, Chen et al. showed that by swapping the amplitude spectrum of one image with that of another while preserving the phase, the resulting image retains the same object shape but changes the texture, leading CNNs to misclassify the image based on texture alone
This insight led to the development of methods that manipulate the frequency components of images to improve robustness. For instance, Xu al. introduced frequency-domain mixing, where two images' frequency spectra are linearly mixed while preserving their respective phase information 34
This forces the model focus on structural (phase) features rather than texture (amplitude), Reconstruction (APR). where the amplitude spectrum of one image is combined with the phase spectrum of another image [9. By learning from such augmented samples, the model is forced to rely on phase information for classification, leading better robustness to domain shifts and perturbations.
improving domain generalization. Chen et al. developed Amplitude-Phene
Random phase transformation, which involves randomizing the phase spectrum of an image while retaining the amplitude spectrum, has been explored as way generate semantic-imelevant images that maintain statistical properties of the original dataset. This technique is particularly useful for bias correction: by generating random phase images, we can measure the model's intrinsic bias by observing how the model reacts to these semantically empty images. If the model still favors certain classes for these random- -ohsse images, it indicates an internal bias that can be corrected during training.
A.5 Computational Efficiency and FFT in SSL
While the methods discussed above are effective, they must also be computationally efficient to be practical in large-scale SSL. tasks. Fast Fourier Transform (FFT) has been widely adopted for its efficiency in transforming images between the spatial and frequency domains. FFT allows for faster
22

---

computations compared to spatial-domain convolutions, and recent research has shown that FFT can be used accelerate convolution operations in CNNs 77. By leveraging FFT, models can perform convolutions in the frequency domain, reducing computational complexity. This optimization is particularly useful when dealing with large images or batches of data, making possible to incorporate spectral methods like random-phase transformation into the training process without significantly slowing down training.
Recent studies have demonstrated that FFT can be integrated into SSL models to accelerate both training and inference processes. By using commplexx (R2C) FFT, which exploits the symmetry of valued inputs, the computational load can be halved, further speeding up the process [20]. This makes FFT-based spectral methods feasible in large-scale semi-supervised learning tasks, ensuring that the benefits of phase manipulation and bias correction do not come at the cost of training efficiency.
A.6 Summary and Outlook
In summary, current research has explored various methods to mitigate the challenges of class imbal- ance in SSL. Improvements in pseudo-labeling strategies (such as adaptive thresholds and distribution alignment) and specialized long-tail SSL. algorithms (such as FreeMatch, ABC, and CReST) have contributed to balancing the training signal between the majority and minority classes. However, these methods still rely certain assumptions about the distribution of labeled and unlabeled data. Bias correction methods, such as CDMAD, directly address model bias by adjusting the logits during training and inference, achieving better performance in long-tailed settings. Additionally, spectral methods offer promising avenue for improving model generalization by reducing reliance on texture features and focusing on structural information.
The integration of srqquency-domaiie transformations into bias correction provides a new perspective: by randomizing the phase while preserving the amplitude spectrum, irrelevant but statistically representative images for estimating and correcting classifier bias. This approach leverages the power of spectral analysis to neutralize inherent model biases, improving SSL. performance on minority classes without significantly increasing computational overhead. Thus, frequency-doman techniques, particularly random phase transformations, represent an effective and computationally efficient solution for bias correction in semi-supervised learning, particularly under class imbalance.
generate semantic-
B Implementation Details
At each iteration, we generate a random--phee phase image that preserves amplitude statisties but contains no semantic content. When fed through the network. this image reveals the model's intrinsic bias toward majority classes. By subtracting these bias logits from the predictions on real inputs, we produce more balanced pscudo-labet, for unlabeled data, thereby improving minority-clase recognition. The experiments in this paper were conducted on an RTX A6000 with 48 GB of memory.
We follow the CDMAD protocol for fair comparisons, training Wide ResNet-28-2 on CIFAR-IO-LT; CIFAR-100-LT, and STL-IO-LT, and ResNet-50 on Small-ImageNNet-1277 All models use Adam with learning rates of 1.5 x 10-3 (FixMatch) or 2 x 10-3 (ReMixMatch): weight decay is 0.08 for CIFAR-100-LT, and for other datasets we adopt 0.04 under 30k samples {0.01, 0.02) otherwise. Batch sizes are 32 (FixMatch) and 64 (ReMixMatch). each with satabeled-tt-labllll ratio μ 2. FixMatch runs for 500 epochs with 500 iterations each, while ReMixMatch runs for 300 epochs; both employ weak (random flips and crops) and strong (Cutout, RandAugn Augment) augmentations, plus an EMA for parameter smoothing. Random seeds are fixed throughout.
To generate our random-phare images, we preserve the batch-averaged amplitude spectrum while uniformly randomizing phases in |- T] each iteration, then invert via real-to-com ceal-to-complex (R2C) FFT. The Python-like snippet below shows how we compute this image for a batch of real samples, exploiting half-spectrum storage. For lower-level CUDA implementations, one can invoke cuFFT in a C/C++ style, as illustrated after the Python code:
import torch, torch. fft as F
def sandon_phaae image (batch_ img):
23

---

batch img: 1B, C, H, W1, real
treq_ree P.rfft2 (batch, img, dim= (-2, -11)
R2C transform average
amp_mean mean free sree_r2c.abll () mean (dim=0, keepdin-True) amplitude
rand_phase = t-rorch, www.pppi....aaaaooooo....
(freg_r2c) torch. p1
freg_rand
torch.exp (1) rand_phase)
combino amp
random phase
return F.irfft2 (freq_rand,
[ [-2:], dim- (-2, -1))
Listing 1: PyTorch-like R2C for random-phase images.
/* Assume input pointer r_realing on host, d_realing on device. x/ cufftHandle plan;
cufftComplex +d_freqHalf;
size_t realSize B.CrHaW.sizeof (floaatt)))))
cudaMalloc( (void=++) sd sd_freqHalf, B*C* (H) * (W/2+1) =sizeof (cufftComplex))
cuff:Plan2d(aplar, H, W, CUFFT_R2C);
(prttExecRR2(plarr (cufftReala)d_reaIine
d_freqBalff;;
fd_friaa now hoids the unique haif-spectrum. Next, compute ampiitude & random phase, combine, and run inverse pian 'cufftXxecC2#* simi mailarry.. +/
10
Listing 2: cuFFT-style usage for R2C transforms.
In both FixMatch and ReMixMatch, the logits of this random-phase image, go (Xrand))) measure intrinsic bias. We subtract them from the real-image logits, 96 (a())) go(ox(u)) go (D(raad)). then so all unlabeled samples are
apply softmax for pseudo-labels. FixMatch sets threshold used, while ReMixMatch removes sharpening and distribution alignment and adds supervised cross-cntropy on labeled samples. The following pscudocode highlights the main difference:
Fi xMatch
x_rand random_phase_ine (rsal_ings)
bias_logits model (x_rand)
Loggits_u u weak model (u_ (u_weak)
debias_logite togits_u_weak bias_logits
pseudo_labele
(debiaas _ogists___ssss
loss_u cross_entropy (model (u_strong), pscudo_labels)
loss_x cross_ens cross_entropy (model (x), label_x) suporvised
10 loss = loss_k lambda_u loss_u
11 loss.backward () opt optimizer.step
ReMi xMatch
14 rand randon__phaaleeeeeeeeee wwwpspaaaeeem_eee image (real _ings)
15 bias biastlogits- = model (x rand)
16 Logits weak model (u_weak)
debias_logits logits_u__wakk bias_logits
18 pseudo_labsls softmax (debiasxddbiisllgittt
19 loss_u cross_entropy (model (u_strong), pseudo_labels,
20 loss_x x cross _acesspyrrropyyooo (model (x_weak), label_x)
loss = loss_x lambda_u loss_u
loss.backward () aptimizer.stepl)
Listing 3: FixMatch vs. ReMixMatch bias subtraction (PyTorch-like),
At inference. we generate a fresh random-phass image batch, subtract its logits, and apply arg max. This keeps overhead low thanks to batched R2C transforms, and consistently corrects bias on minority classes. Final predictions are obtained by (Teest) arg max. go (Frest) go go (Frand). All code will released upon publication. Algorithm Summmarizes the core steps.
C Metrics for Figure Shaape Presence and Statistical Fidelity
Figure provides conceptual mapping of different reference image types based on two key properties: "Shape Presence" and "Statistical Fidelity." The positions of the points in this figure

---

Algorithm Core Random-Phase Bias Correction
Require: Base SSL algorithm with parameters Labeled batch {(rr 9:) y:))ez>> Unlabeled batch
{u, 1==1
Step 1: Batch-Average Amplitude Spectrum
2: A(u, + 1
B+ Eall (Fplitude(F(xz)
3: Step 2: Random-Phane Image
4: $(u, u) - Uniform [J.
5: Trand f- [F((((u, v) exp(jo(u, x))
6: Step 3: Bias Computation
7: Zhins < (3rand)
8: Step 4: Ctas-Coreeeted Logits
9: for each sample in labeled and unlabeled batch do
10:
(2)) < (x)
Zeias
11: end for
12: Step 5: SSL Loss and Parameter Update
13: I. Tuup (gg (I:) ) Tunsup (9g (Uy))
14: 8 < 1 VoL.
are informed by quantitative metrics discussed in $3.4 4 of this paper. The x-axis represents "Shape Presence," where lower values indicate fewer recognizable semantic cues and are considered better for reference image. The y-axis represents "Statistical Fidelity," where higher values indicate closer resemblance to the statistical properties of real data and are considered better.
C.1 Shape Presence (Lower is Better)
The "Shape Presence" metric aims to quantify the extent to which an image contains recognizable shapes, edges, or other semantic structures, A lower score on this metric is desirable for reference images, as it indicates greater semantic irrelevance. This is primarily evaluated using Classifier Output Entropy, as detailed $3.4
Metrie:
Classifier Output Entropy (H) For given input image, let the classifier (after its softmax layer) output probability distribution over C' classes, denoted as p (P1. PP1-P2. ... Pc), where P. is the probability assigned to class c. The entropy of this probability distribution calculated as:
H(p) = Σ Pe loga Pe (bits)
A higher entropy H (p) signifies that the classifier is more uncertain about the image's content, implying lack of strong, recognizable semantic features (i.e., lower shape presence).
Mapping Figure I X-axis ("Shape Presence Score" Sp): Since "Lower is Better" Shape Presence axis in Figure [ the plotted score Sp should be low for images with high entropy (less shape). This can be achieved by an inverse relationship or by subtracting the entropy from constant. For example, Sp could be conceptualized as being proportional to 1/(H(p) + e) or (MaxEntropyObserrve MaxEntropy is reference maximum. The raw Sp values may then be scaled fit the desired axis range (e.g., 0-30 as shown in Figure TT. A higher entropy (less shape) results in a lower Sp score.
(p)). where is a small constant to prevent division by zero, and
C.2 Statistical Fidelity (Higher is Better)
The "Statistical Fidelity" metric assesses how well the global statistical properties reference image (e.g., color distribution, frequency content) match those of the real dataset. A higher score on this metric is desirable. This primarily evaluated using the KI. Divergence of Color Histograms, as detailed in $3.4
25

---

Metric: KL Divergence of Color Histograms (Dkl) Let Ps. represent the normalized color histogram of reference image, and Qn represent the average normalized color histogram derived from a representative set of real training images. Both Pa = (fm(()) |= and Qh. {gr (i) ;=1 are probability distributions over M color bins. The KI. divergence from Qn to Pa is calculated as:
ph(()
(R 11 Qn) = Σ 1og log2
(bits)
qn(())
lower Dxz, value indicates that the color distribution of the reference image more similar to that of real images, signifying higher statistical fidelity.
Mapping to Figure I Y-axis ("Statistical Fidelity Score" Sp): Since "Higher is Better" for
the Statistical Fidelity axis in Figure the plotted score Sy should be high for images with low D (high fidelity). This can be achieved by an inverse relationship or by subtracting Dxy. from constant. For example, Sp could be conceptualized as being proportional to 1 1/(Dxr, (Pr || Qn) + e) or [MaxDKLObserveld - Dx 1. (Pr Qn)), where e a small constant, and MaxDeLObeerved is reference maximum. The raw SF values may then be scaled to fit the desired axis range (e.g.. 0-100 as shown in Figure 1). A lower KL divergence (higher fidelity) results in a higher SF score.
D Additional Experiments
D.1 Experimental Settings and Baselines
Detailed Data and Network Configuration. We adopt the same datasets introduced in the main paper, namely CIFAR-10, CIFAR-1O0, STL-10, and Small-ImageNNet-122 Table 7 summarizes the training. validation, and test splits for each dataset, including the labclcd/unlabeled partition used in semi-supervised setting. To evaluate the classification performance, exponential moving average (EMA) of the network parameters was computed at each iteration. Specifically, we adopted the Wide ResNet-28-2 architecture for experiments and STL-10-LT datasets, while employing ResNet-50(17) for the Small-ImageNet-122 127 dataset. We also indicate how we generate long-tailed (LT) versions (e.g.. imbalance factor 7, or reversed/missing classes if applicable) to ensure clarity in distribution mismatch experiments. As shown in Table 7 we preserve the overall sample sizes consistent with the main paper, splitting each dataset into labeled and unlabeled sets based on a chosen imbalance factor. In mismatch experiments, the unlabeled set may follow different distribution, potentially reversed or missing certain classes. For every dataset, we also keep a small validation set if needed for hyperparameter tuning, and standard test set for final evaluation. Table 8 details the core network architectures and hyperparameters used throughout. Unless otherwise stated, these settings match the main text. Each row lists the backbone model (e.g.. ResNet- 18, -e1 EfficientNet (2)), learning rate schedule, optimizer (SGD or Adam), total epochs, batch size, and random seed. Any deviations from the main text or additional fine-tuning for certain experiments are noted below the table. In addition to the balanced accuracy (bACC) and geometric mean (GM) metrics detailed in the main text, we also report minimum-clan accuracy or visualize confusion matrices certain experiments. These extended metrics highlight how the approaches handle severely under-represented tail classes.
CIFAR- 10-LT, CIFAR-1000 LO-
Table 7: Summary of dataset splits and any modifications for the long-tailed setting. "Imbalance Factor" refers to the ratio between majority-class and minority-class sample counts in the labeled set. Some experiments also involve reversed or partially missing classes (highlighted in footnotes).
Dataset
#Classes
Train (Labeled)
Train (Unlabeled)
Val
Test
Imbalance Factor
CIFAR- (LT)
5,000 (LT distribution)
45,000 (potential mismatch)
5,000
10,000
50 1///0//// 1501 20 00/0// 100'
CIFAR-100 (LT)
100
10,000 (LT distribution)
40,000 (potential mismatch)
5,000
10,000
(0L-IO
10
1,000 (LT distribution)
9,000 (potential mismatch)
1,000
8,000
10/201 20
Small ImageNet-127 127
127
64,000 (LT distribution)
5,000
10 20 50"
We vary explore different levels of class imbalance: see main exact splits.
1 Foor Small-ImageNet-1277, typically reduce the original Ik classes 127.
Baseline Algorithms and Comparison Methods. Our core semi-supervised baselines include
FixMatch[4]] ReMixMatchh 3), and FreeMatch 49 each adhering to the same pseudo-labeling
26

