Long-Tailed Recognition via Information-Preservable Two-Stage Learning
Authors: Fudong Lin
Xu Yuan
Venue: NeurIPS 2025 spotlight ()
Keywords: Long-Tailed Recognition
Two-Stage Learning
Information Theory
Primary area: general_machine_learning

Abstract: The imbalance (or long-tail) is the nature of many real-world data distributions, which often induces the undesirable bias of deep classification models toward frequent classes, resulting in poor performance for tail classes. In this paper, we propose a novel two-stage learning approach to mitigate such a majority-biased tendency while preserving valuable information within datasets. Specifically, the first stage proposes a new representation learning technique from the information theory perspective. This approach is theoretically equivalent to minimizing intra-class distance, yielding an effective and well-separated feature space. The second stage develops a novel sampling strategy that selects mathematically informative instances, able to rectify majority-biased decision boundaries without compromising a model’s overall performance. As a result, our approach achieves the state-of-the-art performance across various long-tailed benchmark datasets, validated via extensive experiments. Our code is available at https://github.com/fudong03/BNS_IPDPP.

Long-Tailed Recognition via Informatiin-PPessrvabbl Two-Stage Learning
Fudong Lin Xu Yuan'
Department of Computer & Information Sciences University of Delaware Newark, DE 19711
Abstract
The imbalance (or long-tail) is the nature of many worldworld data distributions, which often induces the undesirable bias of deep classification models toward frequent classes, resulting in poor performance for tail classes. this paper, we propose novel two-stage learning approach to mitigate such majority-biased tendency while preserving valuable information within datasets. Specifically, the first stage proposes new representation learning technique from the information theory perspective. This approach is theoretically equivalent to minimizing intra- class distance, yielding an effective and well-separated feature space. The second stage develops novel sampling strategy that selects mathematically informative in- stances, able to rectify majority-biaser decision boundaries without compromising a model's overall performance. As a result, our approach achieves kate-of-the--ee performance across various long-tailed benchmark datasets. Our code available at https: //g thub. com/ com/fudongOS Brooongo3/BSS... _BNSS_SST IPDPP.
Introduction
Class imbalance naturally arises in real-world scenarios, spanning across such wide applications as online transactions, medical diagnoses, social networks spam detection, among many others. Such data in real scenarios usually follows a long-tailed distribution, i.e., few head classes dominate the entire dataset while some tail classes account for only small portion. When encountered with such long-tailed data, deep neural networks (DNNs) suffer from majority-biased decision boundaries [31, 53, 24, 57, 13, 41, 16, 42], undesirably favoring frequent classes and leading poor performance in tail classes. Misclassifying tail classes may yield catastrophic consequences, e.g., failing to identify lung cancer from millions of biomedical images may result in fatalities. To this end, building functional classifiers with unbiased decision boundaries when tackling the long-tailed data critical but remains open.
So far, the mainstream strategies addressing long-tailed recognition primarily fall into two categories: one-stage learning and two-stage learning methods. One-stage learning approaches include: re- weighting strategies [38, 10, 3, 9, 47], which prevent dominant head classes from overwhelming the training process by reducing the weight of loss functions for majority samples; and ii) sampling techniques [4, 21, 58, 25, 15], which create balanced training subsets either by downsampling majority samples by synthesizing minority samples. However, such strategies struggle achieve effective performance across both head and tail classes due to their limited representation learning capabilities. On the other hand, two-stage learning methods [28. 51, 11, 62. 30, 39. 34, 35, 27] decouple the training process into "representation learning" and "classification" stages. The former stage focuses on learning effective and generalizable feature spaces, while the latter stage aims to rectify majority- -aiase biased decision boundaries caused by highly skewed data distributions.
"Coresponning author: Dr. Xu Yuan kyyan@udel.ccuu
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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However, previous two-stage learning approaches fail to deliver satisfactory performance in long- tailed recognition, primarily due to the following two limitations. First, in the first stage, conventional representation learning methods [5, 23, 1, 22, 36] face significant challenges in handling long-tailed data, resulting in poor feature representations for both head and tail classes. Recent advancements targeting long-tailed recognition [39, 34, 35, 27] have mitigated these limitations to some extent, able to produce higher-quality feature spaces. Unfortunately, they struggle to learn effective feature spaces for securing clear separation between head and tail classes, ultimately limiting their overall performance. Besides, the second stage is hindered by the absence of effective sampling techniques. Oversampling methods [4, 49, 58, 25, 15] often suffer from mode collapse, producing synthetic mi- nority samples with limited diversity. Conversely, undersampling approaches [21, 40, 61] frequently lead to significant information loss due to the removal of substantial portion of majority samples, severely compromising the model's overall performance.
In this work, we advance two-stage learning through two key contributions: i) novel represen- tation learning strategy, namely Balanced Negative Sampling (BNS), able to learn effective and well-separated feature spaces; and ii) new sampling technique called Information Prrtionn-reeeabblle Determinantal Point Process (IP-DPP), able to balance the training subsets with informative in- stances, for mitigating the majority- tendency. Our key idea involves first training an effective feature extractor using our BNS method. This feature extractor is then fine-tuned on balanced subsets sampled through our IP-DPP approach to handle imbalanced classification. Specifically, our BNS approach constructs an effective and well-separated representation space by maximizing mutual information between two augmented views of the original data. Formal proof is provided showing our solution is theoretically equivalent to minimizing intra-class distance. This allows the model to capture instance-level semantics, ensuring high-quality feature representations, while also learning class-level semantics to achieve well-separated feature spaces. Besides, our IP-DPP method, inspired by Shannon information theory, is designed sample balanced subsets that prioritize mathematically informative instances. As a result, it excels in rectifying majority-biased decision boundaries while maintaining the model's 's overall performance.
2 Related Work
This work adopts a two-stage learning paradigm, in which Balanced Negative Sampling (BNS) is designed to learn an effective representation space, while the Information-Presevvable Determinantal Point Process (IP-DPP) is introduced to select mathematically informative samples. We next discuss how our approaches relate to, and differ from, prior solutions.
Representation learning methods are typically employed in the first stage construct high-quality feature spaces. Previous studies include KCL [34], which devises k-positive contrastive learning for learning balanced feature spaces, TSC [35]. which improves the uniformity of feature spaces via contrastive targeted supervised contrastive learning, SBCL [27]. which proposes subclass-balancion
learning for instance- and subclass-balanced feature spaces, among many others [5, 6, 23, 7, 1, 22. 36, 37J. However, conventional representation learning methods result in suboptimal representations for both head and tail classes, while those specifically designed for long-tailed recognition struggle to achieve well-separated feature spaces. Differently, our BNS method frames representation learning from the information theory perspective. This solution is mathematically equivalent to minimizing intra-class distance, thereby resulting in effective and well-separated feature spaces.
Sampling methods commonly used in the second stage to rectify biased decision boundaries. Existing techniques can be roughly categorized into oversampling and undersampling approaches. Oversampling methods balance class priors by generating minority samples. Traditional methods [4, 19, 20, 54] apply linear combinations of existing instances to synthesize the minority data, but they fail to respect the non-linear structure of synthetic data. Recent solutions [49, 17, 58, 50, 25, 15] leverage deep generative models to capture the non-linear structures in data synthesis. However, these methods suffer from mode collapse, resulting in synthesized minority samples with limited diversity. Undersampling approaches [21, 40, 61]. by contrast, create balanced subsets by removing a substantial portion of majority samples. However, this process causes significant information loss, severely impacting the model's 'ss overall performance. Our IP-DPP method falls into this category. It addresses information loss by sampling mathematically informative instances, thereby effectively rectifying biased decision boundaries while maintaining the model's 's overall performance.

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3 Preliminary
Mutual Information (MI) refers to a measure of the mutual dependence between two random variables. It quantifies the amount of information that knowing one random variable reduces the uncertainty of the other variable. Given two jointly discrete random variables X and Y for their mutual information, we have:
ΣΣ pxr y) log (x y) DEX
MI
px (z)
(1)
Here, (ar, y) represents the joint probability of X and Y, while (x) and (y) are the marginal probabilities of X and Y, respectively.
Information Content (IC), also known Shannon information or seff-information, measures the degree of "surprise" associated with particular outcome. Given ground set X, for any event æ e X with probability p(z), the information content I(-) is defined as follows:
[(x: log [p(=))]
By definition, information content has three key properties: i) an event with 100% probability yields no information; ii) less probable events are more surprising and contain more information; and iii) given set of independent events, its total self-information equals the sum of each event's individual self-information, i.e., (xx) = ExEx (xr).
4
Our Approaches
4.1 Problem Statement
Consider set N samples for training, i.e., x= {(I, y.) where data point æ; is labeled with y:- Suppose the training set has C classes, i.e., y E {1, (1.2.... C}2 and let Ne (c = 1, 2, 2.. C) be the number training data for the e-th class. Without loss of generality, we consider all classes to be sorted in the decreasing order, i.e., N 2 No+1. Naturally, VNe, we have N 2 Nc. Here, we consider a long-tailed setting, ie.,, N1 >> Nc. indicating that head and tail classes are highly skewed.
Deep neural networks (DNNs) struggle with such long-tailed data, where they perform poorly on tail classes. This can be attributed to two primary factors. First, when using conventional representation learning methods, the inherent "label bias" in long-tailed data leads to poor feature spaces. Second, majority class samples tend dominate the training process, resulting in biased decision boundaries that unfairly favor the head classes. These two factors call for the development of innovative solutions in both representation learning and classification stages.
4.2 Stage 1: Representation Learning via Balanced Negative Sampling
Our first stage aims to learn effective and well- well-separated feature space for long-tailed recognition. We resort to maximizing mutual information between instances sharing the same label-a process mathematically equivalent minimizing intra-class distances.
Our Objective. Given any image of the ground set æ E X, we employ data augmentation modules to transform it into two different views of the same sample, denoted as æ; Xq and æ; E Xx. Let fo(-) be a DNN parameterized by 0, used for encoding feature representations. Here, we regard Q(x;; 0) and V (Xy; O) feature spaces corresponding to XQ and Xv. respectively. In this work, our goal is to learn high-quality feature representations for imbalanced classification by maximizing the mutual information between two feature spaces:
arg max MI(Q(XQ 0), ()):
(3)
In the rest of the paper, we abbreviate Qxx:: 0) and (Xx;; as Q and V. respectively. Then, given any images of x; E Xg and x, E Xv, their representations can be expressed respectively as qi and v, V. where q, fo(æ:;) and by fo (xz
CL-bbased Representation Learning. However, directly maximizing the mutual information between two representation spaces Q and V is computationally intractable. Worse still, prior studies [60. 39]
3

