Synthesizing Minority Samples for Long-tailed Classification via Distribution Matching
Authors: Zhuo Li
He Zhao
Jinke Ren
Anningzhe Gao
Dandan Guo
Xiang Wan
Hongyuan Zha
Venue: Accepted by TMLR ()

Abstract: In many real-world applications, deep neural networks (DNNs) often perform poorly on datasets with long-tailed distributions. To address this issue, a promising approach is to propose an optimization objective to transform real majority samples into synthetic minority samples. However, this objective is designed only from the classification perspective. To this end, we propose a novel framework that synthesizes minority samples from the majority by considering both classification and distribution matching. Specifically, our method adjusts the distribution of synthetic minority samples to closely align with that of the true minority class, while enforcing the synthetic samples to learn more generalizable and discriminative features of the minority class. Experimental results on several standard benchmark datasets demonstrate the effectiveness of our method in both long-tailed classification and synthesizing high-quality synthetic minority samples.

Published in Transactions on Machine Learning Research (06/2025)
Synthesizing Minority Samples for Long-tailed Classification via Distribution Matching
Zhuo Li
zhuoli3@link cuhk. edu.co
Shenzhen International Center for Industrial and Applied Mathematics, Shenchen Research Institute of Big Duta, The Chinese University of Hong Kong, Shenzhen,
He Zhao
he he.zhuo hec.zhoo@@daafll.csir...o
CSIRO's s Data61, Australia
Jinke Ren
jinkeren&ccuk..ed..com
Shenzhen Future Network of Intelligence Institute
School Science and Engineering. The Chinese University of Hong Kong, Shenzhen Guangdong Provincial Key Laboratory of Future Networks of Intelligence
Anningzhe Gao
anninoghegoo@@@allloom
Shenzhen Research Institute of Big Dala
Dandan Guo Jilin University
guodandam Adjt.edu..
Xiang Wan Shenchen Research Institute of Big Data
wonciang@srridd.cn
Hongyuan Zha
zhahy@cuhk.c edu.co
The Chinese University of Hong Kong, Shenzhen
Reviewed on OpenReview: https: openreview. net/ forum? VgLe8tPb2z
Abstract
In many real-world applications, deep neural networks (DNNs) often perform poorly datasets with long-tailed distributions. To address this issue, promising approach is to propose an optimization objective to transform real majority samples into synthetic minority samples. However, this objective is designed only from the classification perspective. To this end, we propose a novel framework that synthesizes minority samples from the majority by considering both classification and distribution matching. Specifically, our method adjusts the distribution of synthetic minority samples closely align with that of the true minority class, while enforcing the synthetic samples to learn live features of the minority class. Experimental results on several standard benchmark datasets demonstrate the effectiveness of our method in both long-tailed classification and synthesizing high-quality synthetic minority samples.
generalizable and discrimina-
1 Introduction
The success of deep learning for supervised learning relics on high-quality largo-scale datasets, which are often assumed to have nearly balanced numbers of samples for each class (Russakovsky al. 2015). However, real-world datascts usually suffer from long-tailed problem, where few majority classes occupy most
*Corresponding Author. Cade available on

---

Published in Transactions on Machine Learning Research (06/2025)
data while many minority classes have very few samples (Zhou et al. networks (DNNs) trained on long-tailed datasets have poor generalization performance, especially in minority classes (Zhou et al. 2020 Liu et al. 2019). Therefore, it is of practical importance to develop methods for mitigating the long-taiie tailed problem.
2017 Liu et al. 2015 Deep neural
To alleviate the imbalance issue, several kinds of methods have been proposed in the past decade, which the data-level approach has received significant attention due the simplicity and effectiveness (Yang all 2022). This approach usually aims to achieve balanced training data distribution via re-sampling (i.e.
2023] or data under-sampling (He & Garcia 2009), over-sampling ( (Van Hulse et al., 2007 Gao et al.)
augmentation (Chu et al.] the context of re-sampling, one representative method is the Synthetic Minority Over-sampling Technique (SMOTE) (Chawla et al. 2002), which synthesizes minority samples by interpolating between existing minor- ity samples and their nearest neighbor samples. Recently, Kim et al. (2020) have revisited the over-sampling framework and proposed a new way of to synthesize minority samples, called
2020 Hong ct al. 2022 Li ct al., 2021 Ahn et al. 2023] Gao et al. 2024b). In
(M2m). A unique advantage of M2m over SMOTE-based methods is that M2m utilizes the majority samples to generate minority samples via an optimization process, thus can "cook with much more raw materials". In this way, M2m is able to "transform" majority samples into minority samples to achieve a balanced dataset.
Despite its initial success, M2m only focuses on optimizing the synthetic minority samples from the classifi- cation perspective, whose similarity with the real samples in the concerned minority class is overlooked. In other words, a synthetic sample initialized from a majority class ko is viewed as a sample of the minority class k if a pre- pre-trained classifier g identifies it as class k confidently, and a target classifier f has low confi- dence about it ko, ignoring whether the synthetic sample upholds the genuine characteristics (i.e., feature distribution) of the class k.
In this work, we propose a novel framework for synthesizing minority samples via distribution matching, Our insight is that a desired synthetic minority sample should not only satisfy the classification constraints about g(t) and f()), but also be distributionally close to real samples in the targel minorily class. To satisfy these, we introduce a principled approach that optimizes the synthetic minority samples by enforcing them to satisfy the classification constraints and be close to the distribution real samples, by minimizing the optimal transport (OT) distance (Peyré et al.] 2019 Morcover, to mitigate the harmfulness of unreliable synthetic samples, we define a sample rejection criterion based on the distance between synthetic minority samples and real minority samples. Concurently, the classifier f is jointly trained leveraging these progressively refined synthetic samples, creating a synergistic learning process.
In order to enhance the generality and practical applicability of our method, we introduce an adnal regularization term concerning the "confusing class" within the minority class, which accounts for instances where minority samples are frequently misclassified into a specific class rather than other classes. In this way, we relax the label requirement of majority samples and as result, our proposed method can translate not only Tn-Distribution (TD) majority samples but also Out-of-Distribution (OOD) samples (Wei et al. 2022) into synthetic minority samples, making ours more applicable in practice. Additionally, our method can also be used as a plug-in approach to enhance the performance of other methods, e.g., reweighting loss. Moreover, we conduct extensive experiments on standard benchmark datasets and our method achieves improved long-tailed classification performance. In conclusion, our contributions are summarized as follows:
To address the long-tailed classification problem, we propose a general framework for synthesizing mi- nority samples via distribution matching, where we formulate real samples and synthetic ones as two distributions.
We optimize synthetic minority samples by enforcing them to satisfy the classification constraints and keep close to the real representation distribution by minimizing the OT distance.
We significantly enhance the framework's robustness and versatility by introducing two novel components: a an innovative regularization term focused on "confusing classos", which crucially enables effective syn- thesis from (ID) and (OOD) seed samples; and (b) an effective feature-distance-bsse based sample rejection criterion to ensure the quality of generated instances.
2

---

Published in Transactions on Machine Learning Research (06/2025)
Extensive experiments on standard benchmarks demonstrate the effectiveness of our method, showing that is a promising over-sampling framework for the long-tailed classification problem.
2 Preliminaries
Optimal transport. OT is a widely used measurement for comparing distributions (Peyré et al., 2019) where we only focus on the discrete situation that is more related to our framework. Assuming have two sets of points (features), we formulate the discrete distributions as P = >n=1 unde, and Q Em=1 ym.. where is Dirac function, untuilivll a unit of mass which is infinitely concentrated at location Zt. Dirac function helps convert a collection of raw data into a discrete probability distribution, by modeling each point as an individual "mass". A and AM are the discrete probability vectors that sum to 1. The discrete OT distance between distribution P and Q can be formulated as:
N M
min (T = TEU(P.)
Σ Tum
FumCam
(1)
m
where CER is the cost matrix whose each point denotes the distance between In and ym and transport
probability matrix T E Ry0" satisfies II(,.Q) {TI>>> Tam Um Em Tam Un As directly
optimizing is always time-expensive, Sinkhorn algorithm (Cuturi 2013) introduces an entropic constraint, i.c., (T)= - Emm Tam In Tum for fast optimization.
Long-tailed classification. Assune training dataset Derain {(xi, yi W where E Rd denotes the i-th input and Y: means its corresponding label over K classes. Let N denote the number of the entire training data and N₁ is that of class k, where we assume N1 2 N2 2 2 NK without loss of generality, Following Cao et al. (2019), we define the imbalance factor (IF) as NK which dircctly quantifies the ratio between the largest and smallest class sizes, providing a clear measure dataset skewness. For example, when IF is larger, the training set is more imbalanced, i.e., more challenging. Denote f F: R as the target classifier, which can be learned by empirical risk minimization (ERM) over the imbalanced training set with an appropriate loss function L(f):
Ni
min (r.y)) [f(f: x, v)].
f
However, when Durann exhibits a tailed distribution, the standard ERM objective in Eq. 2 inherently struggles to ensure fairness across classes. Because the expectation E(aay)DDuan is dominated by the high- frequency majority classes, the learned model f becomes heavily biased towards these head categories. This direct consequence of the ERM formulation on imbalanced data usually results in poor generalization performance on underrepresented tail classes (Fang et al. 2021 Han et al. 2023
3 Method
3.1 Motivation
To motivate our method, we first review the most related work in the line of over-sampling, called Major- to-minor (M2m) (Kim et al. 2020). M2m aims to construct a new balanced dataset Dbal from the original dataset Dirain by generating N, - -N synthetic samples for each class k, where the concerned classifier f trained on D is expected to perform better than that trained on D Note that Ni 2 N2 2 ... 2 Nx. Therefore, generating N1 N synthetic samples for each class & enables balanced dataset. Here, synthetic samples in minority classes are generated by translating from other samples in majority classes. In addition to the lo-be-learned classifier ʃ trained on D M2m assumes a baseline classifier pre-trained on the imbalanced dataset Durain with standard ERM training, where f and g have the same structure. Although g may not achieve the optimal performance, training dataset. M2m designs to obtain synthetic sample for a minority class k which translated from real training sample 2n from major class ko in Drrain through several optimized steps.
expected to achiove reasonable performance on the imbalancod
3