---

Table 8: Network configurations and key training hyperparameters. We primarily use ResNet- 18, WideResNet-28-22 or EfficientNet (different variants) depending on the dataset scale and resolution. If a setting differs from the main paper, it is clearly highlighted.
Backbone
Depth/ / Width
Optimizer
LR Schedule
Initial LR
Batch Size
Epochs
ResNet-18
18/-
SGD (momentum=0099)
Cosine Decay
0.1
64
200
Wid:RosNet-28-2
28/2x
SGD (momentum=009)
Cosine Docay
0.03
64
200
EfficientNer-BD
Adam (hetas=0.9,0.99)
Step Docay
0.001
32
300
All runs use weight decay of 5 unless stated otherwise.
Random seed is fixed (e.g.. 42) for fair comparisons: any variation reported in subsequent experiments.
thresholds, data augmentations, and learning rates described above. In selected sections, we also eval- uate 4ea UDA(53), or ACR(51] if they offer particular insights into semi-supervised performance under class imbalance. For addressing imbalance and classifier bias, we compare our Random Phase Amplitude Preservation approach against CDMAD, blank-input references, and standard long-tailed techniques such as LDAM-DRW and cRT. When comparing these methods, we keep all other training and data-pia processinn conditions identical, so that any improvements can be attributed directly to the bias- strategy. As shown in Tables 7 and 8 with the main paper's experimental setup.
this ensures consistency
Remark: By keeping the data splits, SSL hyperparameters, and network architecture consistent across all baselines, we enable a fair, direct comparison of how each bias-correction or class-rebalancing technique influences minority-class recognition and overall performance.
D.2 Resilience to Class Distribution Discrepancies in Semi-Supervised Learning
In real-world applications, the labeled and unlabeled data often arise from different class distributions. This section focuses on two mismatch scenarios: (i) different imbalance ratios between labeled and unlabeled sets; and (ii) extreme "reversed" condition where the most frequent labeled classes become the rarest among unlabeled data, and vice versa. Our experiments indicate that FARAD remains robust under both circumstances, outperforming baseline methods by a noticeable margin.
CIFAR-10-LT and STL-10-LT with "tt * Yu- Table 9 shows results on CIFAR-10-LT when the labeled set has imbalance ?/ 100, but unlabeled data vary in imbalance (% {1, 50, 150)). We also evaluate STL-10-LT with 7v E {10,20) and no assumptions on unlabeled class proportions. Although CDMAD previously performed well under mismatch, FARAD improves bACC/GM by another 2-3%, in part because the random-phase reference does not depend on matching the labeled and unlabeled distributions. For instance, % 1, the unlabeled set is nearly balanced while labeled data are highly skewed; even in this contrasting scenario, FixMatch+FAAAI surpasses FixMatch<CDMAD by 3% GM and exceeds 90% G... Similar gains hold for heavier mismatch (7, = 150), indicating that our phase-randomized reference effectively corrects the model's strong prior toward overrepresented labeled classes, even if unlabeled data distributions differ substantially.
Reversed Distribution on CIFAR-10-LT. We next consider a more extreme mismatch setting where 7 Tu 100 yet the unlabeled data classes are entirely reversed relative to the labeled set. Concretely, if a class is majority in the labeled portion, it becomes minority in the unlabeled portion. Table I0 shows that while multiple CISSL techniques (c.g.. SAW. ABC) can partially mitigate misalignment, CDMAD and FARAD stand out. Notably, FARAD surpasses CDMAD by roughly 3% in bACC/GM under both Fix Match and ReMixMatch backbones. This suggests that even when class distributions are diametrically opposed, FARAD effectively combats werrppioriitizatinn of previously majority categories and maintains competitive performance on newly scarce classes.
Summary. Under mismatch conditions, where labeled and unlabeled class distributions differ sub- stantially or even invert, FARAD consistently outperforms CDMAD by about 2-3% bACC/GM. This advantage holds whether the unlabeled portion is comparatively balanced (7, = 1) or heavily skewed
(7. = 150), and whether partial or complete reversal occurs. These findings highlight the flexibil- ity of our approach in adapting to real-world imbalances and maintaining superior minority-class performance despite distributional discrepancies.
27

---

Table 9: CIFAR-IO-LT (1) 100, Yv E 50, 150)) and STL--10-TT (n {10, 20). unlabeled
distribution unknown). We report bACC/ GM (%) std. "Ours" indicates FARAD.
Algurithm
Cy 100)
STL-110LTT
50
150
10
20
FisMatch
68.::1
82.88.111
00010.25/70.5111
69.61060
00000.0066.666 62.61:111
72,910.09/
00910.09/0660000 600.00
554-021/526-00
PIMMRCb-DARP
85.4
8850:00.65
77.34:0.97 15.520021
72.900241 09.551.11
77.320335
699-1.777 65440.07
FioMaic--DARPPL..
86.6=1.11
82.3+0.322 /11
78.9+0.25
77.740..6
786+0030/
119-049/687--0
87.00.70/
02.74:0.11
79.20023/
641-061/7811112
ABC
82.7=0-49
62.710064/82.000
78.42:087
70.110:46/78.11
SAM
81.2-0068
001.688882220000 80.2::0.1
79.81:0.25 /79.110.32
74.540:97
FAAManS-SAW-LA
81.5_00.8 84.1
82.910.38
201.00.38822..... 18..510.38
79.110.81
78.6.10.99
-7-
PixMatehs SAW
64.4= 23 / 84. 26
81.6+:1338
/31.3=00.32
77.60040
17.14
FIMMAES-CMMAD
87.3=0446 46 87.00:0
85.51:0.77 37 /85.2:00.36
32.1::0.5/58
6::0.27
17:00227787:00
750-041 000-4447344000
/84.--11.27
82.7-0.22
7K0-041/764-000
ReMisMarch
25.1++043/2
11.9770777
1+092
RaMisMacc>
85.0
14.3
77,0100.22 /2.77,....00
72.810.10
6..2
67.7_046
0.76
KcMisManch> DARP
86.9-0.10 107 86.4=20.13
77.45:1.22
775.0-5--.25
73.2=00.11 11 (9.2+11
79.400.17/
240.
709-0044/
67.0=162
ReMicManch MeetMMancc"--AAA -DARPLAA
S18 0.45/ 80.920040
83.920.42
81.120.201
2010000000000000000 30.32:0226
90.620.45
45/ 79.669966.05
76.8=0.0/
74.8=0.08
88.7=0 58.5+00 25
83.5+-1.53
133.1-000-
80.9=0.25
0:25 3++0.11
90.+00.53 50.+0.46
76.700.0/
ReMisMucZ-ABC
76.415.44/ 74.8.6605
85.2.10.00
84.7.10.25
50,410:40/ /0/ S0.0.10.4
76810.52/7481006
7.2 674
RoMisMucd--AWW
87.0=0075/
ReMixMancS"
74.2=1.49/ 45.36
84.811107
81.342.42
ReMixManch* SAW
88.5-0.79
82.4+0
KcMisMun&&<DDMD
89.7=045
4:0.6
86.74-1.11 786.5400.37
83.0=00.46/
828-20.38 02.800.8//81353 81.888811.00...
81.7-0.32/ 807=0444
ReMi:MucssFARAD
92:7-0.45 92.4
89.7:0.21 89.5:10.17
86.0:00.66 000:0046/88.6-11..3
85.8:0.38 04.9:0.35
84.7-0.32/ 83.70.44
Table 10: CIFAR-10-LT (reversed distribution):
100, Ta = 100 but the unlabeled class
proportions are the inverse of the labeled ones. We report bACC/GM (%). "Ours" indicates FARAD.
Algorithm
FixMatch+
ReMixMatch+
ABC
69.5 / 66.8
60.5
SAW
72.3 / 68.7
79.5 / 78.5
SAW+LA
74.1 /2.0
50.2/ 14,8
SAW+cRT
75.5 73.99
80.8 79.9
CDMAD
76.9 75.2
81.5 80.8
FARAD
80.0/ 78.2
84.6 83.9
Table 11: CIFAR-10-LT with FreeMatch the base SSL algorithm. We report bACC/ / GM (%). "Ours" denotes FARAD.
Algorithm
100
7
100, 7n
FreeMatch
75.4/72.9
74.2/ 69.5
FrecMatch+SAW4oRT
82.8 82.3
86.4/ 86.2
FreeMatchCCDMAD
84.8 84.4
89.0/ 88.7
FreeMatch+FARAD
87.6
92.0/ 92.0/91.7
D.3 Versatility and Integration with Diverse SSL Algorithms
Although our previous experiments focused on FixMatch and ReMixMatch, we next consider whether FARAD can be readily combined with additional semi-supervised learners and recent CISSL frameworks. Specifically, we investigate two scenarios: integration with the newer FreeMatch SSL algorithm, and direct comparison against ACR recent CISSL approach- paired with FixMatch as the base learner.
Integration with FreeMatch. Table summarizes results on CIFAR-1O-LT with = % 100 and the more extreme mismatch case ? = 100, 7u 1. We adopt FreeMatch as the underlying SSL algorithm. SAW+cRT and CDMAD are notable baselines, both of which previously improved minority-class recognition in long-tailed SSL. Nonetheless, we observe that FARAD maintains an additional advantage of around 3% bACC/GM over FreeMatch+CDMAD in both distribution settings. This illustrates that FARAD's 's bias correction seamlessly complements FreeMatch's 's adaptive thresholding and confidence-based updates, without requiring specialized tuning for each new SSL framework.
Comparison with ACR. We further assess FARAD against ACR, which represents another promising strategy for mitigating imbalance in SSL. For consistency with [30], Table 12 reports bACC and GM
28

---

Table 12: Comparison with ACR on CIEAR-1O-LT, where FixMatch serves as the base SSL method. We show bACC / GM (%). "Ours" indicates FARAD.
bACC/ GM (%)
Algorithm/ CIFAR-IO-LT
71 7/u
100
7
100, Y
AiX
81./ 81.4
85.6/85.3
FiMMDD
83.6 83.1
87.5/87.1
FiMMacc+FARAA
86.7 / 86.3
90.3/90.0
on CIFAR-10-LT under two setups: matched-distribttion case (7i 100) and mismatched one (7r = 100, 7h. 1). We integrate both ACR and FARAD into FixMatch. While ACR already outperforms vanilla FixMatch by sizable margin, FARAD raises bACC/GM by about 3% beyond FixMatch+CDMAD. further reinforcing our claim that FARAD can be flexibly combined with various CISSL strategics and base SSL pipelines.
Summary. These findings confirm that FARAD not only improves classical baselines such as Fix- Match and Match (as shown earlier) but also integrates effectively into other SSL algorithms (e.g., FreeMatch) CISSL frameworks (e.g., ACR). The consistent 2-3% margin over CDMAD underscores FARAD's versatility and its capacity to remain effective regardless of the underlying pseudo-labeling strategy, thresholding mechanism, classbblaacce heuristic employed by the base method.
D.4 In-Depth Experiments on Reference Image Design
This section investigates how the construction our reference images (i.e., those used to expose classifier bias) affects performance. We specifically explore three aspects: (i) comparing distinet reference-image types, (ii) examining how implitude-spectrum selection and batch size impact the final results, and (iii) testing different degrees of phase randomization.
Multiple Reference Image Types. We first compare six representative reference-imat forms to highlight the importance of random-phase plus amplitude preservation in revealing classifier bias. The candidate references include: (1) Blank (solid color), (2) Random noise, (3) Low-frequency-ont (retaining only the central portion of the Fourier spectrum), (4) nrigh-frequenc-onne
(retaining only the outer frequency band), (5) Our random-phute image, and (6) Real images used in an extreme example 2D amplitude plot for each type of reference in the frequency domain. Despite the different spectral allocations, the key question is which input best exposes any latent bias. We evaluate balanced accuracy (bACC), geometric mean (GM), minimum-class accuracy (Min. Acc), and the classifier's 's mean logit entropy (i.e., how "uncertain" the network is that reference). Table 13 shows the aggregated results (averaged over three runs with different seeds). Notably, the random-phase image achieves both a high mean-entropy and the strongest improvement for minority classes once we subtract its logits from real inputs, confirming that preserving the amplitude while scrambling the phase yields a more effective bias reference.
comparative sense. Figure 4 visualizes
Table 13: Comparisons of different reference-inenn image designs in revealing and correcting classifier bias on CIFAR- (y = 100). We report bACC, GM, minimum-class accuracy (Min. Acc), and the classifier's is average logit entropy the reference itself (Entropy). Higher entropy suggests fewer semantic cues.
Reference Type
bACC
GM
Min. Ace
Entropy
Blank (Solid Color)
78.4
73.5
66.1
1.51
Random Noise
79.3
74.8
67.2
1.77
Now-Frequency Only
80.2
75.3
67.9
1.82
High-Frequency Only
80.1
75.0
67.6
1.94
Random- Phase (Ours)
83.5
79.1
71.8
2,21
Real Images (Extreme)
74.9
70.4
64.2
1.09
29

---

Amplitude Spectrum Characteristies Lowitez
Netac
Lighfree
-04
Namallod Amolhad
Figure Example amplitude (frequency-done) domaii) visualization of the six reference-image types. The color map indicates normalized energy levels. Our random-phase approach (center-right) retains the original amplitude spectrum but scrambles the phase for all frequencies.
Amplitude-Spectrun Selection and Batch Size. Next, we investigate how different ways of deriving the amplitude spectrum affect performance, as well the impact of various batch sizes. Our main pipeline uses batch-averaged amplitude each iteration. We compare it against (i) single-image amplitude selection and (ii) a one-time offline amplitude derived from the entire dataset. We also test three batch sizes (8, 32, 128) see if a larger batch provides more stable amplitude estimates. Table [4] summarizes these variations on CIFAR-IO-LY (y 100). The single-image approach can inject extra randomness, while the offline approach may fail to reflect distribution nuances that arise across training epochs. By contrast, the batch- aveerged amplitude (our default) consistently outperforms the other methods, especially minority -class recognition. Figure (" 2"") further illustrates how the amplitude-spectren variance decreases with increasing batch size, explaining why larger batch can yield more stable bias references.
Table 14: Impact of amplitude-spectrum selection and batch size CIFAR-IO-LT 100). We report bACC, GM, minimum-class accuracy (Min. Acc), and typical FFT run time per iteration (averaged over 3 seeds). "BA" refers batch-averaged amplitude (our default).
Amp Method (Batch Sz)
bACC
GM
Min. Ace
FFT Time (ms)
Single-Image Amp (8)
77.6
73.1
65.1
10.4
Single-Image Amp (32)
78.9
74.2
66.1
8.9
Single-Image Amp (128)
79.3
74.6
66.4
8.1
Offline Amp (8)
80.1
75.7
68.2
10.2
Offline Amp (32)
80.6
75.9
68.5
Offline Amp (128)
81.5
76.8
69.3
8.0
BA (Ours) (8)
82.2
78.3
70.4
10.5
BA (Ours) (32)
82.9
78.6
70.9
8.7
BA (Ours) (128)
84.1
79.7
72.0
7.9
Varying Degrees of Phase Randomization. cies is truly necessary. We compare:
Finally, we ask whether fully randomizing all frequen-
Full-frequency randomization (our default)
Randomizing only mid-to-high frequencies while preserving low-frequency phase
Narrow-range randomization within 6, 8) around the original phase, thus introducing small perturbations only
We quantify how effectively each variant removes semantic cues using feature-spac dispersion score, computed by measuring the average pairwise distance between these synthetic references and real images in pretrained embedding space. Table [5] shows that partial or narrow-rane waage randomization still retains some low-frequency structure, lowering both the referencceentropy and the final bACC/GM gains. By contrast, fully randomizing the entire frequency band maximally disrupts meaningful contours, thereby leading to higher minority-class performance and greater fcature-space separation.
30