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have demonstrated that long- long-tailed data inherently causes "label bias", resulting in poor represen- tations for tail classes. In this work, we reformulate Eq. (3) within the framework of contrastive learning (CI.), enabling the efficient optimization of our objective. Meanwhile, employing CL.-based techniques [5, 23, 29, 37, 27] can also effectively mitigate the "label bias" issue, as highlighted in prior studies [60, 39].
Specifically, our approach leverages binary classifier distinguish the target image from noise samples, drawing inspiration from Noise Contrastive Estimation (NCE) [18]. Given an anchor image 2; E XQ, we have its corresponding target image 3,, E Xv. Here, æ; and 2 are two augmented versions of the same image. Meanwhile, we sample n noise images (having different labels with æ;) from the dataset Xv. denoted {ar
in j==1- Regarding their representations, we have one positive pair
(qi, U (,z) and n negative pairs {(g. yj ) Therefore, there is *TT chance to pick the positive pair and n-T chance to pick the negative pair. Let p(.) be the joint probability of Q and V, and g(-) represent binary classifier, where its output d and d denote the positive and negative pair, respectively. Then, we have:
m4 p(q. ofte) p(q. v,), d
g(q. vy dd)
(4)
m+1
Considering the positive pair only, arrive at:
p plg. v
oiq. d d= 1) p(g: with) v(g;,
(5)
We assume that the distributions between different classes are independent. Then, for any negative pair (qi, 22 ), we have p(g,, Vn p(u,) ). As such, we can rewrite Eq. (5) as follows:
plg.
, my d 1)
(6)
p(g:, w p(g)p(07))
Taking the logarithm of Eq. (6) and rearranging the terms (see Appendix A.1 for detailed derivation), obtain:
p(g),
logg(g., By d 1) log p(q)p(17) log n.
Taking the expectation of p(q,, v (.1) on both sides, we have:
plq: w 2 E planot log logg(q, vy | 1) + log planot log p(q.)((1))
(8)
Combining Eq. (1), Eq. (3), and (8), we have:
MI () Fpla.ott.) log o(q. vs | 1) + log n.
(9)
musimine MI
lower
Here, log is a constant, indicating that maximizing the lower bound in Eq. (9) is equivalent to maximizing the mutual information between the two feature spaces.
Balanced Negative Sampling (BNS). Inspired by prior studies [18, 48, 52], we train a logistic regression classifier to maximize the lower bound in Eq. (9). However, log log,((;, Uy d 1) is computationally intractable. To address this issue, we approximate the classifier's output with sigmoid function (-), expressed as below:
g(g. ny (d) = Ta(
9.
d=1 d
(10)
Here, T is the temperature parameter that controls the sharpness of the similarity scores. As such, our NS-based contrastive learning, designed for learning high-quality representations, is formulated as follows:
Las
log logo 0 qi U
Σ log logo(-
qi
(11)
Our NS-based contrastive learning, i.e., (11), can mitigate "label bias" inherent to long-tailed data, yielding higher quality of representations, However, it is ineffective in learning well-separated

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representation space. This is because positive pairs of head classes dominate the representation space. We then propose a novel Balanced Negative Sampling (BNS) to learn an effective and well-separated representation space. That is, for given anchor image æ; E Xq. we sample additional set of m images from X that share the same label as 2;, denoted as {ox) Let representations for the addition set of images. Then, we have m + 1 positive pairs {(q.. 25, 9. {q:} U Q and n + 1) negative pairs {(g.. v,) Mathematically, our BNS technique can be expressed as:
Q denote
E { Qt,m and j 1, 2, n.
CaNS
Σ
log (
q.
E go Q
log o
q.
(12)
+
(.E(0)
In practice, m is set to a small value due to the limited number of samples in minority classes Eq. (12) effectively enhances the quality of feature representations by maximizing the mutual information shown in Eq. (3). This maximization of mutual information directly corresponds to minimizing intra-class distances, as stated next.
Theorem 4.1. (Intra- Class Distance Mutual Information Theorem) Let X2 and X3 be two sets of images with the same label e, obtained by different data augmentation techniques. Given a feature extractor fa(.), define Q and V as the representation spuces for %22 and Xy,, respectively. Then, any pair of qf Q and v V is positive pain. Let M-(-) and D(-) respectively denote the mutual information and a distance metric, we have:
max MI(Q", (IIQQVV"
V*) ) x min D
(13)
where D(Q V can be considered as the intra-class distance because they have the same label.
The proof of Theorem 4.1 is deferred Appendix A.2.
Instance-Level and Class-Level Semantics. An effective representation space must capture two key aspects: instance-level semantics to ensure high-quality feature representations and class-level semantics to achieve well-separated feature spaces. To understand how our BNS technique achieves this, decompose Eq. (12) as follows:
Cass + log a
q.
log
9.
Σ nE k.m
loga(
9h Siiy + log o( 9h yj
(14)
incancr-evell
According to Theorem 4.1, our BNS approach naturally minimizes distances at both the instance and class levels. Specifically, the pair (q,, vJ.)) originates from the same instance, while the pairs (qk. U v7,;) are from the same class. This dual-level minimization naturally encourages the emergence of both instance-level and class-level semantics within the representation space, leading to improved representation quality and better separation of feature spaces.
4.3 Stage 2: Information- InformationnPPessrvvabe Preservable Determinantal Point Process
Preservable Determinantal Point Process (IP-DPP) aiming to rectify majority- biased classification decision boundaries while main- taining the model's 's overall performance. Specifically, our approach builds on the Determinantal Point Process (DPP) [33], a stochastic process that captures global negative correlations, as outlined below.
Next, propose a new sampling solution, namely Information-Pressen
Definition 4.2 (Determinantal Point Process). Given a ground set with N items, point process P in this ground set is a distribution over discrete and finite subsets of X. Let K E R N be real, symmetric marginal kernel matrix indexed by the elements of X. point process P is called a DPP only if, for every random subset Y C X drawn according to P. we have:
P(Y) det(Ks Kx),
(15)
where Ky = [K [Ky]aey is the principle submatrix of K, indexed by elements of Y.
Since P is probability measurement, iee,, 0 5 P < 1, the marginal kernel matrix K must satisfy specific structural properties, as stated next.
5

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Remark 4.3 (Properties of Marginal Kernel Matrix). The marginal kernel matrix K for a DPP must satisfy: K is a positive semidefinite matrix; and ii) All eigenvalues of K are bounded in the interval 10,1).e.. 0 K 3
However, it is very difficult to construct a DPP through the marginal kernel matrix K in real long- tailed settings. In this work, we follow the prior study [33] by constructing DPP based on the L-ensemble framework [2]. Specifically, consider an image 2; E X with ground truth label ys. let p(i) = ( (z;) denote the probability of correctly predicting y given 2,, where the classifier is parameterized by Φ. For any two distinct elements i, E X. let j(i, denote the joint probability of correctly classifying both elements. Assuming independence between classifications, we have p(i,j) p())p(j). Let S be N x N matrix, where each element S is defined as follows:
p(Op())
i
S
Eu
p(k)p N
i=
(16)
As such, a symmetric stochastic matrix where each row (or column) sums 1. The symmetric stochastic matrix S is positive semi-definite, and all its eigenvalues are bounded in [0, 1], as stated in Lemmas 4.4 and 4.5, respectively.
Lemma 4.4. (Positive Semi-definitenes) Let p(i) p(y((z,) represent the probability of yi given E, S E R N XN is a symmetric stochastic matrix, where each row (or column) sums to Then, we have:
v TSv 2 O, Yu E
(17)
In other words, S is positive semi-definite Lemma 4.5. (Bounds on Eigenvalues) Let N be the eigenvalues of the symmetric stochastic matrix S E N we have:
0 0 11, Ad..
(18)
The proofs of Lemmata 4.4 and 4.5 are deferred to Appendix A.3 and Appendix A.4, respectively.
As such, the symmetric stochastic matrix S satisfies the two properties stated in Remark 4.3. Hence, it can be used to construct DPP through L-ensemble, expressed as below:
(Sy)
Ps(Y) det(S + I)
(19)
where I is an N X N identity matrix. Since PY( is probability measurement, it needs to be bounded [o, 1]. The DPP defined in Eq. (19) is valid, i.e., S PY( S 1, as outlined below.
Theorem 4.6. (Bounded Determinant Probability Measurement) Let % be a ground set with N items and S E N denote a symmetric stochastic matrix, indexed by elements in X. Here, S is positive semi-definite and satisfies 0 < S < I, where is the N x N identity matrix. Let Sy denote the principal submatrix of S corresponding to Y, for any subset Y C X, the following holds:
det(S)
0 S det(S S1
(20)
In other words. Ps(Y)=
SSy Se+ defines a valid probability measurement.
The proof of Theorem 4.6 is deferred Appendix A.5.
Enoormation-Prrsevvable Property. In the long- lailed settings, our DPP method defined in Eq. (19) effectively preserve valuable information, discussed next.
Remark 4.7 Ciformation-Preeevviii information content of item æ relevant to its correct classification. (s U (x)) denotes the probability that item æ is sampled by our DPP approach, which prioritizes sampling elements with higher information content, as expressed by:
Sampling Principle). Let T(a:) log (og(p(((a))) [=)] denote the
Ps (YU (x}) x ((x).
(21)
Here, we use simple example illustrate how Remark 4.7 holds. Let A {i,j} be a subset of X sampled by our DPP approach. For the given ground set, det(S + I) is a constant. Then, we arrive (see Appendix A.6 for details):
dSt(SA)
(A( = del(S + I)
det(SA) = 11-10 -p(1) p()).
(22)

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Algorithm IP-DPP
1: Input: ground set & fat )a its symmetric stochastic matrix S, and sample size &
2: Initialize: standard basis vectors {ei and pairs of orthonormal eigenvalues and eigenvectors
[()i, bx)) for
V
4: for == 1,2, ... ,N do
5: if U(, 1) X,41 then
6:
V+ V UU
k
end
9: i = 0 then
10: break
11: end if
end
13: Y x-D 0
V 0 do 15: for 1,2,2
14: while
N do p(i) IV Evev(uTed e,
16:
17:
end for
18:
argmax p(i)
Y YU [
20: V+V, Vi // Update to an orthonormal basis for the subspace orthogonal to e,-
21:
end while
22: Return: subset Y
Therefore, we obtain Ps(ti,j)) & -p()) p(().. In this work, we have p(i) = p(y (z;). implying that images less likely to correctly classified are more likely to be sampled by our DPP approach. According to information content (see Eq. (2) for details), we have [(x,) - log[p(3: [.))]. Thus, Ps((x;) x [xx,).
Balanced Sample Size. To effectively rectify biased decision boundaries, the cardinality of sampled subsets must be carefully balanced. A subset with large sample size risks preserving the original imbalance, whereas an overly small subset may lead to significant information loss. Given a ground set with N items, we theoretically demonstrate that that the expected sample size of DPP defined in Eq. (19) is (11 2). Due to the page limit, the details of this theorem are deferred to Appendix A.7.
This reduction roughly one-third of the original size is inadequate for balancing the class priors in highly imbalanced setting. To address this issue, we propose Information- Informationn-Preevvvbll Determinantal Point Process (IP-DPP) to sample balanced subsets by selecting a fixed cardinality i instances from each majority class, as defined below:
(St
(
Eir det(Sy.)
(23)
As such, our IP-DPP approach can effectively sample balanced subsets to rectify decision boundaries while preserving valuable information.
Effective Sampling Strategy. However, directly applying Eq. (23) for sampling entails signifi- cant computational costs. Drawing inspiration from prior studies [32, 33], we devise a novel and computationally efficient sampling strategy for our IP-DPP method. Specifically, given the symmet- ric stochastic matrix S, its spectral decomposition yields orthonormal eigenvectors {w;) corresponding eigenvalues {Mr), such that:
J. with
S Taaoy
(24)
Let e; E N denote the i-th standard basis vector, which contains single 1 in its i-th entry and 0's elsewhere. U(0. 1) the standard uniform distribution. Then, Algorithm outlines an efficient sampling strategy for our IP-DPP approach.
7

