Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Authors: Zi-Hao Zhou
Jun-Jie Wang
Tong Wei
Min-Ling Zhang
Venue: ICML 2025 spotlightposter (2025)
Keywords: Weakly-supervised learning
Contrastive learning
Noisy label learning
Partial label learning
Primary area: general_machine_learning->unsupervised_and_semisupervised_learning

Abstract: Contrastive learning has achieved remarkable success in learning effective representations, with supervised contrastive learning often outperforming self-supervised approaches. However, in real-world scenarios, data annotations are often ambiguous or inaccurate, meaning that class labels may not reliably indicate whether two examples belong to the same class. This limitation restricts the applicability of supervised contrastive learning. To address this challenge, we introduce the concept of ``continuous semantic similarity'' to define positive and negative pairs. Instead of directly relying on imprecise class labels, we measure the semantic similarity between example pairs, which quantifies how closely they belong to the same category by iteratively refining weak supervisory signals. Based on this concept, we propose a graph-theoretic framework for weakly-supervised contrastive learning, where semantic similarity serves as the graph weights. Our framework is highly versatile and can be applied to many weakly-supervised learning scenarios. We demonstrate its effectiveness through experiments in two common settings, i.e., noisy label and partial label learning, where existing methods can be easily integrated to significantly improve performance. Theoretically, we establish an error bound for our approach, showing that it can approximate supervised contrastive learning under mild conditions. The implementation code is available at [https://github.com/Speechless-10308/WSC](https://github.com/Speechless-10308/WSC).

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Zi-Hao Zhou 11
Jun-Jie Wang"
12 Tong Wei 12 Min-Ling Zhang
Abstract
1. Introduction
In recent years, there has been a resurgence of research in contrastive learning, which has led to significant advance- ments in the field of representation learning (He et al., 2020; Oord et al., 2018; Chen et al., 2020; Caron et al., 2020; Zbontar et al., 2021; Khosla et al., 2020). These works share a common underlying principle: they aim to pull an anchor and its corresponding "positive" examples closer to- gether in the embedding space, while simultaneously push- ing the anchor away from a set of "negative" examples. In self-supervised contrastive learning, positive examples are generated through various data augmentation techniques applied to the original samples, thereby creating different views or representations of the same data instance. Negative examples, on the other hand, are randomly selected from different samples, as illustrated in Figure ((a). In contrast, contrastive learning utilizes additional su- pervisory information by treating samples from the same class positive pairs, thereby constructing multiple pos- itive examples (Khosla et al., 2020), as shown in Figure ((b). However, woal-world supervisory information is often inaccurate and ambiguous (Lin et al., 2023; Zhang et al., 2021b; Wang et al., 2022; Zhang et al., 2022; Guo et al., 2020; 2025), manifesting as weakly-supervised information such as noisy labels (Amsaleg et al., 2017: Yan et al., 2013) and partial labels (Luo and Orabona, 2010; Chen al., 2018; Dong et al., 2023), which cannot directly indicate class membership between samples. As a result, traditional methods fail to effectively utilize weak supervisory infor- mation, significantly limiting the applicability of supervised contrastive learning in real-world tasks.
Contrastive learning has achieved remarkable suc- cess in learning effective representations, with supervised contrastive learning often outperform- ing self--uuprrised approaches. However, in real- world scenarios, data annotations are often am- biguous inaccurate, meaning that class labels may not reliably indicate whether two examples belong to the same class. This limitation restricts the applicability of supervised contrastive learn- ing. To address this challenge, we introduce the concept of "continuous semantic similarity" to de- fine positive and negative pairs. Instead of directly relying on imprecise class labels, measure the semantic similarity between example pairs, which quantifies how closely they belong to the same category by iteratively refining weak su- pervisory signals. Based on this concept, we propose graph-theoretis framework for weakly- supervised contrastive learning, where semantic similarity serves the graph weights. Our frame- work is highly versatile and can be applied to many seakly-supervised learning scenarios. We demonstrate its effectiveness through experiments in two common settings, i... noisy label and par- tial label learning, where existing methods can be easily integrated significantly improve perfor- mance. Theoretically, we establish error bound for approach, showing that it can approximate supervised contrastive learning under mild con- ditions. The implementation code is available at https//ggttubhcooosppeecceesssss 10308/WSC.
To address the limitations of existing methods, we introduce more general concept of positive and negative examples: continous semantic similarity. Samples with a higher likeli- hood of belonging to the same category are assigned higher semantic similarity, where values of and 0 correspond to positive and negative pairs in traditional contrastive learning, respectively. This can be viewed as a continuous extension of the conventional positive-negative example framework. Furthermore, illustrated in Figure 1(c), we extend the contrastive learning objective to align the feature similarity of two samples with their corresponding semantic similarity.
'Equal contribution 'School of Computer Science and Engi- neering. Southeast University. Nanjing, China *Key Laboratory of Computer Network and Information Integration (Southeast Uni- versity), Ministry of Education, China. Correspondence to: Tong Wei <weit scu.edu.cn>.
Proceedings of the 4 International Conference on Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Specifically, we formulate a graph-theoretic framework for

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Contrastive Learning for Imprecise Class Labels
T(t)
Negetve
Pooirve
Similoray::
Audior
Anchor
Anchor
(a) Self-supervised Contrastive
(b) Supervised Contrastive
(c) Weakly-supervised Contrastive
Figure 1. (a) Self-supervised contrastive learning constructs positive example pairs by using different views of the same image and constructs negative example pairs by using different images. (b) Supervised contrastive learning further regards different images of the same class as additional positive example pairs. (c) Our proposed weakly-supervised contrastive learning abandons the concepts of discrete positive and negative examples.
weakly-supervine representation learning, where vertices represent augmented data points and edges are defined by their corresponding semantic similarity. The semantic simi- larity is constructed from two perspectives. First, data points derived from different views of the same instance are as- signed a relatively high semantic similarity. Second, we leverage the corresponding weakly-supervised information to approximate whether two samples belong to the same class, thereby constructing the semantic similarity between data points from different instances after augmentation.
concept of semantic similarity as generalization of positive and negative examples, our framework offers new insights into how contrastive learning can effec- tively utilize ambiguous supervisory information.
The loss derived from our framework is highly versa- tile and can be seamlessly integrated into a wide range of weakly-supent supervised learning settings, consistently leading to performance improvements. Notably, our approach shows substantial gains in challenging set- tings with high noise and partial label rates, achieving improvements of 6.8% and 7.8% respectively, under- scoring its robustness and effectiveness in addressing real-world weak supervision challenges.
By modeling semantic similarities using the provided weakly-supervin information, our method is highly ver- satile and applicable to various weakly-supervised learning scenarios, including noisy label and partial label settings. The contrastive loss derived from our framework is straight- forward to implement, stable during training, and can be seamlessly integrated plug-and-play component, con- sistently enhancing the performance of existing methods in weakly-supervised learning.
Our theoretical analysis offers rigorous performance guarantee for the proposed framework, showing that it can approximate supervised contrastive learning un- der mild conditions and elucidating how and to what extent greaklysuppervvn information can enhance the effectiveness of representation learning.
Additionally, we provide a comprehensive theoretical analy- sis of the method's performance, deriving the downstream error bound for feature learning using our contrastive loss under various weakly-superviser settings. The analysis shows that, under mild conditions, framework can ap- proximate the performance achievable with sully-sue information, thus theoretically ensuring its effectiveness.
2. Wcakly-Supervised Contrastive Learning
We describe the setup and goal of weakly-supervised learn- ing. Let X' denote instance space and y = 1 .......,, v'} denote label space. D denotes the joint distribution (X, Furthermore, given = (q'.42....,. notes weakly-supervised information space. We de- note De the joint distribution (X, Q) which gener- ated form D with process De(x) D(x), Delg x) T(x)D(y D(y | x) denote [De(g" |s),..,, |=)]] and [D(y* xx),..., D(y x)] respectively and T(x)
de-
In summary, our main contributions are summarized below:
z), where Do(a x) and
We propose a unified framework for contrastive learn- ing that leverages cluding noisy and partial labels. By introducing the
information, in-
2

---

Weakly- Contrastive Learning for Imprecise Class Labels
Ruxe is the transition matrix with T5 Dely' ,,y"). In this paper, we consider two popular weakly- supervised settings with different data generation processes, namely restance-indepeddeer and former satisfies the condition of T(x) is independent of the samples, whereas the latter does not. The objective of learning with weak supervision is to derive a multi-class predictive model h from weakly-supervised training set ". For brevity, denote P
However, in weakly-supervised learning settings, the afore- mentioned edge weights cannot be directly derived due to the absence of true labels. In the next subsection, we pro- pose framework that leverages weakly-supervisee infor- mation to establish additional connectivity, thereby enabling weakly representation learning.
The
2.2. Representation and Semantic Similarity Matching
D = {(x;, ]:)] probability of joint distribution (X 2) induced by 'D and Dg in the reminder of this paper.
The main idea of our framework is to approximate w in Equation 3 using weakly-supervin information. The following proposition demonstrates the existence of such an approximation.
2.1. A Graph-Theoren View
Proposition 2.1. For any S X Rex that satisfies the condition: IP(y xx) S(z) I x) holds almost everywhere in X. the following equation holds:
To investigate representation learning in weakly-supervised settings, we adopt a novel view of contrastive learning (HaoChen et al., 2021), framing it as graph spectral cluster- ing problem where data points serve as vertices and classes correspond to connected sub-graphs. Preliminarily, we con- sider representation learning under supervuud condition by modeling an augmentation graph G where the vertices represent all augmented samples, and the edge weights are constructed based on self-supervised connectivity (Chen et al., 2020; HaoChen et al., 2021):
w 22 (S) A B(..D..F
[S ((0, () (7', (1 (') A(x ()) [3)] = w a,a''
where S ( (), (7) 77) ( S(
Remark: Proposition 2.1 reformulates the goal of ap- proximating supervised connections w/y-. using weakly- supervised information into constructing an S that satisfies specific properties. In the next subsection, we present meth- ods for constructing such S that meets, or approximately meets these properties. Furthermore, S ((ž, q). (H', ,')) in Equation 4 can be viewed as estimation of semantic simi- larity between two samples according to the corresponding weakly-superrvise information. In this context, the graph in Equation 2 is discrete, edge existing only if they belong to the same class, while its approximation Equation 4 rep- resents continuous graph, where the connection strength corresponds to the estimated semantic similarity. The proof of Proposition 2.1 is in Appendix B.1.
Wx,x"' (E [A(x ()A(x² {)],
where A(. i) denotes distribution of augmentations such as Gaussian blur, color distortion and random cropping from natural data ă E X.
The optimal features are obtained through graph spectral clustering of the given graph, where dense intra-class con- nectivity plays key role in ensuring good clustering proper- ties (HaoChen et al., 2021; Sun et al.. 2023). However, when the graph is constructed purely based on self-supervised con- nectivity, the assumption of dense intra-class connections implies that most distinct data points within the same class should share same augmented samples, a condition that may not hold in practice.
Based on the approximate w (S) provided in Proposi- tion 2.1, we now present a unified framework for weakly- supervised representation learning. To this end, we begin by providing a formal definition of the perturbation augmenta- tion graph.
Therefore, it is natural to consider incorporating additional intra-class connections by leveraging label information to further improve the quality of the graph, as follows:
Definition 2.2. (Perturbation augmentation graph) We refer a graph where the vertices represent all augmented samples and adjacency matrix A constructed by perturbation con- nectivity as perturbation augmentation graph. Specifically, giving two augmentation data points xr, x':
(2)
- = 0') A(x A(x 1 [8),,
where I[ [+] denotes the indicator function.
A = -ww" + Bw (S).
(5)
With the additional graph connectivity defined above, we can construct improved augmentation graph with perturba- tion edge weight:
Let F be an embedding matrix with Fz.; E Rd is embed- ding of sample x, we consider optimal features derived from the graph spectral clustering of the given such graph in Definition 2.2 (Chung, 1997: HaoChen al., 2021):
aouik Self-supervised
Bw Supervised.
(3)
F² = arg min ||
A F FF
(6)
3