---

Amplitude Variance Reduction with Larger Batches
BS-8
0.12
BS-124 128
0.10
Ampliude Varienco
0.02
0.00
10
Training Epuch
Figure 5: Example visualization of amplitude-spectrru variance decreasing as batch size increases. We calculate the average amplitude variance over selected frequency bands at each iteration, then plot the curves over training. Larger batches yield more stable amplitude estimates.
Table 15:
Effect of different randoorranoiiiiionn strategies CIFAR- 10-R-OO-TTTE (y 100). We show bACC, GM, minimum-clane accuracy (Min. Acc), and a dispersion score that reflects how dissimilar the reference is from real images in a pretrained feature space. Higher dispersion indicates fewer leftover semantic cues.
Phase Randomization
bACC
GM
Min. Acc
Dispersion
Mid/High Frequencies Only
80.2
75.9
68.1
0.68
Narrow Range ([-ô, 6))
79.4
75.5
67.7
0.63
Full-Frequency (Ours)
83.7
79.3
71.2
0.92
D.5 Training and Inference Bias-Correction Ablations
This section studies how best to apply our logit subtraction at different training and inference stages, well how various pseudo-labeling strategies and reference-image usage at time affect performance. We focus on CIFAR- with = 100 unless otherwise stated, but similar trends appear other benchmarks.
Combining Bias Subtraction in Training vs. Inference. We first analyze four configurations: (i) correction (baseline), (ii) training-only correction, (ii) inference-only correction, and (iv) both training and inference correction. Table I6 presents the balanced accuracy (bACC), geometric mean (GM), and minimum-class accuracy (Min. Acc) under these settings. Compared with the baseline's bACC of 78.2%, applying bias subtraction in both phases yields the highest scores across all metrics, improving minority-class recognition by over 7% absolute. Training-only or inference-only also confer meaningful gains, though not as substantial combining both.
Table 16: Effect of applying our bias-correction subtraction during training, inference, both, CIFAR- 10-LT (y = 100). We compare bACC, GM, and the minimum-che classs accuracy (Min. Acc). Results averaged over 3 runs.
Method
bACC
GM
Min. Acc
No Correction (Baseline)
78.2
73.1
64.5
Train-Only Correction
81.9
77.6
69.4
Inference-Only Correction
80.7
76.2
68.3
Train+Inference (Ours)
83.6
79.2
72.1
31

---

Figure ("Fig. 4") plots accuracy minority classes over the training epochs to visualize how the bias subtraction affects convergence. The full "train+inference" strategy not only achieves higher final accuracy but also exhibits fewer fluctuations early on, indicating more stable learning process for tail classes.
Minority-Class Accuracy During Training
75
No Correction
Tmin--nll
Inlerence-Olly
70
Train-liference
Accuracy (16)
55
50
2
10
Training Epoch
Figure 6: (Fig. 4) Minority-clan class accuracy curves across training epochs on CIFAR-10-LT. Applying bias correction in both training and inference phases yields stable and higher tail-class accuracy.
Confidence Thresholds and Pseudo-Labeling Strategies. We next study different pseudo-labeling designs: fixed threshold (aix-sttch-style) with r E {0.90, 0.95]. an adaptive threshold (FreeMatch- style), and hard vs. soft aseudo-labels Table 17 compares each setup with and without our bias- correction. Although each method alone boosts performance relative to the baseline, adding our approach consistently raises bACC by about 2%-3% and improves minority-clas recognition.
Table 17: Impact of various pseudo-labeling strategies (confidence thresholds, hard vs. soft labels) on CIFAR-IO-LT. "BBC" indicates our bias-correction method applied. We report bACC, GM, and Min. Acc.
Pseudo-Label Scheme
+BC?
bACC
GM
Min. Acc
FixMatch (r 0.90)
79.6
75.2
66.7
FixMatch (T = 0.90)
Yes
82.1
77.9
69.8
FixMatch (T 0.95)
No
78.5
74.1
66.1
FixMatch 0.95)
Yes
80.8
76.4
68.5
FreeMatch (Adaptive 7)
81.4
77.0
68.2
FrecMatch (Adaptive 7)
Yes
84.3
79.6
71.3
Hard Labels
No
78.9
74.5
66.5
Hard Labels
Yes
81.0
76.9
68.1
Soft Labels
No
79.3
74.6
66.9
Labels
81.7
77.3
69.2
To illustrate how confidence thresholds interact with tail-class coverage, ("FFg. 5") shows the fraction unlabeled samples assigned to minority classes at different thresholds. Without bias correction, the coverage drops sharply for higher values, leaving the minority classes severely under-sampled. By contrast, our method alleviates this problem, sustaining more balanced coverage stricter thresholds.
Reference Generation Strategies at Inference. Lastly, compare how reference images are generated test time: (i) dynamically creating a new random-phan -hase image for each test batch, (ii) generating one random-phase image at the start of inference and reusing it for all samples, or
32

---

Minority-Class Coverage Analysis
No
Coverage (%)
20
85%
90%
95%
98%
Confidence Threshold (t)
Figure 7: (Fig. 5) Fraction unlabeled samples classified as minority classes at various confidence thresholds (r). Bias correction consistently increases the coverage, mitigating overconfidence in majority classes.
(iii) using a single offline reference image. Table 18 18 reports average accuracy, standard deviation (over 3 runs), and inference time. Although all yield decent gains over no correction, generating fresh references per batch slightly improves bACC while maintaining stable runtime. In contrast, using a single static image can lead mild performance fluctuations, likely due occasional bias interactions with that fixed phase pattern.
Table 18: Comparison of test-time reference usage CIFAR-IO-LT (y 100). We list average
bACC, standard deviation (ts) and relative inference time. PPer-Bachh' is our default.
Test-Time Strategy
bACC (+stdd)
Min. Acc
Relative Inference Time
No Correction
78,2 + 0.3
64.5
1.00
Per-Batch Random-Phass
83.6 0.4
72.1
1.20
Single Shot (Reused)
82.9 +
71.2
1.10
Offline Fixed Reference
82.6 + 0.5
70.8
1.05
These observations confirm that, while using one static reference already helps, dynamically regenerat- ing the random-phee -hasse image for each batch provides both consistent gains and only modest overhead. Overall, bias correction proves effective across different training/inference splits, pscudo-label strate- gies, and reference- gennrcttonnrttinnn schemes, with the best results emerging when we consistently apply logit subtraction in both training and test stages.
D.6 Extended Experiments on Extreme Imbalance and Distribution Mismatch
This section evaluates our method under more challenging conditions: (i) extremely high imbalance factors (IF) and (ii) severe mismatches between labeled and unlabeled distributions. By pushing each scenario greater extremes, we further validate how robustly our bias-correction scheme handles real-world long-tailed settings.
Performance under Extreme Imbalance Factors. We begin by testing imbalance factors IF 200 and 250 on CIFAR-10-LT, and, in an even more extreme scenario, limiting certain classes to only 1-5 labeled samples. Table 19 (referred to as "Table 9") highlights how quickly minority-class accuracy can drop in the presence of such extreme tails, and how various methods cope with this reduction. We compare our proposed Random-Plon against CDMAD, ABC, and SAW.
Amplitude Preservation (RP+Amp) method
33

---

Table 19: (Table 9) Comparison of different bias-correction strategies under extreme imbalance on LEFAR-1--LI We report balanced accuracy (bACC), geometric mean (GM), and minimum-clan accuracy (Min. Acc) averaged over three runs.
Method
Imbalance
bACC
GM
Min. Acc
CDMAD
IF=200
74.3
70.8
62.4
ABC
1F=200
75.1
71.4
63.2
SAW
IF=200
76.0
72.1
64.8
RP+Amp (Ours)
IF=200
78.6
74.3
66.9
CDMAD
IF=250
70.1
66.5
ABC
IF=250
71.7
67.6
59.4
SAW
IF=250
71.9
67.9
59.2
RP+Amp (Ours)
IF=250
74.2
70.8
61.9
CDMAD
1 1-5 labels
67.5
64.2
53.3
ABC
1-5 labels
68.4
64.9
54.7
SAW
labels
RP+Amp (Ours)
1-5 labels
71.6
67.8
57.5
As Table 19 demonstrates, all methods inevitably see a drop in performance with ever-larger IFs or extremely few labeled examples; however, approach continues outperform the alternatives. In particular, the minimum-class accuracy (Min. Acc) remains substantially higher, reflecting that subtracting the random-phase reference logits consistently prevents the classifier from discarding the rarest samples.
Multiple Mismatch Scenarios. We also examine scenarios where the unlabeled set diverges drastically from the labeled set. Table 20 ("Table 10"" summarizes four representative mismatch patterns, including partial vs. full class reversal, random class removal in either the labeled or unlabeled portion, and different imbalance factors between labeled and unlabeled subsets (e.g.. 100 vs. Tu 1, 50, 150). We provide bACC, GM, and Min. Acc for each setting.
Table 20: (Table 10) bACC, GM, and minimum cllass accuracy for various labelcd/unlabeled mismatch scenarios on CIFAR-10. "Partial Reversal" only flips the top-5 majority classes to minority status, whereas "Full Reversal" flips all classes. "Class Removal" indicates random omission either labeled or unlabeled data. Each result is averaged over 3 seeds.
BACC
GM
Scenario
Method
Min.
The
7a =
7/1
150
The = 50
Yo 150
(1) Partial Reversal: Only top-5 labeled classes become minority among unlabeled
CDMAD
76.1
77.3
74.2
71.6
72.5
68.5
64.2
ABC
77.2
78.5
75.3
71.9
73.2
69.4
65.0
RP+Amp (O.)
79.6
80.8
78.2
73.5
75.1
72.3
67.1
(2) Full Reversal: Labeled majority becomes unlabeled minority (and vice versa)
CDMAD
74.8
76.9
73.5
69.9
71.1
67.4
63.7
SAW
75.5
77.1
74.1
70.3
72.0
68.0
64.0
RP+Amp (0.)
78.7
79.9
76.5
72.6
74.2
71.0
66.3
(3) Class Removal (Labeled); Certain classes missing from labeled datu
CDMAD
66.3
70.2
69.5
62.0
65.1
64.2
56.9
ABC
67.9
71.3
70.6
62.8
66.7
65.4
58.7
RP+Amp (O.)
70.6
73.8
72.1
65.1
68.9
67.9
61.6
(4) Class Removal (Unlabeled): Certain classes missing from unlabeled data
CDMAD
72.1
74.5
72.7
68.9
70.2
66.0
61.0
ABC
73.4
74.9
73.1
69.3
70.9
66.7
62.3
RP+Amp (0.)
75.8
77.4
75.0
70.8
72.6
69.2
64.1
We also illustrate an example mismatch configuration in Figure g ("Fig. 6'), where classes 1-3 are more prevalent in the labeled set but become relatively scarce or even absent the unlabeled set, while classes 7-9 do the opposite. Such reversed partially missing structures often arise in
34

---

real-world scenarios when labeled samples come from a different distribution or domain than do unlabeled ones.
Labeled vs. Unlabeled Distribution Mismatch
350
Labeled Uniabeled
310
300
300
280
Instance Count
150
100
C1
C4
C6
C9
c10
Class Index
Figure 8: (Fig. 6) Example schematic of distribution mismatch, illustrating which classes shift between majority and minority status across labeled vs. unlabeled sets. Classes in blue are heavily represented in the labeled set but are rare or missing in the unlabeled portion; classes in orange show the reverse.
Overall, these experiments confirm that our random-phase reference framework generalizes well even under severe class -mballanceeeee expansions (e.g.. IF=250) and diverse mismatch configurations. The logit subtraction consistently moderates the classifier's inherent bias, preventing it from overfitting to majority patterns and preserving recognition capability the most underrepresented classes.
D.7 Extension to More Networks and SSL Algorithms
We now explore how our bias-correction approach generalizes across different backbone architectures and additional SSL frameworks. All experiments here use CIFAR-IO-LD unless otherwise stated, with an imbalance factor y 100. We focus on balanced accuracy (bACC), geometric mean (GM), and minimum class accuracy (Min. Acc) to gauge performance on minority classes.
Adaptation to Different Backbone Networks. To verify that our method ("RP+Amp") is not tied to particular network, replace the default ResNet-18 with either WideResNet-22--2 EfficientNet- BO and evaluate under the same semi-supervised setting. Table[T] the results alongside competing bias-correction baselines. Across all three architectures. RP+Amp consistently outperforms CDMAD, especially in terms of improving the minority-class recognition (Min. Acc).
to as "Table 11") lists
Table 21: (Table 11) Comparison different bias-correction approaches on three backbone networks under CIFAR-1O-LT ( 100). We report bACC, GM, and Min. Acc, each averaged over three seeds.
Backbone
Method
bACC
GM
Min. Acc
CDMAD
79.4
75.2
67.3
ResNet-18
RP+Amp (Ours)
82.3
78.6
70.2
CDMAD
80.1
76.0
68.5
WideResNee-28-2
RP+Amp (Ours)
83.5
79.4
71.4
CDMAD
78.6
74.8
66.7
EfficientNet-B0
RP+Amp (Ours)
81.7
77.5
69.1
35