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Table I: Experimental results CIFAR-IO-LT and CIFAR-100-LT datasets, with the best results
shown in bold
CIFAR 10-LT
CIFAR-10 100-TT
Methods
Many-shot
Medium-shet
Pew-shet
Overall
Many-shot
Medium-shot
Few-shot
Overall
Focal Loss
86.3
60.6
71.1
43,9
43.5
LDAM Loas
85.8
64.8
51.9
71.5
71.4
44,5
11.7
44.1
7-norm
85.2
64.4
51.7
70.9
60.7
54,4
14.8
44.7
RIDE
86.2
73.1
47.6
KCL
83.7
63.8
53.6
71.7
72.3
46.1
14.8
45.8
71.3
SBCL
81.6
72.4
57.6
72.6
72.7
48.5
20.0
48.5
OTmix
87.9
67.8
47.3
73.8
73.1
48.0
19.1
48.1
DisA
86.1
68.3
50.3
73.6
72.4
49.3
21.9
49.2
62.4
5 Experimental Results
5.1 Experimental Setup
Datasets. We conduct experiments on four artificially induced or real-world long-tailed datasets:
CIFAR-IO-LT and ii) CIFAR-100-LT: we follow the setting in [3] by sampling long- tailed
datasets respectively from the original CIFAR-10 and CIFAR- 100 datasets; iii) ImageNet-LT [43]: truneated version of ImageNet [12] with a total of 1, 000 classes; and iv) iNaturalist 2018 [26]:
naturally tailed dataset containing 8, 142 species around the world. The imbalanced factor (IF)
for CIFAR-IO-LT and CIFAR-I00-LT, if not specified, is set to 100 (i.e., Nuu 100).
New
Compared Approaches. We compare our approach to nine state-of-the-arts for long-tailed recogni- tion: Focal Loss [38], LDAM Loss [3], 7-norm [30], RIDE [59], KCL [34], TSC [35], SBCL [27],
OTmix [15], and DisÅ [14].
Metrics. We evaluate long-tailed recognition performance using four metrics: many-shot, medium- shot, few-shot, and overall accuracies. Many-shot, medium-shot, and few-shot assess model performance in head, medium, and tail classes, respectively. All results are averaged over 5 trials.
Additional experimental settings, including thresholds for defining the above metries and hyperpa- rameters, are provided in Appendix B.1 to conserve space.
5.2 Comparisons to State-of-the-Arts
small-scale long-tailed datasets, i.e., CIFAR-1O-LT and CIFAR-100-11 to compare our approach with nine counterparts mentioned in Section 5.1. Table 1 presents comparative results. On LIFAR-IO-LT. our approach achieves the best overall accuracy of 76.4%. outperforming all counterparts by 2.6% at least. This performance improvement can be attributed two key aspects. First, our BNS approach effectively captures both instance-level and class-level semantics, facilitating the learning of high-quality representations and the creation of well-separated feature spaces, respectively. Second, our IP-DPP method addresses biased decision boundaries by sampling relatively balanced subsets while preserving valuable in- formation. This can mitigate the majority-bian biased tendency while maintaining the model's overall performance. On the other hand, although our approach lags behind prior studies in many-shot accuracy, these methods consistently struggle with biased decision boundaries. They prioritize performance on head classes, resulting in significantly diminished accuracy for medium and tail classes. For instance, while our method falls short of OTmix by 3.9% in many-shot accuracy, surpasses OTmix with significantly higher medium-shov and few-shot accuracies, improving by 8.5% and 19.9%, respectively. These results demonstrate that our approach effectively mitigates biased decision boundaries while maintaining the model's overall performance.
Small-Scale Datasets. We first conduct experiments
We observe similar trends on CIFAR- 1IFAR-100-LT. First, our approach achieves the highest overall accuracy of 52.4%, surpassing prior -f -fhee---tthhh-ttt.. i.e., DisA, by a notable margin of 3.2%. Second, while our approach lags behind OTmix by 10.7 7% many-shot accuracy, it achieves improvements of 11.7% in medium-shie accuracy and 12.8% in few-shot accuracy, as well as an improvement of 4.3% in overall accuracy. These results further confirm that our approach effectively mitigates majority-biased tendencies while preserving the model's overall performance.

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Table 2: Experimental results ImageNet-LT and iNaturalist 2018 datasets, with the best results highlighted in bold
ImageNett-L LT
iNaturalist 2018
Methods
Many-shot
Medium-shett
Pew-shet
Overall
Many-shot
Medium-shot
Few-shot
Overall
Focal Loss
51.4
41.2
16.0
41.7
61.2
62.7
64.4
63.2
LDAM Loas
55.0
46.4
16.7
45.7
65.1
66.8
61.7
64.6
y-norm
56.6
44.2
27.4
46.7
71.3
65.8
69.1
67.7
RIDE
56.7
46.4
67.5
68.6
68.8
KCL
$5.0
42.6
25.4
45.0
61.2
62.7
64.4
63.2
TSC
66.4
65.7
SBCL
55.8
45.7
27.1
47.1
73.4
70.2
69.8
70.4
OTmix
50.9
46.0
25.7
45.1
70.1
70.9
68.6
69.9
DisA
61.0
47.0
25.3
49.4
70.7
70.8
68.4
69.8
72.7
Large-Scale Datasets. Next, we conduct experiments on ImageNet-LT and iNaturalist 2018 to assess the effectiveness of our approach on large-scale, long-tailed datasets. Table 2 provides the comprehensive results. We make two key observations. First, our approach achieves the highest accuracies on both ImageNet-LT (i.e., 51.7%) and iNaturalist 2018 ( (ie., 74.0%), outperforming all competing methods. For instance, on ImageNet-LT, our approach surpasses DisA, the best baseline method, by 2.3%. Similarly, iNaturalist 2018, it outperforms SBCL, the best counterpart, by 3.6%. These results highlight the strong generalizability Moreover, while our approach lags behind some baseline methods in many-shot accuracy, it achieves the highest medium-shie and few-shot accuracies across both datasets. This is because prior methods result in majority- biased decision boundaries, which disproportionately favor head classes.
our method to large-scale, long-tailed datasets.
5.3 Evaluation on Representation Learning
Quantitative Evaluation. Next, we quantitatively evaluate the performance of our BNS method for representation learning. Specifically, we compare it against three state-of-the-art contrastive learning methods for long-tailed recognition: KCL, TSC, and SBCL. To assess the quality of the learned feature representations, we use linear probing accuracy, which involves fine-tuning linear classifier on a pre-trained feature extractor with frozen weights.
Figures 1a and 1b illustrate linear probing accuracies CIFAR-10-LT and CIFAR-100-LT, re- spectively. On CIFAR-IO-LT, our approach achieves the best overall accuracy of 68.2% (see the pink bar), outperforming KCL, TSC, and SBCL by 7.2%, 4.4%, and 3.5%, respectively. This is because maximizing the mutual informa- tion expressed in Eq. (3) equivalent mini- mizing the intra-class distance, which, in turn, enhances the quality of feature representations. Existing methods for long-taie sail-tailed data often ex- hibit an undesirable bias toward head classes, leading to poor performance on tail classes, as evidenced by significant disparities between many-shot and few-shot accuracies: 52.7% for KCL, 45.1% for TSC, and 43.5% for SBCL. In contrast, our BNS method enjoys unbiased representation space, exhibiting similar perfor- mance on many-shar and few-shot accuracies (i.e., 69.4% vs. 67.6%). Similarly, our approach achieves the highest linear probing accuracy of 47.1% on CIFAR- -00-LT, surpassing KCL, TSC, and SBCL by 6.7%, 5.7%, and 2.9%, respectively. Furthermore, tendency. For instance, compared to SBCL, our method achieves substantial
(a) CIFAR-IO-LT
(b) CIFAR-100-LT
Figure 1: Linear probing accuracies on (a) CIFAR-
10-LT and (b) CIFAR-100-LL datasets.
approach effectively mitigates
the majority- improvements of 11.5% in medium-shot accuracy and 6.1% in few-shot accuracy, with only modest 8.5% reduction in many-shot accuracy.
Qualitative Evaluation. To gain a deeper understanding of our BNS method's 's role in representation learning, we utilize t-SNE [56] to visualize the feature spaces learned by SBCL and our BNS approach. Figure 2 illustrates the feature spaces for the CIFAR-10 training and test sets. The training representation space learned by SBCL. exhibits poor separation for medium and tail classes (see Figure 2a), leading to overlapping boundaries among these classes in the test representation space

---

Traniing
Training Set
Text Set
Pooe
Misel Isaaskii
(a) SBCL
(b) SBCL
(c) Ours
(d) Ours
Figure 2: t-SNE visualization of CIFAR-10 10 feature space. (a) and (b): visual representation learned
by SBCL, as well as (c) and (d): visual representation captured by our approach.
Table 3: Experimental results under various imbalanced factors (IF), with the best results shown in bold
CIFAR-IO-LT
CIFAT-100-LT
Methods
IF=10
IF=20
IF=50
IF=100
IF=200
IF=10
IF=20
IF=50
IF=100
IF=200
TSC
80.6
76.4
72.9
71.9
63.7
57.3
51.1
48.3
43.5
40.3
78.6
72.6
58.3
OTmix
81.3
80.3
74.4
73.8
65.8
59.1
55.7
52.2
48.1
42.7
81.0
75.7
73.6
67.2
60.4
53.3
52.3
49.2
42.9
Ours 83.7
82.4
81.9
76.4
73.5
62.6
59.8
55.9
52.4
46.7
(see Figure 2b). In contrast, our approach achieves improved separation for medium and tail classes in the training representation space (see Figure 2c), leading clear and well-defined boundaries for these classes in the test representation space (see Figure 2d). This improvement stems from our method's ability to capture class-level semantics, which promotes the development of well-separated representation spaces. The differences in decision boundaries between SBCL and our approach elucidate why SBCL achieves poor linear probing accuracies on medium and tail classes, whereas our approach demonstrates superior performance on these classes (see Figures la and 1b for details).
5.4 Performance Results under Various Imbalanced Factors
This section presents performance results under various imbalance factors (IF). Here, we consider five IF values ranging from 10 to 200. Table 3 shows comparative results on CIFAR-1O-LT and CIFAR-100-LB 1T0-LT. where compare our approach with TSC, SBCL, OTmix, and DisA. On both datasets, our approach achieves the highest overall accuracies across all IF values. For instance, when the IF is set to 200, our method achieves accuracy of 73.5% on CIFAR-IO-LE and 46. 46.7% on CIFAR-100-LLT Moreover, our method demonstrates greater robustness to large imbalance factors. For example, when the IF value increases from 10 200, our approach experiences performance drop of 10.2% (i.e., 83.7% vs. 73.5%) on CIFAR- 10-LT, whereas baseline methods suffer larger decreases, with at least 15.2 15.2% drop (see DisA, 82.4% vs. 67.2%).
Additional experimental results are provided in Appendices B.2-B.6. These include adaptability of our approach, applicability across different model architectures, evaluation of computational overhead, ablation studies, and hyperparameter sensitivity.
6 Conclusion
This work has addressed the challenging long-tailed data classification problem, by proposing a novel information- informattion-presevvbbl two-stage learning approach. Our key contributions include: i) Balanced Negative Sampling (BNS), a new representation learning strategy that effectively captures both instance-level and class-level semantics, facilitating the creation of high-quality feature representa- tions and well-separated feature spaces; and ii) Information-Presse Preservable Determinantal Point Process (IP-DPP), novel sampling technique designed to select mathematically informative instances, effectively rectifying majority-biased decision boundaries while maintaining the model's overall performance. As such, our approach achieves state-of-the-a performance across various long-tailed datasets by preserving the valuable information within the entire dataset, allowing it to consistently and decisively outperform its counterparts.
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Acknowledgments
This work was supported in part by NSF under Grants 2315613, 2438898, and 2348452. Any opinions and findings expressed in the paper are those of the authors and do not necessarily reflect the views of funding agencies.
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Outline
This document serves as supplementary material the main paper, providing additional support in two key aspects, First, Appendix A presents detailed proofs for the theorems and lemmas presented in the main paper. Second, Appendix B includes additional experimental results that reinforce the findings. Third, Appendix C discusses both potential positive societal impacts and negative societal impacts of this work.
A Proofs of Theoretical Results
This section provides comprehensive proofs for the theoretical results discussed in the main paper. To enhance clarity and facilitate understanding of the mathematical details, we begin by restating theoretical conclusions-suce the Theorems and Lemmas from the main paper- and then proceed with their detailed proofs.
A.1 Detailed Derivation Eq. (7)
This subsection supports Section 4.2 of the main paper by providing the detailed derivation of Eq. (7),
as outlined below:
p(g. v..))
ggg. Uy 1) log
log
p(q. of + n ((())
nxp(q,)p(v,)
1 p(a.v)).)
p(ge, 07,e) p(q:)p(1))
1
(25)
log
log
axplq, (p(
p(g: v3..) log n.
log
p(g))((9))
A.2 Proof of Theorem 4.1
Theorem A.1. (Intra-Class Distance Mutual Information Theorem) Let X2 and Xiy be two sets of images with the same label obtained by different data augmentation techniques. Given feature extractor fo(.), we define Q and V as the representation spaces for X and Xy, respectively. Then, any pair of qf E Q and of V a positive pair. Let MI(-) and D(-) respectively denote the mutual information and distance metric, we have:
max MI(Q", V") x min D(Q", V"),
(26)
where D(Q5, V9) can be considered the intra-class distance because they have the same label.
Proof. According to Variation of Information (VI) [46], we have:
D(Q Vo) H(Q Vo) MI(QS, Ve)
(27)
= ((QQ)) + H(V---VV)-- ( 2MI(Q2, VO).
Here, H--) represents the entropy. For the given Q and Vo, (QQ") and H(V*) are two con- stants. Therefore, we have max MT(Q", Vo) & min D(Q", V*). That is, maximizing the mutual information between Q' and V proportional minimizing their intra-class distance.
A.3 Proof of Lemma 4.4
Lemma A.2. (Positive Semi-definiteness) Let p(i) = p(us(a)) represent the probability of y given T₁. S IR N is symmetric stochastic matrix, where each row (or column) sums to Then, have:
So 00 Vo N
(28)
In other words, S is positive semi-definite
21