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
where A is normalized adjacency matrix of A with A
Algorithm 1 The calculation of Lusse in uniform class dis-
A, d A2 Ex'' An,z'.
tribution case
VA.A_
Input: features of two augmentation views of batch data with corresponding weakly-supervised information X Xg. X E BR xd Q E Q features of two augmen- tation view of batch unlabeled data X X Bu Buxx xd S in Proposition 2.1, coefficients az Output: batch data estimated loss Luse
Now, if we view each row of F as scaled version of learned feature embedding f X + R the above optimal features can be obtained through minimize an end-to-end contrastive learning loss. We formalize this connection in following proposition.
Proposition 2.3. We define Fs; = -VAsz and weakly- supervised contrastive loss Lune as follow:
Compute (X) E R ((X) :. x (X 2
(z)),
Compute م Tr [ g (x
④)
+T BQ By
Compute C2 [|((())' S(X)
& (x(x))
(f _ -f) 2ac:(()-2862(1
B
+ +a²Rz())) + (()() + 20)BLs(1)
Take X = [x2 Qi U1
Xi)
X²²
x Q:
x
where
Compute Lg
(Bq+Bu)*
Return: Cwee -2aL1 -28L2 + +a B//)
Li(S) A B [1,22]
C2(f) A B,00 [S((E,)), (7' (f'
as follows: Li and L2 push the features of positive pairs to be closer, with L1 pulling closer features of two views of the same image, and La pulling closer features of different (10) images with different weights which equal continuous ad- (11) jacency matrix (X)) in Algorithm 1 of the sampled sub-graph. L3, L1, La are regularization terms to prevents feature collapse. Finally. Luse depends solely on S, hence it can be adapted to multiple seakly-supervised setting by (12) constructing the corresponding S. The proof is provided in Appendix B.2.
(a(l) A Px (
(a()) A (1.4).1)88011..10010.
[s' (Fa
Cs(f) A B(a.))-((0)))
(5 (fa
where S ((t,)), (F (( Then the following equation holds:
S()))
2.3. Instantiating under Various Weakly Information
In this paper, we consider two typical paradigms of weakly supervision: noisy label learning (NLL), partial label Iearn ing (PLL). For noisy label leaning, we have Q = y but mis- takes exist D A Iexe) > O. For partial label learn- ing, we have Q = 2 \ 0 and DD q x) 1, Va X. In the following text, we show how to instantiate our frame- work into these two settings.
A - 11-FF") || = (f + Const.
(13)
Furthermore, when assuming uniform class distribution P(y) = [_..... _/7 LJ) can be rewritten as:
Reviewing Proposition 2.1, a good S ought to satisfy or ap- proximately satisfy the condition Pyy | x) = ((z) x)) In this subsection, we provide methodologies for construct- ing it in various settings, thereby completing the instanti- ation of the framework. Next, we will separately discuss mustance-independee and instance-dependent settings.
B (fs a -2aLi(f) - + (a (a+?) =)
(s(
(14)
Remark: By the Eckart- Sontt--ouug--isss theorem (Eckart and Young, 1936), F in Equation 6 consists of the top-d eigenvectors of A. This property enables clustering perfor- mance analysis via the connectivity of A. Proposition 2.3 outlines an end-to-end method for training neural network to approximate these optimal features. The pseudo code for computing Luse in Equation 7 with batch data under a uniform class distribution is given in Algorithm 1, with the general version (without the uniform class assumption) provided in Appendix F. We can intuitively interpret Luse
Instance-Indeeeedent Setting, For this setting, we have instance-independent transition matrix T: there exists a T satisfies T = T(xx holding almost everywhere X. The following proposition illustrates that S can be constructed by estimating the transition matrix T.
Proposition 2.4. Under instance innnnceeeddeeeddeett assumption, we have the sufficient condition for S = 5, Vr X to
4

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Weakly-Supervised Contrastive Learning for Imprecise Class Labels
satisfy condition in Proposition 2.1 as follows:
2. Then given naturally data ã. E X, we ensemble the predictions augmented data and predict:
ST Lexe
(15)
h (i)
arg max (2-1(-))) [I [1), i]] (18)
iEy
where Texe denote the identity matrix of size e x c.
We define linear probe enror of representation f as follow:
[I [ + s]] (19)
s( =
min ((..))((x))))) BE
Proof. When S satisfies the above condition, we have fol- lowing equation holds almost everywhere in X::
Regarding the given constructed S and training set D De J Du. where De is set of training samples with corre- sponding weakly-supervise information and Du is set of unlabeled samples, we learn f through:
(q x) = ST x) = P2 x).
(16)
F(D) = arg min (f),
(20)
Inspired by Proposition 2.4, we can use any matrix S for Swee that satisfies ST = where T is an estimate of T. In the noisy label setting, methods such as Liu and Tao 2016; Patrini et al. 2017; Xia et al. 2019; Li et al. 2021; Lin al. 2023 can estimate T, and S is its inverse. In the partial label setting. T E 1)xx is hard to estimate due to the curse of dimensionality, but with the assumption where labels are independent adopted into the candidate set (Cour et al., 201 1; Lv et al., 2020; Feng et al., 2020b), T can be estimated by e times of mixing proportion estimation (Garg et al., 2021). and S can be derived through pseudo-inversy or constructive methods. See more details in Appendix F.
where (f is the estimate of (f) using data De and Du with constructed S.
In this section, we examine e ef(()) through three steps:
We consider e(f"), where f derived from the spec- tral clustering of the perturbation graph A.
We consider the gap between (wsc(f" and Luse (fD)). Specifically, such a gap is caused by two aspects. On the one hand, there the estimation error resulting from the limited samples. On the other hand, the constructed S cannot fully satisfy the properties in Proposition 2.1.
Instence Setting. In this setting, a universal transition matrix T applicable to all samples does not exist, thereby rendering the methods discussed previously ineffee- tive. However, according to Bayes theorem, we have:
We control e (f()) e(J*) through ((D)) (fsc(f*) to obtain the minimum downstream error.
(y x))
Σ P(u.q q.uqll x) = Σ Ply x. (q x)) REQ
3.1. How Can Supervised Information Help Representation Learning
NEQ
(17) Thus, we can construct S as S(x) = Pply x. () where P(v 2. q) can be any estimation of (yy | q). Specifically, it is obtained through the current predictions of the model. This process is also called self-labeling and has been widely discussed. Specifically, in the context of noisy labels, related studies include Liu et al. 2020; Li et al. 2022; Ortego al. 2021; Liu et al. 2022; Xiao et al. 2023; Qiao al. 2024; for partial labels, related studies are conducted by Xu al. 2021; Wang et al. 2022; Wu et al. 2022; Zhang et al. 2022;
In this subsection, we examine e(f*) where f is de- rived from graph spectral clustering of A with entries Au, ow"".. + (wwt The two properties of a graph serve as crucial elements that have significant im- pact on performance. Specifically, these properties are the density of connections within a class and the sparsity of con- nections between classes. The following several definitions respectively characterize these properties of the graph.
Definition 3.1. (Data points from the different class hardly share augmented data) We refer data augmentation A as "-consisteet under joint distribution ((X,))) if there exist a pseudo labeler j(xx) for augmentation data such that:
Qiao et al. 2023a; 2024. See more details in Appendix F.
3. Theoretical Understanding
E ((x.)))
[I [5 = [(x)] S 7.
(21)
We now present a theoretical analysis of the performance of features learned from weakly-supervisee data. Following the traditional theoretical framework (Chen et al., 2020; HaoChen et al., 2021), we use the linear probe to evaluate the performance of representation learning. Concretely, we use a linear classifier with weights B E dxe and predict h (x) = arg maxiey B.T B,, for an augmented data
Small y in Definition 3.1 implies the sparsity of connec- tions between different classes in A" which constructed by self-supervised connectivity. Next, to formulate density of connections within a class, we introduce Dirichlet conduc- tance and sparsest partition which are standard in spectral graph theory.
5

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Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Definition 3.2. (Dirichlet conductance) For graph ((X,, w) and subset 52 C X, we define the Dirichlet con- ductance of 52 as:
3.2. How Weakty-Supervised Information Approximates Supervised Information
In this subsection, first consider the gap between Luse (fD)) and (fss(f*). ((D) an optimal feature that minimizes empirical loss [f D) defined as:
(92) Exer War
(22)
Ezen We
Lusc (f: D) = X2-
Definition 3.3. (Sparsest partition) For graph ( w) and an integer i E 2. |2]], we define the sparsest i-partition:
[Eusc(t: X x x S.Q)
(25)
Ps = min max(#g(S)))...... Lg(())) 01....
(23)
where S is constructed to approximately satisfy the property in Proposition 2.1 and Lwee in expectation is computed through Algorithm 1.
where .I...., Qt are non-empty sets that form partition of X'
To this end, we decouple Lmse(f(D)) Lusee - into two parts, the finite sample error and the approximation error caused by the constructed S. To analyze the first part, we introduce the Rademacher complexity which is a standard concept in generalization error analysis.
Intuitively, Dirichlet conductance represents the fraction of edges from f to its complement, and sparsest partition rep- resent the number of edges between many disjoint subsets. When sparsity of connections between classes holds, we might expect Pe Rs 0. Furthermore, any Pa where d > needs to break one class into many pieces, hence a large pd reflects density of connections within class. Giving such definition for two main properties of graph, the following theorem bounds linear probe error of feature derived from perturbation graph.
Definition 3.5. (Maximal possible empirical Rademacher
complexity for feature extractor) Let JF be hypothesis class of feature extractors from X' to R4, for n E Z+, we define Maximal possible empirical Rademacher complexity R, (F) for feature extractor under X' as:
1 n n
Theorem 3.4. Given augmentation graph 9 with pertur- wl bation adjacency matrix A aA" " + A." (S) where S stratify properties in Proposition 2.1, if A 7coosisstent with y >> O and representation dimension d > 2c, we have:
(x) max max /E|d] [n)
Eg
sup fer
Σ
(26)
where a = fol,..., Fn} are n Rademacher variables with o; independently uniform variable taking value in (+1.-1}.
e(f*) (0)=0(1 07 + (1- 2c
)
-2
201
(24)
To analyze the second part, we introduce expected bias for approximated S.
where pa is sparsest partition of self-super connectivity graph A", and ax a+37e 3
Definition 3.6. (Expected bias for 3) We defined expected bias ASS) under D(X)) as follow:
B/c
a+D
Remark: Theorem 3.4 bounds linear probe error of fea- ture derived from perturbation graph through properties of self-supervised connectivity graph and perturbation coef- ficients. Theorem 3.4 demonstrates that the perturbation graph increases the density of intra-class connections with- out adding extra-class connections, thereby improving the performance of feature. Specifically, when 8* in Theorem 3.4 equals to zero, the theorem degenerates into Theorem 3.8. in HaoChen et al. 2021 which bounds linear probe error of feature derived from self-supervised connectivity graph. When assuming Pay2 1 - 2r. bounds in Equation 24 decrease as 3" increases. Such a condition is almost always satisfied in reality, due to p < Õ(1 }) for every graph and density of the graph constructed using self-connections is usually much smaller than this upper bound (HaoChen et al., 2021). The proof of Theorem 3.4 is in Appendix C.I.
A ( = Ex-F(X)
[| [P x) (()) =) =)]1]] (27)
Utilizing the above two definitions, we have following theo- rem bounds the gap Luee((()) - (f*)
Theorem 3.7. Given perturbation coefficients o, 8 satisfy a + +3/c 3/c//c=1 1, and assuming ||F1 < +oc, for random training dataset D = De U Du with ny and n. samples respectively. Then with probability at least - 6, we have:
Lwac (e Lwee S (a 1)moR muy
Ranta (F)
+ (a + 1) + ng. 6) + ((5)((5)
+ 3 sup xEx
((x) (()) (I (F) + mun((nn 0)) (28)
0)).
6

---

Contrastive Learning for Imprecise Class Labels
where fjo "h, 122 Vs 1/1 are constants related to |F and feature dimension d and 17,n, 6) v 10g2/6 2'
Remark: Corollary 3.8 demonstrates that the performance of the features learned within our framework is determined by two aspects. Firstly, the quality of the clustering structure of the constructed augmented graph. Secondly, the error arising from approximating the constructed graph using fi- nite samples with weakly-supervised information. As 3 increases, the clustering structure of the constructed graph improves, but the approximation error also increase accord- ingly. This suggests that in practical representation learning, selecting appropriate 3 is crucial for effectively integrat- ing self-supervised and supervised information.
Theorem 3.7 demonstrates that in our framework, the features learned through finite samples and weak su- pervision information can approximate the features derived from the perturbation graph which incorporates supervision information and thus has been proved in the Theorem 3.4 to possess good clustering properties. Specifically, the ap- proximation error can be regarded as two parts. The first part, comprising the first and second terms in Equation 28, is the finite sample error when using all the training samples to approximate the self-supervised graph, which is derived through the standard finite sample approximation analy- sis. The second part is the error when using the samples with weakly supervision information to approximate the perturbed part. The third and fourth terms in the Equation 28 describe two aspects of this error respectively. The third term describes the estimation error brought about by the bias of S which can be regarded as the bias term of the error. The fourth term describes the approximation error due to limited samples and can be regarded as the variance term of the error. supzex (z))((z)) illustrates that the presence of weakly-supervised information exacerbates the learning difficulty, even when an unbiased estimate of S is utilized. From this perspective, the two terms can be regarded as the trade-off between variance and bias. The proof of Theorem 3.7 is in Appendix C.2.
Remark:
4. Experiment
In this section, we empirically validate the efficacy of our framework across two paradigms of weakly-supervised learning, namely noisy label learning (NLL), partial label learning (PLL.). For implementation, our weakly-supervised contrastive (WSC) loss is combined only with the simplest baseline. Detailed implementation can be found in Ap- pendix E.
4.1. Learning with Noisy Labels
Dataset and Experimental Setting, For the case of noisy label learning, following the settings in Li et al. 2020; Liu et al. 2020, we verify the effectiveness of our method on CIFAR-I0 10 and CIFAR-100 (Krizhevsky, 2012) with two types of label noise: symmetric and asymmetric. Symmetric noise refers to the random reassignment of labels within the training set according to a predefined noise ratio. In con- trast, asymmetric noise is designed to replicate real-world label noise, where labels are replaced exclusively by those of similar classes (e.g., dog <> cat). In order to fully ver- ify the effectiveness of the proposed method, we select ten baselines for NLL comparison under the same experimen- tal setting: DivideMix (Li et al., 2020), ELR (Liu et al., 2020), SOP (Liu et al., 2022), GFWS (Chen et al., 2024) and ProMix (Xiao et al., 2023), which do not incorporate contrastive learning, as well as MOIT (Ortego et al., 2021), Sel-CL (Li et al., 2022), and TCL (Huang et al., 2023), which leverage contrastive learning module. Due to space limitations, the implementation details, along with addi- tional comparisons the instance instanceddependett dependent noisy dataset CIFAR- (Wei et al., 2022) and the realistic, larger-scale instance-dependen noisy dataset Clothing IM (Xiao et al., 2015), are provided in Appendix E.
Theorem D.7 in HaoChen et al. 2021 characterize the er- ror propagation from pre- pre-training the downstream task, combine with our Theorems 3.4 and 3.7, we can obtain final e(((D) as follows:
Corollary 3.8. (Main results: linear probe error of feature learned from our framework) Given representution dimen- sion d > 4r, and a, 3 satisfy a _ = 1, A is 7-comsistent with 7 > 0, then with probability at least 1 - 5, we have:
<(f(D)) s δ 7 a [T]
00
d 2²) )
+ 2c
+ (a + 1) ( (F) + viv((1,0) 0)) + ((S)
[(t) SS() ((x) 11 (16 4p (F) + rhnn(ng 5)
+ 3 sup
=EX
(29)
where Wao Th 1/2 1/2, v/dl 27 Ino ah 72, 73, Mul with Ax
the eigenvalue gap between the 3 d-th and the d-th eigen- value of perturbation graph, and ny + ng.
Experimental Result. Table shows the classification accuracy for each comparative approach. Most of the ex- perimental results original paper, except for ProMix, which we reproduce due to differences in our settings. The proposed method demon- strates competitive performance across various settings and datasets. It outperforms existing approaches in most sce-
the same those reported in their
Proof. By substituting Theorems 3.4 and 3.7 into Theorem D.7 in HaoChen et al. 2021, the corollary can be directly obtained. The restatement of Theorem D.7 in HaoChen et al. 2021 can been found in Appendix C.3.
LI
7