---

Although the absolute accuracy varies with model capacity (EfficientNet, for instance, can sometimes yield higher overall accuracy given sufficient training), we observe consistent gap favoring our mugit-subtraction strategy. In particular, the improvement in Min. Acc underscores how effectively our pandom-phass reference continues to expose biases, regardless of the underlying network depth or width.
Additional SSL or Combination Frameworks. In addition FixMatch, ReMixMatch, and FreeMatch, we test our method on other SSL paradigms such Mean Teacher, UDA, and ACR to confirm broader compatibility. Table 22 ("Table 12") shows that, across these diverse semi- supervised routines, appending our bias-correction step (denoted "+ RP+Amp") consistently yields a 2-3% bACC boost on CIFAR-10-LE The gains are most pronounced for the minority classes, reflecting our method's ' general applicability even when pscudo-labeling consistency regularization differ from the FixMatch family.
Table 22: (Table 12) Extending our bias-correction to additional SSL methods on CIFAR-IO-LT (7 100). "+RP+Amp" indicates that we incorporate our random- -hasse reference strategy alongside each baseline.
SSL Method
+RP+Amp?
bACC
GM
Min. Acc
Mcan Teacher
No
77.5
72.8
65.9
Mcan Teacher
Yes
80.2
76.6
68.7
UDA
No
78.9
74.1
66.3
UDA
Yes
81.4
77.2
69.4
ACR
No
79.1
75.4
67.8
ACR
Yes
82.7
78.9
71.6
D.8 Domain Shift under Mismatched Labeled and Unlabeled Data
Although our primary mismatch experiments ($D.2) focus on differences in class distributions, real-world SSL often encounters more severe domain shifts between labeled and unlabeled sets (e.g., differing capture conditions, image styles, or acquisition domains). To evaluate whether our random-phur phasse image approach remains robust under such shifts, we construct scenario where the labeled set is derived from CIFAR- (n) 100) in its standard color version, while the wnlabeled set is drawn from a gray-scale variant of CIFAR-10 with a different imbalance ratio (7v 50). This setup combines both class-distribution mismatch and a non-trivial domain (color vs. gray-scale) shift.
Table 23 reports balanced accuracy (bACC), geometric mean (GM), and minimum-class accuracy (Min. Acc). We compare baseline FixMatch as well as CiXMatchhCDMMD with our Fix- Match+ (Ours). Although the overall accuracy drops compared to purely in-domain unlabeled data (see $D.2). our mandom-phasee image subtraction continues to deliver notable gains especially improving 2.5% bACC and 3.2% GM over the strongest baseline. These results suggest that even when unlabeled images exhibit a different style or capture modality, random- references still effectively reveal and correct the classifier's intrinsic majority-class bias.
Table 23: Domain Shift under Mismatch. Labeled set from CIFAR-IO-LT (color), unlabeled set from CIFAR-10-LD scaly) with different imbalance ratios. We show bACC/ GM / Min, Acc.
Method
bACC
GM
Min. Acc
FixMatch (baseline)
68. 4
63.5
51.0
FiXMatch+CDMAD
70.2
65.9
53,4
FixMatch+RRP4Amp (Ours)
72.7
69.1
55.5
D.9 Interaction with Strong Data Augmentation
integrate standard strong augmentations (RandAugment, Cutout, etc.) in algorithms such as FixMarch or ReMixMatch. One might wonder whether such augmentations overlap or conflict with random-phase transformations, potentially reducing benefits. To investigate this, we design two additional settings on CIFAR- 10-LT (? 100):
Our core experiments ($ 3.2 D.3
36

---

No Strong Augment: We remove RandAngmeni/Ctoou and retain only standard weak flips and crops.
Partial Strong Augment: We keep RandAugment but disable Mixup or other advanced
augmentations.
We compare these against the default Full Strong Augment pipeline. Table 24 shows that even without advanced data augmentation, our bias-correction still yields consistent gains (2-3% in bACC). Additionally, enabling partial or full strong augmentations further boosts overall accuracy, with random-
aundom-phase phase subtraction continuing provide additional minority-class improvements. We see no detrimental "double-counting" effect between random-phase transformations and typical data augmentations. Indeed, our than direct training augmentation applied to real samples.
references serve bias-estimation tool rather
Table 24: Random-PPie vs. Strong Augmentation. We compare no or partial strong augmentation to the default full strong augmentation in FixMatch under CIFAR-IO-LT (y 100). We show bACC (%).
FixMatch RP+Amp
Augment Setting
No Strong
Partial
Full
No Correction
72.4
77.1
79.9
+RP+Amp (Ours)
75.1
79.0
82.3
D.10 Scaling to Higher-Resolution Images
Finally, we examine whether random- phase references remain feasible and beneficial as image resolution grows. While most of experiments use images of size 32 32 64 x 64, many real-world tasks involve resolutions of 224 x 224 or higher. Table 25 compares training epoch time and inference latency (per image) on single GPU for three resolutions: 64 64, 224 x 224, and 384 x 384. We measure speed with/without the R2C-based FFT optimization described in $2.6) Even 224 224, R2C acceleration preserves roughly 40% speedup over naive FFT, allowing each epoch to finish in comparable timeframe to smaller resolutions. At 384 x 384, we observe increased overhead but still find that our method remains practically viable for moderate batch sizes (μ4 unlabeled ratio). In more resource-constrainge scenario, one could generate single random-phase image per epoch reuse small set of references inference further amortize the cost, while still capturing most bias-correction benefits.
Table 25: Performance on Higher Resolutions. Training epoch time (min) and inference latency
(ms/image) on single GPU for different resolutions (64 x 64, 224 224, 384 384). We show the effect of R2C FFT.
Train/Epoch
Inference
Resolution
FFT Type
Time (min)
A%
Latency (ms)
A%
64 x 64
No FFT
6.8
1.1
R2C FFT
3.3
50.0%
0.6
45.5%
224 224
FFT FFT
19.5
3.8
R2C FFT
11.5
41.0%
2.2
-2.1%
384 384
No FFT
28.6
5.7
R2C FFT
17.9
33.4%
3.5
-38.6%
These measurements confirm that while higher-resolution images do increase FFT cost, our R2C optimization makes random-phase bias subtraction remain tractable. Moreover, one can reduce overhead by generating fewer references at test time (e.g., one random-phase image per batch or per epoch), retaining the bias-correction benefits without excessive computational burden.
37

---

E 2D Fourier Transform Basics
A solid grasp of the two-dimensional discrete Fourier transform (2D-DFT) is vital for understanding how we manipulate amplitude and phase in our method. In this section, we provide detailed definitions, derivations, and proofs, following standard references such 21 and [6]
E.1 Notation and Coordinate Systems
Throughout this paper, we adopt zero-based indexing convention commonly used in deep learning frameworks (e.g., PyTorch, NumPy). Specifically, let
æ {0, 1..., H 1}, denote spatial coordinates, and
y {0, 1.......,
W - 1),
u 10. H 1}. v 10, W 1)
denote frequency-domain coordinates. Alternatively, some texts shift indices to center the frequency {
!/ # 1} x {- W W 1]- Here, maintain the simpler O-to- H 1) range at (- 2....
(-...... .2 layout for clarity and direct compatibility with most FFT libraries.
E.2 2D-DFT and Its Inverse
Let f(z, y) be a real-valued image of size H x W. Its 2D Discrete Fourier Transform (2D-DFT) is defined by
F(u, v) = ((f
H-1W-1
ΣΣ
f(z exp -j2n
)))
(10)
where j -II is the imaginary unit. To recover f from its spectrum F, we apply the inverse 2D-DFT:
#
W-1
f(z,8) = HW
Σ
Σ
F(u,v)
(11)
u=0 0=0
x exp
(
In some literature, the factor HW is placed in the forward transform or split evenly between the forward/inverse transforms [21]. Our choice here (normalizing only in the inverse) is common in many signal-processing contexts and does not affect correctness.
E.3 Amplitude and Phase Decomposition
write each dremann coefficient Flu, v) in polar form:
F(u, u) A(u, v) exp(i ((u,u)).
A(u, u) - (M((F(,,U)]''
o)
(12)
o(u, v) ((u,
where A(u, u) is the amplitude spectrum (or magnitude) and o(u, v) the phase spectrum. Intu- itively, A(u, v) reflects how much energy is present at frequency (u, v),, while ø(u, v) captures the alignment of that frequency component in the spatial domain 66]
E.4 Conjugate Symmetry for Real Inputs
key property arises when [xx y) is real-valued: its 2D-DFT satisfies conjugate symmetry. Formally,
F(u,u) v) = F((-u) H, (-r) mod W),
(13)
where the overline denotes complex conjugation. This means each coefficient F(u, v) has partner ((-u)mor mod H, mod W) that is its complex conjugate. As result, need only com- pute or store roughly half the spectrum, a fact exploited by many "real-to-complex" (R2C) FFT implementations.
38

---

DETAILED PROOF OF CONJUGATE SYMMETRY
Starting from the definition in (10), we have
#-1 W-1
Fuu, v) Σ
Σ
(x. exp(-jjrt ur + w
2m0
y
Since f is real-valued, define u' (-u) mod and v' = (-v) mod W. Then,
H-1 W
F(d(,')=
Σ
Σ
f(x, v) expl 2T
11'x v'y)
H
+
W
2=0
y=0
Using (-u) mod H (H u) when u + 0 (or 0 if 0) and rearranging terms, one show
F(u', F(u,v) v)
hence establishing
113 For full details, standard references like 21 66.
E.5
Practical Implications: R2C FFT Optimization
Since real-valued images inherently generate conjugate-symmeetii spectra, modern libraries (e.g.. NumPy, cuFFT, MKL) provide specialized R2C interfaces. These compute only the unique half of the frequency grid (for instance, E {O..... H }. thus reducing both runtime and memory usage without altering the underlying amplitude/phase information.
F
Limitations.
FARAD improves minority-class recognition across diverse long-tailed SSL, settings, yet several caveats merit attention. First, the method presumes that test images share broadly similar global colour-frequee statistics with the training data; under an extreme domain shift (e.g., visible-light training versus infra-red testing) the bias estimate may weaken. Second, although the batched real- to-complex FFT used to construct random- phase references adds only & 5-8% wall-clock overhead 32x32-224xx
inputs, very high resolutions could require additional engineering such as reference caching or spectrum truncation. Third, the current design is image-centric; adapting the "semantic- free yet statistically representative" reference concept to other modalities (e.g., raw audio) remains future work. Finally, uncommon aata-auumentttion pipelines that radically distort frequency spectra might reduce reference fidelity, so practitioners should verify compatibility when deploying such augmentations. These limitations do not undermine the core insight of FARAD, but outline scenarios where further refinement would be valuable.
G
Why Randomizing Phase Ensures Semantic Irrelevance?
While classical Fourier analysis show that randomizing the phase of image leads to "cloud- like" appearance devoid of coherent structures K1, such proofs often focus on spatial autocorrela- tion alone. Here, propose new measure of semantic overlap that generalizes beyond simple autocorrelation and aligns more directly with the notion of "semantic shapes or patterns." We then prove Vanishing Semantic Overlap Theorem, showing that random-phase images have negligible overlap with any shape template in a large dictionary, with high probability.
G.1 Setup and Notation
Let f: fO...., H - 1} x {O...., ...,,,W' W - 1} + R be any real-valued image of size H x W. We denote
its 2D discrete Fourier transform (2D-DFT) by
F(u,v)
Σ
Σ
f(z, y) exp jm
(14)
z=0
39

---

where u E HO - 1}, 11,,,-1-- DE {0, .. W -11 and V-1. form:
We can write each F(u, v) in polar
F(u,u) = A(u, exp(j v(u, where A(u, v) 2 O is the amplitude and ((u, v) E [--,,, the phase.
(15)
G.1.1 Random-Phase Images
random-phase image f is obtained by preserving the amplitude A(u, v) of f but replacing its phase 6(u, v) by random variable ((u, v) ~ [niformi-T,, n., i.i.d for all (u, v). Formally, define
F(u, v) A(u, v) exp(i((u.u))
(16)
and invert it via the standard 2D inverse DFT:
H
W-1
f(z.v) H H
Σ a=0
Σ
v) exp(j2r
ue
mw W
(17)
1=0
Because (uu, v) are i.i.d. uniform, any coherent phase relationships originally present in f are eradicated, causing f to exhibit no discernible edges or shapes.
G.2 A Novel Measure of Semantic Overlap
To formalize "semantic irrelevance," we introduce dictionary of shape templates ( & J each V (o,,.... ,,. H 1} x {O..... W - 1} + R. We do not assume these templates are orthogonal or disjoint; they can be arbitrarily overlapping, capturing edges, contours, textures, text glyphs, small "prototypical" shapes.
Definition (Semantic Overlap w.r.t. Shape Dictionary). Given an image g(x, y), define its correla- tion with the k-th shape template V as
H-1W-1
Corr(g, 12 l 1
Σ y=0
g(2, v) (x(2,)).
(18)
The semantic overlap measure (SOM) of g is then
SOM(g) = max ISRSM
Corr(g, Wa)
(19)
Intuitively, SOM(g) reveals whether g aligns strongly with any of the templates. SOM() = 1, then g matches least one V almost perfectly; ifg)) = then g is essentially unrecognizable w.r.t. this dictionary.
By choosing { } be a large and diverse of morphological or learned shapes, we can capture broad sense of "semantic structure" in images.
G.3 Main Result: The Vanishing Semantic Overlap Theorem
We now state our key theorem, showing that random-pher images have negligibly small SOM with high probability, meaning they cannot semantically match any template in {
Theorem 1 (Vanishing Semantic Overlap Under Random Phases). Let f (0...., H 7 1} x (0...., 1} + IR. be any real-valued image of size H x W, with amplitude spectrum A A(u, v) not identically zero. Consider any finite dictionary of shape templates bk each with || O(VHW). Construct a aandom-phase image by replacing Øu,, v) with i..d. uniform random phases 1 (u, v) ~ Uniform[- -T,, n. n), as in [6 17
Then there exist positive constants (0,,3), depending only on M and max Alu, v), such that for any O,
Pf[SOM(F) 2 e]
expl - 3 HW).
(20)
for all sufficiently large H, W. In particular, SOM(J) -> in probability as H, W -> co. Hence is semantically irrelevant to all shapes in the dictionary with overwhelming probability.
40

---

G.3.1 Interpretation and Consequence
The theorem implies that no matter which shape (edge, silhouette, contour, etc.) we try to find in f. the correlation is exceedingly small with high probability as H, W grow. In other words, random-phase images simply cannol host stable semantic patterns. Therefore, if classifier assigns significant confidence to particular class upon seeing f. that confidence must come from intrinsic bias, not from genuine semantic features.
G.4 Proof of Theorem
We provide a full proof below, structured in four steps: (1) rewriting Corr in terms of the random phases, (2) establishing mean zero behavior, (3) deriving concentration bounds, and (4) applying a union bound over all k.
Proof of Theorem Step 1: Expressing Corr(f, 1x in Random-Phase Form. From 18). let
H-1W-1
Corr (f,
ΣΣ
f(x, (x(2.v).
(21)
1l la
Substitute the inverse DFT representation
(7 for f(x y):
H- W-
j(x,y)
1
Σ
A(u, u) exp( D v))
HW
===
m=0
F(u,v)
exp
(22)
Plugging this into 21 yields large sum of terms, each involving the random exponential exp(.j (u, v)).
Step Mean-Zero and Off-Diagonal Cancellation.
Because (u,, v) Uniform[-Be т are independemt for different (u, v). standard Fourier-analytic arguments show
E expp( (u, r))] = 0,
(23)
and cross-terms the expanded sum vanish in expectation whenever (u, v) A ('), This implies
E Corr(J V&)]
for each k.
(24)
Hence, the random-phase image f does not align with V on average. However, we need a high- probability statement, not just expectation result.
Step 3: Concentration Bound via Bernstein/Hoeffdine Inequalities.
We now examine the variance and higher moments of Corr (f. Wz). Each term in the correlation sum is product of a bounded (by A(u, v)) complex exponential factor and (2 y). Under mild assumptions A(u, v) (i.e. A(u, v) < Amax < 0o), these can be treated bounded random variables. Applying Bernstein's 's or Hoeffding's inequality for sums of independent (or weakly dependent) random variables yields that, for each fixed k;
Pr Pr[ Fr[Coor(F Fx)))) Z
< C, cxp C2 HW).
(25)
for constants C₁. Cg > depending on | Amaxs and numerical factors. Intuitively, the HW
terms in the sum 22 destructively interfere due to random phases, so the net correlation rarely exceeds
Step 4: Union Bound Across All Templates.
Since we have templates, a simple union bound states
Pr
[Crr(f, Vx)) 2 MKKSM
max
<
Pr Pr[[Con(F ( 2e.
(26)
41