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Proof. According to the basic decomposition of symmetric matrices, we have S = QAQ", where Q is an orthogonal matrix QQ I, and A is diagonal matrix, with diagonal entities containing all eigenvalues of S. Given any vector E R. we let c Q v. In other words, Qc. Then, we arrive at:
N
v Su - c Q'Q'qq Qc= A c= Σ A.c.c.
(29)
where { represents the eigenvalues of S, and we have c c 0. Let Tr(:) denote a matrix's trace. Then, regarding the trace of S, we have:
Tr(S) = Tr(QAQT) = TH(QQQ)) Tr(A) = Σ A 2 0.
(30)
Combining Eq. (29) and Eq. (30), we obtain o. Sr 2 0 for any vector E R. Therefore, S is positive semi-definite.
A.4 Proof of Lemma 4.5
Lemma A.3. (Bounds Eigenvalues) Let { be the eigenvalues of the symmetric stochastic
matrix S E R NxN, We have:
0 0.A S1, Aλg
(31)
Proof. Since S E NxN is a symmetric stochastic matrix, the following conditions always hold:
S.2 0,
visijs
(32a)
N Σ
S =1, PisiSM N. j=1
(32b)
Let be an eigenvalue of S, with its corresponding eigenvector denoted as v = [V1, 02, ": UN] T Then, obtain:
Sv = Av.
(33)
Suppose Us E v has the largest absolute value, i.e., looe 2 |1s| for all < S N. According to Eq. (33), for the k-th row of S, we have:
S + Sk,2 + ..... + SK,NUN = AUA-
(34)
Combining Eq. (32a), Eq. (32b), and Eq. (34), we arrive at:
| |ua
= | = |S + Su,202 + ...... SR,NON
< 1 + |S
SK.NUNl
S 1 + 1. +... NK
N
(35)
-Ivkl Σ Ski
1=1
|vxel-
According to Eq. (35), we obtain [\] |vyl < |wgl. Therefore, for all < S N, we always have |&] < Meanwhile, according to Lemma A.2, we obtain A 2 for all < i < N. Finally,
combining both scenarios, we arrive at the following:
0 A 11, Aλ
(36)
22

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A.5 Proof of Theorem 4.6
Theorem A.4. (Bounded Determinant Probability Measure) Let X be a ground set with N items, and let S E R NxN denote its similarity matrix. Here, S is positive semidefinite and satisfies z S = I. where I the N identity matrix. For any subset Y C %, let Sy denote the principal submatrix of S corresponding to Y. Then, the following holds:
det dSy 1.
0 S (S I
(37)
In other words, the value Ps(YY)
detiSy Sy) defines a valid probability measure. det(S+1)
Proof. Since S is positive semidefinite, both the principal submatrix Sy and the matrix S + I are also positive semidefnite. If matrix positive semidefinite, all its eigenvalues are non-negative. det(S,) Then, we have det( Sy) 2 0 and det(S + r > 0. Therefore, det(S+ 2 always holds.
To establish that dec(S+1) < 1, we aim to prove the key identity:
det( (y)
Σ
det(Sx) det(S Ix).
(38)
FCX
For any S X. Eq. (38) is a special case of the following equation:
Σ
det(Sy)
det(S + IA),
(39)
ACYEX
where I is diagonal matrix with the following properties: entries are 1 for indices corresponding elements in A X A, and entries are for all other positions. Therefore, if Eq. (39) is satisfied, follows that det( (et(S+1]
(St
I 1. This result is motivated by Theorem in the prior work [33].
Specifically, for the case = X. Eq. (39) always holds. For the case where A X, we assume Eq. (39) holds whenever A has cardinality less than & (where k >> 0), i.., |A| k. Let i be an arbitrary element of A, so E Á. By partitioning the ground set into {i} and {i), we can decompose the problem follows:
S + I =
-3',,
S4 +1 S
S R
(40)
Sx-(1) + (
Here, S; denote the subcolumn of the i-th column of S, restricted to rows corresponding to elements in T. Similarly, S represents the corresponding subcolumn but transposed. By leveraging the multilincarity property of determinants, we observe:
S S
S
0
det(S + (A) =
+
Sx-() + x- (1)
S
Sx + Ix-(-)-
(1)
(41)
= det S + I AD(iT) + det (Sx- + Ix- Ix-1)--)
By applying the inductive hypothesis to each term separately, we obtain:
dct(S +IA)=
Σ
dot(Sx)
Σ
dct(Sx)
AUDICYCX
ACYCX-
(42)
Σ
det(Sx).
ACYCX
We observe that each subset Y falls into exactly one of mutually exclusive categories: either Y contains the element i (contributing to the first sum) or Y does not contain i (contributing to the second sum). According to Eq. (38), Eq. (39), and Eq, (42), we have det(Sy) ( < 1.
Finally, combining both scenarios, arrive at 0
det(Sy) des
< 1.
23

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Algorithm 2 Efficient Sampling Algorithm for a Standard DPP
1: Input: ground set X {a.) and its symmetric stochastic matrix $
2: Initialize: standard basis vectors {ei N. and pairs of orthonormal eigenvalues and eigenvectors
[()i, bz)) for
V
4: for 1or1=122 1,2, ...,, do
5: if U(, 1) <x,
A; then
6:
end 8: end for
9: <-D 0
10: while V 0 do
11: for 1,
N do p(i) IV Lver e.)²
13: end for
4 arg max p(1)
15: YU
16: V+ V1 // Update V to an orthonormal basis for the subspace orthogonal to ei-
17:
end while
18: Return: subset
A.6 Detailed Derivation of Eq. (22)
This subsection supports Section 4.3 by providing the detailed derivation of Eq. (22). Given that A {i,j} is subset with two elements, have N 2. Then, the derivation follows:
(A(A) = del (SA)
1 p(t)
p())p())
p(0)p()
det(S + +1)
x det(SA)=
p(()
p())-p())
p())
p()
(1 p(1) p(() )2 (p()) (1()p
2
2
1 - p(i) p(j).
(43)
A.7 Expected Sample Size of Determinantal Point Process (DPP)
This subsection complements the main paper by presenting the theorem and its corresponding proof for the expected sample size of a DPP. To facilitate understanding of the theoretical results, Algorithm 2 presents an efficient sampling method for a standard DPP, i.e., DPP without fixed sample size
Theorem A.5. (Expected Sample Size of DPP) Let be a ground set containing N elements. For any subset Y C X sampled by Determinantal Point Process (DPP), the expected size Y is given by:
E| = N(1 - In
(44)
Proof. The size of the sampled subset Y from Algorithm 2 is determined by the cardinality of the selected eigenvector set VV.. This cardinality follows Poisson- -innomall distribution (see Line in Algorithm 2 for details), where each of the N independent Bernoulli trials succeeds with probability p; X. corresponding to the i-th eigenvalue A of the kemel matrix. Thus, | ~ E Bernoull(pp.).
According to Lemma A.4, we have 0 S A; < YA, Now, we assume that A; is random variable distributed over the interval [0, 1). Denote this random variable as X. The expected value is given by:
L A f(A)aa
E
(45)
X+1
24

---

where f(A) is the probability density function (PDF) of A. Without loss of generality, suppose is uniformly distributed over (0, 1]. the PDF f() for À E [0, 1]. The expected value becomes:
A
dA.
(46)
A 1
Let = A so du = dA and when X = 0, u = 1; when À = 1, = 2. The integral becomes:
L. X+1 dx
u du {"² (1-
2
du
Idu { -du [u]] In u|}
(47)
1 - 1n
Therefore, we have p; =
d
= 1 ln2. Finally, the expected size of the set can be
x,+1
approximated follows:
N (1 In N(1 - 2n.
|
Σ 1=1
(48)
Notably, we assume uniform distribution of eigenvalues to simplify the above proof. Although this assumption is idealized, the empirical results (see Appendix B.7) on the subset sample size after DPP demonstrate that it does not substantially distort the practical behavior of the process.
B
Supplementary Experimental Results
B.1 Additional Experimental Settings
Metric Threshold. We define thresholds for many-shot, medium-shot, and few-shot accuracies. Specifically, for CIFAR-10-LE many-shot refers to classes with more than 500 images, medium-shot to classes with 200 to 500 images, and few-shot to classes with fewer than 200 images. For other those with 20 to
datasets, many-shot refers classes with more than 100 images, medium-sher 100 images, and few-shot to those with fewer than 20 images.
Hyperparameters. ResNet-18 18, ResNet-34, and ResNet-50 [24] are used for CIFAR-IO-LT, CIFAR- 100-LT, and ImageNet-LTT LT (or iNaturalist 2018), respectively. In the first stage, the feature extractor is trained using the AdamW [45] optimizer with 31 0.9. B2 = 0.95, and weight decay of 0.05. The training process consists of 1, .000 epochs, including 20 warm-up epochs, with a batch size of 1, 024. cosine learning rate decay schedule [44] applied, starting with base learning rate 10c and incorporating layer-wise learning rate decay [8] of 0.75. Data augmentation strategies, including random cropping, random color distortions, and random Gaussian blur, are used, followed by those reported in SimCLR [5]. The number of additional positive pairs was set to 6 by default. In the second stage, the pre-trained feature extractor is fine-tuned for 100 epochs, including 5 warm-up epochs, with batch size of 64. The sample size k is set to 10Nc. where No represents the sample size of the smallest class. All experiments were conducted on workstation equipped with an RTX 4090 GPU.
B.2 Adaptability of Our Two-Stage Learning Approach
This work introduces novel two-stage learning approach, incorporating Balanced Negative Sampling (BNS) for representation learning in the first stage and information-Presrvvvble Determinantal Point Process (IP-DPP) for sampling the balanced training set in the second stage. this subsection, we conduct experiments to demonstrate that our BNS and IP-DPP approaches can be easily adapted by existing studies. Specifically, BNS is combined with re-weighting methods, i.e., Focal Loss and LDAM Loss, while IP-DPP is combined with prior representation learning approaches, i.e., KCL, TSC, and SBCL. Table presents the experimental results on CIFAR-10-LT and CIFAR-100--LT, where the performance outcomes of existing methods are also included for a better comparison.
25