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Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Table Comparisons with each methods simulated NI.. datascts. Each has heen three times with different randomly generated noise, and the mean of the last five epochs are reported.
Dataset
CIFAR-10
CIFAR-100
Noise Type
Sym.
Asym.
Sym.
Asym.
Noise Ratio
0.5
0.8
0.9
0.4
0.2
0.5
0.8
0.9
0.4
CE
80.70
65,80
42.70
82.20
58.10
47.10
23.80
3.50
43.34
DivideMix
94.60
93.20
76.00
93.40
77.10
74.60
60.20
31.00
55.57
ELR
94.80
93.30
78.70
93.00
77.90
73.80
60.80
33.40
69.94
SOP
95.50
94.00
93.80
78.80
75.90
63.30
69.53
GFWS
96.60
94.12
84,22
94.75
77.49
75,51
66,46
45.82
75.82
ULAREF
94.31
91.47
92.56
76.16
72.39
54.72
76.11
ProMix
93.23
83.11
89.83
75.43
71.64
43.35
72.13
MOIT
90.00
79.00
69.60
92.00
73.00
64.60
46.50
36.00
71.55
Sel-CI.
93.90
89.20
81,90
93.40
76.50
72.40
57.70
29.30
74.20
TCL
93.90
92.50
89.40
92.60
78.00
73.30
65.00
54.50
73.10
WSC (Ours)
95. 19
94.62
90.93a0.14
95.22a0.99
79.21
77.51
71.92au.17
61.32an.15
76.31s0.32
Table 2. Comparisons with each methods on simulated PLJ. datasets, Each run has heen repeated three times with different randomly generated partial labels, and mean and standard deviation values of the five epochs are reported.
Dataset
CIFAR-10
CIFAR- 100
CUB-200
Partial Ratio
0,5
0.6
0.7
0.8
0.05
0.1
0.2
0.3
0.05
LWS
85.3020336
80.33
72. 20..8
58.4920.33
24.78.0.2
50.4420.38
32.4420.45
25.4920.58
39.7420.43
PRODEN
89.82+0.3
87.44+0.35
86.44+0.0
85.78+0.55
72.65+nm9
71.05+025
68.4440.9
55.21 +0.44
62.56-0.1
CC
82.30+0.28
80.08+0.77
77.15+004
75.94+0.8
63.7444 14.744
57.55+0.19
10.41+047
40.28+0.29
55.61
MSE
75.47.0.85
23.6410.28
69.0940.67
64.32.0.33
51.1710.25
47.3320.94
42.5510.33
20.19
22.07.2.36
GFWS
95.22+0.08
95.0
94.21
93.58=0.21
76.89+0.32
75.95+0.10
18.18
60.25+0.37
70.77=0.20
RCR
95.01
94.37=0.07
93.28sa.05
91.67=0.10
77.01
75.85
72.58+0.31
57. 97.48
PiCO
94,63s0.21
94.25
93.6810.06
92.01
74.192028
72.7420.4
70.89
61.3520088
72.12.0.4
WSC (Ours)
95.4110.08
95.22:0.10
95.0310.04
94.41
77.88x0.19
77.26x0.30
75.13
69.15
74.55
narios, including those based on other contrastive learning techniques. Notably, our method shows significant improve- ments under high noise rates. For instance, on CIFAR- 100 with 90% noise rate, it surpasses the previous best TCL by 6.85%, highlighting the substantial performance gains our framework offers for NLL.
2024), which do not incorporate contrastive learning, as well as PiCO (Wang et al., 2022), which employs con- trastive learning module. The implementation details, along with additional comparisons using different partial label ra- tios on CUB-200 and the hierarchical-genae -ennrrtte partial label dataset CIFAR-100-HE (Wang et al., 2022), are provided in Appendix E.2.
4.2. Learning with Partial Labels
Experimental Result. Table 2 shows the main results for PLL. Our proposed method achieves the best performance across almost all settings compared to the bascline methods. It is worth noting that our method achieves a larger perfor- mance gap among the previous methods when the partial rate is large. Especially CIFAR-100 with partial ratio of 0.3, the proposed method outperforms previous best method by 7.8%, %, which is a strong evidence that our method can achieve better results on these more difficult datasets.
Dataset and Experimental Setting. For the case of par- tial label learning, following Wang et al. 2022, verify the effectiveness of our method CIFAR- 10 (Krizhevsky, 2012), CIFAR-100 (Krizhevsky, 2012) and CUB-200 (Wah et al., 201 2011b) Ib) with different partial ratios. The partial label datasets are generated by flipping the negative labels to can- didate labels with a specified partial ratio. In other words, all negative labels can uniformly aggregated into the ground truth label form the set of candidate labels. We choose seven baselines for PLL using same experiment setting for comparison: LWS (Wen et al., 2021), PRODEN (Ly et al., 2020), CC (Feng et al., 2020b), MSE (Feng et al., 2020a), RCR (Wu et al., 2022) and GFWS (Chen et al.,
5. Conclusion
This paper introduces novel approach for contrastive learning that incorporates weakly-supervised information
8

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Weakly-Supervised Contrastive Learning for Imprecise Class Labels
through graph spectral theory. Extensive experiments and theoretical analysis validate its effectiveness. We propose semantic similarity continuous measure of class mem- bership, offering new insights into utilizing ambiguous su- pervision in contrastive learning. Our framework has the potential to be extended to a wider range of scenarios, includ- ing bag-level weak supervision and multi-modal matching.
Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. In International con- ference on machine learning (ICML). 2020.
Fan RK Chung. Spectral Graph Theory, volume 92. Ameri- can Mathematical Soc., 1997.
Stéphan Clémençon, Gábor Lugosi, and Nicolas Vayatis. Ranking and empirical minimization of u-statistics. The Annals of Statistics, 2008
Acknowledgements
This work was supported by the National Science Foun- dation of China (62206049, 62176055), and the Big Data Computing Center of Southeast University. We would like to thank anonymous reviewers for their constructive sugges- tions.
Timothee Cour, Ben Sapp, and Ben Taskar. Learning from partial labels. Journal of Machine Learning Research, 2011.
Jingyi Cui, Weiran Huang, Yifei Wang, and Yisen Wang. Rethinking weak supervision in helping contrastive learn- ing. In International conference machine learning (ICML), 2023.
Impact Statement
Our work may have broader implications, including the po- tential for increased unemployment among traditional data annotators as annotation quality standards decline. Addition- ally, careful attention is needed to address privacy concerns, as using raw data from web scraping can effectively train high-accuracy models.
Ruo-Jing Dong, Jun-Yi Hang, Tong Wei, and Min-Ling Zhang. Can label-speeific features help partial-label learning? In AAAI Conference on Artificial Intelligence (AAAI), 2023.
Carl Eckart and Gale Young. The approximation of one matrix by another of lower rank. Psychometrika, 1936.
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Supplementary Material
A. Notation
Table 3. List of common mathematical symbols used in this paper.
Symbol
Definition
N.J (
}
Instance space and label space respectively
Q = (8
Weakly-supervised information space
D
Joint distribution on (x' y) for target pattern
D
Joint distribution on (X', Q) for weakly-supervised pattern which generated form D Transition matrix with T(r),.) = (4 0" x,y²)
T(x) E Roxe
D (T)
The abbreviation of vector [D(y* |x), [))...,, D(y x)]
Dolq
The abbreviation of vector (De 1,) De(a" x)]
P
The abbreviation probability on joint distribution (X 3/ Q) induced by D and De
Do
Set of training samples with corresponding weakly-supervise information
Du
Set of unlabeled training samples Final training dataset
D Do Du
g. Au (T, M), (F',7') 17' I, A A.... ))
Size of D, De. Du. respectively Two naturally data points with their label sampling form D Two augmentation data points form 전 and E' respectively Distribution of the data augmentation
Wazw w
The edge weight of the edge connecting the two augmented samples constructed The edge weight constructed through self-supervised connectivity The edge weight constructed through fully-supervised The edge weight constructed through weakly-supervisee connectivity perturbation coefficients for integrating self-supervised and supervised information. Adjacency matrix of constructed augmentation graph
wl girr
a,
A
À
Normalized adjacency matrix of constructed augmentation graph
S:X + gexv
The recovery matrix used for compute the semantic similarity of samples Computed semantic similarity samples with corresponding weakly-supervised information Constructed approximate recovery matrix
S (( (z',
S:X X + Rexv
F
Embedding matrix with Fu.: E Id embedding of sample Learned feature embedding Linear classifier weight
f:X- - Rdd
BE Raxe
hz,
Prediction for augmentation data with feature and classifier B Ensemble prediction for naturally data ø with feature f and classifier Posterior category probability give æ predicted by neural networks Linear probe еиrоr of feature embedding
(ī)
g(z)
e(f)
P
Sparsest partition of graph in Definition. 3.3
nd
Sparsest partition of self-supervised connectivity graph A"
no: .o.....7
Constant only related to F.. and features dimension
Ax
Eigenvalue gap between the [3d/4] Bd/4|-th and the d-th eigenvalue of graph A Proportional amplification of constant 7/o, 74
vo....,7 1/4
13

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
B. Proof of Section. 2
In this section, we provide the proof details the proposition in the Section. 2.
B.1. Proof of Proposition. 2.1
Proposition B.1. (Recap of Proposition 2.1). For any S X' - RRxx that satisfies the condition: Ply x)) = S(x)P(q x))
holds almost everywhere in X, the following equation holds:
Wa a (S) - (F ( () (6', (f', A(r A) [')] 2
(30)
where S
= S
Proof. The proof is direct, on the one hand, expand w and obtain:
w E. = -E
(tpiaaaaa-la-------- = 77] A(r A) x²)]
=/ E P(C. / A(r A))(" #')dadr'
P(t)P(E didd'
Σ
F( 2)P(() ID = AT ())((a() (2)
U.g'eya
(31)
(E)P(i')ddd T) A))(z a)A(3' (8)
JEY
P(3)p(E'dddd (P(y Pa) E\)) A(x ())((' (5)
Ea [Ply Ply d')A(a a)A(d' E')]
On the other hand, we can expand the ward and obtain:
(( ($..) A( 1E)((² |
I Σ ((E.))((' (# A( Z)A(d' (')did
- P(3)P(3)dddd Σ P(ğ () () ())(sa (1)
(32)
/ ((O)(())ddd 2) (() P(g F)A(() ()A(z² (T)
= E2
[P ;)7 (((T))((EE) P(q a A(= A))( {0)]
When S satisfies the condition of (Pyl z) = ((z) x) holds almost everywhere in X, we have follow equation holds almost everywhere in 22:
P(g T ((E)) Pg 3' (S a)) (S(*') E¹)) P T x¹)
(33)
Combining Equations 31, 32, and 33, we finish the proof.
B.2. Proof of Proposition. 2.3
Proposition B.2. (Recap of Proposition. 2.3). We define Fz.: V Vala and weakly-superviseed contrastivee loss Lusse as follows:
Luse -f)Li(f) - 23C2(f) + Lz()) --0-0010000000000000-00--------0 + 2o.
(34)
where:
L (T( A FB- [faf]]
(35)
14

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
C2(J) - Buggl/eapperrr......xk.......... ( (s [fN
(7)
(36)
Cz(f) - Ba (faft)
(37)
Cx(f) & B(E.0. 112.0 (1 (fafz)
(38)
(s()) - -,.00
(5(() (fatt)
(39)
where S ((E,q), (F) S(a) s()) S -
Then the following equation holds:
A - FF" L + Const,
(40)
Where Az,z' = -ow" + fwoy
a,a and A
A.. VA VA.
Furthermore, when assuming uniform class distribution P(y) ....... 21" above Cwee can also rewrite as:
(a+2) Lz(s).
(41)
| FFi
Proof. First we can expand
as follows:
FF
=
VANVA Ar - FF...
Ar, -F,
2
F
2,z
F Fr. 1/; ))²² -2 VA.VA, Ag,
Asx F F + Const
(42)
Σ A,Ax (Ja1z)² 2 Sx Const.
Next, We analyze the two terms in the RHS of Equation 42 separately. We first expand the second term RHS of Equation 42 as follows:
Σ 2A
-2
2A
faft
Σ
2aw
Buuld Doff
n,z!
20
Ee A)A(I' [a)ffff
23 Σ
()))xx)) (S ((z,g) (8', g')) A(z 2)A(d | faft
(43)
-2aaExF(X))
A |()<\>\\<>' |8)faf
2) Σ S(E s)A(d 18)sa
--2aEz-F(X) E a) [a
- (0 (S ②) (#', (')) Be-AA1B) [faft]]
By substituting the definitions of Li( and C2(J) into the Equation 43, we have:
Σ Saf - 23Cc(f)
(44)
20.0
15