---

Applying (25 to each term yields
Pr Pr[sOM(f) 2 $ s M C exp(- HW).
(27)
Set a == M M and = C2; we obtain the desired exponential decay (20p. Since M and C1 may depend on the maximum amplitude Amax but not H or W, the probability goes zero rapidly as H, W grow. Hence with high probability,
SOM(f) max Corr(f, vx) E,
k
for arbitrarily small This completes the proof.
G.5 Extensions to Multi- Channel Images and Large Template Classes
RGB or Multi-Channel. If f has m > channels (e.g. m for RGB), we simply define independent random phases for each channel: &e(u, v) Uniform[- n). (l...., m]. The same proof applies, except we view each Vx as having dimension mWW or define separate shape dictionaries per channel. The result remains: the semantic overlap of f to any shape is negligible with high probability.
Infinite (or Very Large) Template Families. A finite dictionary { is adequate for many practical shape sets, but might also consider an uncountably large family of templates with bounded complexity (e.g., certain parametric shapes or wavelet-based expansions).
such scenarios, advanced chaining arguments from empirical process theory [29 45 can show a similar phenomenon: the supremum of the correlation over that (potentially huge) family remains small under random- -happee transformations, provided the family has finite VC dimension covers compact set of shapes. detailed presentation is beyond our scope, but the principle is analogous.
G.6 Implications for Bias Correction
Theorem offers powerful guarantee: semantic pultern from the dictionary can be found in random-phase image. Consequently, any significant classifier response on f must reflect intrinsic bias rather than legitimate shape or texture cues. This underpins the motivation for subtracting logits on random- madoe- references to remove biased responses, especially in long-tailed SSL (22).
Summary. By introducing the Semantic Overlap Measure (SOM) and proving it converges to zero under random-pin phase transformations (Theorem D. we establish a rigorous foundation for the claim that phase randomization yields semantic irrelevance. This theoretical result justifies using random- phase images bias-reference in challenging learning scenarios, ensuring that any non-trin non-trivial classifier activation such images cannot arise from actual semantic content.
H
Why Preserving the Amplitude Spectrum Reflects Real Data Statistics?
Preserving the amplitude spectrum of image is often described as "retaining second-order statistics" 337. but this explanation can feel like basic collage of well-known theorems. In this appendix, we introduce Global Spectral Alignment (GSA) framework that directly quantifies how well an image's frequency-energy distribution matches that of real data. We then prove a Vanishing Spectral Divergence result, ensuring that the avee-avveeeged amplitude converges the true spectral profile of the dataset. This viewpoint clarifies why amplitude preservation is so effective at capturing "global" image statistics-ewt even we randomize phase to destroy semantic structure.
H.1 Global Spectral Alignment (GSA) and Spectral Divergence
Let f (O..... Hl 1} x (O...., W - 1} - R a real-valued image. Its 2D discrete Fourier transform (2D-DFT) is
W Σ
F(u,v)
Σ
f(z, v) exp -j -j2n 2T
+
my W
(28)
y=0
42

---

where 0 I a H - and 0 I u W - 1. We write
F(u,v) = Ay(u, u) exp(j (5( v)),
(29)
where Ay(u, 2 is the amplitude and d (u,, v) E [- m] the phase.
Amplitudes in a Dataset. Suppose we have a dataset D of real images, each of size H W. For each image x D, its Fourier transform has amplitude Az(u, We define the (unknown) population amplitude distribulion as the joint distribution of {An(u, (u..) (u.)) when is sampled
from D.
Definition 2 (Global Spectral Alignment Measure (GSA)), Let Ay(u, v) be the amplitude spectrum of an image f, and p(u, v) be a target mean amplitude profile (e.g., the dataset-average). We define a global alignment measure:
W-1 GSA(F:n) Σ
Wayv Af(,,)) u(u,v))
(30)
where p 2 1 is a chosen exponent (e.g., p= 2) and {Wae} are nonnegative weights (e.g., all ones, emphasizing particular frequencies). low GSA(f: μ1) means Ay(u, u) is close to ulu, 2) across all frequencies, indicating strong alignment with the desired global amplitude profile.
If juu, v) represents the true average amplitude from the dataset D. then (f: μ)) measures how closely f's amplitude matches the global frequency-energy allocation found in real data. Crucially, GSA is agnostic to phase: two images with the same amplitude spectrum yield the same value of GSA relative to μ..
H.2 Preserving Amplitude + Preserving Second-Order Energy Distribution
Classical theory often highlights that sharing the same amplitude spectrum implies sharing the same autocorrelation magnitude via the Wiener-Khinchin theorem 22. Below restate stronger viewpoint using our GSA measure to clarify that amplitude preservation is precisely what preserves second-order frequency statistics.
Theorem 2 (Amplitude Preservation and Second-Order Alignment). Let fi and fz be two real-valued images with amplitude spectra A,, and A₁2- Suppose A,, (u, v) Af (u, v) for all (u,v). Then:
(i) GSA(Si; Asa) = 0 and GSA(f2; Ani) O, meaning and f2 are perfectly aligned in amplitude space.
(ii) fi and f2 have the same power spectrum, hence the same autocorrelation magnitude up to any global phase shift.
the sum GSA is zero. if A (u, = A iz (u, v) for all (u, v), then | (u, A,,
(i) From 30
(u, w)] = 0, so
(ii) By definition, [F, =))²= ( 2) implying that fi and f2 share the same power spectrum. Classical Khinern arguments then show that fi and f2 induce identical autocorrelation magnitudes 337 2]
This result illustrates that merely fixing Ay(u, v) ensures strong second-order statistical match, even we arbitrarily rearrange or randomize the phase (u, v). As result, phase-randomizer images preserve the data's global energy patterns while discarding recognizable spatial cues.
H.3 Vanishing Spectral Divergence from Batch-Averaged Amplitude
Let 'D be dataset (or distribution) of real-valued images of size H x W. We draw B samples {x(b) BB Tim1 from D and compute each image's amplitude Ay(m) (u, v). We then form the batch-averaged amplitude:
1 Σ
Ab(
2 (u, v).
(31)
B
43

---

Intuitively, Ag(u, v) estimates the mean frequency-eners that images in D allocate at (u, v). We now formalize this through a new Vanishing Spectral Divergence theorem, showing that Ag converges to the true amplitude distribution of DD in the sense of our GSA measure.
Theorem 3 (Vanishing Spectral Divergence of Batch-Amplitudes). Let AD(u, v) be the (random) amplitude of an image sampled from DD Assume AE[ ri]l < oo for all (u, u) and define
pD(u, v) = [[A
i.e. the true mean amplitude. For batch of B samples {-1² H
Tbz from D, define AB(u, v) as in (31).
and let fn be any image whose amplitude is exactly (u v) (with arbitrary phase). Then under suitable boundedness assumptions on AD.
GSA PD)
G.S.
(32)
where GSA defined in 30). Hence the batch-averaged amplitude converges (almost surely) to the dataset's s true mean amplitude profile.
PROOF SKETCH
Step Pointwise Convergence of An(u, v). By the strong law of large numbers, for each fixed (u, v) we have
B
A
Σ (u, = uD(u,v), B +
B
This holds almost surely whenever E [ll,, w)ll co.
Step 2: Summation over Frequencies. Since there are finitely many frequency bins (u, v) {O..... 1} fo..... 1}. the pointwise convergence of AB(u, v) + polu, v) implies
max (B po(u, _a, 0. M0
Step 3: Convergence GSA. From 30).
GSA(SB: PD)
Σ Wa. AB(U, uD(u,u)|*
The maximum absolute difference above goes to zero almost surely, so the entire sum converges to zero. Hence,
GSA (D) - 0.
Thus, fB's amplitude distribution globally aligns with the dataset's mean amplitude profile.
Interpretation. In simpler terms, if you sample enough images from real dataset, the average amplitude each frequency almost surely converges to the dataset's true average amplitude. Con- structing an image fa by plugging in Ap(u, v) (with random phases) yields image whose global frequency-enene -nnegyy distribution matches that of the dataset (i.e., GSA - 0).
H.4 Implications for Bias Correction in SSL
By Theorem 3 random-phase image whose amplitude is batch-averaged faithfully reflects the dataset's overall energy composition-cont colors, brightness, texture cetpuenciis-wthhor retaining any shape cues. Thus:
Statistical Representativeness:
The amplitude matches real data aggregate, does not introduce artificial color or frequency biases.
(2) Semantic Irrelevance: Phase randomization (see Appendix G removes coherent edges or shapes, leaving recognizable semantic content.
Together, these points ensure that if a classifier responds strongly to such a random-phase image, must be expressing learned bias rather than detecting genuine semantic structure. This directly
44

---

justifies using amplitude- umplitude-preserved, phase-randomized images references for bias correction in long-tailed imbalanced SSL.
H.S Conclusion and Forward-Looking Remarks
Summary. We presented a new viewpoint on amplitude preservation via the Global Spectral Alignment (GSA) measure, bridging classical second-order statistics with intuitive metric on how close an image's s amplitude is to a desired spectral profile. The Vanishing Spectral Divergence Theorem (Theorem B shows that batch-averaged amplitudes converge the dataset's mean fre- queney distribution, yielding random-phars images that accurately capture the dataset's s global energy composition.
Takeaway for SSL and Beyond. By preserving the amplitude spectrum while randomizing phase, we guarantce that crucial low-level statistics (second-orde) or "global requency-energy" remain realistic, yet all higher-order shapes edges are obliterated. This makes random-phas -hase images an ideal reference for bias subtraction, ensuring that any classifier response to them reflects intrinsic bias rather than legitimate semantic evidence.
Overall, these results provide a principled theoretical basis for why amplitude preservation captures "global" dataset statistics even under severe phase corruption, thus aligning with real data distributions while remaining free of any recognizable structure.
I Why Random-Phase Image Logits Reveal the Classifier's "Default Bias"?
models exhibit default biases toward certain classes, even in the absence of semantic cues [4]. While this phenomenon can be illustrated empirically, we provide here rigorous new framework that explains why random-phuse images (RPI) expose those biases. Our presentation adopts a novel Bias Disentanglement Measure (BDM) to quantify how much a classifier's s logits reflect true feature evidence vs. intrinsic priors. We then prove that RPI, which lacks meaningful semantic correlations, forces the classifier to rely solely on its internal priors thus isolating the "default bias" vector logit space.
In many classification settings- partscularlllllllyy under class imbalance -mounlsee-
I.1 Classifier Logits and Bayesian Factorization
Let go X > I be a C-class logit function, parameterized by 6, and
exp(go(z)c)
Pa = (x) = D 2k-1 exp(go(2)))
(33)
We think of ge(z) as approximating log Poly c x) (up to shared offsett 4 decomposition often separates log Po(y I) into two parts:
A Bayesian
log Po(y = e x) log Poly = c) + E(r,)),
(34)
prior term
evidence term
where log Po(y e) captures the classifier's internal class bias, and E(x, c) is the log-likelihood ratio or "feature evidence" that x genuinely belongs to class e [16]
L.2 A New Bias Disentanglement Measure (BDM)
While (34) is conceptually standard, we introduce a measure that directly gauges how much of a logit vector arises from bias evidence:
Definition (Bias Disentanglement Measure (BDM)). For an input x and classifier go. define
BDM( x) (1))(x) [log Poly = 1)....,,oog Poly C)]]l,,
(35)
where || Ilp a norm (e.g. p = 1 or parp- = 2). BDM(gg: x) R2 0, then ge close to the pure prior vector [log Po = c)]E= indicating that x contributed little or no feature- evidence.
In practice, log Po e) is not directly accessible. But as we now show, an RPI prompts the classifier to produce logits arbitrarily close these prior terms, driving BDM near-zero.
45

---

I.3 Random-Phase Images (RPI) Contain No Discriminative Evidence
Let Frand be an image constructed by preserving only the amplitude spectrum but randomizing the phase (Appendix G Hn. Since random-phasn phase images exhibit no recognizable shapes or edges, we assume that Frand fails to provide any meaningful evidence about class membership. Formally:
E {(rradd c) O, Vc (I,...,C).
(36)
Thus, if (2) approximates log Poly z)) the classifier's ' logit vector Zrand must be near: go (FFrad)) s log (1 = c)2 exposing the model's default bias or "prior" toward class
(37)
1.4 Main Theorem: Random-Phase Logits Reveal Default Biases
We now give a theoretical guarantee that if Frand is uninformative about y, then go (Frand) converges to the classifier's 's internal prior logits. turn, the difference go(x) -g([rraa)) disentangles the true feature-based evidence.
Theorem 4 (RPI Logits Classifier Prior). Let go be C-class logit function. Suppose Frand is drawn from a distribution that contains no class-relevant features (i.e., BDM Frand) ) + 0 in
probability). Then as Frand becomes increasingly phase-randomizer
go (Frand)
[log Poly = 1)....,,e, log
= Po
prob 0.
(38)
Consequently.
go (z) go (Fraad)))
((x,,))...,,(,,,
(39)
"bios subtraction" which is purely the evidence contributed by x.
SKETCH OF PROOF
From the theory of random-phase images (Appendix G). if Frand has no spatial structure or shape cues, then it lacks any discriminative features for ga. In Bayesian terms, (Irand RJ O.
Step 1: Random Phase => No Class-Specific Features.
Step 2: Logit Vector Converges to Class Priors. By (34).
go (Frand)) log Poly = e) + El (Frand, c) log Poly = c).
R00
Hence go (Trann))) converges to log Pa(y = c) That is, as Trand becomes increasingly
"shape-free," the classifier's logits become purely default bias terms.
Step 3: Bias Disentanglement, From Definition 3 BDM(go: Zrand) - 0 means go (Zrand) is arbitrarily close to the prior vector. For any real input -r subtracting these RPI logits yields
ge
go(x) go(
[[(x; 1), E(x, C)].
Thus we isolate the log-likelihood difference that æ provides relative to the prior. This completes the subtaaubtn mechanism.
1.5 Further Remarks and Practical Consequences
(a) Links to Class Imbalance: When training data are imbalanced, the classifier's ' learned log Po(y c) becomes skewed toward majority classes. An RPI then reveals that imbalance directly in the logit outputs (often strongly favoring certain classes), and subtracting it effectively "de-biases" the model.
(b) Enhanced SSL Labeling: supervupe learning, subtracting the RPI logits helps produce more balanced pseudo-labels, mitigating the tendency to over-assign unlabeled samples to majority classes.
(c) Comparison with Uniform Gray References: uniform gray image can also reveal bias to some extent 33), but it does not preserve any real amplitude statistics and can inadvertently overestimate bias. In contrast, a random-phase image matches global color/frequency distributions (Appendix while remaining shape-free, leading to more faithful estimate of prior logits.
46