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Table Experimental results on CIFAR-IO-LT and CIFAR-100-LI1 with the best results shown in bold
CIFAR 10-LT
CIFAR 100-IT
Methods
Many-shot
Medium-shal
Many-shol
Mediumn-sot
Few-shot
Overail
Focal Loss
86.3
60.6
46.3
69.2
71.1
43.9
10.5
43.5
LDAM Loss
85.8
64.8
51.9
71.5
71.4
44.5
11.7
44,1
KCL TSC
83.7
63.8
53.6
71.7
72.3
46.1
14.8
45.8
71,9
71,9
43.3
SBCL
81.6
72.4
57.6
72.6
72.7
48.5
20.0
48.5
BNS Focal Loss
82.2
73.6
57.1
72.9
63.6
57.7
27.8
50.8
BNS LDAM Loss
82.6
71.8
61.9
74,2
63.3
58.7
27.8
51.0
KCL IP-DPP
79.2
75.7
66.6
74.7
64.2
58.3
26.7
50.8
TSC IP-DPP
78.9
76.8
63.4
73.8
63.2
58.3
30.1
51.5
SBCT. TP-DPP
80.0
73.5
66.8
74.8
62.7
57.8
29.7
51.2
Table 5: Experimental results on ImageNet-LT and iNaturalist 2018 using various model architectures. Best results are highlighted bold
Models
ReaNet-50
VII-Base
Deil-Base
Swin-Base
ImageNet-LT
$1.7
50.1
50.8
51.2
iNaturalist 2018
74.0
72.6
74.3
74.6
Table Comparative analysis of computational overhead. The sampling time, measured in seconds, is reported.
Datasets
CIFAR-IO-IT
ImageNat-I.T
Random
12.4
118.2
IP-DPP (wlo efficient sampling)
19.0
36.8
399.8
IP-DPP
13.2
15.8
144.0
Our approach effectively enhances the performance of existing methods. For instance, combining SBCL with IP-DPP (i.e., "SBCL + IP-DPP") achieves the highest overall accuracy of 74.8% on IO-AR- Similarly, integrating TSC with IP-DPP (i.e.. "TSC + IP-DPP") yields the best overall accuracy of 51.5% on CIFAR- 100-TT. Furthermore, integrating our approach with existing methods consistently improves their performance compared the original methods. For instance, combining BNS with Focal Loss (i.e., "BNS Focal Loss") outperforms "Focal Loss" by 3.7% (i.e., 72.9% vs. 69. 2%) on CIFAR 10-LT and by 7.3% (i.e., 50.8% vs. 43.5%) on LIFAR-100-LT These results demonstrate that our BNS and IP-DPP approaches can be effectively integrated into other methods to enhance performance in imbalanced classification tasks.
B.3 Applicability Across Various Model Architectures
In this subsection, we conduct experiments to evaluate whether our approach can be applied effectively across different model architectures. Table presents the results on ImageNet-LTT and iNaturalist 2018 using four architectures: ResNet-50 [24], ViT-Base [13], DeiT-Base [55], and Swin-Bas Base [41]. Our approach demonstrates consistent performance across different model architectures. For instance, on ImageNet-LT, it achieves 51.7% accuracy using ResNet and 51.2% accuracy using Swin-Base. Similarly, on iNaturalist 2018, ResNet-50 achieves an overall accuracy of 74.0%, while Swin-Base achieves slightly higher accuracy of 74.6%. These results highlight the generalizability of our approach to a variety model architectures, confirming its robustness and flexibility.
B.4 Evaluation Computational Overhead
This subsection supports the main paper by presenting an evaluation of the computational overhead introduced by our IP-DPP method. We first examine the additional sampling time incurred by IP-DPP. Specifically, we evaluate three scenarios: the training set sampled using Random Undersampling [21]. IP-DPP without the efficient sampling strategy, and IP-DPP with the proposed strategy. Table provides comparative analysis of these methods across three long-tailed datasets: CIFAR-IO-LT. CIFAR-100-LT. and ImageNet-LT. The results reveal two key observations, First, while our IP-DPP approach introduces additional computational overhead compared to Random Undersampling, the
26

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Table Analysis of computational overhead. The total sampling time and training time are reported in seconds. The baseline method does not involve additional sampling, so its sampling time costs are not applicable.
CIFAR-IO-LE
CIFAR-I00-LT
Datasets
Sampling Time Training Time Sampling Time Training Time
Baseline TP-DPP
537.2
781.6
132.0
484.4
158.0
491.8
Table Linear probing accuracy on CIFAR-IO-LT and CIFAR-I00-LT, with the best results high- lighted in bold
CIFAR I0
CIFAR 100.TT
Datasets
shot Medium-shol Fow-shot Overall Many-shot Medium-shot Few-shot Overall
Buseline
66.7
63.5
34.8
56.5
55.1
50.2
10.1
39.9
BNS (w/o additional positive pairs)
68.3
64.77
43.3
60.1
56.7
53.1
20.9
44.7
BNS
67.6
68.2
26.6
47.1
difference is minimal, with IP-DPP being only 0.8 seconds slower CIFAR- 0.8 seconds slower on CIFAR-100-LT, and 25.8 seconds slower on ImageNet-LT, Second, the effective sampling strategy of IP-DPP significantly reduces computational overhead. Compared to its counterparts without the efficient sampling strategy, IP-DPP achieves speedups of 1.4x on CIFAR-10-LT, 2.3x on CIFAR-100-LT, and 2.8x ImageNet-LT.
Next, we conduct experiments to evaluate the impact of our IP-DPP approach on total training time. Table 7 summarizes the total training time for CIFAR-10-LI and CIFAR-100-L3 over 100 epochs, with the total training cost of Focal Loss serving as the baseline. practice, the training set re-sampled using our IP-DPP approach every 10 epochs, strategy designed to mitigate overfitting. Therefore, Table reports the cost of performing ten sampling iterations as the total sampling time. Although our IP-DPP approach incurs additional sampling time, it results in lower total training time compared to the baseline method. This reduction is attributed to the significant decrease the sample size of the training set after using IP-DPP.
B.5
Comprehensive Ablation Studies
Ablation Studies on Our BNS Approach. We conduct experiments on long-tailed datasets to evalu- ate the effectiveness of our BNS approach in representation learning. For this, we use SimCLR [5], a conventional representation learning method, as the baseline. We examine two variants of our approach: one without additional positive pairs and another with them. Table 8 presents the linear probing accuracies on CIFAR-1O-LT and CIFAR- 100-LT datasets. We make three key observations. First, the conventional method struggles to learn high-quality feature spaces for long-tailed datasets, leading to poor linear probing accuracy in tail classes, e.g., 34.8% on CIFAR-IO-LT and 10.1% on CIFAR-100-L3 Second, both variants of our BNS approach significantly outperform the baseline method. This improvement is attributed to their ability to effectively capture instance-leve semantics, thereby enhancing the quality of feature representations. Third, compared its variant without addi- tional positive pairs, our BNS approach with additional positive pairs achieves superior linear probing accuracy, particularly in tail classes (e.g., 67.6% vs. 43.3% on CIFAR- CIFAR-IO-LT),, This improvement due to the inclusion of multiple positive pairs, which enables our BNS method to capture class-level semantics, thereby promoting a well-separated feature space.
Ablation Studies on Our IP-DPP Approach. We conduct experiments to evaluate the necessity and significance of the innovative designs within our IP-DPP approach. In these experiments, Random Undersampling [21] serves as the baseline method. For our IP-DPP approach, we examine its variants: IP-DPP without the symmetric stochastic matrix, IP-DPP without the fixed sample size, and the complete IP- method. Table presents the comparative results on CIFAR- 10-LT and SIFAR-100-LT- We have three observations. First, compared to the baseline method, all IP-DPP variants achieve better overall accuracies on both datasets. This is because the baseline method incurs significant information loss due to its random undersampling strategy. Second, removing either the symmetric stochastic matrix or the fixed sample size results in significant performance degradation.
27