---

Published in Transactions on Machine Learning Research (06/2025)
Although M2m achieve promising results, it translates è from zy purely in the view of classification and ignores the similarity between & and the corresponding real samples in the concerned minority class k. In this scenario, the synthetic sample might mislead the model and cause inaccurate predictions.
3.2 Learning Synthetic Minority Samples with Distribution Matching
To address the above issue, we aim to learn high-quality synthetic samples that not only satisfy the classifi- cation constraints about g( and f()) but also follow the distribution of real samples in the target minority class k. We achieve the first goal by solving the optimization problem within steps, as shown below:
(0) = arg min L(g(t), + Afka (T),
where & is initialized with +e and is standard Gaussian noise, i.e., 2! +- to a(0) denotes the synthelic sample at 1-th optimization step within 7 iterations. Next, we focus on the second goal. Taking the k-th class in Dirain as an example, we denote Dx = {TTp yn)] j as the set of real samples, and Dr = {frm 2m) M as the to be-beelearned syuthetic set, where Mr = Nı - N is the number of synthetic samples of class k. Then the empirical distributions of Dr and Dr can be formulated as:
No
Ma
PR =
Q
82m-
(4)
N&
M
n=1
m=1
Note that label is omitted since In = yn = k. Moving beyond Eq. 3 which only utilizes the classification loss to learn minority synthetic samples, we further introduce distribution matching loss to enforce the to-be-learned distribution Qx to stay close to the real distribution PA of class k. Let (ik Qk) denote the distance between the distributions P& and Qt- Here we adopt the principled approach of OT to define Dist Qx), although other approaches are also available, such as Maximum Mean Discrepancy (MMD) Grreton et al. 2012) and Energy Distance (ED) (Rizzo Székely 2016). We defer the implementation of Dist (kk, Q+) with other measures such as Maximum Mean Discrepancy (MMD) (Gretton et al. 2012 and
Energy Distance (ED) Rizzo & Székely 2016) to Appendix A
Since the training images are high dimensional, minimizing the distribution distance in the image space is expensive and inaccurate. Therefore, we assume an embedding function the Rd + Rd parameterized with and compute the distribution distance Disto Qx) in the feature space. Specifically, we define using the entropic OT:
Dista (Pr, Qx)
min TEl(PL.Q.) (T. -
where 7 > 0 is hyper-parameter for the entropy constraint H H(T). The transport plan satisfies:
M
II Qk) =
Tam 1/ME,
Σ
T T 1/Nx
(6)
and the cost function Can measures the distance between the real sample x, and synthetic sample o..- Cum can be viewed as distance metric in the embedding space. Although theoretically it is possible to use any reasonable distance metric, we use the cosine similarity, i.e., Crm = 1 cos(we (o(2m)), which gives the best performance in this work.
3.3 Embedding Function the and Optimization Problem
In order to achieve efficient computation of distribution distance, the parameterization of an embedding function vhe is necessary and important. Commonly, we can employ the feature extractor in or f as the embedding function. However, g a biased model that carries class-specific biases relevant to the long- tailed distribution and f learned during each training itcration, whose parameters may not be optimal for computing the DPs Qk). In order to create diverse and unbiased projections in a computationally efficient. manner, motivated by Zhao & Bilen (2023 that computes feature distance based family
4

---

Published in Transactions on Machine Learning Research (06/2025)
of models, also match Ps and Qs in many sampled embedding spaces. Specifically, when computing Dista Pk, Qx) each time, employ a neural network that keeps the same architecture with the encoder of our target classifier f to serve as our embedding function de, which thus can be initialized by commonly used Kaiming Initialization (He et al. 2015). Morcover, we experimentally validate that the family of randomly initialized embedding spaces can produce better results than using one embedking space.
To summarize,, we can minimize Eg [Diste (Pa, Qx)] such that the synthetic samples are optimized to match the original data distribution in various embedding spaces. The overall optimization objective is formulated as:
ME Σ (fim ) m=
m = arg min
[(g(ūm),)) + Y1 A1 (2m)) + A2 [Dist Qx)),
(7)
My
where each fm is initialized by a randomly sampled image I' from a major class and Cm is the randomly sampled Gaussian noise, ie., Im + 2m + Em. Similarly, optimize êm through T iterations and use i' to denote &. at f-th optimization step.
3.4 Leveraging
Data
Beyond translating majority samples in the ID setting (e.g., samples in Dirain) to achieve efficient over- sampling and data augmentation for minority classes, it is more practical and valuable by leveraging OOD data to achieve the balance of a long-tailed dataset. Let Doud = {2; No denote the OOD datasct, where the label information is
Ty E Rdd denotes i-th sample. We assume that the OOD dataset is unlabeled not useful due to the large distribution shift from the ID dataset Dusin Now can initialize the minority sample as fm := mood. + 6m, where Food,m is randomly sampled image from Dood for 2...
Recall that M2m restricts the larget classifier f to have lower confidence on the original class kn of xo by adding a regularization term in Eq. 3 However, the introduction of the OOD dataset brings a challenge as there is no corresponding ko for each Zood,m in Dood. Therefore, we replace the constraint of the synthetic samples about kg by introducing a confusing class kg Specifically, we obtain the confusion matrix A E R using randomly sampled balanced subset from Drrain and the pre-trained classifier 9, whose element. Aij denotes the probability that sample belongs to class but is predicted as class j. Then, for the target minority class k, k, is its most confusing class if AR.k. 2 Au,i where i E [1, K] and i + k. Finally, we design the constraint on the confusing class for an optimized sample imm as fu. (ăm) and rewrite Eq. 7 as:
Me Σ
$00) m
arg min (dim Pood m mm m=1
(C(g(Em)) k)) + AISk, (fm)) + A2 (Disto Qx)),
(8)
where &m is initialize by an OOD sample Food with noise e, and (fm) restricts I to have lower confidence say, we should avoid the synthetic samples to contain significant information of the confusing class in the viewpoint of target classifier In addition to addressing the issuc
on the confusing class kg. That
of exploiting Dood, this regularization term can not only address the issue of exploiting OOD but also be added to the training loss in the ID) scenario.
3.5 Implementation Details
Mini-batch lcarning. We adopt the stochastic gradient descent (SGD) Ruder 2016) to learn the target classifier f and optimize the synthetic samples based on a batch-wise re-sampling. More specifically, we use standard over-sampling (Huang et al. 2016) to obtain a class-balanced mini-batch (( (e) v) To stimulate the generation of N samples for any k, for each sample x, in the mini-batch, we use probability MI-N,, to decide whether learning a synthetic sample ④; to replace 24. Following this, a joint learning step occurs within the mini-batch: we re-sample a randomly initialized the to serve as an embedding function, then optimize the synthetic samples guided by our method, which are immediately used to update the target classifier f. We give a whole training paradigm in Alg. g in Appendix. B
N1
5

---

Published in Transactions on Machine Learning Research (06/2025)
Algorithm 1: Oversampling Minority Samples via Our Method (In-Distribution).
Input Dirain classifier f, pre-trained classifier and lyper-parametenn
Initialize Dost +- Drraini
for k == .,.,, K do
3 Compute Ma +- Ni - Nui
4 Initialize D2 < 0;
for 1,,., Mr do
// Step 1. Sample selection
Sample a majority class ko with p= g( N,)+
Sample a from kn
Initialize $m Im 6m with Gaussian noise 6m-i
Update DR t Dr ((5m)) fm = k)};
9
10
end
11
Build Qx = Em= Mr 88m and Ps = Eme 1 Nr
drm according to Eq.
12
for = 1,... 1 do
// Step 2.
Optimize &
a(0)
arg min Em= (L(o(Em)) k) + (&m + AeBomPy [Disto Qu)l according to Eq.
13
2m
14
end
15
Updale Dr with optimized synthetic samples.;
16
for im in Dr do if [(g(2m),k) 2 r or Reject 1 then
// Step 3. Sample rejection for a
17
P. + with random sample from class k in Drraini end
19
Update Daat Doat U ((fm,
end
22 end
Sample selection criteria for zo. We choose seed sample to learn ft for æı with class k. In an OOD setting, we just randomly sample an image from Dood as In ID setting, we first choose ko with the probability ko ~ ((No-N,)* in the current mini-batch, where (+)+ := m), and I E [0, 1) is a hyper-parameter. After that, zo sampled uniformly among samples in class kig- Once we choose the seed sample Tn for the minority class k, we start learn Rather than using all N₂ samples within class k we randomly sample a subset of the real samples from class k construct Ps each iteration to reduce computation. Besides, we use the to-be-optimized samples for class k in the current mini-batch to build Qk- Finally, optimize å using Eq. 7 or Eq. by performing T iterations with step size of n, depending on ID or OOD, respectively.
Sample rejection criteria for To reduce the harmfulness of unreliable synthetic samples, it is neces- sary to design sample rejection criteria to discard unsatisfactory synthetic samples. Here, we consider two conditions that can determine a reliable synthetic sample. Following M2m, the first one is setting a threshold and rejecting the resultant synthetic sample for k-th class if C (g k) for stability. For the second factor, M2m designs the rejection probability as IP (Reject & No, k) (MMoo -N.)* Diffcrent from M2m that utilizes the class frequency of ko and larget class l to decide the reliability of G, we introduce more general sample-level criteria to reject &o (i.e., Reject & x=1) 1) if it satisfies:
Σ
d d(vo( (0(mm))
N
N
where d()),()) (x,)), (ho,)) indicates the distance between & and Fn and can be defined by cosine similarity, d.e., 1- - (os())(())) the (zx)). This rejection criterion can avoid the requirement for NNa, which also be applied to the OOD scenario. The underlying intuition is that the synthetic samples are expected to have a smaller distance from real samples in class k than the intra-class distance. We replace æ; in the current mini-batch by & if it satisfies the above two factors. If dose not satisfy the condition (i.e., Reject = 1) or the pro-trained biased classifier still produces a high classification confidence (i.c., L [g; &, k) > 7), we will replace with real minority sample from D We summarize the synthetic process for the ID setting in A Algorithm 1