---

1.6 Conclusion
Key Insight. A random random-phase image Frand with no dlas-discriminnttve cues draws out the classifier's default biases. In logit space, (o(Trand) approximates [log Po = any difference ge (z) go(Trand) pure evidence term for æ. Thus, RPI-based bias subtraction precisely disentangles the prior from the discriminative signal.
Po = C)}, making
By theorem and novel measures (BDM), elevate what is often explained purely intuitive level-i.e., "random-phase images have no shape, so they reveal bias"-into more formal statement about Bayesian decomposition and logit-space geometry. This offers robust theoretical founda- tion for using random-phee -haae references in long-tailed or semi-supervised learning contexts, where controlling classifier priors is crucial.
J
Why FFT Acceleration Does Not Violate Theoretical Assumptions?
Practical FFT implementations employ real-to-complex (R2C) shortcuts, batched parallelism, and floating-point arithmetic. A natural concern is whether these approximations undermine the theoretical guarantces of random-phase irrelevance ( or amplitude preservation (HH). We present here a new framework that quantifies potential distortions via two novel measures- Bounded Phase Distortion (BPD) and Amplitude Fidelity Divergence (AFD)-a and prove they remain negligible for typical image sizes and precision levels. This shows that efficient FFT computations preserve all essential theoretical assumptions with overwhelming probability.
J.1 Bounded Phase Distortion (BPD) Under Floating-Point Arithmetic
Let f: (o,...., - x (O..... Fourier transform (2D-DFT):
W 1} + E be an H x W image, and let Flu, v) be its discrete
H-1W-1
v(u,u)
ΣΣ
[(x, () expl
expp-jjjeee -j2n
vW
random-phase image replaces F(u, v)'s phase with some i.i.d. random ((u, Uniform[ m, m). In prating-point arithmetic, we denote the computed phase as
(u, v) 5u,, v) + 6u,07
(40)
where o, represents roundoff errors or parallel-summation discrepancies. We measure these errors by: Definition (Bounded Phase Distortion (BPD). Define
BPD( max 60,0
(41)
0.y
the maximum absolute deviation of the computed phase from the ideal ((u, v). IJBPD(5) is small (e.g. < To of radian), then the phase remains effectively uniform on |, at all (u, v) in practice.
A standard floating-poin analysis 18 implies that BPD(8) O( log HW)) in typical GPU- based FFT libraries, where e machine precision (&x 10-7 in single precision or 10-15 in double). Hence the random-phase distribution cannot deviate by more than small fraction of radian in any frequency bin.
Why Small Phase Distortion Does Not Reintroduce Structure. Even perfect random phase can theoretically be off by integer multiples of 2T without changing exp(j (buu,) u)). Thus, small deviations under floating-point rounding are vanishingly unlikely to "re-synchronize" phases across different (u, v) in a way that reconstructs edges or shapes. Formally, as long as BPD()) & म, the mandom-phase arguments in still hold with high probability.
J.2 Amplitude Fidelity Divergence (AFD) in R2C Transforms
Real-to-Complex (R2C) Compression. For real f(z y). the 2D-DFT F(u, v) satisfies conjugate symmetry:
P((-ummod H, (-v) w) F(u, v),
47

---

so modern libraries compute and store only the "unique half" of F(u, v). We denote this partial storage by FRec(u, v) for (u, v) E Q S {O...., H 1] (O..... W 1}. Reconstructing F(u, v) outside St is straightforward: Fu, v) Fuzc(u', v²) for an appropriate (u', o') E Q.
Amplitude Fidelity Divergence. Since amplitude preservation is key to retaining global statistics, we define:
Definition (Amplitude Fidelity Divergence (AFD)), Let A(u, v) be the computed amplitude after an R2C FFT (and inverse FFT if needed), and let A(u, u) be the ideal amplitude in exact arithmetic. We define
AFD(F)
Σ a=0
Σ
A(u,v) - A(u,v).
(42)
==0
A small AFD(f) means the R2C compression and floating-poit roundoff introduce negligible amplitude distortion relative to the ideal A(u, v).
can relate AFD(J) to well-known bounds on FFT stability
18
Theorem 5 (Amplitude Preservation in R2C Computations). Consider an X W real image fxz y) with N HW and amplitude A(u, v) |F(u, t)). Suppose R2C FFT (in single- or double-precision) produces a partial spectrum Frze from which F(u, u) is reconstructed. Let A(u, v) = [(,, v)l be the computed amplitude. Then with high probability,
AFD()) S (log N) N max [(x,,))) 2.y
(43)
where C 0 is small constant. Consequently, AFD(S) scales linearly with N, log N, and meaning fidlitude remains robust for typical image sizes and standard hardware precision.
SKETCH
We leverage standard backward-error analyses 18 44 of FFT computations: each butterfly operation accumulates floating-point error Oee) times local partial sums. Summing across O(N log N) operations yields O(e N log N) final error in the worst case. Since amplitude Ri (RFF22 + (F)² second-order expansion shows amplitude errors are similarly bounded by O N log N). Summing over all (u, v) yields (43) up to a small constant C' from union-bounding over H x W bins.
J.3 Combined Theorem: Negligible Global Impact on Random- Phase Images
Finally, we show that even when both BPD and AFD are considered, random-phase images remain shape-free and amplitude-realisie with overwhelming probability:
Theorem 6 (Global FFT Stability for Random-Piin Phase Construction). Let f be an H W real image, N HW, and e be the floating-point unit roundoff. Suppose we create a random-phan image J by:
(a) R2C-FFT to get FR2C.
(b) Replace phases by Ō(u, u) in floaaing-poinn to get ((u, v) TB v) + 8a,
(c) Inverse R2C-FFT to yield? f.
Then:
(i) The phase distortion satisfies BPD()) S O(E log N) with high probability.
(ii) The amplitude fidelity deviation satisfies AFD(F) S O( N log N) with high probability.
Hence none of the random-phens phase irrelevance or implitude-statistee arguments are broken by these approximations, as N logg N e remains small for typical image sizes (e.g. S 107) and standard single/double precision.
48

---

(i) Follows from bounding floating-point roundoff within each frequency bin: each 6,, ( log N) [18] (ii) Directly from Theorem [ which applies identically for the forward and inverse R2C transforms. Since small BPD and small AFD imply that neither the phase randomization nor amplitude magnitude is significantly altered, the theoretical properties
established B and remain valid.
J.4 Conclusion and Outlook
Key Insight. Real-to-complex FFT shortcuts, parallel batched transforms, and finite precision all induce small deviations in computed amplitude/phase. However, these are rigorously bounded by O(N log N E), which remains negligible for practical image sizes and hardware. Consequently:
Bounded Phase Distortion (BPD) ensures madomm-paase angles remain effectively uniform, preventing any reappearance of semantic structure.
Amplitude Fidelity Divergence (AFD) remains low enough that global energy distributions are faithfully preserved.
Takeaway. Far from violating our theoretical foundations, modern FFT accelerations simply compress the transform domain by exploiting conjugate symmetry (R2C) and reorder summations in parallel. They do not meaningfully alter the amplitude or phase randomization at the levels required to reintroduce shape
degrade amplitude-based statistics. Hence all the key arguments about random-phan irrelevance, bias subtraction, and amplitude realism still hold in floating-point large- batch FFT practice.
In short, efficient FFT implementations well-aligned with the high-level theory, ensuring the random-phan phase approach retains its mathematical integrity under real-world computational con- straints.
49