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Table 9: Ablation studies IP-DPP technique, with the best results shown in bold
CIFAR-IOA
CIFAR-100-1 100-LT
Methods
Many-shot
Medium shot
Few-shol Overall
Many-shot
Medium shot
Few shot Overall
Baseline
75.5
72.8
61.9
70.8
56.7
56.4
24.1
47.8
matrix]
72.7
IP: DPP (wlo fixed sample size)
82.7
74.7
63.8
75.4
65.5
57.0
26.7
50.8
IP-DPP
82.0
76.3
67.2
76.4
62.4
59.7
31,9
52.4
(a) Many-shot
(b) Medium-shiee shot
(c) Few-shot
(d) Overall
Figure 3: Linear probing accuracy CIFAR-1O-LT using different numbers of additional positive
pairs (i.e., m.)
(a) Many-shot
(b) Medium-shot
(c) Few-shot
(d) Overall
Figure 4: Impact of the fixed sample size (i.e., k) on imbalanced classification, where many-shot, medium-shot, feo-shot, and overall accuracies CIFAR-100 are reported.
This is because the symmetric stochastic matrix is essential for preserving mathematically informative samples, while the fixed sample size plays crucial role in maintaining data balance after applying IP-DPP. These statistical results validate the importance of the novel designs within IP-DPP in rectifying majority- -bissd decision boundaries while preserving the model's overall performance.
B.6 Hyperparameter Sensitivity
Effects of Additional Positive Pairs. Here, we investigate the impact of the number positive pairs in our BNS approach on representation learning. Notably, given any anchor image, we have at least one positive pair, i.e.. two augmented images from the anchor image. Here, we conduct experiments by gradually increasing the number of additional positive pairs, i.e., m in Eq. (12). Figure illustrate the linear probing accuracy on CIFAR 10-LT, where m gradually increased from 10. As shown in Figure 3d, when the value of m is small (m < 6), increasing m leads to higher overall accuracy. However, for larger values of m (m > 6), further increases m negatively impact overall accuracy. A similar trend is observed in the many-shot, medium-shot, and few-shot accuracies (see Figures 3a, 3b, and 3c for details). This occurs because small value m enhances the model's ability to capture class-level semantics, whereas large value introduces data imbalance within the feature spaces. Based on these experimental results, the value m, by default, set to 6 in the real long-tailed scenario.
Impact of Fixed Sample Size. We conduct experiments on CIFAR-100-LT to explore the impact of fixed sample size (i.e., k) on imbalanced classification. In the experiments, we gradually increase the value of l from 100. Note that the value of i: can be to larger than the sample size of the minority class. In this case, the actual sample size is set to min(k: Ne), where N is the sample size of the minority class. Figure shows the experimental results. It is observed that large value of & consistently benefits the many-shot accuracy (see Figure 4a) but hurts the few-shot accuracy (see Figure 4c). This is because increasing the value of & corresponds to adding additional majority samples, but it also enlarges the data imbalance between the majority and minority classes. Figure 4d
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Table 10: Performance results CIFAR-10 under various temperature parameters, with the best results shown in bold
Temperature Parameter
Many shot
Medium-shat
Few-
Overall
66.2
67.7
0.3
69.4
66.0
67.6
68.2
T= 0.5
72.3
64.7
57.4
66.3
0.7
73.1
62.6
56.8
66.1
Table 11: Average sample size after DPP sampling across different ground set sizes
Ground Sot Size (N)
100
1000
2000
5000
Sample of Subset
31.4
147.0
308.5
606.9
1548.4
Percentage of N
31.4%
29.4%
30.9%
30.3%
31.0%i
demonstrates that the best trade-off is achieved when the value of k is set to 50, resulting in the highest overall accuracy of 52.4%. We emphasize that when k is set to 50, the ratio & = 10, Ne
where No = 5 represents the sample size of the smallest class in CIFAR-100-LT. Thus, in practical scenarios, the value of k is typically set to 10Nc by default.
Influence of Temperature Parameter. This section investigates the influence of the BNS temperature parameter on representation learning. Specifically, we conduct experiments CIFAR-IO-LT with an imbalance factor of 100, varying temperature parameters from 0.1 to 0.7 in steps of 0.2. The results, summarized in Table 10, are reported linear probing accuracy.
We observe that smaller values of r (e.g., 0.1) improve performance on tail classes (Few-shot) but reduce accuracy on head classes (Many-shot), Conversely, larger values of increase Many-shot accuracy at the expense of Few-shot performance. This behavior occurs because larger т values soften the contrastive loss, giving more weight to hard negative head-class instances insten favoring head classes during representation learning. The best trade-off' between head and tail performance is achieved when is set to 0.3, where the accuracy gap between Many-shot and Few-shot categories is minimized (i.e., 1.8%).
dominated by
B.7 The Expected Subset Sample Size after DPP
This section supports Appendix A.7 by presenting the empirical results the expected subset sample size after Determinantal Point Process (DPP). The Theorem A.5 states that for ground set containing N samples, its expected number of elements selected by DPP under uniform eigenvalue distribution assumption is approximately N - In 2) x 30.7%. To empirically validate this claim, we ran the DPP over the CIFAR-IO-LT dataset with its ground set size ranging from N = 100 to 5000. Table 11 presents the empirical results (averaging over trials). observed that the resulting subset sizes consistently align with the theoretical prediction, i.e., N - In across all tested values of N. These results suggest that the uniform eigenvalue assumption-Nwe while idealized does not significantly distort the practical behavior of DPP.
C Broader Impact
Our proposed methods address the challenging problems regarding the long-tailed data distributions, envisioned to make significant contributions machine learning (ML. research and advance its applications across various critical domains. First, real-world data often exhibit long-tailed or imbalanced distributions, posing significant challenges for conventional ML algorithms. These issues are prevalent in critical applications such as social network spam detection, online transaction fraud detection, medical diagnosis, and others. Misclassifications in these domains can lead to catastrophic consequences, such as undetected fraudulent activities or delayed medical treatments. By effectively addressing the challenge of long-tail data distributions, our method enables state-of-the-ar ML techniques to perform reliably in critical domains, fostering improved Decision-making and delivering meaningful societal benefits. Second, our proposed two-stage learning framework comprises Balanced Negative Sampling (BNS) for representation learning and Information- Prreeevvabeee Determinantal Point Process (IP-DPP) for rectifying biased classifiers. These components are not only effective on their own but can also be seamlessly integrated with existing state-of-the-ar methods. Our
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experiments reveal that integrating BNS or IP-DPP into existing methods significantly improves their performance on long-tailed data. This broad adaptability enables more inclusive application of ML techniques across diverse datasets and tasks, providing a practical tool for practitioners in various industries.
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Reviews:
[{"review_id": "6s7CQLaQep", "reviewer": "Reviewer_8YK9", "summary": "This paper proposes a novel two-stage learning framework to address the challenges of long-tailed data distributions in classification tasks. The authors identify two major issues in long-tailed settings: representation learning bias and classifier bias. To tackle these, they introduce:\n\nBalanced Negative Sampling (BNS): A contrastive representation learning strategy that constructs semantically rich and balanced feature spaces by carefully selecting informative negative samples. BNS captures both instance- and class-level semantics, enabling better feature separation, especially for medium- and few-shot classes.\n\nInformation-Preservable Determinantal Point Process (IP-DPP): A sampling method used during classifier training to select a diverse and representative subset of the training data. IP-DPP maintains a balance between classes while preserving informative instances and rectifying biased decision boundaries.\n\nExtensive experiments are conducted on CIFAR-10-LT, CIFAR-100-LT, ImageNet-LT, and iNaturalist 2018 datasets. The proposed method achieves state-of-the-art performance and shows robustness across different imbalance levels and model architectures. The paper also includes theoretical insights, ablation studies, computational cost analysis, and discussions on hyperparameter sensitivity. The framework’s modularity and compatibility with existing methods make it a promising solution for real-world imbalanced learning scenarios.", "questions": "1. Clarification on the Statistical Assumption in DPP Derivation\n\nIn Theorem A.5, the expected sample size of a DPP is derived under the assumption that eigenvalues follow a uniform distribution in [0, 1]. This is a strong assumption and may not reflect the actual eigenvalue distribution in practice. Could the authors provide empirical evidence (e.g., histogram of eigenvalues) to support the validity of this assumption on real datasets? Alternatively, could the authors comment on how sensitive the performance is to this assumption?\n\n2. Ablation on the Interaction Between BNS and IP-DPP\n\nThe paper presents ablations for BNS and IP-DPP independently. However, it is unclear how these two components interact. Could the authors provide an ablation showing the incremental benefit of BNS and IP-DPP when added sequentially (e.g., base → base+BNS → base+BNS+IP-DPP)? This would help isolate their individual and combined contributions more clearly.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\nThis paper proposes a novel two-stage learning framework to address long-tailed recognition, integrating Balanced Negative Sampling (BNS) for improved representation learning and Information-Preservable Determinantal Point Process (IP-DPP) for balanced classifier training. The approach is both theoretically grounded and empirically validated. The authors provide rigorous mathematical derivations, detailed ablation studies, and comprehensive evaluations on multiple standard benchmarks, including CIFAR-10-LT, CIFAR-100-LT, ImageNet-LT, and iNaturalist 2018. Notably, the method consistently outperforms strong baselines and can be easily combined with existing techniques, demonstrating broad applicability. The paper is well-written and clearly structured, with useful visualizations and thorough supplementary material that supports the main contributions. The use of IP-DPP for informative and class-balanced subsampling is particularly innovative, and the combination of DPP theory with long-tailed learning is a valuable contribution.\n\nWeaknesses:\n\nWhile the theoretical contributions are strong, some derivations (e.g., those involving eigenvalue distributions in DPP) rely on simplifying assumptions (e.g., uniform distribution) that may not always hold in practice, limiting their generality. Some notation in the theoretical sections is dense, and parts of the methodology (e.g., details of the BNS selection mechanism) could benefit from more intuitive explanation.\n\nOverall Assessment:\n\nThis is a high-quality, well-motivated, and thorough piece of work that offers solid improvements and insights into long-tailed recognition problems.", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for your constructive feedback and your appreciation of our work.\n\n**Q1. Theorem A.5 assumes a uniform distribution of eigenvalues. Could the authors provide empirical evidence (e.g., histogram of eigenvalues) to support the validity of this assumption on real datasets? Or could the authors comment on how sensitive the performance is to this assumption?**\n\n**R1.** Thank you for your question. We appreciate your close examination of our theoretical assumptions.\n\nDue to the NeurIPS rebuttal policy, which prohibits the inclusion of new figures via anonymous links, we are unable to provide a histogram of eigenvalues at this time. However, we provide more direct empirical evidence to support the practical validity of the uniform eigenvalue assumption used in Theorem A.5.\n\nTheorem A.5 states that for a ground set of size $N$, the expected number of elements selected by a DPP (Determinantal Point Process) under a uniform eigenvalue distribution assumption is approximately $N(1 - \\ln 2) \\approx \\mathbf{30.7 \\\\%}$. To empirically validate this claim, we ran DPP sampling over the CIFAR-10-LT dataset with ground set sizes ranging from $N=100$ to $N=5000$, averaging the results over 10 trials per setting.\n\nAs shown in Table A below, the resulting subset sizes consistently align with the theoretical prediction across all tested values of $N$, suggesting that the uniform eigenvalue assumption—while idealized—does not significantly distort the practical behavior of the DPP. We will include additional empirical analyses, including eigenvalue histograms on real datasets, in the camera-ready version to further substantiate this assumption.\n\n\n\n**Table A. Empirical average sample size after DPP sampling across different ground set sizes**\n\n| Ground Set Size ($N$) |  100  |  500  | 1000  | 2000  |  5000  |\n| :-------------------: | :---: | :---: | :---: | :---: | :----: |\n| Sample Size of Subset | 31.4  | 147.0 | 308.5 | 606.9 | 1548.4 |\n|   Percentage of $N$   | 31.4% | 29.4% | 30.9% | 30.3% | 31.0%  |\n\n\n\n\n**Q2. Could the authors provide an ablation showing the incremental benefit of BNS and IP-DPP?**\n\n**R2.** Thank you for your comments. We separated BNS and IP-DPP in the original ablation study because they serve different purposes and are evaluated under different metrics. Specifically, BNS is designed for representation learning and is evaluated via linear probing accuracy, whereas IP-DPP is a sampling strategy applied during the classifier training stage and evaluated using classification accuracy.\n\nTo directly address your question, we conducted an additional ablation study on CIFAR-100-LT with an imbalance factor (IF) of 100. All methods follow a standard two-stage learning strategy:\n\n- **Stage 1**: Representation learning\n- **Stage 2**: Classifier training to mitigate decision boundary bias\n\nFor the baseline method, we use SimCLR for Stage 1 and Random Undersampling for Stage 2. We then incrementally improve this setup by: (i) replacing SimCLR with BNS (Stage 1), and  (ii) replacing Random Undersampling with IP-DPP (Stage 2). Finally, we report the performance of our full approach, which combines both BNS and IP-DPP.\n\nThe results are shown in Table B below. We make four key observations:\n\nFirst, **Baseline (SimCLR + Random)** performs worst overall because SimCLR fails to learn effective representations for tail classes, while Random Undersampling discards valuable information during re-balancing.\n\nSecond, **BNS + Random** improves Few-shot and Medium-shot performance by learning more structured representations, but still suffers in Many-shot accuracy due to suboptimal classifier training.\n\nThird, **SimCLR + IP-DPP** improves Many-shot performance by preserving informative head-class samples, but continues to struggle with tail classes due to SimCLR’s poor representations.\n\nFourth, **our method (BNS + IP-DPP)** achieves the best overall performance by combining strong, balanced representations from BNS and informative sample selection from IP-DPP to reduce bias during classifier training.\n\n\n\n**Table B: Ablation studies on the incremental benefit of BNS and DP-DPP (CIFAR-100-LT, IF = 100)**\n\n|          Methods           | Many-shot | Medium-shot | Few-shot | Overall |\n| :------------------------: | :-------: | :---------: | :------: | :-----: |\n| Baseline (SimCLR + Random) |   58.1    |    56.6     |   28.1   |  48.6   |\n|        BNS + Random        |   57.8    |    59.6     |   31.4   |  50.5   |\n|      SimCLR + IP-DPP       |   62.7    |    56.9     |   28.5   |  50.4   |\n|    Ours (BNS + IP-DPP)     |   62.4    |    59.7     |   31.9   |  52.4   |"}, {"review_id": "zfkXMc0E0e", "reviewer": "Reviewer_bC3v", "summary": "This paper addresses the challenge of class imbalance in real-world datasets by introducing a novel two-stage learning framework. The first stage employs an information-theoretic representation learning method that effectively minimizes intra-class distances to create well-separated feature spaces. The second stage introduces an innovative undersampling technique that selects informative samples to correct biased decision boundaries caused by majority classes, without sacrificing overall model performance. Extensive experiments on long-tailed benchmark datasets demonstrate that this approach achieves state-of-the-art results.", "questions": "1.How do the limitations of traditional representation learning manifest? Why can't the feature learning stage effectively distinguish between head and tail classes? \n\n2.Is it possible to visualize how Balanced Negative Sampling (BNS) improves representation specifically for tail classes?\n\n If the authors are able to address the above issues, I would be willing to raise my score.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths :\nThe author's reasoning is very clear and easy to understand, with a well-defined motivation. The overall flow is smooth, and the ablation studies are comprehensive.\n\nWeaknesses:\n\nTables 1 and 2 do not show results under different class balance ratios. The appendix only reports overall performance on CIFAR datasets. It would be beneficial to present more detailed results showing the trade-off between head and tail class performance at varying balance ratios.\n\nThe effect of the initial value of the temperature parameter τ on results is not investigated (e.g., how changes in τ impact accuracy).", "was_revised": true, "final_justification": "The author's answers to my questions are very clear, including the ablation experiments on temperature and other supplementary experiments. At the same time, the author also resolved my confusion in the theoretical part.I believe these methods are very important for evaluating the completeness of this paper, so I have decided to raise my score.", "rebuttal": "We sincerely appreciate your thoughtful feedback and recognition of our work.\n\n**Q1. How do the limitations of traditional representation learning manifest? Why can't the feature learning stage effectively distinguish between head and tail classes?**\n\n**R1.** Effective representation learning for long-tailed data must capture two key aspects: (i) **instance-level semantics** for high-quality feature representations; and (ii) **class-level semantics** for well-separated feature spaces. Traditional contrastive learning methods such as SimCLR and MoCo focus solely on instance-level discrimination, treating each sample as its own class. While this works well under balanced distributions, it lacks mechanisms for class-level alignment, which is crucial in long-tailed scenarios. As a result, head-class features dominate the space, while tail-class instances—due to data scarcity—fail to form coherent clusters, resulting in poor generalization.\n\nMoreover, these methods lack effective sampling strategies and often fail to retrieve sufficient positive pairs for tail classes, further degrading their representation quality and pushing them to ambiguous regions of the feature space.\n\nOur proposed BNS method addresses these limitations by jointly modeling both instance-level and class-level semantics. The BNS loss, i.e., Eq. (12) in the paper, can be decomposed as:\n$$\n\\mathcal{L}_{\\textrm{BNS}} = - \\frac{1}{m+1} \\left [\\underbrace{ \\log \\sigma (\\frac{\\mathbf{q}\\_{i}^{\\top} \\mathbf{v}\\_{j,i}^{+}}{\\tau})         + \\sum\\_{j=1}^{n} \\log \\sigma ( - \\frac{\\mathbf{q}\\_{i}^{\\top} \\mathbf{v}\\_{j}^{-}}{\\tau})}\\_{\\textrm{instance-level}}  +  \\underbrace{ \\sum\\_{q\\_{k} \\in \\mathbf{Q}\\_{i,m}^{+}}         \\left [ \\log \\sigma (\\frac{\\mathbf{q}\\_{k}^{\\top} \\mathbf{v}\\_{j,i}^{+}}{\\tau})        + \\sum\\_{j=1}^{n} \\log \\sigma ( - \\frac{\\mathbf{q}\\_{k}^{\\top} \\mathbf{v}\\_{j}^{-}}{\\tau}) \\right] }\\_{\\textrm{class-level}}\\right]\n$$\n\n- The **instance-level** term maintains fine-grained distinctions between individual samples.\n\n- The **class-level** term promotes intra-class cohesion, especially important for tail classes.\n\nFurthermore, BNS constructs a balanced neighborhood by sampling an additional $m$ positive pairs from the same class, ensuring sufficient positive context even for underrepresented classes. This dual-level alignment results in a more structured, balanced feature space that improves generalization across both head and tail classes.