---

Published in Transactions on Machine Learning Research (06/202))
Table 1: Test top-1 errors (%) of ResNct-32 on CIFAR-LT-10 / CIFAR-LT-100 under different imbalance factors on the ID setting, where i, $ and denote the results from the original paper, our reproduction and
MetaSAug (Li et al. 2021). respoctively. Results of SMOTE are from Kim et al. (2020 The methods are trained with CE loss unless otherwise stated.
Method
CIFAR-LT-10
CIFAR- -T- 100
Imbalance Factor
200
100
50
10
200
100
50
10
CE Loss*
34.13
29.86
25.06
13.82
65.30
61.34
55.98
44.27
Focal Loss*
34.71
29.62
23.29
13.34
64.38
61.59
55.68
44.22
CB,CE, Loss"
31.23
13.10
64.44
61
55.21
42.43
LDAM-DRW'
25.26
21.88
18.73
11.63
61.55
57.11
52.03
41.22
AetaSAngt
23.11
19.46
15.97
10.56
60.06
53.13 13
48.10
38.27
RSG
17.2
55.4
51.5
MBJI
19.0
13.4
54.2
47.4
39.3
CB-SAFA:
27.18
23.68
19.79
12.07
60.34
54.13
52.04
39.77
CUDA
57.3=0.4
52.8#0.4
40.420.66
CMO:
25.43
19.59
16.47
11.50
63.47
56.13
51.71
40.49
OTMixt
16.6
53.6
49.3
38.4
M2mf
25.34:10.461
21. 11.11
18.81=0.76]
12:5::0.13
03.7720.333
57.11:0.16
40/18:10,43
44.8=00051
OURS
0.185+00112
18.20+0.21
12.96+0.11
9.34+0.11
61.28 0.221
52.95+0.188
47.02-00226
0..57+0.32
4 Experiments
In this section, we present experimental results to show the effectiveness of the proposed method. The detailed experiment settings and hyper-parameters are provided in Appendix C.1
Datasets. We evaluate our method on CIFAR-LT-10 / CIFAR-LT-100, ImageNet-LT and Places-LT. We build CIFAR-LT-10 / CIFAR-LT-100 from the standard CIFAR 10 CIFAR- 100 datasets (Krizhessky et al. 2009) with IF E (50, 100, 200} (Kim al. 2020 Kang et al. 2019 et al. 2021). ImageNNet-ITT a subset
of the ImageNet-2212 dataset (Deng et al. 2009) with 1000 classes and IF 1280/5 (Kim et al.) 2020 Ren
ct al.
2020 Places-LJ LT is subset from the Places-365 dataset (Zhou et al. 2017 with 365 classes and
IF
4980/5
Cao et al.
2019 Ren et al. 2020
Baselines. We compare with five types of baselines: (1) Cross-entropy (CE). (2) Re-weighting loss, including Focal loss
Lin et al.) 2017), Class- BallaneB CB)
loss (Cui et al. 2019 Palanced-Softman (BS)
loss (Ren et al. 2020) and LDAM-DRV loss (Cao et al. 2019
(3) Feature based augmentation methods,
including MctaSAug [Li al.
2021
SAFA (Hong et al.
2022
CUDA (Ahn et al. 2023), RSG (Wang et al.
2021) and MBJ (Liu et al.] 2022).
(4) Minorily over-sampling methods, including SMOTE CChawla
et al. 2002) M2m (Kim et al.
2020 , CMO (Park et al. 2022 and OTMix (Gao et al. 2024b (5) OOD
methods, i.e. Open-Sampling (Wei et al. 2022).
4.1 Experiments on Long-tailed CIFAR
Results with the ID setting. Tab. I summarizes the average results of our method for three independent runs with standard deviation CIFAR-LT-10 CIFAR-LT-100 under different settings. We find that method outperforms the CE baseline and re-weighting methods by large margin. Moreover, our met hod achieves significant improvement than both feature- and sample- based data augnentation methods, except for IF = 200 with CIFAR-LT-106 when compared with MetaSAug. Remarkably, the comparison between ours and the minority sample syuthetic method, i.e., M2m, confirms the validity of introducing the distribution matching loss when transferring the majority samples to the minority classes. Besides, we use MMD and ED to implement Distg and report results on CIFAR-LT-10 and time complexity in Section 4.8
Results with the OOD setting. To validate whether proposed method can translate OOD instances, we employ 300,000 random images[ (Hendrycks et al.] 2018) as the OOD datasct Dood for CIFAR-LTT-0) CIFAR-LT-100 by following
(Wei et al. 2022). We report the performance in the case of IF 100 and IF = 50. As shown in Tab. [ we find that our method is significantly better than Open-Sampling (OS), which utilizes open-set noisy labels to re-balance the long-tailed training dataset. It is reasonable since we optimize the OOD samples from the view of classification and distribution matching rather than

---

Published in Transactions on Machine Learning Research (06/2025)
Table 2: Test top-1 errors (%) of ResNet- 32 with different imbalance factors on the OOD setting.
Table 3: Test top-1 errors (%) of ResNct-50 on ImageNet-ITT (INLT) and ResNet-152 on Places-LT (PLT).
Method
INLT
PLT
Method
INLT
PLT
Method
CIFAR-I.T-10
CE
58.4
70.1
FSA
63.6
Focal Loss
65.4
MBJI
61.9
IF
100
50
LDAM-DRW BS
50.2
CMO RIDE
43.8
OS
22.38+0.28
18.24=0.51
49.0
61.3
OTMix CE
48.0
RIDE experts) BCL
45.1
OTMix+BS
44.4
OURS
20.17 16.39+0.22
44.0
OTMix RIDE
42.7
IF
100
50
M2m + CE
55.40
63.27
OURS+CE
53.77
61.68
Over-Sampling + CE*
35.34
64.27
OURS+ BS
47.31
60.37
OS
59.74+0.65
55.23::0.25
RSG + LDAM-DRW
60.7
OURS OURS+RIDE
43.22
OURS
55.09:0.223
49.3810.27
MisLA8
47.3
OURS+BCI.
41.89
endowing them with noisy labels without re-labeling OOD samples. Besides, we perform experiments by combining both OOD and ID settings, which produce better performance than that in the ID setting. The detailed results are provided in the supplementary materials.
Boosting other methods. To investigate whether our method can be combined with other long-lailed methods under ID and OOD settings, we consider several clas- sical re-weighting losses, including CB loss (Cui al. 2019) and BS loss (Ren al., 2020). As shown in Tab. 4 our method significantly improves the performance of re- weighting methods under the ID setting and performs bet- ter than M2m. Under the OOD setting, M2m is not us-
Table 4: Test top-1 errors (%) of ResNet-32 CIFAR-LT-10 dataset under both ID and OOD settings when combined with different re- weighting methods.
Method
CIFAR-LT-10 (ID)
CIFAR-LT-10 (OOD)
IF
200
100
50
200
100
50
CE
31.13
29.86
25.06
able, while the performance of Open-Sampling is worse
M2m
25.34
21.70
18.81
than our method combined with different re-weighting losses. These results indicate the effectiveness and flexi- bility of our method when combined with other methods under both ID and OOD settings.
+ 0S
28.28
22.38
18.24
OURS
22.85
18.20
15.96
23.43
20.03
16.39
2.24
4.30 1.5.0
2.15
4.85
4 2.35
1.85
CB-DRW
31.23
27.32
21.87
+ M2m
25.24
19.33
18.25
+
29.77
24.23
19.90
OURS
21.19
18.07
16.30
22.69
20.16
16.70
4.2 Experiments on ImageNet-LT and Places-LT LT
A
4.95
1.22
1.995
7.088
4.00
4.320
BS
21.97
18.37
Results. As summarized in Tab. 3 we perform expori- ments on ImageNee-LLT and Places-LT. We can see that our method using CE loss outperforms the vanilla CE, over-sampling and M2m, which indicates the effectiveness of generating the minority samples from the view of the distribution matching. Furthermore, our method can also be combined with other losses, where we take the BS loss as an example and obtain improvements by 1.69% and 0.93% compared with the BS loss on ImageNet-LT and Places-LT, respectively. These re- sults show that our proposed data augmentation method is effective on largo-scale complicated long-tailcs datasets.
+ M2m
25.16
23.43
19.96
+ OS
28.59
20.95
17.24
OURS
20.98
16.13
14.22
23.08
19.81
17.06
4 4.18
4.7. 7.30
4 5.74
3.51
1.14
4.0.8
4.3 Ablation Study
We conduct a series of ablation studies to analyze the influence of each term on CIFAR-LT-10, begining with brief recall of the key components: a) L(g(ām),)) evaluates how strongly pre-trained classifier g still associates synthetic sample with its target (e.g., minority) class k. b) fro (2m) represents term measuring the confidence of our target classifier f that the synthetic sample è correctly aligns with its original majority label kg- c) fke, (fm) denotes term reflecting the likelihood, according to our target classifier f, that the helth sample & is misclassified into a known "confusing" class k d) Ea-P, [Disto (Ps, Q+)l signifies our core distribution matching objective, which aims to minimize the discrepancy (e.g., using Optimal Transport distance) between the feature distributions of real and synthetic samples for given class For brevity the subsequent ablation discussion, we will refer to these components as a) L(k), b) L(ko). c) L(ke), and d) T.(D), respectively.
8

---

Published in Transactions on Machine Learning Research (06/2025)
M2m: (28 Lin OURS: AKLl
24-
25p2
We cxmple rejection
20
MT cample rejoction
Toxt porterron rata (%)
OURS: 60M 11M-1001+-00 (X))
24-
24-
22-
1654 15.95
1Gp
68
200
1te
203
201
293
Factor
Imbalance Facter
(a) ID setting
(b) OOD setting
(c) OOD setting
Figure 1: Ablation studies of our method on CIFAR-LT-10 with varying imbalance factors.
Distribution Matching Loss (L(D)). To verify the effect of our proposed distribution matching loss, L(D), we first compare M2m (which utilizes an objective based on L(k) + L(ko)) with our method (using L(k) + [(k)) LLD) in Fig. [(a] under the ID setting. It is clear that our method consistently outperforms M2m across different imbalance factors. This improvement stems from L(D) guiding the synthetic sample distribution to closely match that of the real samples, thereby enabling our method to generate more effective synthetic samples. Additionally, M2m's reliance on the [(ko) term (target class confidence for fm) makes it challenging to apply directly to OOD samples where the target label hig for an OOD input is not predefined. For OOD scenarios, we therefore compare an objective using only L(k: against one that combines L(ke) L(D) for minority sample synthesis. As shown in Fig. 1(b) the latter configuration, incorporating our L(D) term, achieves better performance.
Confusing Class Regularization (I(kz)). While M2m is primarily designed for ID settings, our method's applicability can extend to OOD scemarios, partly facilitated by introducing the L(ka) term. This term addresses potential misclassification of a synthetic sample (intended for a target minority class kj) into a froquently confused class kg As shown in Fig. [(b] for OOD scttings, our method incorporating an objective of ([k))<[(k)))) + L(D) achieves superior performance compared to an objective with only ((k))L((D)) across various imbalance factors. Notably, this [kka) regularization is also beneficial in the ID setting. Fig. [(a) demonstrates that our method gains further performance improvements in ID scenarios with the inclusion of L(ke). These results illustrate that the L(ke) term not only aids OOD applicability but also enhances ID performance, by promoting the generation of higher-quality synthetic minority samples that are more discriminable from their confusing classes.
Sample rejection in the OOD setting. As specified in previous section, the unreliable generation quality of synthetic samples urges us to propose the sample rejection criteria, especially in the OOD setting. To validate our proposed rejection strategy Eq. we perform an ablation study on the CIFAR-LT-10 with IF = 10. Results in Fig, I(e) present that using our proposed rejection strategy consistently improves the performance when leveraging the OOD samples for CIFAR-LT-IE with different imbalance factors. That is to say, our method can fully use OOD samples to generate minority samples while alleviating the toxicity of the distribution shift brought about by OOD samples.
Embedding spaces Eg-ral)) []] We investigate the effect of the embedding for computing the distribu-
Table 5: Test top-1 errors (%) with different embed- dings.
tion matching loss deseribed in Eq. 7 We use the encoder in the to-be-learned model f and that in the pre-trained model g as the baselines, where we also discuss the performance of a randomly initial- ized encoder and a CLIP vision encoder (Radford let al. 2021p. p. All encoders have the same architec- ture for a fair comparison, except for CLIP. Besides, also consider non-neural network model, Prin- cipal Component Analysis (PCA). As summarized in Tab. 5 the PCA serves as the worst embedding function. The possible reason behind this is the principal components derived from samples can be quite variable and may not always capture the most discriminative features relevant to the overall distribution, especially compared to the features extracted by neural networks. Besides, using the encoder in the imperfect pre-trained g and f also achieves inferior results. is reasonable since g is a biased model and cannot extract satisfactory features, and the optimization of is coupled with that of synthetic samples. Interestingly, we
CIFAR-LT-10(DD
Feature Extractor
200
100
50
PCA
28.37
23.82
21.98
Model Model
25.60
21.24
16.55
25.72
20.03
16.88
A randomaly initialized encoder
25.32
19.78
16.57
CLIP (RN50)
23.32
18.94
16.21
The family of random encoders
22.85
18.20
15.96
9