Reviews:
[{"review_id": "Ev2c3YxNqx", "reviewer": "Reviewer_jUz2", "summary": "The paper proposes FARAD, a new method for bias correction in semi-supervised learning\n(SSL) under class imbalance. FARAD generates random-phase images by preserving the\namplitude spectrum of real images while randomizing the phase. This design removes semantic\ncues (no meaningful shapes) and preserves statistical representativeness (realistic\ncolor/frequency statistics). By feeding these random-phase images into the classifier, FARAD\nestimates its intrinsic class bias and subtracts this bias from real predictions during training and\ninference.", "questions": "NA", "limitations": "NA", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strength:\n\n1) Achieves improved performance over state-of-the-art baselines across multiple SSL\nalgorithms and datasets.\n2) Demonstrates compatibility with multiple SSL frameworks (FixMatch, ReMixMatch,\nFreeMatch), showing generalizability.\n3) The paper also provides theoretical insights showing that random-phase images reliably\neliminate semantic content while preserving dataset-level statistics.\n\nWeaknesses:\n1) The method relies on the amplitude spectrum of training images matching test data.\nExtreme domain shifts may weaken bias correction.\n2) The paper does not report training and inference time compared with other baselines\nwhich is important to realize the scalability of the proposed method compared to other\napproaches.\n3) All benchmarks in the paper (CIFAR, STL, Small-ImageNet) are natural image datasets.\nThere&#39;s no evaluation on other types of datasets (e.g., medical images) where class\nimbalances are more pronounced.\n4) While the paper shows that batched R2C FFT makes the method efficient, it assumes\navailability of GPU FFT libraries. This can be a barrier for deployment on constrained or\nnon-GPU devices.", "was_revised": true, "final_justification": "Thank you for providing a detailed rebuttal. The additional explanation and experimental results have addressed my initial concerns. As a result, I have updated my final rating accordingly.", "rebuttal": "#### **Regarding Weakness 1: Robustness to Extreme Domain Shift**\n\nWe agree that robustness under extreme domain shift is a key measure of generalization. To validate FARAD's performance in such challenging scenarios, we conducted a dedicated experiment in **Appendix D.8** of our paper. In this experiment, we created a harsh setting that combines both class distribution mismatch and domain feature shift: we used the color version of CIFAR-10-LT ($\\gamma_l=100$) as the labeled dataset and a grayscale version with a different imbalance ratio ($\\gamma_u=50$) as the unlabeled dataset.\n\nAs shown in **Table 23** of the appendix, while all methods' performance decreased due to the domain shift, our FARAD (72.7% bACC) still significantly outperformed CDMAD (70.2% bACC) and the FixMatch baseline (68.4% bACC). This result provides strong evidence of FARAD's robustness. The underlying reason is that even when the image color space changes, the frequency energy distribution (i.e., the amplitude spectrum) corresponding to intrinsic textures and structures is largely preserved. FARAD captures these more fundamental statistics to estimate bias, making it more robust than methods like CDMAD that rely on simpler color-based cues.\n\n---\n\n#### **Regarding Weakness 2: Lack of Training and Inference Time Comparison**\n\nWe thank the reviewer for this important question regarding computational efficiency. We acknowledge this comparison was missing from the original manuscript. To address this, we have conducted a new experiment and will add the following detailed time comparison to the final version of our paper.\n\n**Table A: Training and Inference Time on CIFAR-10-LT (single NVIDIA A6000)**\n| Method                                    | Avg. Time per Epoch (min) | Increase vs. Baseline | Inference Latency (ms/img) |\n| :---------------------------------------- | :------------------------ | :-------------------- | :------------------------- |\n| FixMatch (Baseline)                       | 6.5                       | -                     | 0.5                        |\n| FixMatch + CDMAD                          | 6.6                       | +1.5%                 | 0.5                        |\n| FixMatch + FARAD (Naive FFT)              | 9.8                       | +50.8%                | 1.1                        |\n| **FixMatch + FARAD (R2C FFT Optimized)** | **6.9** | **+6.2%** | **0.6** |\n\nAs shown in Table A, while a naive FFT implementation introduces overhead, our **batch-wise Real-to-Complex (R2C) FFT optimization** (detailed in Section 2.6 and Figure 2) is highly effective. With this optimization, FARAD adds **only about 6.2%** to the training time per epoch. We believe this minor cost is an excellent trade-off for the consistent ~3% bACC performance improvement across all imbalance scenarios. This result demonstrates that our method achieves significant gains while maintaining high scalability.\n\n---\n\n#### **Regarding Weakness 3: Lack of Evaluation on Other Datasets like Medical Images**\n\nThis is an excellent forward-looking suggestion. To address the concern about generalization to different domains, we have followed the reviewer's advice and conducted a new experiment on a **real-world, challenging, long-tailed medical imaging dataset: NIH ChestX-ray14**. This dataset is well-known for its natural and severe class imbalance, making it an ideal testbed for long-tail learning algorithms.\n\nWe constructed a semi-supervised long-tailed classification task on this dataset. The results are as follows:\n\n**Table B: Performance on Long-Tailed NIH ChestX-ray (Mean AUC)**\n| Method                     | Mean AUC (Area Under the Curve) |\n| :------------------------- | :------------------------------ |\n| Baseline (FixMatch)        | 0.781                           |\n| + CDMAD                    | 0.805                           |\n| **+ FARAD (Our Method)** | **0.823** |\n\nThe results show that FARAD not only outperforms the baseline but also achieves a **1.8% improvement over CDMAD**. This provides strong evidence that FARAD's effectiveness is not limited to natural images. This advantage stems from the nature of medical imaging, where diagnoses often rely on **textures, tissue morphology, and frequency patterns**. By preserving the amplitude spectrum, FARAD generates reference images that are statistically consistent with real pathological textures, allowing it to more accurately probe and correct the model's bias towards common diseases (majority classes). We will add this experiment to our final version to fully demonstrate our method's generalization capabilities.\n\n---\n\n#### **Regarding Weakness 4: Dependence on GPU FFT Libraries**\n\nThank you for this practical question about deployment. We acknowledge that our efficient implementation benefits from modern GPU-accelerated libraries, which are standard for deep learning **training**.\n\nHowever, for resource-constrained **inference** scenarios, we have already evaluated a solution in **Appendix D.5 (Table 18)**. The **\"Offline Fixed Reference\"** strategy addresses this perfectly:\n1.  After training, we pre-generate one or more fixed reference images from the training set's overall statistics.\n2.  During inference, we use these static images directly, **requiring no FFT calculations on the deployment device**.\n\nAs shown in Table 18, this approach maintains a **strong performance** (e.g., 82.6% bACC vs. 83.6% for the online version), still significantly outperforming all baselines while completely removing the dependency on real-time FFT computation. This provides a practical and efficient pathway for deploying FARAD across diverse hardware environments."}, {"review_id": "eheF2MHIj3", "reviewer": "Reviewer_imHP", "summary": "This paper introduces FARAD, a new method for bias correction in the challenging context of long-tailed semi-supervised learning. The core idea is to estimate and subtract classifier bias by using a specially designed reference image, termed a \"random-phase image\". This reference is constructed by preserving the batch-averaged amplitude spectrum of real data while randomizing the phase spectrum. Through extensive experiments on standard benchmarks like CIFAR-10/100-LT and Small-ImageNet-127, the paper demonstrates that FARAD significantly outperforms state-of-the-art methods.", "questions": "- The proposed bias correction mechanism appears to be quite general. Could the authors comment on its applicability to the standard fully-supervised long-tailed learning setting? For instance, has the method been tested as a post-hoc adjustment for a pre-trained supervised model, or integrated directly into supervised training? Such experiments could further demonstrate the method's generality beyond the SSL context.\n- The paper states that a new random-phase image is generated for each test mini-batch, which implies that the prediction for a single test sample is not deterministic and can vary depending on the other samples in its batch. This is a potential drawback for practical deployment. Could the authors clarify the rationale behind this design choice versus using a single, fixed random-phase image (e.g., generated from the training set statistics) for all test-time inference, which would ensure deterministic predictions?\n- In Table 1, several results for baseline methods  are missing for certain imbalance factors.", "limitations": "Yes.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n- The paper is well-motivated, identifying a clear and plausible limitation in existing reference-based debiasing methods. The proposed solution is a principled and logical response to this identified gap.\n- The method achieves impressive state-of-the-art results across multiple challenging long-tailed SSL benchmarks. The significant gains over strong baselines convincingly demonstrate its effectiveness.\n- The experimental evaluation is thorough, featuring extensive ablation studies, robustness checks under various conditions, and detailed analysis. The work is well-supported by its appendices, enhancing its rigor and reproducibility.\n\n## Weaknesses\n- The title, \"Fourier Clouds,\" is catchy but its connection to the paper's content is tenuous. The term \"Fourier Clouds\" is not defined, explained, or used within the body of the paper. This creates a slight confusion.\n- While the empirical gains are clear, the conceptual contribution could be viewed as an incremental improvement over the reference-based debiasing paradigm established by CDMAD. \n- The paper provides extensive theoretical analysis of random-phase images but surprisingly lacks any visual examples. Including a figure that shows what these generated images look like for different datasets would provide invaluable intuition for the reader. It would help to directly supporting the central claims of the paper.\n- The related work section omits several recent and highly relevant works in Imbalanced SSL. A more complete literature review should include and discuss methods such as:\n 1. DASO: Distribution-Aware Semantics-Oriented Pseudo-label  for Imbalanced Semi-Supervised Learning\n 2. Towards Realistic Long-Tailed Semi-Supervised Learning: Consistency Is All You Need\n 3. Twice Class Bias Correction for Imbalanced Semi-Supervised Learning.\n 4. InPL: Pseudo-labeling the Inliers First for Imbalanced Semi-supervised Learning", "was_revised": true, "final_justification": "All my concern has be address.", "rebuttal": "### **Regarding Weaknesses**\n\n#### **1. On the Connection of the Title \"Fourier Clouds\" to the Content**\n\nWe agree that the term \"Fourier Clouds\" could be more clearly explained in the manuscript. We chose this title as a concise and descriptive summary of our method's core.\n* **\"Fourier\"** refers to the core technique we use: the Fourier transform.\n* **\"Clouds\"** comes from the visual appearance of our generated random-phase images. As we state on lines 141 and 1106 of the paper, these images visually resemble \"turbulent or cloud-like patterns.\"\n\nIn the final version of the paper, we will add a visual example (as mentioned in our response to Weakness 3) to make this \"cloud-like\" appearance clear. We will also add a sentence to the introduction to explicitly explain the origin of the term.\n\n---\n\n#### **2. On the Conceptual Contribution Being an Incremental Improvement over CDMAD**\n\nWe believe FARAD represents a fundamental paradigm shift, not an incremental improvement over CDMAD.\n\n1.  **Problem Reframing: From \"Zero-Information\" to \"Optimal-Information\".** CDMAD's paradigm asks, \"How can we measure bias with a zero-information reference?\" Our work asks a deeper question: \"What is the *optimal* information content for a reference image to most accurately measure classifier bias?\" Our answer—semantically blank but statistically representative—marks a shift from eliminating all information to preserving only the critical statistical information.\n\n2.  **Theoretical Grounding: Addressing the Core Bias of CNNs.** It is well-established that CNNs have a strong \"texture bias.\" CDMAD's blank reference completely ignores this. FARAD is the **first** reference-image debiasing method to directly address this. By preserving the amplitude spectrum, which encodes texture and frequency statistics, our method precisely probes and corrects this core bias inherent to CNNs.\n\n3.  **Validation Through Performance:** The consistent and significant ~3% performance gain across all scenarios validates our new conceptual framework. A simple incremental change would struggle to explain such a robust and universal advantage.\n\nTherefore, FARAD's contribution is a new, more theoretically profound, and practically effective paradigm for bias estimation, not just a simple replacement of the reference image.\n\n---\n\n#### **3. On the Lack of Visual Examples for Random-Phase Images**\n\nThis is a very pertinent criticism. We agree that a visual example is crucial for understanding our method, and we apologize for this oversight.\n\nWe will add a high-quality comparative figure to the final version of the paper. This figure will show (a) an original image, (b) the blank reference image used by CDMAD, and (c) our generated \"Fourier Cloud.\" This will provide valuable intuition and support our core claims of \"semantic-agnosticism\" and \"statistical-representativeness.\"\n\n---\n\n#### **4. On Missing Important Literature in the Related Work Section**\n\nThank you for pointing out these important related works. We will expand our literature review in Appendix A of the final version to discuss these four papers and clarify the connections and differences.\n\n* **vs. DASO:** DASO focuses on optimizing the pseudo-labeling process. Our method, FARAD, offers a more upstream correction by debiasing the model's raw logits **before** any pseudo-labels are generated. FARAD addresses the root cause of biased pseudo-labels, making it complementary to methods like DASO.\n* **vs. ACR (Towards Realistic...):** ACR is an elegant **adaptive** method that estimates the unlabeled data's true class distribution. In contrast, FARAD is an **estimation-free** approach. We probe the classifier's internal bias directly with a statistically representative reference, avoiding the need to explicitly estimate the target distribution, which may make our method more robust when distribution estimation is difficult.\n* **vs. Twice Class Bias Correction (TCBC):** TCBC uses a complex, two-stage correction based on distribution estimation. FARAD provides a simpler, unified mechanism that corrects based on the classifier's **internal** learned state, rather than relying on an estimate of the **external** data distribution.\n* **vs. InPL:** InPL uses energy scores to select reliable \"in-distribution\" samples for pseudo-labeling. FARAD's contribution is **orthogonal and complementary**. We focus on improving the logit quality of **all** samples, not on selecting *which* samples to use. In principle, FARAD's logit debiasing could be combined with InPL's sample selection to achieve further gains.\n\n---\n### **Regarding Questions**\n\n#### **1. On the Method's Applicability in a Fully-Supervised Long-Tailed Setting**\n\nThis is an excellent question about our method's versatility. FARAD's bias correction mechanism is general and can be seamlessly applied to fully-supervised long-tailed learning.\n\nThe application is straightforward: during standard supervised training, for each batch of labeled data, we generate a reference image, compute the bias logits, and subtract them from the original logits before calculating the standard cross-entropy loss. This actively counteracts the model's tendency to accumulate bias towards majority classes.\n\nTo provide empirical evidence, we conducted a new experiment on CIFAR-100-LT ($\\gamma=100$):\n\n**Table C: Performance of FARAD in Fully-Supervised Long-Tailed Learning (Accuracy %)**\n| Training Method                     | Accuracy (%) |\n| :---------------------------------- | :----------- |\n| Cross-Entropy (CE) - Baseline       | 38.5         |\n| CE + Logit Adjustment (LA)          | 42.7         |\n| **CE + FARAD (Our Method)** | **44.1** |\n\nThe results show that FARAD not only significantly outperforms the baseline but also surpasses the strong and classic Logit Adjustment method, demonstrating that it is a powerful and general bias correction technique.\n\n---\n\n#### **2. On the Uncertainty of Test-Time Predictions**\n\nThis is a key question for practical deployment. Our default strategy of dynamically generating a reference image for each test batch does introduce prediction uncertainty, which is designed to adapt to the statistics of the current batch.\n\nHowever, we recognize the importance of deterministic predictions. We have already evaluated a solution for this in **Appendix D.5 and Table 18**. The **\"Offline Fixed Reference\"** strategy provides a perfect solution:\n* It uses a single, **pre-computed, fixed reference image** (based on the entire training set's statistics) for all test samples.\n* As shown in Table 18, this approach ensures **fully deterministic predictions** with only a negligible drop in performance (e.g., 82.6% vs. 83.6% bACC), while still far outperforming all baselines.\n\nUsers can therefore choose the inference strategy that best fits their deployment needs.\n\n---\n\n#### **3. On the Missing Baseline Results in Table 1**\n\nWe apologize for the missing entries in Table 1 and thank the reviewer for pointing this out.\n\nOur experimental setup strictly follows that of our main baseline, **CDMAD**, to ensure a fair comparison. The entries that are missing in our table are also missing in the original CDMAD paper. Our own attempts to reproduce these specific baselines did not yield stable and reliable results.\n\nTo maintain the rigor of our comparison, we decided to only report results that we could either directly cite from CDMAD or successfully reproduce under the exact same experimental conditions. We assure the reviewer that all **presented** data was obtained under identical settings, ensuring a fair comparison to our method. We will add a footnote to the final version to clarify this."}, {"review_id": "PPORuuIHSP", "reviewer": "Reviewer_Cove", "summary": "This paper introduces FARAD (Fourier-Adapted Reference for Accurate Debiasing), a novel debiasing method for long-tailed semi-supervised learning (LTSSL). FARAD operates by generating a random-phase reference image, computed by retaining the average amplitude spectrum of a batch of images while randomizing the phase spectrum. This semantic-agnostic but dataset-representative reference is passed through the model to estimate long-tailed bias. The resulting bias vector is subtracted from the model's logits to produce more balanced predictions. During training, FARAD is used to correct pseudo-labels under consistency regularization objectives (e.g., FixMatch, ReMixMatch), and during inference, it adaptively estimates the bias in a batchwise manner. The method is evaluated across several LTSSL benchmarks, showing improvements over prior baselines such as CDMAD.", "questions": "Q1 - Please provide a consistent explanation of how the FARAD correction modifies either supervised and/or unsupervised objectives. Is it applied as a bias term in logits (left-hand of CE), or pseudo-label correction (right-hand side)? The confusion between Algorithm 1 and Listing 3 in the Appendix should be resolved.\n\nQ2 - FARAD’s test-time strategy violates inductive assumptions by using other test samples for bias estimation. Could the authors explore per-sample test-time reference estimation? What would be the performance/inference-time trade-offs of this strategy? Also, are there realistic scenarios where test-time batches are available?\n\nQ3 - Please correct the citation of CDMAD ([34] vs [36]) and contextualize ABC/RECD, FixMatch, and ReMixMatch properly. Additional relevant SSL debiasing methods should be discussed in more detail.\n\nQ4 - Since the reference image is computed from a batch, how sensitive is FARAD to the specific images used? Could dataset distribution in test batches skew the reference image (e.g., imbalance or co-occurrence of classes)?\n\nQ5 - Could the authors show the average class probabilities (aka empirical marginal distribution) of FARAD (similar to Figure 3 in CDMAD [34]) to provide further evidence that FARAD achieves balanced predictions?\n\nAddressing W1 (loss formulation) and W2 (transductive setting) with clear implementation and evaluation details could raise my score from borderline accept (4) to accept (5).", "limitations": "1) The authors address limitations of their work in Section 4 (Conclusion). \n\n2) The discussion about “Broader Impact” is missing in Section 4, although the checklist in (10) reads:\nQuestion: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?\nAnswer: [Yes]\nJustification: Sec. 4 – “Broader Impact” paragraph discusses positive impacts on fair model training and notes potential misuse for adversarial debiasing with mitigation strategies.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n* S1 - Novelty: The paper proposes a creative use of Fourier-transformed images to construct semantically meaningless yet statistically representative reference images, offering a fresh perspective on model debiasing for semi-supervised learning approaches, while improving the work introduced by CDMAD.\n\n* S2 - Conceptual Motivation: The method is grounded in a compelling hypothesis: semantic irrelevance + statistical representativeness leads to accurate bias estimation. Both theoretical rationale and empirical studies support this.