---

Seakly-Superrised Contrastive Learning for Imprecise Class Labels
On the other hand, we have:
A
Σ An = a
Σ
w² W2,s'' + Σ
w a
ul
= Σ A(e 8)A(((( (3)
s (E..)) ( Σ
A))((=) 18)
(45)
= [A(x x)]
+ -)
Σ S ((0, (#', a²'
(#' (2), A(r 1))((a (0)
By the definition of ((#.0) 9). (#', a))), we can rewrite the second term of RHS of Equation 45 as follows:
Σ S ((t,)) (3',)) 2))(z² |x
A(x 2)S( 1
((()))(()))))) E')l Σ A(d' A(d'1*')
(46)
[A(z ())( K (P ||)||
((2,0))(Xx.2) [A(z] ()) P(D)
((x [S ()x a)
where we defined () = S P(y) before. By substituting Equation 46 into Equation 45, we have:
A, = aB P()) [A(z 2 + 3 (x.0))x)))) [S" () []].
(47)
Using results of Equation 47, we can expand the first term in RHS of Equation 42 as follows:
EA.AN Ar (JaST)²
(aBz)z()) [A(x a)] + 3 F (7. E))
(aE (A(c E')] + + (S I (&))) (faF3)²
= =a'Eg.e~P(39)
Σ
A(x A(}( (t)) (Jaf
2,
(48)
+ (
Σ a.a!!
1))((e) (o'd)z (fafz)
+ 20.
(' A(x | A) (8) (Jal2)²
A(<r
2,3
By substituting the definitions of Ca(s), La( and Ls) into RHS of Equation 48, we rewrite its three terms as follows, respectively:
a a² ('
Σ |
())((' (1)) = -. [ CoL)) (49)
PE S AA 1
B)A(E' (+')
(50)
1000:00PP000-00000000-000- ( (( (F'.7)))) ( =BP(a
16

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
20 2x (' ①) Σ A(x 8)A(d (3) (Jastt)
A(x 2,00
(51)
() (faft) = =f)
(a.0 -.
By combining the results of Equations 48, 49, 50, and 51, obtain:
A.A o C3
2a.Ls.
(52)
By substituting the results of Equations 44 and 52 into Equation 42, can obtain the Equation 40. Furthermore, when assume uniform class distribution Pyy) = [ _ above Li(f) can be rewritten as follows:
(a( E (. (S ( (
(( 8)
Σ
(( (8) (fusz)²
iEQ.I'Y
T'EY
(53)
((()) 8))B Σ
(( (')) (JnSZ
2
BEJ
Jeyy
(faftt = ca(l)
1
Similarly, (f) can be rewritten as follows:
La(s) = (. ()(t, (Ja17)²
Σ S(T) 8) (Jar+)²
EEQ.I'Ey
(54)
Σ (S(i)P(q x))₂ (fasz)) JEY
C (1
[foff) - -
=B -.
Combining the results of Equations 53 and 54, L.see (f) can be rewritten follows thus finishing the proof of Equation 41.
Lwse( -2oLi (f) 23Lx(l) ( (L(( + 203Ls(s)
(
9 2
a8
--2oLi (1) -3C2( + a2 +
+
Es(
(55)
=-2oLi(f 2322(f) - + (a+ B
fa(s).
17

---

Weakly-Supervise Contrastive Learning for Imprecise Class Labels
C. Proof of Section. 3
In this section, we provide the proof detail of theorems in the Section. Our theoretical framework mainly follow theoretical framework HaoChen et al. 2021. In order to eliminate technical complexity. we assume vertices of augmentation graph have be a finite but exponentially large set. This is reasonable assumption given that modern computers store data with finite bits the possible number of all data has to be finite, and results can been easier generalized the infinite through Theorem F.2 in HaoChen et al. 2021. Moreover, without loss of generality, we assume a uniform class distribution to simplify the proof.
C.1. Proof of Theorem 3.4
Theorem C.I. (Recup of Theorem 3.4) Given augmentation graph G with perturbation adjacency matrix A aAA + 3AAh (S) where S stratify properties in Proposition. 2.1, if A is c-coosistent with yy > O and representation dimension
> 2c, we have:
=ff) (1 + 8 (1 1 (1-2)
20 1-))))
(56)
where p² is sparsest partition of self-supervised connectivity gruph A", and a*
at 3 a+Bfcc
Lemma C.2. For any proportionality constant > 0, then linear probe error of features derived from A is equal to features derived from A.
Proof. Assuming that g" is features derived form AA, then through results of Proposition. B.2, we have 92 G define as follows:
G
where
vnAz
G = arg min n GG
(57)
Since g
DA
we have:
VnA.VIA.
VALAA.
G
arg min À GG
F
(58)
Hence features derived form 7 can be rewritten as:
G
F
fz V1j
g VnA2
(59)
V1AA
For any linear classifier B Rdxe we have:
1
h B (x) = arg max B " (x) = arg max B (x) h.),
(60)
TE
icy
ViI
In the end, we finish the proof with:
[I [h + r0]
[T [ [T BB) # b]] e(f))
E(g") min BeRaxe
((
min BeRaxe
(()))))()))
(61)
Using results of Lemma C.2, we can assume + 8 = in Theorem C.I (When + & 1, we consider perturbation graph with A B+8/C A "
3 A
a+B/c
We are now ready to formally prove the theorem. Our proof relies partly on the theoretical results HaoChen et al. 2021. For the sake of completeness, will restate some of the theoretical results in HaoChen et al. 2021 in this section.
Lemma C.3. If A is consissent augmentation distribution, and ((x) corresponding pseudo labeler for augmentation data such that satisfy condition in Definition 3.1, then for any linear classifier B and feature f, we have:
((2,))~((),)) [T [hy. (T) # =0] < (BZ-E(X), [1 + [xx)]] + +7)
(62)
18

---

Weakly-Supervise Contrastive Learning for Imprecise Class Labels
Proof. For any that satisfy h (T) + D. claim:
Be ((),n(()) # al 2 2
(63)
Hence, we have:
((x,)) [ (.),(()) & b/] - (,, ( # ill
-2aa))) [1) (1 +(] + + (())))
< -(B [- + 5] + + # ((=)]]
3 2 BB [[) + /
(64)
Using results Lemma C.3, the problem has been transformed into analyzing Cxx [[),,(()))))))))))]))((())))())) + [x)]]. We proceed with the following analysis.
Lemma C.A. Given real symmetric matrix with size of N, and = d, N be the top-d smallest unit-norm eigenvector of L with eigenvalue (make them orthogonal in case of repeated eigenvalue), then for any vector p R there exists vector b RRd such that:
R(4)
hau
S
11/112.
(65)
dd+1
Y=1
where R(n)
LL p
is the Rayleigh quotient ofu.
Proof. We first decompose μ on the basis of the space spanned by the eigenvectors μ EE b;1;, then we have:
N Di= X
Ad+1 2i=d+1
by
Ad+1
R(x)
ba00
(66)
1
111112
|
Lemma C.5. Given the pseudo-label indicator function U define Un.i VA.1 (f(x) i] and feature F* derived form
perturbation graph A with perturbation coefficients and B satisfy B/C == 1, then have:
27
mün BERIXE
F'B
(67)
A4t1
where A4t1 is (d+1)-th smallest eigenvalue of normalized Laplacian matrix I - À of perturbation augmentation graph.
Proof. According to the Eckart- SosarttYuung--irsee theorem (Eckart and Young, 1936), F* contains scaling of the top-d smaller eigenvectors of I - A. Using results of Lemma C.4, we have:
min Be
UU F F'B2
min U0 Fbbll
-UT U. U A)U
Ad+1
1=1
Σ Amll(()) = - - A =
Xd41
i=1
(68)
Σ A = =in(()) [ 1 +
A4t1
20,00
Ae. + ((x')
Adt1 2,30
19

---

Weakly-Superiie Contrastive Learning for Imprecise Class Labels
Furthermore, we have:
Σ A + ;(x0)= = a Σ 2,
W MÍ(x)) # 0(x()] + 8 Σ
W Te (S)T((()) + j(z')
2,
2 (((()) + 0222)2 3 Σ - + ((22)
a PB
Σ T((()) + (a'))A( 1 (T)
Σ 110()) + OA 8))((') 18))
+ +.
< <aBaggP(X.)
Σ (1 + #D) AC (3)
+ -
Σ (1(((z) # 0] +10+')) 1リ 87 A(x 1))A(E( (3
+ 20
[T [x))]
- T'll
- + (1/)+//))+ = 27,
(69)
here the first inequality holds because:
[Í(()) + 0(''(()) S ()xx) + M +10(2')) # 5].
(70)
10(x) + ( 5/1 < (L)(2) + ग +
(71)
Finally, we substitute Formula 69 into Formula 68 to complete the proof.
Lemma C.6. Given the pseudo-label indicator function V define as V a, = [}(x)) = i] and feature (*(x) = vA. then we have:
- (F x. T
min Be
[1)).,B(z)) + [x))]] 2 min BER
VT.
z.: - B" B" fx)
4y
(72)
2
Ad+1
Proof. Firstly, we prove the first inequality in the Formula 72. For any hy- hy-,((x) + ((x) there exists y' such that (B" ((()), > (B" (x))))())) hence, we have:
VT. f²)
2 (1 (B" (())) + (BB f2))))
2
(73)
2 (1 ( f' (B" ( 2
(B"
VT - BT (x) and obtain:
Next, we can expand Eg-P(X) 20 M6 (8)
E2.P(x))
V B" r²
Σ P(x) PP(r)
VT
B² r 2
(74)
20

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
We claim P(x) Ag when a + 8/c= 1:
A Σ "a 3 Σ w
aEE=F(x)(A(1 (3) E [3]] (...
I = A))((1 (8) A(d'
A(x' ( 18
(75)
aLE [)] + -. 11 = A(x x)]
aP(x) + P(x) = P(x)
By substituting Equation 75 into Equation 74, we have:
=x- | BB (z)) 2
= As
Ax
VT z:
B" (x(x)
Σ
U 2i
B² (FZ.))
(76)
U F B 1
Using the results of Lemma C.5, we finish the proof of the second inequality in the Formula 72.
Combining Lemmas C.2, C.3, C.6, we have prove under condition Theorem C.1:
s(() 2 (Notn +1) 7,.
(77)
where Ad+1 is d + 1-th smallest eigenvalue of normalized Laplacian matrix of perturbation graph.
The following lemma establishes the relationship between this eigenvalue and graph connectivity, allowing us to get rid of the dependency on Ad+1-
Lemma C.7. (Higher-order Cheeger's inequality, Proposition. 1.2 in Louis and Makarychev 2014) Ler g = (V, E) be weight graph with [V] = N. Then for any t E [N] and Ç > O such that (1 + <)L E (N), there exist partition Dt,-.. Qn of V with:
( 1 poly (t) V AAL+s) logt, Vi [t].
(78)
In Lemma C.7, take 1 and (1 + C)t 2t d 1, there exist partition 2,,..., S [a/2] such that ma/2e[(4/2]] (9,)) I / log (d + 1). We take p as sparest partition perturbation graph. By Definition 3.3, have P[a/2] < maxig[[a/2]] (L,) Vdet1 log (d (d+1) which leads to there exist constant ry(d) such that Aati S (dd) [//2) The following lemma bounds the sparest partition of perturbation graph.
Lemma C.8. Given perturbation graph with Ar.z Wa,ar oow + Bw (S) where + B/c 1, and self- connectivity graph A with sparest partition p", then we have lower bounds of sparest partition p of graph A as
follows:
Pa 2 ap' + (1 9
(79)
21

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Proof. Firstly, for any subset S2 of augmentation data points, we have:
(91) = Exer. 40 Wa,x'
Exeruere aw" + Bw 2, (S)
Eren W2
Enen We
Exet 4n 000 n.a" + Bw'
Exer ow + Bw 2.x
Ezea Wx
P(12)
Enena "a.a
Enerxyen w
(80)
a
P(2))
P(2))
E Gxfl.x'09 En W
ExED, w/ a,
a
Exen W
c
Emen w/
= a A- (12) + 2 (1)),
here the 4-th and 6-th equality derived from Equation 75.
Hence have lower bounds of Pa as follows:
Pa = min max x (2)), 4 (
= min max Ar (22) + (.a + _ x Qu
(81)
2 min max 1 A~ (S4), ...., (1x))))) + 8 min max ((1))...,0..(( fa......
aa.....
8
= ap'
c Pda.
here the inequality holds because taking the minimum separately is always less than taking the minimum simultaneously.
In the following text, we prove that p'a 2 (1 - to finish the proof of Lemma C.8. For any subset Q we can compute t A: (52) as follows:
A (2.) = Exer whe,
Exererg "a,a" P(N)/c
ExEr Wx
Σ m gn I = 87 A(x 8)A(x' xz)
c
P( ū = c)
1=
o7
> ngn 1 = 7/ A(x ())((( E3)
Σ
P(N j 1-=0 c)
(82)
2 Ei B(2,29)
EvER A(x {))(d' [89]
E
P(2 1 = c)
Σ 1
E B
M 1)2
Exeox En A(r A))(' |8)
P( j=
E P(2 i) P( 1 = i))
PP 5 -)
For any partition .I...., Qg of augmentation graph, we claim that they must satisfy conditions as follows:
Ee P(R. ū= j) =1, 1, P(9 1 セー > 0.
Vjj [c] vi jd.
22