---

Published in Transactions on Machine Learning Research (06/2025)
Table 6: Test top-1 errors (%) on CIFAR-LT-10 with IF E [100, 50] under the ID and OOD settings. For the left part, We evaluate the influence of sample rejection criteria and sample selection criteria, where 7 = 0.9 and 3 = 0.999. For the right part, we evaluate the influence of different T and B. All the experiments are conducted by using sample selection criteria and sample rejection criteria.
Selection
Rejection
100
50
100
50
ID
ko
31.16
27.25
ID
0.9
0.999
18.20
15.96
ID
ko Ko
L(p)
20.73
20.86
ID
0.6
0.999
18.69
16.33
ID
L(p) + L(d)
18.20
15.96
ID
0.3
0.999
19.12
16.82
ID
Random
L(p) + L(d)
20.07
16.32
ID
0.9
0.888
18.53
16.27
OOD
Random
L(p)
20.32
16.56
ID
0.9
0.777
18.97
16.59
0OD
Random
+ L(d)
20.03
16.39
ID
0.9
0.666
18.77
16.30
find that only using a randomly initialized encoder during the entire training process can produce acceptable performance, proving the effectiveness of the appropriate and unbiased embedding function for the loss of distribution matching. Moreover, randomly initializing the embedding function at each training iteration outperforms other scttings. It shows that the family of embodding spaces can be obtained by sampling randomly initialized DNNs, and is effective in computing the distance between real and synthetic samples. Moreover, we find that even the CLIP (Radford et al. 2021 visual encoder can be used for our distribution matching purposes.
4.4 Detailed ablation study on sample selection and rejection criteria
In this section, we present a detailed ablation study on sample selection strategies and two key rejection criteria for synthesized samples, which we denote L(p) and L(d) for brevity in this discussion. The L(p) criterion, is proposed by M2m from probability perspective, which rejects synthetic sample & if a biased pre-trained classifier 9 can not confidently classify it as its target minority label k (requiring its confidence score L(g: &t. k) > r). Our proposed L(d) criterion, with its precise mathematical condition detailed in Eq. g drawn from a representation distance view, evaluates whether & is sufficienlly close geometrically to the real minorily samples of the target class to be accepted for training. We now examine the impact of applying these criteria.
As shown in the left part of Tab. G the model performs worst under the ID setting when using only the ko selection criterion and no rejection criteria, regardless of IF values being 50 or 100. However, the int of the Lpp rejection criterion for synthesized samples results in a marked improvement, with test errors decrcasing from 31.16% to 20.73% for IF = 100 and from 27.25% to 20.86% for IF = 50. This finding underscores the efficacy of Lpp) in discerning and filtering out less reliable syuthesized samples, leading to improved model performance. Furthermore, the subsequent incorporation of our proposed L(d) criterion contributes an even more pronounced decrease in test errors, underscoring its effectiveness in further refining the selection of high-quality synthetic samples. These conclusions are corroborated by results in the OOD setting, which demonstrate that our combined rejection strategy (I(p) + (d)) elfectively ident ifies high- quality, credible samples for model training, thereby catalyzing performance gains across different distribution settings. Finally, when maintaining L(p) + L(d) as the rejection mechanism and evaluating different sample selection criteria, observe that the Ko ~ g(No.N.)* approact surpassing the alternative of randomly selecting a seed sample for initialization.
7 demonstrates superior performance,
In the right part of Tab. 6 we evaluate the influence of different values for the threshold Tр (for the L(p) criterion) and / (related to ko sample selection) on performance, using the combined L(p) + L(d) rejection criteria. While an excessively lenient Tp (a very high value) could theoretically allow more lower-quality synthetic samples
pass (those L(p) aims to filter) and potentially lead to suboptimal performance, our experiments reveal a nuanced trend within the tested range. Specifically, with 3 fixed at 0.999, we observe that test error decreases as Tp increases from 0.3 to 0.9. This suggests that, within this range and in conjunc- tion with our L(d) criterion effectively filtering samples by their proximity to the target class distribution, a less stringent T(p) (higher Tp) allows for a beneficial set of samples to be considered. This indicates that our
2 AAssuming No here refers class count relevant to the ky selection, ensure consistency with its definition elsewhere.
10

---

Published in Transactions on Machine Learning Research (06/2025)
Table 7: Test top-1 errors (%) of ResNet-32 on CIFAR- LT-10 under different. imbalance factors, where fand tdenote the results from the original paper and our re- production, respectively. The methods are trained with CE loss unless otherwise stated.
Initialization
Meshod
CIEAR-LTT10
Source
Imbalance Factor
100
50
ID
M2n
25.34t
21.7]
16.81
Real
Syothri
OOD
Open-Sampling
28.28]
22.381
18.21f
Visualization of features of synthetic/real samples on CIFAR-LT-10 (IF 100) with ResNet- 32, where 'o' and the star indicate real and synthetic samples in the same class, respectively.
Figure 2:
OOD
OURS
23.43
20.03
16.39
ID
OURS
22.85
18,20
16.96
00D to ID)
OURS
21.83) 1.02
17.96), 0.24
15.32)
0.64
overall rejection framework (L(p) + L(d)) effectively manages sample quality across different configurations of Tp. Regarding B, which influences the diversity of synthetic sample initialization, larger 3 (when Tp is 0.9) yields the best performance, highlighting that greater initialization diversity aids generalization on long-tailed datasets.
4.5 Combining OOD and ID.
Beyond leveraging the OOD samples to replace the majority samples in our framework, further, we explore whether combining the OOD setting and ID setting can produce better performance. To this end, we conduct experiments on CIFAR-LT-10 with IF E (200, 100, 50) on ResNet-32. Specifically, we firstly optimize synthetic samples and train the target classifier f using Dood. Then we save the best checkpoint and employ an additional 20 epochs further train f under the ID sotting, where we initialize the minority samples with majorily samples, using the Alg. [ In other words, we first use the OOD setting to train the target classifier f and then further train f under ID setting.
As shown in Tab. 7 our method in ID and OOD settings outperform the M2m and Open-Sampling, respec- tively. Furthermore, introducing the OOD dataset into the ID setting, our method further achieves 1.02%, 0.24% and 0.64% gains with IF E {200, 100, 50), respectively. These demonstrate that our framework in ID or OOD setting can achieve better performance than corresponding baselines. Besides, the OOD samples can be utilized to further improve the performance of our proposed method under the ID setting.
4.6 Visualizations
Visualization of synthetic samples in feature space. As shown in Fig. 2 we visualize synthetic minority samples and real minority samples in CIFAR-LT-10 (IF 100) in feature space, using (-SNE (Van der Maaten & Hinton, 2008). We show classes 7, 8, 9, and 10 (descending ranked by the number of their samples), cach of which has 232, 139, 83, and 50 real samples. We randomly seleet 50 synthetic samples for each class after using the sample rejection criteria. In terms of M2m, synthetic samples from each class are difficult to capture the corresponding real distribution. Besides, synthetic samples from different classes are seriously coupled together. As expected, our synthetic samples can effectively capture the real distribution of each class. Therefore, it reveals why our method can generate more beneficial synthetic minority samples than M2m.
Visualization of synthetic samples in pixel space. We use M2m and ours to optimize a sample xo with Kn car to k deer the same pre-trained model g. Figure 3(a) shows that 2n is correctly classified with probability of 0.99 g. Then, after optimization, ours and M2m produce different synthetic samples & and corresponding noise, even though the two å are visually indistinguishable. At this time, the probability of & optimized by ours being classified as its original class ko on is 0.07, and the probability of being classified to is 0.91, while the corresponding probabilities of M2m are 0.19 and 0.73. This shows that ours successfully pushes the synthesized sample away from its original label ko on g, and makes it closer to our target label k. At the same time, ours also makes f believe that & is a sample from k with a higher probability (0.57, 0.32 larger than M2m) on the target classifier f, and its classification probability on kn is
11

---

Published in Transactions on Machine Learning Research (06/2025)
07
Prly curlk:g) 0.07
Pr deer( 0.1
HMO
+
Text eenrr Rate (%)
SAug
- carlR;f) = 0.13
Pr(y dear[ 0.57
Pry curlR:g) 0.19
Pr (R)) 0.73
Pr(y = car(::)) = 0.49
Pr(y dear[
Pr car (+s:)) 0.99
Taining Time (min)
(b) TTpp1 test errors (%) with the in- creased training time on CIFAR-LT-10 with IF 100 trained ResNet-32.
(a) An illustration of syuthetic minority sample by our method and M2m, where g assumed to be ReaNet-32 trained by standard ERM. The noise image amplified by 20 for better visibility.
Figure 3: Visualization of synthetic sample (a) and time-consuming analysis (b).
Table 8: Test top-1 errors (%) ResNet-32 on CIFAR-LT-10 CIFAR-LT-100 under different imbalance factors, where fand fdenote the results from the original paper and our reproduction, respectively. The methods are trained with CE loss unless otherwise stated.
Method
CIFAR-LT-10
CIFAR-LT-I00
Imbalance Factor
200
100
50
200
100
50
M2mf
25.344:0.461
21.7+0.16
18.81=0.76;
63.77++0.33
57.110.16
50.48+0.433
MUR
23.01=0.37
18.91+0.29
06.42+0.115
62.28
53.84+0.15
47.25++0.17
OURS+ED
22.93
18.75:00.15
16.16:0.22
61.97:10110
53.01=0.22
47.3311.22
OURS+OT
22.85:0.12
18.20:0.21
15.96=1.11
61.28=0.21
52.95+0.18
47.02:0.26
significantly lower than the corresponding results of M2m by 0.30. This shows that the samples synthesized by our method are more credible and realistic for f.
In fact, we do not optimize majority sample zo into a picture that is similar to minority sample in the pixel space, like generator. Instead, we simply optimize the sample zn directly to confuse models y and f, making the models believe that our synthetic samples & are indeed from real minority class, which helps the network to generalize on the minority class. Recall the visualization of feature space, which shows that the feature distributions of our synthetic samples are closer to real samples æ of class k. This indicates that the features of our synthetic samples are more realistic and credible, where ʃ regards the features of the real samples and the synthetic samples are from the same distribution. On the other hand, Section C.6 shows that the probability of our synthetic samples being correctly classified the target class k the larget classifier f is siguificantly higher than that of M2m. From the classification perspective, our method can produce better synthetic minority samples and reduce the difficulty of the model learning.
In summary, our starting point is to generate synthetic samples that are more realistic and credible for the model. From the perspective of pixel space, our method and M2m have no obvious difference. However, from the classification probability and feature matching perspectives, our synthetic samples are more realistic and credible for the network and, therefore, more effective than M2m.
4.7 Comparison of different implements of Disto Qx)
To prove the generality and effectiveness of our method, we conduct experiments on CIFAR-LT-10 and CIFAR-LT-100 with different imbalanced factors (IF) by using different implements of Distø Dista(Pe, Qx), where the results of different methods are summarized in Tab. g We implement Dista Qx) using MMD, ED and OT, denoted as OURS+MMD, OURS+ED and OURS+OT, respectively. We report the average results of our method for three runs with the standard deviation independently. We can find that all of them have superior performance compared to the M2m baseline by a large margin. Besides, we can observe that ours+OT performs best, which might benefit the more accurate characterization and measurement of the distance between distributions brought by OT. other words, OT Iearns an optimal transport plan which endows each cost element with corresponding importance T.j. demonstrating the effectiveness and generality of our proposed method.
12