\n\n* S3 - Comprehensive Evaluation: The authors conduct an extensive suite of experiments, testing FARAD across multiple datasets, varying the degree of imbalance severity and distribution mismatch. The paper includes extensive ablations and comparisons to LTSSL methods, clearly demonstrating the effectiveness of FARAD, especially over CDMAD.\n\nWeaknesses:\n\n* W1 - Ambiguous Loss Integration: The paper lacks a consistent and precise explanation of where the FARAD bias correction enters the training losses. It remains unclear whether the bias is applied on the input (logit) side of the loss (like margin-based debiasing [32,43]) or on the pseudo-label target side of consistency losses (more likely as in CDMAD). The mismatch between Algorithm 1 (affects supervised & unsupervised losses, L13) and Listing 3 (pseudo-label correction, L725-726, L735-736) adds to the confusion.\n\n* W2 - Transductive Test-Time Bias Estimation: FARAD's test-time strategy uses a batch of samples to compute the reference image, effectively treating test data in a transductive manner. This contradicts the inductive assumption common in SSL benchmarks and may yield unfair advantages. The authors discuss some alternatives to per-batch random-phase images in Appendix Table 18, but these come with limitations: 1) a single static reference image can lead to performance fluctuations (L949-950), 2) reusing single-shot images is a breach of the inductive protocol, and 3) adaptive single-shot images would probably have an impact on inference performance.\n\n* W3 - Incomplete/Incorrect Related Work Discussion (Appendix, section A):\nW3.1) Citation [36] is fabricated/hallucinated. The correct attribution for CDMAD appears in [34].\nW3.2) Section A.2 cites FixMatch (L570-571) and ReMixMatch (L577-578) to discuss their strengths/limitations in the context of long-tailed settings, which is inaccurate; both were originally proposed for balanced SSL.\nW3.3) Section A.3 overlooks a significant body of work on SSL debiasing (e.g., [32,41,43,52,56]). These works are indeed cited in the introduction, but without clear context. ABC/RECD [35,41], approaches that are not based on the principle of debiasing with reference images, are cited out of context in L68-69.", "was_revised": true, "final_justification": "The authors introduce FARAD, a novel and effective debiasing method for long-tailed semi-supervised learning that combines training- and inference-stage debiasing.\n\nThey’ve addressed my initial concerns by proposing a fully inductive alternative to their original transductive methodology and validating it experimentally. They’ve also clarified the loss-function inconsistencies and demonstrated a balanced marginal distribution. Furthermore, they’ve committed to strengthening the related-work section, contextualizing previous work in a better way.\n\nIn response to other reviewers' concerns, they’ve added a run-time comparison to highlight their approach’s efficiency and experiments under domain shift, medical-imaging datasets, and the fully-supervised LT setting, demonstrating FARAD's flexibility.\n\nOverall, I’m inclined to raise my score and recommend accepting this paper.", "rebuttal": "### **Regarding Weaknesses**\n\n#### **1. On the Vague Loss Integration (W1/Q1)**\n\nWe apologize for the inconsistency and lack of clarity in describing the loss integration. The reviewer is correct: **our method directly corrects the logits before they are passed to the loss functions, and this correction is applied to both the supervised and unsupervised losses.**\n\n **Implementation Details:** **Listing 3** shows the exact implementation. For the **unsupervised loss**, we first compute the original logits for an unlabeled sample, subtract the bias logits to get debiased logits, and then apply softmax to these debiased logits to generate higher-quality pseudo-labels for the consistency loss. For the **supervised loss**, we similarly use the debiased logits as the direct input to the cross-entropy function.\n\n**Correction to Algorithm 1:** We acknowledge that **Algorithm 1**, as a high-level overview, was not precise and caused this confusion. We will thoroughly revise Algorithm 1 in the final version to be perfectly consistent with the logic in Listing 3, clearly showing how the debiased logits `g*` are used for both `L_sup` and `L_unsup`.\n\nWe hope this clarification and the planned correction fully resolve the reviewer's concern.\n\n---\n\n#### **2. On the Transductive Test-Time Bias Estimation (W2/Q2)**\n\nWe sincerely thank the reviewer for these crucial and insightful questions. We completely agree that a strict inductive setting is vital for fair comparison, and we appreciate the opportunity to clarify our analysis and provide a more rigorous presentation. We will address every point thoroughly.\n\n#### 1. Analysis of a Per-Sample Inductive Strategy and its Trade-offs\n\nYou asked if we could explore a per-sample test-time reference. We did investigate this strategy, and our findings are presented in **Appendix Table 14** (\"Single-Image Amp\").\n\n* **Performance:** This approach proved to be suboptimal. Using a single image's statistics to generate its own reference introduced too much noise for a stable bias estimation, resulting in significantly lower performance than our other inductive strategies.\n* **Inference Time:** A per-sample execution is also the least efficient, as it cannot leverage the amortization benefits of batched FFT computations, adding a non-trivial latency to each inference call.\n\n#### 2. Practical Scenarios for Batch-Based Test-Time Estimation\n\nYou also asked about realistic scenarios where test-time batches are available. Such scenarios are common in many real-world applications. For instance, in **medical imaging**, it is standard practice to analyze a batch of image slices from a single patient's CT or MRI scan. Our batch-based strategy was designed with these practical applications in mind.\n\nHowever, we fully agree this does not justify its use for standard benchmark comparisons, which must adhere to a strict inductive protocol.\n\n#### 3. A Rigorous Inductive Evaluation Demonstrating FARAD's Core Contribution\n\nTo fully address your core concern, we now present a comprehensive analysis of our method under a **strict inductive protocol**. This new ablation across different imbalance ratios not only resolves the fairness issue but also shows that our method's advantages become even more pronounced as the problem difficulty increases.\n\n**Ablation: Inductive Strategies across Different Imbalance Ratios (CIFAR-10-LT)**\n\n| Inductive Setting                   | bACC (%) at `γ=100` (Standard) | bACC (%) at `γ=200` (Extreme)  |\n| :---------------------------------- | :----------------------------- | :----------------------------- |\n| 1. Baseline (No Correction)         | 78.2%                          | 70.5%                          |\n| 2. **FARAD (Train-Only)**           | **81.9%** (+3.7% vs. baseline) | **75.5%** (+5.0% vs. baseline) |\n| 3. Test w/ Blank Image Ref.         | **82.2%**                      | **75.9%**                      |\n| 4. **Test w/ FARAD Ref. (Offline)** | **82.6%**                      | **76.5%**                      |\n\nThis analysis yields two powerful takeaways:\n\n* **The core value of our training-phase correction is amplified by difficulty.** The performance gain from our training-only correction grows from a significant **+3.7%** in the standard setting to a massive **+5.0%** in the extreme setting.\n* **The superiority of our FARAD reference design becomes more pronounced in challenging scenarios.** In a fair inductive test, the performance gap between using our proposed **FARAD Reference** and a simple **Blank Image** widens as imbalance increases (from a 0.4% advantage at `γ=100` to a 0.6% advantage at `γ=200`).\n\n#### **Commitment for Manuscript Revision**\n\nBased on your invaluable feedback, we will make the following major revisions:\n\n1.  **All main results will be replaced with results from the strict inductive protocol** (using the Offline FARAD Reference).\n2.  The original transductive results will be moved to the appendix and explicitly framed as a secondary analysis.\n3.  This new, comprehensive ablation table will be added to the paper to transparently demonstrate our method's value under a fair evaluation protocol.\n\nWe are confident these changes fully address your concerns and more accurately highlight the robust contributions of our work. Thank you again.\n\n---\n\n#### **3. On Incomplete/Incorrect Related Work and Citations (W3/Q3)**\n\nWe sincerely apologize for the errors and omissions in our related work section and thank the reviewer for the careful proofreading. We will make the following corrections in the final version:\n\n* **W3.1 (CDMAD Citation):** We apologize for this significant citation error. The correct citation for CDMAD is indeed [34], and we will correct this throughout the manuscript.\n* **W3.2 (FixMatch/ReMixMatch Context):** The reviewer's criticism is correct. We will revise our wording to clarify that we are discussing the limitations of these methods *when applied to long-tailed scenarios*.\n* **W3.3 (Related Work Discussion):** We will expand the related work section to include a more detailed discussion of the important works mentioned [32,41,43,52,56] and will better contextualize methods like ABC/RECD to highlight the uniqueness of our reference-image-based paradigm.\n\n---\n### **Regarding Questions**\n\n#### **4. On Sensitivity to Test Batch Distribution**\n\nThis is a great question. Our method mitigates sensitivity to the specific composition of a test batch through two key designs:\n\n1.  **Batch Averaging:** By averaging the amplitude spectrum across the entire batch, we get a smoothed, statistically stable reference that is robust to the influence of a few outlier or specific-class images.\n2.  **Statistics vs. Semantics:** The amplitude spectrum primarily encodes low-level statistical properties (e.g., textures), not high-level semantic class information. Critically, the phase randomization step is what destroys the semantics, and its effect is overwhelming and independent of the input distribution.\n\nOur experiments in **Appendix D.2 (Tables 9 and 10)** provide strong empirical proof of this robustness. The method performs well even under extreme scenarios where the training and test distributions are severely mismatched. If the method were highly sensitive to the batch composition, its performance would have collapsed in these settings, which it did not.\n\n---\n\n#### **5. On Showing Average Class Probabilities (Empirical Marginal Distribution)**\n\nThis is an excellent suggestion for visually demonstrating our method's debiasing effect. We will add this analysis to the final paper. Since we cannot include new figures in this rebuttal, we present the core data in the table below. This table shows the empirical marginal distribution of predictions on a balanced CIFAR-10 test set after training on a long-tailed set ($\\gamma=100$).\n\n**Table D: Predicted Empirical Marginal Distribution on Balanced CIFAR-10 Test Set**\n| Class Grouping      | Ideal Probability | FixMatch Predicted Prob. | **FARAD Predicted Prob.** |\n| :------------------ | :---------------- | :----------------------- | :------------------------ |\n| Head Classes (0-2)  | 30%               | ~71%                     | **~35%** |\n| Medium Classes (3-6)| 40%               | ~24%                     | **~38%** |\n| Tail Classes (7-9)  | 30%               | ~5%                      | **~27%** |\n\nAs the table shows, the FixMatch baseline's predictions are heavily skewed towards the majority classes. In contrast, **FARAD's prediction distribution is significantly flatter and much closer to the ideal uniform distribution**. This provides direct numerical evidence that our method successfully balances the model's predictions.\n\n---\n\n#### **Regarding the Discussion of Limitations and Broader Impact**\n\nThank you for pointing out the discrepancy between our checklist and the main text. The reviewer is correct that we omitted a discussion of broader impact in Section 4, and we apologize for this oversight.\n\nWe will add a dedicated **\"Broader Impact\"** paragraph to our Conclusion (Section 4) in the final version. As promised in the checklist, this section will discuss the positive societal impacts of our work (e.g., improving AI fairness on underrepresented data like rare diseases or minority groups) as well as potential risks."}, {"review_id": "Zlk61hl0Ys", "reviewer": "Reviewer_abUx", "summary": "This paper proposes a novel method for addressing class imbalance in semi-supervised learning by estimating and correcting class-wise prediction bias. The key idea is to generate reference images that preserve low-level statistical features while removing semantic information, which is achieved through Fourier transform manipulation. Specifically, the method randomizes the phase spectrum while maintaining a batch-averaged amplitude spectrum. These reference images are then used to estimate and rectify class-wise prediction bias in pseudo-labeling. The approach is simple yet effective, demonstrating superior performance over existing methods on multiple benchmark datasets.", "questions": "1. How does using a batch-averaged amplitude spectrum affect the bias estimation, particularly for majority vs minority classes? Have you considered using more balanced averaging approaches?\n\n2. Please clarify the statement \"we do not discard high-frequency components of x during randomization\". What is the technical significance and how does it impact the method?\n\n3. Why are the improvements more pronounced when using ReMixMatch compared to FixMatch as the baseline? This warrants more analysis.\n\n4. Could you provide more details about the pseudo-labeling design choices and threshold selection criteria?\n\n5. Why were the results for ReMixMatch+FARAD omitted from Table 2? Including these would help complete the comparison.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n\nFrom a technical perspective, the authors present an innovative approach that cleverly leverages Fourier transform properties to disentangle semantic content from statistical features. The mathematical formulation for bias estimation using reference images is particularly elegant, representing a creative fusion of signal processing principles with modern machine learning techniques.\n\nThe method itself is remarkably well-designed, striking an excellent balance between simplicity and effectiveness. The authors have crafted an approach that requires minimal modifications to existing architectures while maintaining computational efficiency. Each component of the method is clearly justified and serves a specific purpose in the overall framework.\n\nThe empirical validation is thorough and convincing. The authors conduct extensive experiments across multiple standard benchmarks including CIFAR-10, CIFAR-100, and Small-ImageNet-127, demonstrating consistent performance improvements over competitive baselines. The comprehensive ablation studies effectively validate the contribution of each component. The analysis is particularly strong, offering deep insights into semantic irrelevance and statistical representativeness, supported by clear visualizations that help readers understand the method's effects.\n\nWeaknesses: \n\nA key technical limitation lies in the use of batch-averaged amplitude spectra, which could potentially inherit biases from majority classes. \n\nThe handling of high-frequency components and the justification for various pseudo-labeling design choices remain somewhat unclear. \n\nThe experimental evaluation, while generally strong, shows some inconsistencies in performance gains across different baseline methods, and notably omits comparison with ReMixMatch+FARAD. The analysis of failure cases could also be more comprehensive.", "was_revised": true, "final_justification": "Most of my concerns have been addressed by the author in the rebuttal.", "rebuttal": "### **Regarding Weaknesses**\n\n#### **1. On Batch-Averaged Amplitude Spectrum Potentially Inheriting Majority-Class Bias (W1 & Q1)**\n\nThis is a deep technical question that gets to the core of our method. We argue that having the reference image's amplitude spectrum inherit the statistical properties of the training data is not a flaw, but a crucial and **deliberate design choice** that is key to its effectiveness.\n\n1.  **Unifying the Source of Bias and its Measurement:** A classifier's bias is learned from the biased statistics of the training distribution itself. To accurately measure this learned bias, the \"probe\" (our reference image) **must** share the same statistical properties as the data that created the bias. Using a \"balanced\" reference to measure a bias learned from an \"imbalanced\" distribution would create a mismatch, leading to an inaccurate and misleading bias estimate.\n\n2.  **Why a \"Balanced Average\" is Sub-optimal:** If we were to use a balanced average (e.g., by re-sampling classes), we would create a reference image whose statistics do not match the distribution the model was actually trained on. Probing the model with a statistical distribution it is \"unfamiliar\" with would weaken the accuracy of the bias correction.\n\nOur experiments in **Appendix D.2**, which cover severe distribution mismatch scenarios, provide strong empirical support for this design. The method remains robust even when the batch statistics differ greatly from the labeled set, proving that our adaptive estimation of bias is effective.\n\n---\n\n#### **2. On the Handling of High-Frequency Components and its Technical Meaning (W2 & Q2)**\n\nThank you for asking for clarification on this key technical detail from lines 271-273 of our paper, where we state that we do not discard high-frequency components.\n\n1.  **Technical Meaning:** In image signals, high-frequency components correspond to **fine textures, sharp edges, and intricate details**. These details are often critical for accurate classification in tasks ranging from natural images to medical imaging.\n2.  **Impact on Our Method:** If we were to discard high-frequency components (e.g., via a low-pass filter), our reference image would become blurry and lose the rich textures present in real images. This would make it **impossible to measure the classifier's bias** in response to these critical high-frequency features. By **retaining the complete amplitude spectrum**, our reference image is statistically realistic across all frequencies. This ensures our bias estimation is comprehensive, accounting for the model's response to both coarse structures (low frequencies) and fine details (high frequencies), leading to a more accurate overall correction.\n\n---\n\n#### **3. On Why the Improvement is More Significant on ReMixMatch (W3 & Q3)**\n\nThis is a very keen observation. Our analysis indicates this phenomenon is primarily due to the internal mechanics of ReMixMatch and its particular vulnerability in long-tailed settings.\n\n* **ReMixMatch's \"Distribution Alignment\":** A core feature of ReMixMatch is that it forces the distribution of its pseudo-labels to match the (imbalanced) distribution of the labeled data.\n* **Negative Effect under Distribution Mismatch:** As shown by prior work and our results (Table 9), this mechanism has a **significant negative impact** when the labeled and unlabeled class distributions are severely mismatched. It warps the pseudo-labels towards an incorrect prior, making ReMixMatch a relatively \"weaker\" baseline in these challenging scenarios.\n* **FARAD's \"Rescue\" Effect:** FixMatch lacks this distribution alignment module and is therefore more stable. FARAD's direct and robust debiasing mechanism effectively \"rescues\" the struggling ReMixMatch, leading to a more dramatic *relative* performance gain. While the absolute improvement on the more stable FixMatch is also large, the relative increase is smaller.\n\n---\n\n### **Regarding Questions**\n\n#### **4. On Pseudo-Labeling Design Choices and Thresholding Standards**\n\nOur design choices, detailed in **Appendix B (lines 1048-1051)**, were made to ensure a **fair comparison** with our primary baseline, CDMAD, and to adhere to the principles of long-tailed learning.\n\nSpecifically, we set the confidence threshold `τ=0` (using all pseudo-labels) because, under severe class imbalance, any fixed high-confidence threshold would inevitably filter out nearly all samples from tail classes. This would defeat the purpose of using unlabeled data to boost minority class performance.\n\nThe core philosophy of our method is not to *filter* uncertain samples via thresholding, but to **improve the quality of all logits** (especially for minority classes) through bias correction, making them more reliable targets for consistency regularization.\n\n---\n\n#### **5. On the Missing ReMixMatch+FARAD Results in Table 2 (W3 & Q5)**\n\nWe sincerely apologize for omitting the ReMixMatch+FARAD results in **Table 2 (CIFAR-100-LT)**. This was an oversight during the preparation of our results.\n\nWe will **add this missing data** to the final version of the paper to ensure the comparison is complete. For your reference, the full results on the ReMixMatch baseline are provided below.\n\n**Supplementary Data: ReMixMatch+FARAD on CIFAR-100-LT (bACC %)**\n| Imbalance Factor ($\\gamma$) | ReMixMatch+CDMAD | **ReMixMatch+FARAD (Ours)** |\n| :--- | :--- | :--- |\n| $\\gamma=20$ | 57.0 ± 0.32 | **59.8 ± 0.39** |\n| $\\gamma=50$ | 51.1 ± 0.46 | **54.0 ± 0.48** |\n| $\\gamma=100$ | 44.9 ± 0.42 | **48.1 ± 0.45** |\n\nThese results are fully consistent with our paper's overall conclusions: FARAD provides a significant and stable performance improvement over CDMAD, regardless of the base SSL algorithm, especially on the more challenging CIFAR-100-LT dataset. Thank you for pointing out this omission."}]

Author rebuttal:

Meta-review: This paper proposes a novel method for addressing class-imbalanced semi-supervised learning by estimating and correcting class-wise prediction bias. The approach is simple yet effective, demonstrating superior performance over existing methods on multiple benchmark datasets.

After rebuttal, the paper received four positive ratings. All reviewers acknowledged the paper's contributions and expressed satisfaction with its technical merits.

The AC concurs with the reviewers' assessments and recommends acceptance. 

However, one potential weakness of the paper may be the missing comparison with stronger baselines, such as CCL [1]. The authors are suggested to discuss and include more recent competing methods in their experiments in the next version, if possible. 

[1] Continuous Contrastive Learning for Long-Tailed Semi-Supervised Recognition. NeurIPS 2024
Decision: Accept (poster)