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Hence, when take P(N. i j) Wi, We have:
Ej Wis(1 - Whns)
p'a 2 min (w) = min max iEId
2j=1 Wis
S.t.
Wiy = 1, Ej
(83)
Σ j=1
W1, > 0,
eiel [d]
0 Wi 11
Vi [c], E [d].
We claim that the w' that minimizes the constrained minimization problem defined by Formula 83 should satisfy the condition: there exists i",, j = arg jwn. 20 wi, such that w = 0, Vj E [c] \ and ('w)) = 1 w
To view that, one the one hand, for any break this condition, we can construct w' such that v((w)) < v (w) as follows:
0, wh.g- Wi." + we w/ whos Wadi otherwise,
(84)
here we assume W1," > O. order to verify Vi ((()) < M(ww,, we first notice that:
w/- Ejj wir, - wi Ej=1 wi wi - -wre))
(85)
Ejer Weit
Ej=l Wir
here the inequality holds because for any j jr, we have Win S Wi Furthermore, we also notice that:
Σj= "1,5 w(
Ejtr W1,(1 w1.s) + (wisg" + wi .t )11 - W1,j" wi wir
ビリ= w/.jj
Ejes" Wis + W1." + w
(86)
Ener Wln wi.)) + win' (1 1 w1..) Σم Wl,1 (1 wh,s)
Ejer Wi,j Wi,j
Σ ンزZز ررب
Combining Formulas 85 and 86, we have V(w)) ((W)..
On the other hand, when the condition is met, for any i + it, we have:
Σ Wi,s (1 Wir))
S 11---1;- -Wi+
(87)
ΣØ= Wi.j
here the inequality holds because for any j with Wij # O, we have 1 - Wi.j < 1 Wi J
Utilizing the above results, we can rewrite the constrained optimization problem as follows:
min 1 - Wi-
w
s.t.
Σ Wii =1, Vjelc [c]
1=1
(88)
Σ Wij > 0, Vi [d] j=1
0 w.j 1, vi [cl.j e [d]
Wir = Vj \
where i" = arg ming Wiy- Furthermore, for any w that satisfies the constraint of the above Formula 88, we have c= Ein Whit 2 d= through there are at least d non-zero elements in wi. for any E [d]. Substituting this result into Equation 83, We have proved that P/ 2 1 द. therehy completing the proof.
23

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Finally, we combine the results of Lemma C.8 and Equation 77 to complete the final proof of Theorem C.I.
Proof of Theorem C.I. By substituting Formula 79 into Formula 77 using Lemma C.7, we obtain the following bound, thereby completing the proof of Theorem C.I.
4_ x==+1) v/d) 7
{(*) 2 Aat1
Pial2j
(89)
S n(dd] (7 (7( + _ -1- - 20 /,)
-2)
C.2. Proof of Theorem 3.7
Theorem C.9. (Recap of Theorem Given F, assuming || < toc, for random training dataset D with ny and nu samples respectively. Then with probability at least 1 8, we have:
Do
log 2/5
8
Lwwe ((D, Luss( S (a + (a + B/c)3)
n1 ton (F) + +m
+
By + nhu
2
(90)
log 2/5 8 1)
sup =
((x)))(())
R2 7 (F)
B1)AA(S),,
where 7/u 3 Rd + x2d?: 1 = 3 Rd. m3. 7/4 3 R2d are constants related 11 feature dimension d.
Definition C.10. (Empirical weakly- supee--sueer spectral contrastive loss under uniform class distribution) Given constructed Th
S X + Rexn and considering training dataset with corresponding veakly-supervised information De {(1ra, q)); and training dataset without any supervised information Du {(=:)] "t }. we define empirical weakly-supervised
spectral contrastive under uniform class distribution follows:
Emso DD = Exx X (2) XX- [EuooXX] X
x2 x x2 (,,3)
tuttn
ne + nn The +ig
= Eeg A(
- 2a
f(t)){(a))T + +a+ 3/c)²
Σ
Σ
((()))(22))))
Thu 14g
(nu + mn)
1
ng
12g
+ +
-28
(xx)).
12
j=1
(91)
Giving F(D.S) mig Ewsc (f: D) in Definition C.10. The follow lemma decompose the error bound of Lowsel ((D,S) (f*)
Lemma C.11. We define Ewse S) expected weakly-superrised spectral contrastive loss using constructed S, then huve:
Luuse ((D,)))))))) - Lwse((") < sup fEF
(Zuss(J; D, - (:)) + 2 sup
(): - E.wscls) (92)
SEF
Proof. On the one hand, we have:
(Cusc(f(D,S)) Lwoelft)
((D))((D) (f" :3))
Lusee ((D,5)) - Emsc( (D, S); 3) + [maal [mso(
(93)
< sup feF
Euse(f;
(f: 5) - Lweelf)
24

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
On the other hand, we have:
((D, - L(,c 3
+ Eweel (FD.) D, (f D, D) + Ewsclf (f: D, () (f
(94)
< 2 sup
(Cuse D, S) [f)
FEF
Combining Formulas 93 and 94 completes the proof.
We first bounds the first term in Formula 92. It direct to see that (f; D.S) an unbiased estimator of we (f:; S), make of the Rademacher complexity theory given generalize bound of this term, we convert the non of i.i.d pairwise function to sum of i.id form by using perturbations in U-process (Clémençon et al., 2008).
Definition C.12. (Sub-sampling for empirical supeevisd spectral contrastive loss) We sample tuples to calculate the empirical weakly- weakly-supervised spectral contrastive loss as follows:
,t,0, A(- 1x,), i= .....,,.. My + nbu
(95)
2 n/2
Dwsc({:DDS)
= 2a
Σ
fx + (a (a+ 3/0)2 Σ (1E)))(
2 ng/2
(96)
2,3
S ((x)) (1144a12) { (x)) (2
1=1
Cu (f: D. S) + (f:
The following lemma reveals the relationship between the Rademacher complexity of feature extractors and the Rademacher complexity of the loss defined on U-process sub-sampling.
Lemma C.13. Let FF be a hypothesis class of feature extractors from X to R. Assume || and let Rr(F) be maximal possible empirical Rademacher complexity for feature extractor in Definition 3.5, then we can bounds empirical Rademacher complexity on sub- uub-sampling samples as follows:
n/2
Eg
sup fEF n12
(dkRR//((F)
(97)
n/2
Ea
sup
Σ (H(c))((((tm))))
< 16d x2 (F)
(98)
SEF
11/2
n2/2 /2
Eg
sup FEF
0 (xx). (, (a)
sup zet
((x) 8dkRn,,2(F)) (99)
nny 2
Proof. We first proof Formula 97 as follows:
1 n/2
n/2
Eg
sup |EF
Σ (.((())))))))
E, sup HEF n/2
1
Σ o Σ ((DT(()
n/2
(100)
n/2
d max max E, # at sEld]
sup f,FF, 12/
Σ (a
25

---

Weakly-Superviee Contrastive Learning for Imprecise Class Labels
For any ut, un and j. use Talagrand's lemma bound the end term of Formula 100:
n/2
Eg
sup F 12/2
Σ (z
1
1 (
(1)(z)) (
-((3)))((()))))
E,
n1/ i=1 E
sup
di
(101)
n/2
n/2
EE,
sup $YEP, n1/ i=1
1
Σ o5 (fz(z)) fz(x3))²
+ +
sup
Σ
o. (fz(zt) -
EEEP n/2
i=1
n/2
n/2
a=55(1?)) 1)
1
S 4k
sup [yEF,
+ +R
sup f5EF,
n/2
n/2
Combining results of Formulas 100 and 101, we can get Formula 97.
Next, we also use Talagrand's lemma to bound LHS of Formula 98 as follows:
1 n/2
n/2
E sup FEF
Σ oi (161))(()--1227) n/2
2driE,
sup sEF n/2
Σ
1 n/2
(102)
2dh max max E, =| JEld]
(f
n/2
< 16d 16dRkRRn (nn
where the first inequality follows from Talagrand's lemma, and the second and third inequalities result from Formulas 100 and 101, respectively.
In the end, we consider 95. (s. (c))) chitz continuous, hence use Talagrand's lemma to bound LHS of Formula 99;
S(x)) (za)) ( (st)) /² S is sup S(x) ><))>
Lip-
nyy /2
E2
sup
(T ((2:4n122) (2) 1
JEF
ng/
12
1=1
a//2
(103)
< sup =ex
((x) T ((z)
Eg
sup fEF ng/2
o o.f (z)4))((2
< 8dk sup =EX
(())
RF)
where the second inequality is directed results of Formula 97 which we have already proved in the previous text.
Lemma C.14. Let be a hypothesis class of feature extractors from X' to RR Assuming [F = k. Then with probability at least å over random training set D, and D., we have
log 2/8 8
sup fEF
(C:ec(f: - (a (a (a+ (a+8/c)))
(f:
No tot (F) + m (V
Mg + nle
2
(104)
log 2/5
8
sup xe
12R (F) + 173
2
where 7) is constant related to [| and feature dimension d.
26

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Proof. For convenience, we will abbreviate + (a 3/c)² vla,, 9). On one hand, note that fact - -2x (a + (a + 3/c)² (f()))((( takes values in range [-2ax2d, 2ax + (a + d/ we apply standard gener-
alization analysis based on Rademacher complexity and get: with probability at least 1 T over the randomness of D and corresponding sub-sampling tuples, we have, for any f E F:
( (f; D, ED Pascc ( S n(a,8) 32 (kd x2dd) Rm)2((F) (4 xd))
log 2/8
V
(105)
nl + nu
This means with probability at least 1 1- over D, we have: with probability at least 1 1 over corresponding sub-sampling tuples condition on D, Formula 105 holds. Hence, with probability at least probability 1 56/2 over D, we have:
32 (nd + R2d2) Rm/2(F)+ (An + x²d=) (v
log log2/6 2/6
8
(f: ED [ (f: s <ma,
n1 + nu
2
(106)
On the other hand, note that fact (=) ( takes values in range [-23k² 2,3x2 d] and using the exact same analysis process as above, we can obtain: with probability at least 8/2 over the randomness of D, for any f E F:
2 D:: - En (J; < B sup
(J:
(()) ((w) I|o
32ndR my (F) + 4x2d (1
log 2/6
( 2)) (107)
2
xex
By combining Formulas 106 and 107 with facts [wwe(f;S) ED Du (f: + Ems D, Si The proof can be concluded.
We next bound the second term in Formula 92. The following lemma bounds SuP [J: -Juso) -
Lemma C.15. Let F be a hypothesis class of feature extractors from X' to IR with || = K. Then we have:
(w: Lww - 56x dx²d
sup fEF
(108)
Proof. We expand (f: - ()) and obtain for any f E F:
[[walff: (f) s 23 (1).)),.770)))-010000000-,
((5 (( faf
2x²
=2x2d
P(q 6) ((()) P(qq T3)
=2x2d3
Ea
[P a) [)) - 3) [
< E2NE(()) ||| PP () (())P(a 8) = GG))A((),
(109)
here the 5-th inequality holds because we have:
[P(q| | (()) E') - IPy
= '')² (P(y ā) (()) E))
+ | 2)7 (P(y) x') -(E)(()))( &z')
(110)
+ (P(y) 8) (())P(q 8)) (P(y 8') 5(*') 81)
2
Ply 5) - (())P(a क) + Ply (') - -(()) 8')
27

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Finally, we combine the results of Lemmas C.11, C.14, C.15 complete the final proof of Theorem C.9.
Proof of Theorem C.9. Take do = 64(kd+* 1/1 = 3x@d+2k*d 7/2 = 64kd, 77 = Sx²d, 1/1 = d. substituting the results of Lemmas C.14 and C.11 into Formula 92, and then the proof can be completed.
C.3. Detail of Corollary 3.8
By substituting Theorems 3.4 and 3.7 into Theorem D.7 in HaoChen et al. 2021, the corollary can be directly obtained. We restate Theorem D.7 in HaoChen et 2021 as follows:
Theorem C.16. (Theorem D.7 in HaoChen et al. 2021) Assume representation dimension d > 4r + 2, Recall X be the i-th largest eigenvalue of the normalized adjacency matrix. Then, for any > 0 and f E F such that Lwsc( S Lwoe( ) + e, we have:
d
(111)
A
where
A
>130/4) Xa is the eigenvalue gap between the [3d/4] I-th and the d-th eigenvalue.
D. Related Work
Many methods have been developed to solve various weakly-supervisee learning problems. At the same time, research on representation learning has also made significant breakthroughs in recent years. In this section, we revisit these related work and divide them into two parts:
learning and representation learning.
D.I. Weakty-Supervised Learning
In real world, the widely existing supervised information is weakly-supervised information which is either inaccurate or ambiguous. In order to enable effective and accurate learning from this weakly-supervised information, previous work has proposed a series of methods for weakly-supervised learning. The two typical paradigms related to weakly-supervised learning are noisy label learning (NLL) and partial label learning (PLL), which is what we will introduce in this section.
Noisy Label Learning (NLL). Noisy labels may be caused by annotation errors, and overfitting to these label errors will lead to poor model performance (Zhang et al., 202 Wei et al., 2024; Li et al., 2023). Several strategies have been developed to mitigate the label noise. The mainstream of NLL methods cover the following aspects: using robust loss function (Zhang and Sabuneu, 2018; Wang et al., 2019), modeling the noise transfer matrix, performing sample selection and correcting incorrect labels. In particular, label correction methods have shown promising results than other methods in noisy label learning. Liu et al. 2020 prove that the model can accurately predict subset of mislabeled examples during the early learning phase. This observation implies a potential strategy for rectifying the corresponding labels, which can be achieved by generating novel labels tantamount to either soft or hard pseudo-labels estimated via the model (Tanaka et al., 2018). Han et al. 2018 propose Co-teaching which trains two different networks for the label correction. Inspired by the fact that clean samples have a smaller loss the early learning stage, Arazo et al. 2019 apply a mixture model the losses of each sample estimate the probability that sample is mislabeled and correct the loss based the prediction of the network. Similar to the previous two methods, DivideMix (Li et al., 2020), deploys two neural networks to conduct mutual sample selection and apply semi-supervised learning methodology, where the targets are computed based on the average predictions obtained from different data augmentations.
Partial Label Learning (PLL). Partial labels, while preventing the omission of the correct label, introduce greater ambiguity into the labeling process. The prior works can be divided into average-based strategies and identification-base strategies. The average-based methods usually treat all candidate labels equally. Lv al. 2023 discussed and analyzed the robustness performance of different average-based loss. However, those methods may introduce the misleading false positive label into the training process. To overcome these limitations, identification-bas based strategies, which regard the correct label as a latent variable and aim to identify it from the candidate label set, have drawn intensive attention and achieved remarkable progress. PRODEN (Lv et al., 2020) and CC (Feng et al., 2020b) utilize the predictions generated by the predictive model as label information, thus assigning more weights to labels that are more likely to be correct. Wen et al. 2021 propose a family of loss functions for label disambiguation. Wu et al. 2022 perform supervised learning on non candiaate labels and employ consistency regularization on candidate labels.
28