---

Published in Transactions on Machine Learning Research (06/2025)
4.8 Convergence and time complexity
To fairly compare training time-consuming,, conduct experiment on CIFAR-LT-I0 with IF 100 ResNet-32 in the same device environment with one Tes0a-V100 GPU and we run all the methods with the same epochs. As shown in Fig F(g3(() 3(b] our method (OT, ED and MMD) achieves a lower test error rate within the same computation time, regardless of whether Optimal Transport (OT), Maximum Mean Discrepancy (MMD), or Euclidean Distance (ED) is used to calculate Dista(Pz,Qu))
4.9 Additional Analysis
We analyze the impact. of OOD initialization source on synthesis in Appendix C.5 classification confidence of synthetic samples on the target classifier in Appendix C.6 influence of the pre-trained model g in Appendix [C8] visualization of the confusion matrix in Appendix C.9 and more visualization of synthetic samples in Appendix C.10
5 Related Work
Over-sampling methods for long-tailed problem. Data-based methods aim to solve the imbalance problem by building relatively balanced classes from the perspective of data, including under-sampling
majority samples (He & Garcia 2009;
Drummond & Holte 2003). over-sampling minority samples (Shen
et al. 2016 Buda et al. 2018]] Barandela et al. 2004) and data augmentation (Ahn et al. 2023 Park et al.
2022 Yan al., 2019
2020 Gao et al. 20223 2024a Li al. 2025 Guo et al. 2022b). Our method has a close connection with minority over-sampling methods. A related work is Optimal transport over-sampling (OTOS) (Yan et al. 2019), which maps the noise to synthetic ones based on the Wasserstein barycenter. Different from OTOS which generates samples by mapping matrix and is limited to a binary classification, we provide more general and direct optimization objective for generating synthetic samples. By minimizing this objective, we can obtain synthetic samples with reliable classification confidence and high representation similarity, where we can handle multi-class classification task and leverage more practical OOD setting. Another related work, M2m (Kim et al. 2020). translates majority samples to the largel minority class by maximizing the prediction probability. However, our work, we optimize synthetic minority samples from both perspectives of classification confidence and distribution matching, where extend the ID to OOD setting for further versatility.
Kim et al.
Utilizing auxiliary dataset for long-tailed problem. In imbalanced learning, Yang & Xu (2020) leverage unlabeled ID data as additional samples to compensate for the minority classes, while Su et al. 2202 adopts a aunissupervieed learning framework to incorporate out-of-class samples from related classes. Open-Sampling (Wei et al., 2022) explores the benefit of using OOD data in the long-tailed problem. The major difference between ours and Open-Sampling is that we translate OOD samples by introducing an optimization phase and introducing sample rejection strategy but Oamp-Sanpling assigns noisy label to each OOD sample using pre-defined label distribution without filtering the OOD) data.
6 Conclusion
To address the long-tailed classification issue, propose novel framework for translating majority ples into synthetic minority samples by leveraging classification confidence and distribution matching. Our method optimizes the synthetic minority samples by enforcing them to satisfy the classification constraints and being close to the distribution of real samples in the larget minority class. In addition, we introduce an effective regularization term for confusing classes, enabling our framework to better utilize available and rich OOD data to synthesize minority classes. Extensive experiments on benchmark datasets demonstrate that our framework can generate effective minority samples and achieve the desired long- classification performance.
13

---

Published in Transactions on Machine Learning Research (06/2025)
Acknowledgment
This work of Dandan Guo was supported by the National Natural Science Foundation of China (NSFC) under Grant 62306125. This work of Hongyuan Zha was supported by the National Natural Science Foundation of China (NSFC) under Grant 72495131. This work was supported by the Guangxi Key R&D Project (No. AB24010167), the Project (No. 20232ABC03A25), Shenzhen Longgang District Science and Technology Innovation Special Fund (No. L.KCYLWS2023015)) Futian Healthcare Research Project. (No.FTWS002), and Shenzhen Medical Research Fund (No. C2401036), and Hospital University United Fund of The Second Affiliatdd Hospital, School of Medicine, The Chinese University of Hong Kong, Shenzhen (No. HUUF-MS- 202303).
References
Sumyeong Ahn, Jongwoo Ko, and Se- Young Yun. Cuda: Curriculum of data augmentation for long-lailed recognition, arXiv preprint arXiv:230 05499, 2023.
R. Barandela, R. M. Valdovinos, JS Sánchez, and F. J. Ferri. The imbalanced training sample problem: Under over sampling? In Structural, Syntactic, and Statistical Pattern Recognition, Joint IAPR International Workshops, SSPR 2004 and SPR 2004, Lisbon, Portugal, August 18-20, 2004 Proceedings, 2004.
Mateusz Buda, Atsuto Maki, and Maciej A Mazurowski. A systematic study of the class imbalance problem in convolutional neural networks. Neural networks, 106:249 259, 2018
Kaidi Cao, Colin Wei, Adrien Gaidon, Nikos Arechiga, and Tengyu Ma. Learning imbalanced datasets with labol-distrbutiion-waarr margin loss. Advances in neural information processing systems, 32, 2019.
Nilesh V. Chawla, Kevin W. Bowyer, Lawrence O. Hall, and W. Philip Kegelmeyer. SMOTE: synthetic
minority over-sampling technique. J. Artif. Intell. Res., 16:321 357, 2002. doi: 10.1613/jair.955 URL
https://doi. http://66b..rrg110.61/////////
Pong Chu, Xiao Bian, Shaopeng Liu, and Haibin Ling. Feature space augmentation for long-tailed data. In Computer Vision-ECCV 2020: 16th Europeon Conference, Glasgow, UK, August 23-28, 2020, Proceed- ings, Parl XXIX 16, pp. 694-710. Springer, 2020.
Yin Cui, Menglin Jia, Tsung-Yi Lin, Yang Song, and Serge Belongie. Class- loss based on effec- tive number of samples. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp.
2019.
Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. Advances in neural information processing systems, 26, 2013.
Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pp. 248-255. Iece, 2009.
C. Drummond and R. C. Holte. C4.5, class imbalance, and cost sensitivity: Why under under-sampling beats over-sampling. proc of the ieml workshop on learning from imbalanced datasets ii, 2003.
Cong Fang, Hangfeng He, Qi Long, and Weijie Su. Exploring deep neural networks via layer-peeled model: Minority collapse in imbalanced training. Proceedings of the National Academy of Sciences, 118 (433::2103091118, 2021.
Jintong Gao, He Zhao, Zhuo Li, and Dandan Guo. Enhancing minority classes by mixing: An adaptative optimal transport approach for long-taile classification. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (ods.), Advances in Neural Information Processing Systems, volume 36, pp. 60329- 60348. Curran Associates, Inc., 2023. URL https:/pproceaaggs.. wwt.ripps.ccppppr___eeeee////e/e
14

---

Published in Transactions on Machine Learning Research (06/2025)
Jintong Gao, He Zhao, Dan dan Guo, and Hongyuan Zha. Distribution alignment optimization through neural collapse for long-tailed ion. In Forty-first International Conference on Machine Learning, 2024a. URL
Jintong Gao, He Zhao, Zhuo Li, and Dandan Guo. Enhancing minority classes by mixing: An adaptative optimal transport approach for long-tailed classification. Advances in Neural Information Processing Systems, 36, 2024b.
Arthur Gretton, Karsten M Borgwardt, Malte J Rasch, Bernhard Schölkopf, and Alexander Smola. A kernel two-sample test. The Journal of Machine Learning Research, 13(1):723-773, 2012.
Dandan Guo, Zhuo Li, He Zhao, Mingyuan Zhou, Hongyuan Zha, et al. Learning to re-weight examples with optimal transport for imbalanced classification. Advances in Neural Information Processing Systems, 35: 25517 15517-25530, 2022a.
Dandan Guo, Zhuo Li, meixi zheng, He Zhao, Mingyuan Zhou, and Hongyuan Zha. Learning re- weight examples with optimal transport for imbalanced classification. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (eds.), Advances in Neural Information Processing Systems, volume 35, pp. 25517 25530. Curran Associates, Inc., 2022b. URL
neurips. wwtpa/i.....rrpp..pppppprrrrrr cttppp/.ec/appfffeess////// files/ pdf
Zongbo Han, Zhipeng Liang, Fan Yang, Liu Liu, Lanqing Li, Yatao Bian, Peilin Zhao, Bingzhe Wu, Changqing Zhang, and Jianhua Yao. Umix: Improving importance weighting for subpopulation shift via uncertainty-aware mixup, 2023. URL https://arxiv..ort wttpe/iv..rr//bb///00000 aas//bbb/222 08928
Haibo He and Edwardo A Garcia. Learning from imbalanced data. IEEE Transactions knowledge and data engineering, (19):1223314 2009.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving doep into rectifiers: Surpassing human- level performance on imagenet classification, 2015. URL httpe://arriv..rrr/aaa////00..
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition.
Proceedings of the IEFE conference on computer vision and pattern recognition, pp. 770-775 778, 2016.
Dan Hendrycks, Mantas Mazeika, and Thomas Dietterich. Deep anomaly detection with outlier exposure. 04606, 2018.
arXiv preprint
Yan Hong, Jianfu Zhang, Zhongyi Sun, and Ke Yan. Safa: Samplo-adaptive feature augmentation for long- tailed image classification. Computer Vision ECCV 2022: 17th European Conference, Tel Aviu, Israel, October 23 27, 2022, Proceedings, Part XXIV, pp. 587- -03. Springer, 2022.
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 pottern recogrition, pp. 5375-5384, 2016.
Bingyi Kang, Saining Xie, Marcus Rohrbach, Zhicheng Yan, Albert Gordo, Jiashi Feng, and Yannis Kalan- tidis. Decoupling representation and classifier for long-tailed recognition. arXie preprint #XXi:11100009211 2019.
Jachyung Kim, Jongheon Jeong, and Jinwoo Shin. M2m: Imbalanced classification via major-to-minor translation. In Proceedings of the IEEE/CVP conference on computer vision and pattern recognition, pp. 13896-13905, 2020.
Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.
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 IEEE/CVF conference on computer vision and patlern recognilion, pp. 5212-521 2021.
15