\n\n\n\n**Q2. Is it possible to visualize how Balanced Negative Sampling (BNS) improves representation specifically for tail classes?**\n\n**R2.** Thank you for your question. A t-SNE visualization focused solely on tail classes is an effective way to demonstrate this improvement.\n\nWhile the NeurIPS rebuttal policy restricts us from adding new figures via external links, we would like to highlight that Figures 2(a) and 2(c) in our submission already provide relevant insights.\n\nSpecifically, Figure 2(c) shows the feature space learned using BNS. Notably, the top-most (gold) and left-most (cyan) clusters correspond to two tail classes. These clusters exhibit strong intra-class compactness and are well-separated from nearby head classes (depicted in orange and blue), indicating that BNS facilitates more discriminative and robust representations for tail classes.\n\nIn contrast, Figure 2(a), which shows representations from SBCL, displays substantial overlap between head and tail classes. The head classes dominate the feature space, leading to poorer separability for tail classes. This qualitative comparison demonstrates that BNS significantly enhances the representation quality for tail classes by promoting both intra-class compactness and inter-class separability.\n\n\n\n**Q3. It would be beneficial to present more detailed results showing the trade-off between head and tail class performance at varying balance ratios.**\n\n**R3.** Thank you for the valuable suggestion. To provide a more comprehensive evaluation of our method under different imbalance scenarios, we report detailed results on CIFAR-10-LT and CIFAR-100-LT, broken down into Many-shot, Medium-shot, Few-shot, and Overall accuracy. These results are summarized in Tables A and B below.\n\nOur findings show that as the imbalance factor increases, performance on Many-shot (head) classes remains relatively stable, whereas accuracy on Few-shot (tail) classes declines significantly. This trend reflects the growing representation gap between head and tail classes under more severe imbalance, which makes learning effective representations for tail classes increasingly difficult. These extended results further highlight the importance of addressing data imbalance and will be included in the final manuscript.\n\n\n**Table A: Performance on CIFAR-10-LT under varying imbalance factors**\n\n| Imbalanced Factor (IF) | Many-shot | Medium-shot | Few-shot | Overall |\n| :--------------------: | :-------: | :---------: | :------: | :-----: |\n|         IF=50          |   85.3    |    81.4     |   72.2   |  81.9   |\n|         IF=100         |   82.0    |    76.3     |   67.2   |  76.4   |\n|         IF=200         |   81.4    |    75.9     |   64.4   |  73.5   |\n\n\n\n**Table B: Performance on CIFAR-100-LT under varying imbalance factors**\n\n| Imbalanced Factor (IF) | Many-shot | Medium-shot | Few-shot | Overall |\n| :--------------------: | :-------: | :---------: | :------: | :-----: |\n|         IF=50          |   63.7    |    57.4     |   34.7   |  55.9   |\n|         IF-100         |   62.4    |    59.7     |   31.9   |  52.4   |\n|         IF=200         |   62.8    |    58.9     |   24.2   |  46.7   |\n\n\n\n\n\n**Q4. Please include ablation studies on the temperature parameter $\\tau$.**\n\n**R4.** We conducted an ablation study on the temperature parameter $\\tau$, with results presented in Table C below. The study was performed on CIFAR-10-LT with an imbalance factor of 100, varying $\\tau$ from 0.1 to 0.7 in steps of 0.2.\n\nWe observe that smaller values of $\\tau$ (e.g., 0.1) improve performance on tail classes (Few-shot) but reduce accuracy on head classes (Many-shot). Conversely, larger values of $\\tau$ increase Many-shot accuracy at the expense of Few-shot performance. This behavior occurs because larger $\\tau$ values soften the contrastive loss, giving more weight to hard negative samples—typically dominated by head-class instances—thereby favoring head classes during representation learning.\n\nThe best trade-off between head and tail performance is achieved when $\\tau$ is set to 0.3, where the accuracy gap between Many-shot and Few-shot categories is minimized. We will include these findings in our manuscript.\n\n\n\n**Table C: Ablation study on the temperature parameter $\\tau$ (CIFAR-10-LT, IF = 100)**\n\n| Temperature Parameter | Many-shot | Medium-shot | Few-shot | Overall  |\n| :-------------------: | :-------: | :---------: | :------: | :------: |\n|     $\\tau$ = 0.1      |   65.1    |  **66.2**   | **73.0** |   67.7   |\n|     $\\tau$ = 0.3      |   69.4    |    66.0     |   67.6   | **68.2** |\n|     $\\tau$ = 0.5      |   72.3    |    64.7     |   57.4   |   66.3   |\n|     $\\tau$ = 0.7      | **73.1**  |    62.6     |   56.8   |   66.1   |"}, {"review_id": "r7kBD6JMqS", "reviewer": "Reviewer_2ts7", "summary": "To solve the long-tailed imbalance classification problem, the authors propose a novel two-stage learning approach to mitigate such a majority-biased tendency while preserving valuable information within datasets.\n\nIn the first stage, the work aims to learn an effective and well-separated feature space for long-tailed recognition, by maximizing mutual information between instances sharing the same label—a process mathematically equivalent to minimizing intra-class distances. In the second stage, the authors propose Information-Preservable Determinantal Point Process (IP-DPP), aiming to rectify majority-biased classification decision boundaries while maintaining the model’s overall performance.", "questions": "What is the relationship between the first and second stages? Is there any interaction between the two stages? What parameters are trained in the second stage? What is the L-ensemble as stated in the second stage?\n\nHow it performs if the authors compare the method with the clip-based long-tailed methods? Since CLIP is pretrained based on the contrastive learning, it may provide a powerful backbone.\n\nWhat type of long-tailed methods can be combined with the proposed method except for the representation-based methods?\n\n\nAs shown in Table2, the few-shot results in iNaturalist 2018 even outperforms the Many-shot and Medium-shot, what's the reason? How to achieve a balance between these groups in this method?\n\nIn terms of Table 5, why experimental results on ImageNet-LT and iNaturalist 2018 using various model architectures seem similar?", "limitations": "The second stage is somewhat difficult to understand. It would be better if the authors can provide an overall training workflow and the discussion in the end of method. Please see weakness and questions.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 2, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths: This work has addressed the challenging long-tailed data classification problem, by proposing a novel information-preservable two-stage learning approach. \nWeaknesses: It would be better if the authors can provide a comparison with the clip-based long-tailed methods since the vision encoder is learned in a contrastive way.\n\nIt would be better if the authors can discuss the limitations of their proposed work.", "was_revised": true, "final_justification": "The authors have addressed my concerns.", "rebuttal": "We greatly appreciate your constructive feedback and your positive evaluation of our work.\n\n**Q1. What is the relationship between the first and second stages? Is there any interaction between the two stages? What parameters are trained in the second stage? Could the authors provide an overall training workflow?**\n\n**R1.** Our framework follows a two-stage training strategy commonly used in long-tailed learning. A deep neural network (DNN) for classification is typically composed of a feature extractor and a linear classifier, and we leverage this architecture as follows:\n\n- Stage 1 (Representation Learning with BNS):\n  We apply our proposed BNS to pre-train the feature extractor on the **entire imbalanced dataset**. This stage focuses on learning generalizable and well-structured feature representations across both head and tail classes.\n- Stage 2 (Classifier Fine-tuning with IP-DPP):\n  In this stage, we fine-tune **both the feature extractor and the linear classifier** on a **class-balanced subset** sampled via our IP-DPP. This step is designed to correct the biased decision boundary introduced by the imbalance in Stage 1.\n\nAlthough the two stages are not jointly optimized, they are closely related: the success of Stage 2 depends on the quality of the representations learned in Stage 1. Without strong feature representations from BNS, training on a limited balanced subset would not be sufficient to generalize well.\n\nWe summarize the full training workflow below:\n\n1. Input: An imbalanced training dataset.\n2. Stage 1: Train the feature extractor using BNS on the entire dataset.\n3. Stage 2: Use IP-DPP to select a balanced subset. Fine-tune the entire network (feature extractor + classifier) on this subset using cross-entropy loss (or long-tailed learning loss, e.g., Focal Loss).\n\n\n\n**Q2. What is the L-ensemble stated in the second stage?**\n\n**R1.** An L-ensemble is a standard formulation of a Determinantal Point Process (DPP), where the probability of selecting a subset of items is defined using a positive semi-definite matrix called the L matrix. This matrix jointly captures the individual quality of items and their pairwise diversity.\n\nIn the context of our second-stage sampling strategy, the L-ensemble is constructed based on **information content**, allowing our IP-DPP method to prioritize mathematically informative samples. This enables the selection of balanced and representative subsets that effectively correct biased decision boundaries—without sacrificing critical information—especially in long-tailed settings.\n\n\n\n**Q3. Can the authors compare the proposed method with CLIP-based long-tailed methods?**\n\n**R3.** We conducted additional experiments on iNaturalist 2018, comparing our method with **VL-LTR** [1], a recent CLIP-based approach designed for long-tailed recognition. \n\nTable A presents the comparative results. We observe that VL-LTR achieves higher Many-shot accuracy, likely due to its ability to exploit rich visual-textual pairs available for head classes. In contrast, our method outperforms VL-LTR on Medium-shot, Few-shot, and Overall accuracy, with a particularly significant improvement of +5.6% in Few-shot accuracy, demonstrating the strength of our approach in the long-tailed setting.\n\nThese findings suggest that our BNS and IP-DPP methods effectively handle long-tailed distributions without reliance on external textual supervision. Nonetheless, we believe integrating CLIP-based backbones or leveraging textual information could further enhance our method's performance, especially in extreme imbalanced settings. We will explore this promising direction in our future work.\n\n\n\n**Table A: Comparison with CLIP-based long-tailed method on iNaturalist 2018**\n\n| Method | Many-shot | Medium-shot | Few-shot | Overall  |\n| :----: | :-------: | :---------: | :------: | :------: |\n| VL-LTR | **75.3**  |    72.6     |   69.9   |   71.8   |\n|  Ours  |   72.7    |  **72.9**   | **75.5** | **74.0** |\n\n\n\n[1] VL-LTR: Learning Class-wise Visual-Linguistic Representation for Long-Tailed Visual Recognition, ECCV 2022.\n\n\n\n**Q4. What type of long-tailed methods can be combined with the proposed method, except for representation-based methods?**\n\n**R4.** Our proposed methods— BNS and IP-DPP—are highly flexible and can be integrated with a broad range of long-tailed learning techniques beyond representation-based methods.\n\nSpecifically:\n\n- BNS can be readily combined with **re-weighting approaches** such as Focal Loss and LDAM Loss, as well as **sampling-based strategies** like SMOTE and Random Undersampling.\n- IP-DPP can be integrated with standard or long-tailed **contrastive learning frameworks** such as SimCLR, KCL, TSC, and SBCL, as well as aforementioned re-weighting methods.\n\nWe provide empirical evidence in Table 4 of the Appendix, which reports results on CIFAR-10-LT and CIFAR-100-LT using combinations of BNS and IP-DPP with some of the aforementioned approaches. The consistent performance improvements across these combinations demonstrate the broad compatibility and utility of our two key components. \n\n\n\n**Q5. Why does the Few-shot accuracy on iNaturalist 2018 outperform the Many-shot and Medium-shot accuracies? How can a balanced performance across Many-shot, Medium-shot, and Few-shot categories be achieved?**\n\n**R5.** Thank you for your thoughtful question. The observation arises primarily due to the **high imbalance factor (i.e., IF=500)** present in the iNaturalist 2018 dataset. Our IP-DPP method aims to mitigate this imbalance by sampling a more **balanced subset** in the second stage of training.\n\nIn particular, we fix a **uniform sample size $k$** across all classes during IP-DPP sampling. While this strategy helps elevate the performance on **few-shot classes**, it inherently reduces the number of samples from **many-shot** and **medium-shot** classes—leading to a drop in their respective accuracies.\n\nTo achieve a more balanced performance across all three groups, we can fine-tune the per-class sample size $k$. By adjusting $k$ (e.g., through class-aware or adaptive sampling strategies), we can trade off between the extremes and improve overall class-wise balance.\n\nWe plan to explore more adaptive subset selection strategies in future work to dynamically balance performance across many-shot, medium-shot, and few-shot classes.\n\n\n\n**Q6. In Table 5, the experimental results on ImageNet-LT and iNaturalist 2018 appear similar across different model architectures. Could the authors explain why the performance is relatively consistent despite the architectural differences?**\n\n**R6.** Thank you for the insightful observation. The relatively consistent performance across architectures on ImageNet-LT and iNaturalist 2018 can be attributed to two main factors. First, both datasets contain a large number of similar animal categories, e.g., birds, mammals, and insects, which encourages models to learn comparable visual features across architectures. Second, both datasets exhibit similar long-tailed distributions due to their web-based collection process. Common species tend to dominate the head classes, while rare species form the tail, creating similar imbalance patterns that influence model behavior more than architectural differences.\n\n\n\n**Q7. Please discuss the limitations of the proposed work.**\n\n**R7.** The main limitation of our proposed method lies in the additional computational overhead introduced by our IP-DPP sampling strategy, which involves non-trivial matrix operations (see Tables 6 and 7 in the Appendix for computational overhead analysis). While this overhead is manageable in our experiments, it may become a concern in large-scale, real-world datasets.\n\nIn our revision, we will include a dedicated Limitations section to discuss this issue in detail and outline potential directions for improving the efficiency of IP-DPP.\n\n\n\n**Q8. Figure 2 appears before Figure 1 in the paper. Please revise the ordering for consistency.**\n\n**R8.** Thank you for pointing this out. We will fix this in our revision."}, {"review_id": "8poqqSYNe8", "reviewer": "Reviewer_VmnD", "summary": "This paper addresses the challenge of long-tailed recognition by proposing a novel two-stage learning framework that preserves valuable dataset information while mitigating class imbalance. In the first stage, the authors introduce Balanced Negative Sampling (BNS), a representation learning strategy grounded in information theory. BNS is shown to maximize mutual information between augmented views, which is theoretically equivalent to minimizing intra-class distances, thus leading to well-separated feature spaces. In the second stage, the authors propose Information-Preservable Determinantal Point Process (IP-DPP), a novel undersampling technique that prioritizes mathematically informative samples to correct majority-biased decision boundaries without degrading overall model performance. Extensive experiments on four benchmark datasets validate the effectiveness of the proposed method.", "questions": "Please see the \"Strengths And Weaknesses\" part.", "limitations": "yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\n- The authors provide a theoretical analysis of the proposed method, enhancing its soundness and credibility.\n- Extensive experiments are conducted, including performance comparisons and ablation studies, to validate the effectiveness of the method, which strengthens its overall reliability.\n- The results demonstrate that the proposed method is effective, achieving substantial performance gains.\n\n\n\n**Weaknesses:**\n\n- The paper would benefit from more explanation of why the BNS method is particularly effective for learning representations under long-tailed scenarios. It would be helpful to elaborate on the differences and advantages of the proposed implementation compared to standard contrastive learning (CL) methods, as well as other CL-based long-tailed learning methods such as KCL, TSC, and SBCL.\n- More representative long-tailed learning methods, such as PaCo$^1$, BCL$^2$, DDC$^3$, GPaCo$^4$ and DirMixE$^5$, should be included in the related work for a more comprehensive comparison.\n- There are also some typos, such as in line 161, where \"presentation space\" should be \"representation space\".\n\n-----\n\n$^1$ Parametric contrastive learning, ICCV 2021 \n\n$^2$ Balanced Contrastive Learning for Long-Tailed Visual Recognition, CVPR 2022\n\n$^3$ A Unified Generalization Analysis of Re-Weighting and Logit-Adjustment for Imbalanced Learning, NeurIPS 2023\n\n$^4$ Generalized Parametric Contrastive Learning, TPAMI 2023\n\n$^5$ Harnessing Hierarchical Label Distribution Variations in Test Agnostic Long-tail Recognition, ICML 2024", "was_revised": true, "final_justification": "Dear authors,\n\nThank you for your thoughtful response and clarification. I appreciate the explanation regarding how the proposed loss function combines both instance-level and class-level semantics, which makes it more effective than other CL-based methods in long-tailed learning scenarios. Your other responses also satisfactorily address my remaining concerns. All my questions have been resolved, and I will maintain my score of 4.", "rebuttal": "Thank you for your valuable feedback and for recognizing the contributions of our work.\n\n**Q1. Compared to standard contrastive learning (CL) methods and CL-based long-tailed learning approaches, why BNS is particularly effective in long-tailed data?**\n\n**R1.** BNS is particularly effective because it jointly captures instance-level and class-level semantics, both of which are essential for balanced representation learning in long-tailed settings. To illustrate, we decompose the BNS loss, i.e., Eq. (12) in the main paper, as:\n$$\n\\mathcal{L}_{\\textrm{BNS}} = - \\frac{1}{m+1} \\left [\n\\underbrace{ \\log \\sigma (\\frac{\\mathbf{q}\\_{i}^{\\top} \\mathbf{v}\\_{j,i}^{+}}{\\tau}) \n        + \\sum\\_{j=1}^{n} \\log \\sigma ( - \\frac{\\mathbf{q}\\_{i}^{\\top} \\mathbf{v}\\_{j}^{-}}{\\tau})}\\_{\\textrm{instance-level}}  +  \\underbrace{ \\sum\\_{q\\_{k} \\in \\mathbf{Q}\\_{i,m}^{+}} \n        \\left [ \\log \\sigma (\\frac{\\mathbf{q}\\_{k}^{\\top} \\mathbf{v}\\_{j,i}^{+}}{\\tau})\n        + \\sum\\_{j=1}^{n} \\log \\sigma ( - \\frac{\\mathbf{q}\\_{k}^{\\top} \\mathbf{v}\\_{j}^{-}}{\\tau}) \\right] }\\_{\\textrm{class-level}}\n\\right]\n$$\n\n- The instance-level term ensures fine-grained discrimination between individual samples, similar to standard CL.\n- The class-level term enforces intra-class cohesion, which is critical for tail classes with limited samples.\n\nStandard CL methods such as SimCLR lack explicit class-level alignment, while existing CL-based long-tailed approaches such as KCL, TSC, and SBCL emphasize class-level semantics but often let head-class samples dominate the feature space, weakening instance-level distinction for tail classes. BNS directly optimizes both levels, resulting in more robust and balanced representations that generalize well across both head and tail classes.\n\n\n\n**Q2. Please include more representative long-tailed learning methods, such as PaCo, BCL, DDC, GPaCo, and DirMixE, in the Related Work section.**\n\n**R2.** Thank you for the valuable suggestion. In our revision, we will update the Related Work section to discuss how our method relates to, and differs from, the representative long-tailed learning approaches you mentioned—namely PaCo, BCL, DDC, GPaCo, and DirMixE.\n\n\n\n**Q3. There are also some typos, such as in Line 161, where \"presentation space\" should be \"representation space\".**\n\n**R3.** Thank you for pointing this out. We will carefully proofread the manuscript and correct all typos, including the one you identified in Line 161, in the manuscript."}]