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Weakly-Supervised Contrastive Learning for Imprecise Class Labels
In recent years, more practical setting called instance-dependent partial label learning (IDPLL) has also received considerable attention. In IDPLI., the candidate set is generated from labels that have certain semantic similarity with the true label, which also increases the difficulty of disambiguation. Qiao et al. 2023b propose to explicitly model the instance-dependent generation process by decomposing it into separate generation steps. Wu et al. 2024 introduce a novel rectification process to ensure that candidate labels consistently receive higher confidence than non-candidates. He et al. 2023 propose a two-stage framework employing normalized entropy for selective disambiguation of well-disambiguated examples. CEL (Yang et al., 2025) employs a class associative loss to enforce intra-candidate similarity and inter-set dissimilarity among the class-wise embeddings for each sample.
Some research has also turned trying (Qiao et al., 2024) improves the reliability of label refinement by globally detecting the reliability of the prediction model and locally enhancing the supervision signal, thereby improving the performance of the prediction model. GFWS (Chen et al., 2024) treat the ground-truth labels as latent variables and try to model the entire distribution of all possible labeling entailed by weakly-supervised information, thus allowing a unified solution to deal with NLL and PLL. Although these methods try to uniformly exploit the common features of weakly-supervised information to solve the -epervss learning challenges, they all ignore the potential of high-quality representations to achieve label disambiguation.
more unified framework to solve both NLL and PLL problems. ULAREF
D.2. Representation Learning and Weakly-Supervised Representation Learning
In recent years, contrastive learning has become dominant in representation learning because they can learn more distinct representations. A plethora of works has explored the effectiveness of contrastive learning in unsupervised representation learning (He et al., 2020; Oord et al., 2018; Chen et al., 2020; Caron et al., 2020; Zbontar et al., 2021). Meanwhile, there are also many works trying to introduce more information into contrastive learning or bring new insights from different views (Khosla et al., 2020; HaoChen et al., 2021; Sun et al., 2023; Cui et al., 2023; Zhou et al., 2024).
Khosla et al. 2020 attempt to introduce supervised information into contrastive learning. The approach regards samples in same classes as positive samples and achieves significant performance improvements on multiple supervised learning tasks. Due to the success of contrastive learning, many researchers have made a lot of attempts to improve the performance on weakly-supervised learning incorporated with the advantages of contrastive learning. On the line of NLL, the role of contrastive learning is different. MOIT (Ortego et al., 2021) uses the agreement between the features learned by contrastive learning and the original labels to identify mislabeled samples. Sel-CI. (Li et al., 2022) utilizes k-nn nearest neighbors to select confident sample pairs and use these sample pairs for supervised contrastive learning. TCL (Huang et al., 2023) uses a Gaussian mixture model to disambiguate labels, and then uses self-supervised contrastive learning to further learn more robust representations. For PLL, PiCO (Wang et al., 2022) integrates contrastive learning with prototype learning. The former facilitates the formation of well-structured clusters, which in turn enables prototype learning to acquire prototype representations. The latter assists in the selection of positive samples for contrastive learning.
On the other hand, some researchers try to view and cxplain contrastive learning from different perspectives. Inspired by the widespread application of spectral graph theory in the field of machine learning, HaoChen et al. 2021 first regard contrastive learning as problem of graph clustering on augmentation graph and introduce spectral contrastive loss, which greatly promoted the progress of unsupervised contrastive learning. Along this line, Sun et al. 2023 introduce supervised information to this spectral contrastive learning framework, transforming the augmentation graph in the original problem into a perturbation graph, thus achieving great performance improvement on open-world world semi-supervise learning task.
Despite their success in the respective field, they all overlooked the importance of correctly utilizing weakly information. Relevant theories and experiments have verified that simply introducing inaccurate noise labels for contrastive learning is ineffective and even harmful (Cui et al., 2023). Therefore, it is necessary to design special contrastive learning information. Overall, our method is the first approach to utilize weakly-supervised
framework for weakly information for contrastive learning from a graph spectral theory perspective. Sufficient experiments and complete and rigorous theory guarantee the effectiveness of the proposed method.
E. Implementation Details
In this section, we provide more details on the implementation of our approach. In general, our proposed framework contains three training strategies: supervised loss, strong-weak augmentation consistency regularization (Xic et al., 2020), and the proposed weakly-supervised contrastive learning (WSC) loss. Among them, the supervised loss varies with different settings,
29

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Weakly-Supervised Contrastive Learning for Imprecise Class Labels
while the other two losses do not change with the setting.
Specifically, given any sample its supely-sueereieed information can be denoted as q Q. Let the extracted feature embedding be f(x), the final probability prediction of the neural network be g(x),, and the strong augmentation function be (.).) Our method can be formalized as:
((x,q) = Coup(((())))) + (oELo(A + (f)
(112)
where ((x) means the prediction is made by fixed copy of current model, indicating that the gradient is not propagated. The first term in equation 1 112 means the supervised loss which will be introduced next. The second term is the called strong-weak augmentation consistency regularization, which consistently utilize the prediction of the weakly-augmentee data to train the groongl--uugentee data, and it is commonly used in weakly-supervise et al., 2022; Wu et al., 2022; Chen et al., 2024). The third term is the proposed WSC loss, which needs at least three inputs: the embeddings of the features of two different views of the sample and the constructed matrix S. In this paper, we compute this loss through Algorithm 1 which assumes a uniform class distribution, pseudo code of computing WSC loss without this assumption can be found in Algorithm 2. We left method for construct S in next section.
learning (Xie et al., 2020; Wang
E.1. Noisy Label Learning
Setup. For CIFAR-10 and CIFAR-100, an 18-layer PreAct ResNet is employed and stochastic gradient descent (SGD) is utilized for training with momentum value of 0.9, a weight decay factor of 0.001, and batch size of 128 over a total of 250 epochs. The initial learning rate is set to 0.02 and adjusted by a cosine learning rate scheduler. In our framework, both the projection head and the classification head are configured as a two-layer Multilayer Perceptron (MLP) with a dimension of 256 and the number of classes. The parameters are set to = 1, 3 = 12 for CIFAR-10 10 and a = 2, = 300 for C0FAR-100. Additionally, for CIFAR-100, the last term of our loss function is scaled by a factor of 3. In addition, we also conduct experiments on CIFAR- ION and CIFAR-1OON, where we use a 34-layer ResNet as the backbone for feature extraction, others are same as the previous one. Besides, a 50-layer ResNet pretrained with ImageNet-1K was used to train for 15 epochs on Clothing M. The batch size are set to be 64 and the initial learning rate is 0.002 and then multiply by a factor of 0.1 in the 7th epoch. We also set 1, 28. The detailed crpper-prramters are presented in Table 4.
Table 4. Hyper-parameters for noisy label learning used in experiments.
Hyper-parameter
CIFAR-IO (CIFAR-ION)
CIFAR-100 (CIFAR-10ON)
Clothing
Image Size
32
32
224
ResNet-50
Model
PreAct-ResNett (ResNet-34)
PreAct-ResNet-I1 18 (ResNet-34)
(ImageNet-IK Pretrained)
Batch Size
128
128
64
Learning Rate
0.02
0.02
0.002
Weight Decay
le-3
le-3
1e-3
LR Scheduler
Cosine
Cosine
MultiStep
Training Epochs
250
250
15
Classes
10
100
14
1
2
1
B
12(6)
300
28
Baselines. The performance of proposed method for noisy label is compared against ten baselines:
CE, which utilizes the standard cross-entropy loss directly to train the model in a batch.
DivideMix (Li et al., 2020), which regards noisy instances as unlabeled data and employs the strategy involving label co-refinement and co-guessing.
ELR (Liu et al., 2020), which focuses on early learning via regularization to preclude the memorization of incorrect labels.
SOP (Liu et al., 2022), which models the noise with sparse over-parameterrizatin and exploits implicit regularization.
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Weakly- Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Table Comparisons with each methods CIFAR-ION, CIFAR-10ON and Clothing M. Each runs has been repeated times with different raneomly-genertedd
generated noise and we report the best accuracy.
Dataset
CIFAR-ION
CIFAR-100) 100N
Clothing IM
Noisy Type
Randoml
Aggregate
Worst
Clean
Noisy
Ins.
CE
85.02x0.65
87.77=0.38
57.69=155
76. 76.704
55.50=0.66
69.10
Forward
36.88+0.50
88.24+0:22
79.79 49.79:0.46
76. 18+0.37
51.01
Co-teaching
90.3320.13
91.2022013
83.83=0.13
73.4620.09
60.3720.27
ELR
94.43x0.41
94.83+0.10
91.09=1.60
78.57=0.12
66.72a0.07
72.90
GFWS
94.86+0.07
95.30e0.03
93.55=0.14
78.53+0.21
68.07
74.02
CORES
94.45x0.14
95.25
91.6620.09
73.87s0.16
55.7220.42
73.20
SOP
95.28
=5.61
93.24=0.21
18.91
67.81
73.50
WSC (Ours)
96.13s0.52
96.50=a.72
93.60=0.21
81.31
71.00=0.16
74.75+0.18
GFWS (Chen et al., 2024), which is a unified framework that uses Expectation- Expectation-axiiizzaaii -aximizaaiin to model noisy label as
latent variables.
ULAREF (Qiao al., 2024), which trains the predictive model with refined labels through global detection and local enhancement.
ProMix (Xiao et al., 2023), which meticulously selects, dynamically extends, and optimally utilizes clean sample sets within the devised supi-supe learning framework.
MOIT (Ortego al., 2021), which jointly exploits contrastive learning and classification to enhance performance against label noise, with contributions including ICL loss, novel label noise detection method, and fine-tuning strategy.
Sel-CL (Li et al., 2022), which leverages nearest neighbors to select confident pairs for supervised contrastive learning.
TCL (Huang et al., 2023), which disambiguate labels by Gaussian mixture model and uses self-supervised contrastive learning to further learn more robust representations.
Discussion. Recent works on NLL can be categorized into three types, with methods based the noise transition matrix demonstrating empirical effectiveness. In our proposed framework, we use the cross-entropy loss align the model output conditioned by the estimated noisy transition matrix and the noisy label. Besides, in order to better learn from the noisy label, we follow Li et al. 2021 and impose the following regularization constraints on the estimated noise transition matrix T. The above two terms will construct the supervised loss as:
Coup(o(x).g) = LcE(o(=)T, ②) + Allog detg))
(113)
where À is a regularization coefficient.
Additional Experiments. In addition the results we present in the main paper which mainly focuses on the the simulated datasets, also verify the effectiveness of the proposed method on the CIFAR-N dataset (Wei al., 2022), which equips the training samples of CIFAR dataset with human- -anottted noisy label. On CIFAR-ION, we used three noise types: Aggregate, RandomI, and Worst, with corresponding noise rates of 9.03%, 17.23%. and 40.21 %. For CIFAR-10ON, we compared the model performance under Clean and Noisy settings with noise rate of 40.21% %. We also include a full comparison on Clothing M which has more realistic and large-scale instance noise (Xiao et al., 2015).
The results shown in Table show the superiority of our method on these datasets. It is particularly noteworthy that our method performs very well on CIFAR-1OON, which should be due to the fact that our contrastive learning method specially designed for weakly supervised learning helps learn more and better feature representations. It is worth noting that since the noise type of these datasets is not suitable for estimation using noise transition matrix, we ignore the second term of Equation 113 during training.
Qualitative Results. Figure 2 visualizes the learned representations under a high noise ratio. Compared to ELR+, which employs two distinct backbones to obtain more meaningful representations, our method generates well-defined structures with semantically meaningful information, leading to superior accuracy these settings.
31