---

Published in Transactions on Machine Learning Research (06/2025)
Zhuo Li, He Zhao, Anningzhe Gao, Dandan Guo, Tsung-Hui Chang, and Xiang Wan. Prototype-oriented clean subset extraction for noisy long-tailed classification. IEEE Transactions on Circuils and Systems for Video Technology, pp. 1 1, 2025. doi: 10.109//CCSVT.2253335000
Tsung-Yi Yi Lin, Priya Goyal, Ross Girshick, Kaiming Hc, and Piotr Dollár. Focal loss for dense object detection. In Proceedings of the IEEE international conference on compuler vision, pp. 2980 2988, 2017.
Jialun Läu, Wenhui Li, and Yifan Sun. Memory-basee jitter: Improving visual recognition on long-tailed data with diversity in memory. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp. 1720 1728, 2022.
Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In Proceedings of the IEEE international conference on computer vision, pp. 7730-3738, 2015.
Ziwei Liu, Zhongqi Miao, Xiaohang Zhan, Jiayun Wang, Boqing Gong, and Stella X Yu. Large-scale long- tailed recognition in an open world. In Proceedings of the IEEE/CVE conference on computer vision and poltern recognation, pp. 2547-2546, 2019.
Seulki Park, Youngkyu Hong, Bycongho Heo, Sangdoo Yun, and Jin Young Choi. The majority help the minority: Context-rich minority oversampling for long-tailed classification. In Proceedings of the IEEE/CVF Conference on Compuler Vision and Pallern Recognition, pp. 6887-68888 6896, 2022.
Gabriel Peyré, Marco Cuturi, et al. Computational optimal transport: With applications to data science.
Foundations and Trends@ in Machine Learning, 11(5-6):355 607, 2019.
Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. International conference on machine learning, pp. 8748 8763. PMLR, 2021.
Jiawei Ren, Cunjun Yu, Xiao Ma, Haiyu Zhao, Shuai Yi, et al. Balanced meta-softmax for long-tailed visual recognition. Advances in newral information processing systems, 33:4175 4186, 2020.
Maria Rizzo and Gábor Szákely. Energy distance. wiley interdisciplinery reviews: Computational statistics, 8(1):27 38, 2016.
Sebastian Ruder. An overview of gradient descent optimization algorithms. arXiv preprint 6rXiv:1609.04747, 2016.
Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International journal of computer vision, 115:211-253 2015.
Li Shen, Zhouchen Lin, and Qingming Huang, Relay backpropagation for effective learning of deep convo- lutional neural networks. In Computer Vision-ECCV 2016: 14th European Conference, Amsterdam, The Netherlands, Oclober 11-14, 14, 2016, Proceedings, Part VII 14, pp. 467-482. Springer, 2016.
Su, Zezhou Cheng, and Subhransu Maji. A realistic evaluation of semi-supervise learning for fino-grained classification. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12966 12975, 2021.
Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, (11),, 2008
Jason Van Hulso, Taghi M Khoshgoftaar, and Amri Napolitano. Experimental perspectives on learning from imbalanced data. In Proceedings of the 24th international conference on Machine learning, pp. 035-942, 2007.
16

---

Published in Transactions on Machine Learning Research (06/2025)
Jianfeng Wang, Thomas Lukasiewicz, Xiaolin Hu, Jianfei Cai, and Zhenghua Xu. Rsg: A simple but effective module for learning imbalanced datasets. In Proceedings of the JEEE/CVT Conference on Computer Vision and Pattern Recognition, pp. 3784 3793, 2021.
Hongxin Wei, Lue Tao, Renchunzi Xie, Lei Feng, and Bo An. Open-sampling: Exploring out-of-distribution data for ro-balancing long-tailed datascts. In International Conference on Machine Learning, pp. 23615- 23630. PMLR, 2022.
Yuguang Yan, Mingkui Tan, Yanwu Xu, Jiezhang Cao, Michael Ng, Huaqing Min, and Qingyao Wu. Over- sampling for imbalanced data via optinal transport. In Proceedings the AAAT Conference on Artificial Intelligence, volume 33, pp. 5605-5612, 2019.
Lu Yang, He Jiang, Qing Song, and Jun Guo. survey on long-tailed visual recognition. International Journal of Computer Vision, 130(7):183-11872, 2022.
Yuzhe Yang and Zhi Xu. Rethinking the value of labels for improving class-imbalanced learning, Advances in neural information processing systems, 33:19290 19301, 2020.
Bo Zhao and Hakan Bilen. Dataset condensation with distribution matching. In Proceedings of the IEEE/CVP Winter Conference on Applicalions of Compuler Vision, pp. 6514 6523, 2023.
Bolei Zhou, Agata Lapedriza, Aditya Khosla, Aude Oliva, and Autonio Torralba. Places: A 10 million image database for scene recognition. IEEE tronsactions on pattern analysis and machine intelligence, 40(6): 1452-1464, 2017.
Boyan Zhou, Quan Cui, Stu-Shen Wei, and Zhao-Min Chen. Bbm: Bilateral-branch network with cumulative learning for loilet visual recognition. 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), Aug 2020. doi: 10. 10000///rp22200000000000077 URL http://dx. doi. org 10. 1 1109/1110ppeee 20000//cpp42200.2000000 2020
17

---

Published in Transactions on Machine Learning Research (06/2025)
Appendix For Synthesizing Minority Samples for Long-tailed Classification via Distribution Matching
A Alternatives for (Ps Qx)
A.1 Maximum Mean Discrepancy (MMD)
Preliminaries. MMD is effective non-parametric metric for comparing the distributions based on two sets of data (Gretton et al. 2012), where the general MMD between two distributions P and Q is defined as
(M.Q(P. = sup |Ez-P [x)) - Ey-q 1 (v)l 11
(10)
where Ex-p[] denotes the expectation with regard to the distribution Φ is the embedding function, and |1 1 defines set of functions in the unit ball of reproducing kernel Hilbert space (RKHS) H.
Define Dists(Ps: (Pstt( Qk) with MMD. As we do not have access to ground-truth data distributions for syn-
thetic and real samples shown in Eq. 10 we can use a biased empirical estimate of the MMD by replacing the population expectations with empirical expectations (Grettn et al. 2012), which are computed on the synthetic and real samples in Pa and Qs and denoted as
N₂
My
2
1
1
Dista (Px, Q+)
Do (xn)
Σ m=1
she (2m)
(11)
N
MR.
A.2 Energy Distance (ED)
Drawing inspiration from the concept of potential energy between objects in a gravita- tional field, Energy Distance (ED) (Rizzo & Székely 2016) measures the similarity between two probability distributions, P and Q. This can be mathematically expressed follows:
Preliminaries.
ED3 Q) = 2I2 (lll((t))) - <(v)) -E-rllo(z)) -(z))l- Ey 6(/)
(12)
where [. denotes the expectation with respect to the distribution P and || denotes the Euclidean norm (length) of its argument. In addition, x' and yf are independent copies of and y, respectively.
Define Dista Qk) with ED. Here, follows:
define Dist Pk. Qk) based the energy distance (ED)
Ma
2
Disto (Pr, Qx)
Σ
(1m))
Mr
No
(1)a) - (2mm2
(13)
N
2
n,m=1
M
Σ
(0mmm))
M
B Algorithms of our framework
In this section, we give the algorithm processes of our method under OOD settings as shown in Alg. Our jointly learning framework is shown in Alg.
18

---

Published in Transactions on Machine Learning Research (06/2025)
Algorithm 2: Oversampling minority samples via our framework (Out-of-Distruution))
Input Dataset Draain and oodd classifier f and pre-trained classifier 9, a confusion matrix A, (11, X2, 7, T, T, 5}
Output: A class-balanced dalaset Dhal Initialize Dal < P
Randomly sample balanced subset from Drrain and obtain confusion matrix A by evaluate the pre-trained model y using this subset;
for k = 2,... K do
Compute M&- +- N1 - Nk; Initialize Dr 0;
for 1, 1,... MR do
// Step 1. Sample selection for
Sample a 2m from Dood randomly;
Initialize ::10 -- Fm + 6m with standard Gaussian noise cm-i
7
update DD + Dr ((mm (.0mm,, ;));
end
10
Build QN = = Mr
Mr
6em + and Ps En= No
ds, according to Eq
11
Obtain ke if Ak.k: 2 AR.s where i E [1, K] and i iadiik k;
12
for 1 = 1, -,..,, T do
// Step 2. Optimization for &
am Mm/ (L) + An Jn. (6m)) A2 Disto Pky Qk), according to Eq.
14
end
Update Dr with optimized synthetic samples; 15
for im in Dr do if ((g(ām),)) 2 Reject = then
// Step Sample rejection for &b
17
18
im + with a random sample from class is in Diraini end
19
20
Update Daal t Daal U {(0ms k)];
21
end
22
end
Algorithm 3: Joint Training Paradigm with Synthetic Sample Generation
Input: Imbalanced dataset D, warm-up epochs Ewarm; main epochs Emain
Output: Trained classifier f
Randomly initialize classifier f;
for = 1,..., Ewarm do
/+ Stage 1: Warn-up +/
for miná-batch B in D do
4
Update I using B;
// standard CE training
end end
for e= Fwarr + 1,... Fmnin do D do Resample an embedding function the Generate balanced batch Bhat by synthesizing minority samples; Update of using Bbali end 12
/* Stage 2: Joint training (ours) +/
for mini-bated
// See Alg.
11
// train augmented data
13 end
C More details about datasets and experiments
C.1 Settings and Training details
Unless otherwise stated, we set the imbalance factor as IF NN/NK and use = iterations with a step size of 0.1 to optimize the synthetic samples at each training iteration. The hyper-paramett for the OT entropy constraint is 0.1.1 and the maximum iteration number in the Sinkhorn algorithm 200. We use SGD with momentum 0.9 and weight decay 5 and conduct all the experiments 8 Tesla V100 GPUs.
19