Author rebuttal:

Meta-review: This paper proposes a two-stage method to address class imbalance by learning compact representations and refining decision boundaries through informative undersampling. The approach achieves state-of-the-art performance on long-tailed benchmarks.

The reviewers praised the paper for its well-motivated and clearly presented two-stage framework that tackles long-tailed classification using Balanced Negative Sampling (BNS) and a novel Information-Preservable Determinantal Point Process (DPP) approach. They highlighted the strong theoretical grounding, smooth reasoning, and comprehensive ablation studies. The method consistently outperforms strong baselines across several benchmark datasets and is modular enough to integrate with existing techniques. The use of DPPs for informative, class-balanced subsampling was seen as a particularly creative and valuable contribution.

The reviewers pointed out several areas for improvement. Some derivations, especially those involving eigenvalue distributions in DPPs rely on simplifying assumptions like uniformity that may not hold universally, reducing generality.  Notation can be dense, and the BNS mechanism would benefit from more intuitive explanation. Empirically, results are only shown for fixed class imbalance ratios, with limited analysis of head–tail trade-offs or the influence of key hyperparameters like the temperature τ. Reviewers also requested more clarity on why BNS is effective under long-tailed conditions and how it compares to other contrastive learning methods such as KCL, TSC, and SBCL. Additionally, several strong long-tailed baselines (PaCo, BCL, DDC, GPaCo, DirMixE) were missing from related work. Minor presentation issues like typos and a lack of discussion around limitations were also noted.

The paper presents a strong and well-rounded contribution to long-tailed classification, combining theoretical innovation with solid empirical results. All reviewer concerns were addressed during the rebuttal phase, including additional ablations, empirical validations, and new baseline comparisons (e.g., with CLIP-based methods). The authors have also committed to incorporating these updates, such as expanded analysis of their BNS approach, sensitivity to temperature settings, and a clear discussion of limitations, into the final manuscript. With these revisions, the work not only maintains its technical rigor but also improves in clarity, completeness, and relevance, warranting acceptance.
Decision: Accept (spotlight)