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Contrastive Learning for Imprecise Class Labels
(a) WSC on CIFAR-10
(b) ELR+ on CIFAR-10
(c) WSC on CIFAR-IO1 100
(d) ELR+ on CIFAR-100
Figure 2. We present t-SNE visualizations of the learned representations the CIFAR dataset with 90% symmetric noise. Figure 2(a) and Figure 2(b) illustrate the results for WSC and ELR+ on CIFAR-I0, while Figure 2(c) and Figure 2(d) display the results for WSC and ELR+ CIFAR- 100 under the same noise condition, where only 20 categories with the highest accuracy for each model shown.
E.2. Partial Label Learning
Setup. For simulated PI.I settings, we use Wide-ResNet-28-2 trained from scratch and ResNet-18 pre-trained with ImageNet-1K: IK as our feature extractors on the CIFAR dataset and CUB-200 dataset, respectively. Following most of experimental setups conducted on the contrastive network, we use a 2-layer MLP that outputs 128-dimensional embeddings as our projection head. The batch size 256 and the weight decay is 0.0001 across all the PLL settings. We set the initial learning rate to be 0.1 and adjust by MultiStep and Cosine scheduler on CIFAR datasets and CUB-200 respectively. On CIFAR-I0 and CIFAR- 100 CCFFAR-10-HH)
train our model for 200 epochs using parameters
1, B 12 for and
CIFAR-10 and a 2, 300 for CIFAR-100, For the CUB dataset, train for 300 epochs with parameters linearly increasing from 0 to 400. The detailed choice of hyper- parameters is provided in Table 6. Furthermore, for IDPLL settings, we use ResNet-34 pre-trained with ImageNet- 1K as our feature extractors following (Yang et al., 2025). We train our model for 500 epochs and set the initial learning rate be 0.1 and adjust it by Cosine scheduler on both four fine-grained datasets. The detailed choice of hyper-ppranee paramerrrseeters is provided Table 7.
Table Hyper-parameters for simulated partial label learning used in experiments.
Hyper-parameter
CIFAR-10
CIFAR-100 (CIFAR-100H)
CUB-200
Image Size
32
32
224
ResNet-18
Model
Ne2t---2
(ImageNet- IK Pretrained)
Batch Size
256
256
256
Learning Rate
0.1
0.1
0.01
Weight Decay
le-4
le-4
le-5
LR Scheduler
MultiStep
MultiStep
Cosine
Training Epochs
200
200
300
Classes
10
100
200
2
2
8
12
300
0-400
Discussion. We provide the detailed implementation of Coup on PLL setting here. The average- based methods has been widely used in recent PLL works. Given the probability prediction yar)), we can denote the classifier as h((r) = arg min g (x).
TE Besides, the candidate set g can be derived from the weakly supervised information, thus, the family of average partial label losses can be formally writen as:
Lavg (h(x),
Σ (((())
(114)
1q
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Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Table Hyper-parameters for instance-dependeut partial label learning used in experiments.
Sypee--arameter
CUB200
CARS196
DOGSI20
FGVC100
Model
(cssttt-3((P)ttrainee)
RcsNet-34(Pretraineed)) ResNect--((Petrainncc))
(osNott33((retraineed)
Image Size
224
224
224
224
Batch Size
256
256
256
256
Learning Rate
0.01
0.01
0.01
0.01
Weight Decay
le-5
le-5
lc-5
le-5
LR Scheduler
Cosine
Cosine
Cosine
Cosine
Training Epochs
500
500
500
500
Classes
200
196
120
100
O
0-400
0-400
0-240
0-200
where represents the cardinality. There are different loss functions & that can be chosen, and we use the commonly used cross-entropy loss to implement our framework. The supervised loss can then be obtained from the above average partial label loss.
Baselines. We compare the proposed framework with seven methods handling partial labeled data:
LWS (Wen et al., 2021), which weights candidate labels and non-candidate labels through a leverage parameter.
PRODEN (Lv et al., 2020), which progressively identifies correct labels through the model output.
CC (Feng al., 2020b), which derives classifier consistent risk estimator through transition matrix.
MSE (Feng et al., 2020a), which uses Mean Square Error loss to learn with multiple complementary labels.
RCR (Wu et al., 2022), which performs supervised learning non-candidate labels and employ consistency regular-
ization on candidate labels.
PiCO (Wang al., 2022), which utilizes contrastive loss term enhance the model disambiguation ability.
We also include four IDPLL methods to compare:
IDGP (Qiao et al., 2023b), which proposes to explicitly model the instance-dependent generation process by decompos- ing it into separate generation steps.
DIRK (Wu et al., 2024), which introduces a novel rectification process to ensure that candidate labels consistently receive higher confidence than non-candidates.
NPELL (He al., 2023), which proposes two-stage framework employing normalized entropy for selective disam-
biguation of well-disambiguated examples.
CEL (Yang et al., 2025), which employs class associative loss to enforce intra-candidate similarity and inter-set dissimilarity among the class-wise embeddings for each sample.
Additional Experiments. To further investigate the performance of our proposed method on fine-grained image classification tasks, we conduct additional experiments using CIFAR- 100 with hierarchical labels (CIFAR-100-H) (Wang al., 2022). where candidate labels are generated from the 20 superclasses in CIFAR-I00. Furthermore, we extend our experiments on CUB-200, as presented in the main paper, include wider range of partial label ratios, specifically 0.01, 0.05, 0.1. We also extend our experiments on IDPLL setting. Following previous work (Wu et al., 2024; Yang et al., 2025), we conduct experiments on four fine-gratd fine-grained datasets: CUB-200 (Wah et al., 201 Ia), CARS-196 (Krause et al., 2013), DOGS-120 (Khosla et al., 2011), and FGVC-100 (Maji et al., 2013), and we employed the IDPLL noisy label generation method proposed by VALEN (Xu et al., 2021) to generate instance-depe -eeunddde noisy labels which is the standard experimental protocol in IDPLL settings.
33

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Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Table Comparisons with each methods simulated PI.. datasets. Each has heen repeated times with different randomly- generated partial labels and we report the mean and std values of last 5 epochs.
Dataset
CIFAR-100-H
CUB-200
Partial Ratio
0.1
0.5
0.8
0.01
0.05
0.1
PICO
76.55=0.68
74.98s0.42
66.3820.24
74, 1 14
72. 72.174
62.0221.16
LWS
63.88=0.10
29.37=0.25
40.33+0. 19
73. 23.74
39.74+0.43
12.30=0.77
PRODEN
67.4220.91
64.8920.75
50.41
72.3410.04
62.5610010
35.8920.05
CC
58. 1
56,44.0.37
30. 1 1:0.48
56.63+0.01
55.61
17.01
MSE
57.88=0.85
51.13+0.32
28.33+0.44
61. 12+0.51
22.07+2.36
11.40+2.42
GFWS
77.1020.18
76.0820.28
61.35s0.71
73.1920.42
70.7720.20
47.4420.15
RCR
77.38=0.33
75.78+0.25
65.31
WSC (Ours)
78.31
76.67=0.11
66.77=0.05
76.88+0.39
74.55=0.17
13.0343.34
Table 9. Comparisons with each methods on IDPLL settings with four fine-grained datasets. Each runs has been repeated 3 times with different randomly-generann partial labels and report the and std values of last 5 epochs.
Dataset
WSC (Ours)
CEL
DIRK
NEPLL
IDGP
PiCO
PRODEN
CUB-200
68.90
68. 18.6020.10
66.60s1.07
62.88
56.16
58.562090
65.05a0.14
CARS-196
87.88
86.2220.08
80.31
85.0520.17
79.5620.46
83.35a0.05 10.
DOGS-120
79.75s0.32
78. 12.18+s...
75.97=0.29
74.84a0.09
66.79an.38
67.8 67.80+0.06
70.94=0.43
FGVC-100
77.80s0.39
78.36*0.19
76.86+3.54
75.36
72.48a0.86
63.52=0994
69.34
Our method outperforms other methods except the experiment on CUB-200 with 0.1 partial ratio as shown in Table 8. The improvement can be attributed to PiCO's class mototype-based pseudo-labeling mechanism for disambiguation, strategy that has proven effective in several fine-grained classification tasks. Moreover, our approach avoids the need for additional complex techniques. By incorporating our contrastive loss into the training objective, we achieve an 8.69 % improvement over GFWS in this setting, highlighting the potential for even greater performance gains. Similarity, our WSC method demonstrates strong performance in the instance-depencee Table without introducing any complex modules or additional technical components. The results further validate that WSC can effectively address the challenges of instance-dependent label ambiguity while maintaining a straightforward implementation.
partial label learning (IDPLL) setting as shown
E.3. A blation Study
We conduct ablation experiments on CIFAR-100, as shown in Table 10. The results demonstrate that the proposed WSC loss consistently improves performance, especially under high noise rates, where it increases performance by 16.22% % and 8.99%, respectively, further validating the effectiveness of the proposed method.
Table 10. Ablation studies of our proposed algorithm on CIFAR-100 with differrnt ratio of noisy label and partial label.
Components
Noisy Label
Partial Label
Supervised Loss
Consistency Regularizer
WSC Loss
0.5
0.8
0.9
0.1
0.2
0.3
53.89
41.35
20.11
61.50
55.22
35.44
75.43
66.40
45.10
75,80
73.12
60.16
77.51
71.92
61.32
77.26
75.13
69.15
F. Details of Construction of Semantic Similarity under Different Scenarios
In this section, we will introduce more details respectively regarding the construction Š in the scenarios of noisy label learning and partial label learning.
34

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Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Algorithm 2 batch data estimation of Luuee under general scenario
Input: features of two augmentation view of batch data with corresponding weakly-supervine information X R Boxx Q E Q features of two augmentation view of batch data without any supervised information X c. Xi BBu xd S in Proposition. 2.1, class prior P(y) E [Rex1, proportional coefficients a; 3
X
Output: batch data estimated loss Luose
Compute ((X)
R Bo, ((X)) )). = S(x):.
Compute S' (X) R RI (') = PP(y)
Compute 2 =
Tr x (x +Tr (
But BB
Compute E2 (5(x)) " s(X))
x (x))))
Compute Eg xx3 |T||
Compute Σs- Vs'(X) (x) 5' X
B3
Take XQ Q = 1x) ei
x 2 X =
X U
x
U11 =
Compute Es- V Ding 5'(x) x x
tBq x By
Return: Ewso
1 Q:
X² x U x S, -2 - 23L2 + a²L + 3C + 2a3Bs
F.1. Details of Construction of Semantic Similarity under Noisy Label Setting
Using Environment Information. For instance-independent noisy label setting, there exists a T satisfies T = T(x) holding almost everywhere in X. Hence, inspired by our Proposition 2.4, in this setting, we can use S(x) = (T) for any X, where i is estimated noisy transition matrix. Quite a number of studies have already investigated how to estimate the noise transition matrix (Liu and Tao, 2016; Patrini et al., 2017; Xia et al., 2019; Li et al., 2021; Zhang et al., 2021b; Lin et al., 2023). These methods are usually divided into two categories: anchor-based methods and anchor-free methods. The former (Liu and Tao, 2016; Patrini et al., 2017; Xia et al., 2019) usually fits the noise posterior directly, and then estimates the noise matrix by this estimated noise posterior probability
the data of reliable anchor points. In recent years, it is the bb; Lin et al., 2023) that has achieved better results. They obtain the estimated noise
latter (Li et al., 2021; Zhang et al., 2021b; matrix by searching for the matrix that minimizes a specific metric within the matrix family that can linearly represent all noise posteriors. Specifically, Li et al. 2021 uses volume of estimated noisy matrix as this specified metric and Zhang et al. 202 uses total variation as this specified metric. Lin et al. 2023 additionally adopts a bi-level optimization to increase the robustness against the error of the estimation of the noise posterior and thus achieving better performance. All above methods can be used to construct S.
Using Both Environment and Samples Information. However, the aforementioned way of constructing semantic information while ignoring the information of the samples
similarity only takes into account the themselves. Therefore, even if we obtain a completely accurate noise matrix through estimation, such a construction method will still add a variance term to the learning error (sup ((xx)T(()) performance in practice. Recalling Bayes theorem:
Formula 29), thus resulting in sub-optimal
xx) =) = Σ P(y,ol1)
Σ
Ply a. P4 x))
(115)
SE@
9 e
Thus we can take S1x)y. Ply g) where PM(y x. q) is estimation posterior with corresponding weakly-supervised
information. With estimated noisy transition matrix T. we can estimate P(y) x, q) as follows:
P(y x) x P(y. q æ) = P'y x x)TT 9.v (o) q.3-
(116)
where (x) denotes posterior probability predicted by the current neural network.
Such a construction method can be regarded as constructing semantic similarity by integrating the information of the current neural network, the sample information and the estimated environmental information. Therefore, better performance has been achieved in practice.
35

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Using Samples Information. For instance-dependent setting which we do not assume that all samples share the same noise transition matrix, the methods mentioned above become invalid. In such setting, we unable utilize the environmental information and can only rely the properties of the neural network itself as well as the sample information for estimation P(y | Iz q). This type of method has also been widely discussed in noisy label learning (Liu et al., 2020; Li et al., 2022; Ortego et al., 2021; Liu et al., 2022; Xiao et al., 2023; Huang et al., 2023; Qiao et al., 2024) and has achieved higher performance in practice. The main idea this method can be summarized by the following equation:
Ply æ.9) 4) = Ply x, () = 1
I I Dr x D,
(117)
(g(
where the ÿ denotes corresponding noisy label, and D, is a "reliable" set of noisy labels which are screened out using a specific method.
Specifically, such screening methods are usually carried out based on empirical observation that the neural networks tend learn easy (correct) samples first, and then start to fit onto the hard (corrupt) samples in the later phase of training, hence the samples with small loss values are presumed to be reliable, while those with large loss values are not. The specific implementation of screening is diverse. Any screening method can be used in our framework to construct the corresponding S. In this paper, we will not describe these methods in more details.
Construct of Main Experiments. To avoid introducing more details into the methods in the main paper, the S selected in the main experiments of this paper only adopts the simplest form and does not use any screening mechanisms that have been proven to be effective. We simply take S as (xx)y. Pp(y () g (x)))y for any x. More experiment of using different Š can be referred to in the Table 11. For the sake of simplicity in this experiment, the first way to construct is to use fixed real noise matrix. The second way to construct S is the same as the method used in our main paper. the third approach, we construct (yy] Ir, q) as a convex combination of the original noisy label and the model predictions, with the weights predicted by two-component Gaussian mixture model. This method is a commonly used label bootstrap technique, as demonstrated by (Huang et al., 2023). The second method achieves the best results, as it leverages both the overall environmental information and the sample information more effectively. Additionally, due to the accuracy of clean sample screening in the early learning stage, the third method achieves performance similar to the second method. In contrast, using the inverse of the noise transition matrix alone, even with the true noise transition matrix, leads to suboptimal results because it neglects the sample information.
F22 Details of Construction of Semantic Similarity under Partial Label Setting
Using Environment Information. For mnstance-indepentee partial label setting. unlike the situation of noisy label, it is generally quite difficult to directly estimate the transition matrix T due to the curse of dimensionality. However, when using the commonly assumption that labels are independent adopted into the candidate set (Cour et al., 2011; Lv al., 2020; Feng al., 2020b), the problem
be simplified. We formally present this assumption follows:
Assumption F.1. (Selected candidate uniform and independent (SCUT) assumption) The partial label problem satisfies the SCUI assumption if its generation process meets follow equation:
P(a x,,'') = Tly E v)
II MEV.utu'
of P(y & y x, Mow E a xr, x,,5 )
(118)
= Tyy' e al
II
(1 (1-63 (y Ulvetl Houel
where (y E g x, y²) = My for any y # y².
Giving this assumption, we can decompose the partial label problem into c times independent binary classification noisy label problems. When each o, in known, it is easy to see follow S will satisfy ST = I exxe
y E
v.g =
xg. 1-0
y Φ
(119)
Hence, if we can get estimated o, for any y E y, we can take Sy.q through Equation 119,
36