---

Published in Transactions on Machine Learning Research (06/2025)
C.2 CIFAR-10 10 and CIFAR-100 datasets
CIFAR-LT-10 / CIFAR-LT-100 The original CIEAR-10/ 1ARR-11/CIFAR-1O0 datasets Krizhevsky et al. 2009) include 60,000 images and 10/ 100 classes with a size of 32 x 32, where there are 50,000 images for training and 10,000 for testing. By following (Kim et al.] 2020), we create CIFAR-LT-10 and CIFAR-LT-100 by randomly under-sampling in the original datasets with IF = (200, 100, 50). We use the original test dataset to evaluate our method. Training details. Following (Kim et al. 2020 Li et al. 2021 Guo et al. 2022a)
we use ResNet-32 (He et al. 2016) as the backbone. We employ 200 epochs for training f with an initial learning rate a of 0.1, which is decayed by le-2 at 160-th epoch and 180-th epoch. We set batch size as 32 and start our method at 160-th epoch, where we set A1 and 22 as 0.5, 3 as 0.999 and I as 0.9.
C.3 ImageNet-LT and Places-LT
ImageNet-LL. The original ImageNet-20122 dataset (Deng et al.
2009 includes 1,281,167 images and 1000
classes with a max size of 1300 x 732. By following (Kim et al.
2020
Li et al. 2021 Liu et al.) 2019). we
create ImageNet-LT with 115.8K samples in 1000 classes and IF dataset to test our method.
1280/5. 55. We adopt the original validation
Places-LT The original Places-365 dataset Zhou al. 2017 includes 1,803,460 images and 365 classes with a max size of 5000 x 3068. By following (Liu et al. 2019). create Places-LT with 62.5K samples in 1000 classes and the imbalance factor IF 4980/5. We adopt the original test dataset to test our method.
Training details. Following previous works (Kim et al., 2020 Li et al. 2021 Kang et al. 2019), we use ResNet-50 as the backbone for ImageNet-L.T. We employ 200 epochs for training f with an initial learning rate a as 0.1, which will be decayed by le-1 at the 160-th epoch and 180-th epoch. For Places-LT, we employ ResNett-12 pre-trained on the full ImageNett dataset (Russakovsky et al.] 2015) as the backbone following (Guo et al.) 2022aa Li al.) 2021]. We set 200 epochs for training with an initial learning rate a as 0.1, decayed by 1e- every 40 epochs. We start our method at 160-th epoch for ImageNet-LT and 90-th for PLaces-LT. We set A and A2 as 0.5, B as 0.999 and T as 0.3. For all experiments, we initialize batch size as 64 and set it as 32 after deploying our method for training stability.
C.4 Training details about pre-trained model g
CIFAR-LT-10 / CIFAR-LT-100. For CIFAR-LT-10/ / CIFAR-LT-100, use ResNet-32 (He ct al. 2016) as backbone network for pre-training. We employ 200 epochs for training 9 with initial learning rate a of 0.1, which will be decayed by 1e-2 at 160th epoch and 180th epoch. We use SGD with momentum 0.9 and weight decay 54 and set batch size as 128. In the first 160 epochs, we use the original imbalanced dataset to train the model g. For the last 40 epochs, we use the vanilla over-sample technique by inverse class frequency to further train the model g. We save the best checkpoint as our pre-trained model g.
ImageNet--LT & Places-LT For ImageNet-LLT, we use ResNet-50 (He al.) 2016) as backbone network for pre-training. We employ 200 epochs for training g with an initial learning rate a of 0.1, which will be decayed by le-² at 160th epoch and 180th epoch. For Places-LT, we employ ResNet-152 pre-trained on the full ImageNet datasct. We use 120 epochs for training 9 with an initial learning rate a as 0.1, which is decayed by -e-1 every 10 epochs. For both datasets, we use SGD with momentum 0.9 and weight decay 5e and set batch size as 512. Similar to CIFAR-LT-10 / CIFAR-LT-100, before the 160 - th and 90 - th epoch on ImageNet-LT and Places-LT, we use the original imbalanced dataset to train model g. For the last epochs, we adopt the vanilla over-sample technique by inverse class frequency (Drummond & Holte 2003) to further train g. We save the best checkpoint as our pre-trained model g.
C.5 Impact of OOD Initialization Source on Synthesis
To assess the influence of the initial seed on synthetic sample generation, we evaluated several Out-of- Distribution (OOD) data sources for initialization when targeting the CIFAR-LT-10 dataset, with results presented in Tab. g We compared initializations using: (i) Medical images (OrgSnAMNNSTT, (ii) Pure
20

---

Published in Transactions on Machine Learning Research (06/2025)
Noise, and (iii) OOD Natural image against CE-Baseline without synthetic samples. The results clearly demonstrate that our synthesis framework, irrespective of the OOD initialization source, yields substantial improvements over the CE- Baseline across all tested imbalance factors. This underscores the overall benefit of incorporating synthetic samples.
Among the different OOD initialization strategies, a clear performance hierarchy emerged. While initializing with Medical images improved upon the baseline, was generally outperformed by Pure Noise initialization.
This suggests that very large domain gap (as between medical images and CIFAR-LT-10's natural scenes) limit the effectiveness of the initial seed. Notably, initializing with OOD Natural images consistently produced the best results, surpassing both Pure Noise and Medical image initializations across all imbalance factors.
These findings highlight that the choice of OOD initialization source is critical, with closer domain proximity to the target datasot leading superior synthesis outcomes. For natural image classification tasks like CIFAR-LT-10, utilizing OOD natural images as an initialization source is most effective. We attribute this to their ability to provide more relevant foundational features, such common textures and color distributions, which are advantageous for generating high-quality minority class samples tailored to the target domain.
Table 9: Test top-1 crrors (%) of ResNet-32 on CIFAR-LT-10 with IF E [200, 100, 50] and different OOD datasets. Distribution means the initialization of to-be-learned synthetic samples â. e.g., OOD denotes we initialize & with OOD samples. Domain indicates the corresponding domain of OOD dataset.
Domain
200
100
50
CF-Baseline w/o synthetic samples
34.13
29.86
25.06
Pure Noise
23.93
20.40
16.75
Medical
23.95
20.95
17.17
Natural
23.43
20.03
16.39
C.6 Classification confidence of synthetic sample on target classifier f.
To prove the effectiveness of our method in enhancing the generation of high-quality synthetic samples, we compare the classification performance of the synthetic samples on the target classifier for CIFAR-LT-10 (ID) with IF = 100 on ResNet-32. Specifically, we use the larget classifier to output the probability of the synthetic samples in class & being correctly predicted as the & class, where we consider & E (2.... ,10.. Then we compute the average probability for all samples and express it as a percentage. Compared to M2m, Fig. [(a) illustrates that the syuthetic samples generated by our method (based on OT) have higher classification confidence for the corresponding concerned class during the training of target classifier in both ID and OOD settings. This finding suggests that the generated samples by ours are more credible.
C.7 Proportion of Successfully Synthesized Samples
To quantify the contribution of our novel sample generation relative to the fallback mechanism (repeating existing minority samples), Figure A(b) illustrates the proportion of successfully synthesized and accepted minority samples within the total augmented minority slots per epoch on CIFAR-10-LT (ID setting). The results show that a consistent fraction, typically ranging from 15% to 21% depending on the imbalance factors, consists of newly generated synthetic samples that pass our quality criteria. This demonstrates that while the fallback ensures robust minority representation, our synthesis module actively produces a continuous stream of novel data, and the overall performance improvement is attributable to both this baseline rebalancing effect and the diversity introduced by these gemuinely new instances, distinguishing approach from simple upsampling.
21

---

Published in Transactions on Machine Learning Research (06/2025)
522
IF-50
IF-100
IF-200
Ratio (#5yn #Total)
55
0.19
9.18
0.27
0.36
25
OURS (ID)
(OOD)
20
M2n
1
25
30
35
40
100
165
170
175
Epech
180
185
190
196
200
Training Epoch
(a) Probability of synthetic samples correctly classified as the concerned class & by the target classifier f.
(b) Evolution of s'ssthetiic sample ratio under the
ID setting across epochs.
Figure 4: Visualization of classification accuracy and out-of of-distribttion (OOD) sample synthesis behavior.
C.8 Influence of pre-trained model g
In this section, we investigate whether the effectiveness of the proposed method is affected by the performance of pre-trained g. We can see that the better the performance the pre-trained classifier, the better the final classification result ours in general. Although the pre-trained classifier in the 100th epoch and the res--porformed classifier have diffcrent classification performance, they have similar impact on the final classification result. Besides, the ensemble of three best-performed classifiers achieves betler performance than only using one best-performed classifier. In our work, perform the experiments only using best- performed classifier. It demonstrates that the final classification results will increase if we use the ensemble of the pre-trained classifiers to optimize the synthetic samples.
Table 10: Test top-1 errors (%) on CIFAR-LT-10 with IF 100 under the in-distribution sctting. Pre-trained performance indicates the overall performance on the y and Final performance is the corresponding final classification result.
g
Pre-trained performance
Final performance
1-th epoch
66.51
21.16
3-th epoch
59.75
20.23
10-th epoch
44.26
20.17
20-th epoch
41.71
19.07
100-th epoch
30.95
18.33
Best epoch
28.17
18.37
Ensemble
28.77, 30.23, 29.01
18.01
C.9 Visualization of confusion matrix
To demonstrate the offectiveness of our method in improving the performance of minority classos, we visualizo the confusion matrices of CE, M2m and OURS on CIFAR-LT-10 with 1F-200. As shown in Fig 5 CE has poor classification performance in minority classes. Therefore, Although M2m mitigates the problem, still performs poorly on the rarest classcs. Our proposed method achieves better performance than CE and M2m. In particular, ours is superior to strong baseline M2m for almost every class, thereby alleviating the imbalanced classification problem.
necessary to solve the long-tailed problem.
22

---

Published in Transactions on Machine Learning Research (06/2025)
airpanas
M2
OURS
-AA
AAI
C2L
C29
COR
C09
03
ships
327
323
6X
CCB
620
Figure 5: Confusion matrices of the CE, M2m and OURS on CIFAR-LT-10 with the imbalance factor 200.
C.10 Visualization of Synthetic Samples
In this section, shown in Fig. 6 Fig. and Fig. present visual examples of synthetic minority samples generated by method the ImageNet-LT dataset. Similarly, while our synthetic samples and their source images may be difficult to distinguish by eye in pixel space, the subtle, oplimized "noise" introduced during our synthesis process effectively alters the classifier's 's perception and classification of these modified samples.
23

---

Published in Transactions on Machine Learning Research (06/2025)
Source Image: 5
Noise
Synthetic Image: 734
Source Image: 5
Noise
Synthetic Image: 412
Source Image: 4
Noise
Synthetic Image: 541
Source Image:
Noise
Synthetic Image: 898
Source Image: 6
Noise
Synthetic Image: 913
Figure An illustration synthetic minority samples by our method on ImageNet-
24

---

Published in Transactions on Machine Learning Research (06/2025)
Source Image: 5
Noise
Synthetic Image: 696
Source Image: 3
Noise
Synthetic Image: 158
Source Image: 5
Noise
Synthetic Image: 380
Source Image:
Noise
Synthetic Image: 185
Source Image: 6
Noise
Synthetic Image: 544
Figure An illustration synthetic minority samples by our method on ImageNet-Lii
25

---

Published in Transactions on Machine Learning Research (06/2025)
Source Image: 5
Noise
Synthetic Image: 541
Source Image: 4
Noise
Synthetic Image: 64
Source Image: 1
Noise
Synthetic Image: 563
Source Image: 3
Noise
Synthetic Image: 729
Source Image:
Noise
Synthetic Image: 369
Figure An illustration synthetic minority samples by our method on ImageNet-LTT
26


Reviews:
[{"review_id": "xVTyeyK2Il", "reviewer": "Reviewer_jj2N", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The presentation is generally good and motivations behind the choices of loss functions, regularizers and backbones are either logically or empirically validated.\n2. Barring a few confusions, the Mathematical descriptions seem to be lucid.\n3. Experiments are robust and rigorous, especially the ablations justifying the choices of the components. The experimental results suggest substantial improvement compared to the competing approaches.\n\nWeaknesses:\n1. The notations are confusing at certain places. E.g., a) While describing $\\mathbf{C}$ just after eqn. (1), the use of $x_n$ and $y_m$ seems to be confusing. Would they be $u_n$ and $v_m$ instead of $x_n$ and $y_m$? I am asking as the distributions are defined in terms of $\\mathbf{u}$ and $\\mathbf{v}$ above the equation. b) (Last line of second paragraph of section 3.1): Is the condition $k_0 < k$ right? Or are they the number of examples in classes $k_0$ and $k$? c) The use of the acronym IF is never explained. It is first used in section 4. Does it mean Imbalance factor? What does it mean IF to be 4980/5? d) Results with the OOD Setting: In this subsection, it is never mentioned which table should we refer to look at the result. I feel it is table 2, but without proper reference, it is hard to judge the merit. e) The legends in Fig 1(a) and 1(b) are a little confusing when read along with the text in section 4.3. In the text, it is said that the method is compared with M2m with $L(k)+L(k_0)$, but in the legend there is no mention of M2m. So, it is hard to gauge whether M2m is employed or not. Additionally, the caption of Fig 1 says the IF value is 100 while the horizontal axes on each of the subfigures show 3 different IF values.\n2. Section 3.3: Here different options of embedding spaces have been discussed and it has been argued that minimizing the distribution distance in the image space is expensive (section 3.2), thus an embedding function $\\psi_\\theta$ replaces the image space in computing the distribution distance. Though, in the experiments section, the experimental validation is provided for this, still details about the choice of the embedding function is missing. In section 3, it is only said that the feature embedding is obtained by first randomly sampling a randomly initialized network and then passing the images through it. How is the random initialization of the network done and how and why does it require to be resampled? On a related note, What about targeted ways of dimensionality reduction - say PCA or Probabilistic PCA on either the image space or feature space or parameter space? As this step is crucial in having the distribution distances, it is also important to understand, motivate and justify the choices made in this step.\n3. Minor typos and issues: a) Section 2: Subsection Long-tailed classification: This description is pretty generic and applicable to general classification problem (including both long-tailed and non-long-tailed). So why is the subsection name 'Long-tailed classification' is not clear. b) Introduction second paragraph: ‘imbalanced issue’ -> ‘imbalance issue’; Section 3.2 – 4th paragraph: ‘highly dimensional’ -> ‘high dimensional’; Section 3.5 ‘for saving cost consumption’ -> ‘to reduce computation’; Page 6, last line - Section number missing; Page 11 last line – Figure number missing.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses long-tailed classification where examples from many classes are only a few and only a few classes have many training examples. The authors take a familiar but well thought-out route in tackling these. The authors made a level playing field by generating samples which are less in number (minority samples). And these are made from majority samples. The formulation made sure the following.\n - synthetic samples are to be classified as the target minority class with high confidence. It should also have very low confidence to be classified as the source majority class.\n - The feature distribution of the synthesized samples and the original samples of the same class are made similar. The distance function used to measure similarity is the optimal transport distance.\n - Using the same distance criterion, the authors employ a sample rejection criterion where the distance of a synthetic minority sample from a real minority sample of the same class is high.\n - The authors employ another regularization term which accounts for the cases where minority samples are frequently misclassified into a specific class rather than other classes.\n\nThe authors motivated the choices and empirically validated them with adequate experiments. The analysis and ablation of the experiments made the work stronger.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "If I have to make request for some change it would be mostly presentation related. Please refer to the first weaknesses I have mentioned above. At the same time, it would be good to have a small experiment on targeted embedding space by classical dimensionality reduction techniques like PCA or Probabilistic PCA and compare them to the results obtained by random embedding spaces.", "broader_impact_concerns": "None as such.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "DSAqucLet0", "reviewer": "Reviewer_1nRv", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper proposes an effective and intuitive modification to the M2m framework for handling long-tailed classification. The idea is well-motivated, and the authors provide extensive experiments across multiple benchmarks. The method consistently outperforms existing approaches, and the ablation studies help demonstrate the contribution of each component in the objective. The visualizations and empirical breakdowns (e.g., using different distance metrics) are also helpful in understanding the method’s behavior.\n\nHowever certain points need more clarification in the paper:\n\n1.  The training framework described in the main body is somewhat unclear. Although the method builds on M2m, it would be helpful to include a more self-contained description to make the paper standalone. The proposed sample synthesis objective relies on the target classifier $f$, but it is not clearly explained how $f$ is trained jointly with the generation of new samples. The authors partly clarify this in the appendix that $f$ is first trained for 160 epochs before starting the sample synthesis (which I believe is worth mentioning briefly in the main body as well for clarity). However, it's still unclear how the joint updates of $f$ and sample synthesis are handled at each iteration. \n\n2. The rejection condition labeled as \"Reject = 1\" in Algorithm 1 is not clearly explained within the algorithm itself and is only clarified later in the text, which can be confusing.  Also, when a sample is rejected, the method instead repeats existing minority samples. This fallback effectively acts as a form of loss reweighting, similar to class-based upsampling or weighted cross-entropy with weights inversely proportional to class frequencies. Reporting the proportion of synthesized versus repeated samples (e.g., in the setup of Figure 1(c)) would provide useful context for understanding how much of the improvement comes from the proposed synthesis mechanism as opposed to standard rebalancing strategies.\n\n3. I believe, in section 4.4, $L(p)$ and $L(d)$ are used without prior definition. This makes the purpose of this section difficult to follow.\n\n4.  While the paper includes a discussion on the computational cost of the proposed method in Section 4.8, the referenced figure appears to be missing. Moreover, the runtime comparison is limited to M2m, which only differs in the regularization term. For completeness, it would be helpful to include comparisons with other strong baselines—such as MetaSAug—that achieve close performance but follow different methodological approaches.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a method for addressing the long-tail classification problem, where standard training with cross-entropy results in poor performance on minority classes. The authors build on the M2m (Major to Minor) framework (Kim et al. 2020), which addresses class imbalance by synthesizing new minority samples from majority class examples. The method begins by selecting a sample from a majority class $k_0$, then translating it through an optimization procedure to create a new sample that resembles a target minority class $k$. This is done by encouraging a pretrained classifier $g$ to confidently classify the modified sample as class $k$, while also ensuring that a target classifier $f$ has low confidence in predicting it as the original class $k_0$.\n​\n\nThe main contribution of this work is to regularize the sample synthesis objective by encouraging the distribution of generated minority samples to match the distribution of real minority samples in the training set. The paper also shows that this distribution-matching approach can be extended to synthesize minority samples from out-of-distribution (OOD) data and can be combined with other long-tail learning strategies, such as loss reweighting.\n\nThe authors evaluate their method on several long-tailed benchmark datasets (CIFAR10-LT, CIFAR100-LT, ImageNet-LT, and Places-LT). They report improved performance compared to existing methods. Additional ablation studies analyze the impact of each component in the objective, explore different distance metrics for matching distributions, and provide visualizations to illustrate how the method works.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "In addition to the issues raised above, there are some typos throughout the paper (some listed below). A careful proofreading pass would improve the paper.\n\n- In Section 3.1 (second paragraph), it is stated that M2m optimizes the synthetic sample $ \\hat{x} $ using gradient ascent. However, Equation (3) defines a minimization objective, so should this be gradient descent?  \n- *(minor)* In Section 4, the abbreviation IF (imbalance factor) is introduced without prior definition. \n- At the end of page 6, a reference tag (\"??\") is missing when referring to a section.  \n- In Section 4.3, the expression $ L(k_c) $ should be defined as $ f_{k_c}(\\hat{x}_m) $.   \n- At the end of page 11, there is another missing reference (\"??\") when pointing to a figure, and the results of section 4.8 seem missing. \n- Section C.6, the same text is repeated twice.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "vzNWk5dWIN", "reviewer": "Reviewer_dhP2", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**strengths**\n1. The paper provides a solid justification for adding distribution matching.\n2. The framework effectively generalizes to OOD settings.\n3. The method is extensively tested on multiple benchmarks.\n\n**weakness**\n1. The proposed method initializes synthetic samples from majority class examples, but does not explore or justify why this is better than using purely random initialization (e.g., Gaussian noise), which is widely adopted in generative models such as GANs, diffusion models, and DeepDream.\n2. The initialization of synthetic samples from majority class examples is insufficiently justified. This is especially problematic when class-wise semantic gaps are large.\n3. The expression makes it appear that $\\hat{x}$ is deterministic or fixed, which is misleading since $\\hat{x}$ is iteratively optimized.\n4. It is unclear whether $f$ is jointly updated with synthetic samples or kept fixed. \n5. The manuscript contains multiple incomplete citations marked as “??”, e.g., on pages 6 and 11. \n6. The primary results are based on ResNet-32 and ResNet-50. These architectures are relatively outdated; more modern alternatives like ViT or DINO should be considered to verify the generality.\n7. The visualizations in Figure 3 are limited to CIFAR-10, which contains low-resolution images and relatively simple semantics. While the method is also evaluated on ImageNet-LT, a more challenging and high-resolution benchmark, no corresponding qualitative visualizations are provided.\n8. The method adopts OT as the primary tool for distribution matching but lacks a clear rationale for why OT is preferred over other widely used metrics such as KL divergence, JS divergence, or MMD.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a novel oversampling framework for long-tailed classification by synthesizing minority class samples via distribution matching. It additionally enforces the synthetic samples to align with the true distribution of the target class using optimal transport. TExtensive experiments on CIFAR-LT, ImageNet-LT, and Places-LT demonstrate improved performance over existing methods.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "see weaknesses", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)