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
Now consider how to estimate this By. We assume uniform class distribution and encode g L E [0, 1Js, where 4 = indicates y² q. The following equation holds:
P(r 4 1) P(2, Y = y 1 = + 11)
+ y², =
P(4=1)
(1/c)P(z Y = -1' + 1/c)o,P(x |Y x3)
(1/))((1 Y (1/c) (1cc)(1111//)g
(120)
-1/eooo,, 1==0)
(/co)((111/0)))
(1 - 0,) Y -1) + 0.1((1 1 = =0,
where 02 (c-1)a 1)y
1
Using Equation 120, we transform the problem of estimating o; into the problem of estimating 0,. The later is mixture proportion estimation problem to estimate 0, given samples sampled from the P(r 4 1) and P( 4 = 0). Under many assumptions to ensure identifiability, including the irreducibility assumption (Scott et al., 2013; Garg et al., 2021), the anchor point assumption (Scott, 2015; Liu and Tao, 2016), the separability assumption (Ramaswamy et al., 2016), Ag can been estimated through many off-the-shelf methods (Scott et al., 2013; Scott, 2015; Ramaswamy et al., 2016; Garg et al., 2021).
Using Both Environment and Samples Information. Similar to the noisy label setting, the above method for constructing semantic similarity overlooks the information contained in the samples themselves. To address this, we can incorporate sample information by applying Bayes' theorem. The estimation of Pyy xr, 9) can then be expressed as follows:
Ply g) PPy P) x, v) P(y x)) I of' II 0xy) Ily E q) g(Au(x)) Iy g)-
(121)
Such a construction method can be regarded as constructing semantic similarity by integrating the information of the current neural network, the sample information and the estimated environmental information. Therefore, better performance has been achieved in practice. When By is unknown, we can simply assume that σy = a for any E y, Equation 121 can still be used to construct semantic similarity.
Using Samples Information. In the instance-dependent partial label setting proposed in recent years (Xu et al., 2021; Qiao et al., 2023a), the SCUI assumption does not hold, the methods mentioned above become invalid. To estimate F(y1 q) in this situation, Xu et al. 2021 proposes variational label enhancement method which relies solely on current neural network. Additionally, Qiao al. 2023a parameterizes Pg x. y) and uses maximum likelihood estimation to learn these parameters. Given this estimation Pp(g| x, q), we can also estimate iP(y | xr, 4) using the Bayes theorem: Pyy g) x F(yl 1)) P(g x. v)
Construct of Main Experiments. To avoid introducing more details into the methods in the main paper, the S selected in the main experiments of this paper only adopts the simplest form. We simply assume that 0, for any E ) and apply Equation 121. Specifically, we take Sxx)y. Ply x. q) Mly E 9] Eves a(A,))))))
(x),, (x)), for any x. More experiments
using different S can be found in Table 11. For the sake of simplicity in this experiment, the first way to construct S is to use fixed real partial ratio. The second way to construct S is the same as the method used in our main paper. In the third approach, we construct Ppyy Iz q) by using variational label enhancement method proposed in (Xu et al., 2021). The second
Table 1. Comparisons with each methods for constructing S on CIFAR-100 with different ratio of noisy label and partial label. We report the mean and std values of last 5 epochs. Env, Sap, denote the environment information and sample information, respectively.
Type
Noisy Label
Partial Label
Ratio
0.5
0.8
0.9
0.05
0.1
0.2
WSC w/ Env.
75.29+0.15
67.34
55.01
76.31
71.33+1.33 55.85+0.54
WSC w/ Env. & Sap.
77.51
71.92+0.17
61.32+0.15
77.88=n.19
77.2640.30
13..344.24
WSC w/o Env.
76.83+0.1
71.18
59.58+0.38
78.01
76.88+0.2
74.50+a.25
37

---

Weakly-Supervised Contrastive Learning for Imprecise Class Labels
method achieves the best results, it leverages both the overall environmental information and the sample information. Additionally, because Xu al. 2021 also uses the sample information, hence achieves performance similar the second method. In contrast, using the environment information alone, even with the true o, leading to suboptimal results because it neglects the sample information.
38


Reviews:
[{"review_id": "hcWAdp9ilO", "reviewer": "Reviewer_jhEr", "summary": "This work rethinks contrastive learning for noisy real-world settings by replacing rigid class-based positive/negative sampling with adaptive, graph-driven “semantic similarity”. By blending self-supervised augmentations with weak supervision signals (e.g., noisy/partial labels), the method achieves state-of-the-art performance while offering theoretical guarantees.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer jhEr:\n\nThank you for your valuable suggestions. We truly appreciate your review and are committed to addressing your concerns with care and attention. We sincerely look forward to engaging in a more in-depth discussion with you, as your insights are essential in helping us improve and refine our approach.\n\n> **W1. The paper does not provide sufficient evidence that shows its superiority in representation learning.**\n\nIn addition to Figure 2, we also theoretically analyze that the proposed method helps improve the quality of representation learning features.\n\n- Theorem 3.4 bounds linear probe error of feature derived from perturbation graph through properties of self-supervised connectivity graph and perturbation coefficients, and demonstrates that the perturbation graph increases the density of intra-class connections without adding extra-class connections.\n- In Theorem 3.7, we further show that the features learned through finite samples and weakly supervised information can approximate the features learned from supervised information.\n- Finally, we present the linear probe error of the features learned by our framework in Corollary 3.8, which can be regarded as a qualitative analysis of the learned features.\n\n> **Q1. Can the proposed method be applied to weakly-supervised multi-label learning problems?**\n\n- The method we proposed is based on the semantic similarity between samples. The construction of semantic similarity depends on the transfer conditions approximately satisfied by each sample. It is not specially designed for multi-label classification tasks and cannot be simply applied to multi-label classification. \n\n- We believe that by further expanding the concept of semantic similarity, it is possible to apply our method to weakly-supervised multi-label learning problems. We will explore this problem in future work.", "claims_and_evidence": "Claim #1: Instead of treating labels as binary (same/different class), the authors model similarity as a continuous measure derived from label noise patterns or partial candidate sets.\n\nEvidence #1: This avoids brittle assumptions about label correctness, which is crucial for messy real-world data. On Clothing1M (real-world noisy labels), the method achieves best performance, outperforming prior works as reported in experiments.\n\nClaim #2: Augmented samples form nodes in a graph, with edges weighted by both augmentation-based similarity (e.g., two crops of an image) and label-derived signals (e.g., estimated noise patterns).\n\nEvidence #2: By framing the problem through spectral graph theory, the authors prove the learned representations approximate supervised contrastive learning (Corollary 3.8), a novel theoretical bridge.", "methods_and_evaluation_criteria": "Method: Constructs a graph where nodes are augmented samples, and edge weights combine (1) self-supervised similarity (via data augmentations) and (2) weakly supervised similarity (derived from label patterns or noise estimates). From this graph, a contrastive learning objective is derived by solving a spectral clustering problem.\n\nEvaluation: Evaluated on multiple benchmarks spanning synthetic noise (CIFAR), real-world noise (Clothing1M), and fine-grained ambiguity (CUB-200). This paper compares with many strong baselines in LNL and PLL, which demonstrates its superiority.", "theoretical_claims": "I have checked the proof of Proposition 2.3. and Corollary 3.8 which appears valid.", "experimental_designs_or_analyses": "yes", "relation_to_broader_scientific_literature": "The paper builds on prior work (i.e., Provable guarantees for self-supervised deep learning with spectral contrastive loss) and extends it to weak supervision tasks.", "essential_references_not_discussed": "All essential references are discussed.", "other_strengths_and_weaknesses": "**Strengths**\n\nS1. Weak supervision is ubiquitous in real-world data, yet contrastive learning under such conditions remains underexplored. This work fills that void.\nS2. Connects spectral graph theory with representation learning, providing error bounds while delivering strong empirical results.\nS3. Outperforms specialized methods even in extreme noise regimes (e.g., 90% label noise on CIFAR-100).\n\n**Weaknesses**\n\nW1. Since contrastive learning is designed for improving representation learning, however except Figure 2, the paper does not provide sufficient evidence that shows its superiority in representation learning.", "other_comments_or_suggestions": "None.", "questions_for_authors": "Q1. Can the proposed method be applied to weakly-supervised multi-label learning problems?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "y7Ts0j8Y9a", "reviewer": "Reviewer_x5zp", "summary": "This paper tackles a key challenge in contrastive learning: handling real-world datasets with messy or ambiguous labels. The authors propose replacing traditional binary positive/negative pairs with \"continuous semantic similarity,\" modeled via a graph where edge weights reflect how likely two examples belong to the same class. This framework integrates weak supervision (e.g., noisy/partial labels) into contrastive learning and shows strong empirical results across tasks like noisy label learning (NLL) and partial label learning (PLL).", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer x5zp:\n\nThank you for your valuable suggestions. We truly appreciate your review and are committed to addressing your concerns with care and attention. We sincerely look forward to engaging in a more in-depth discussion with you, as your insights are essential in helping us improve and refine our approach.\n\n> **W1 & Q1: It seems this paper relies on uniform class distribution for theoretical guarantees—how does this hold for imbalanced data?**\n\n**(1) Practical Applicability to Imbalanced Data:**  \n\n- Our Algorithm 2 explicitly handles imbalanced scenarios by estimating $\\mathcal{L}_{wsc}$ without any class prior assumption ($\\mathbb{P}(\\boldsymbol{y})$). This ensures practical validity under arbitrary class distributions. Algorithm 1 and Proposition 2.3 are merely simplified special cases under class balance, not prerequisites for imbalance handling.  \n\n**(2) Theoretical Scope:**  \n\n- The theoretical guarantee of unbiased representation learning under weak supervision (as formalized in Theorem 3.7: \"Learning from Weak Supervision Approximates Supervised Learning\") is inherently class-prior-invariant, requiring no assumptions about label distribution.  However, the downstream generalization analysis under supervised information Theorem 3.4 currently requires balanced assumptions for tractability. Hence, the adverse impacts of extreme long-tailed distributions on our framework remain unclear. \n  We will conduct further research on the framework's application under extreme long-tailed conditions in future work.\n\n\n>**W2: While t-SNE plots (Fig 2) suggest better representations, deeper qualitative insights are missing.**\n\nIn addition to Figure 2, we also theoretically analyze that the proposed method helps improve the quality of representation learning features.\n\n- Theorem 3.4 bounds linear probe error of feature derived from perturbation graph through properties of self-supervised connectivity graph and perturbation coefficients, and demonstrates that the perturbation graph increases the density of intra-class connections without adding extra-class connections.\n- In Theorem 3.7, we further show that the features learned through finite samples and weakly supervised information can approximate the features learned from supervised information.\n- Finally, we present the linear probe error of the features learned by our framework in Corollary 3.8, which can be regarded as a qualitative analysis of the learned features.\n\n>**Q2: What’s the computational overhead of building/maintaining the similarity graph compared to standard contrastive learning?**\n\nCompared with standard contrastive learning, our additional computational overhead is mainly reflected in the single-step matrix multiplication used to construct semantic similarity, that is, $S((\\tilde{x},\\tilde{q}), (\\tilde{x}',\\tilde{q}'))$. This step of calculation does not bring too much additional computational cost and is negligible compared to standard contrastive learning. It can be considered that the computational overhead of our proposed method is basically the same as that of standard contrastive learning.\n\n> **Q3: Are there scenarios where discrete labels would still outperform continuous similarity (e.g., clean labeled data)?**\n\n- The concept of continuous semantic similarity we proposed is designed for weakly supervised learning, and our method is very effective under weakly supervised information empirically.\n\n- Theorem 3.7 explains the difference between the two features. The third and fourth terms on the right side of the inequality characterize the error induced by the usage of weakly supervised information, where the third term can be regarded as the estimated error term, and the fourth term describes the extent to which the weakly supervised information affects the learned representation. If the supervision information is completely accurate, it can be seen from the analysis that our proposed framework can be approximately equivalent to supervised contrastive learning.", "claims_and_evidence": "1. Continuous similarity > binary labels for weak supervision\nThe graph-based approach replaces rigid class labels with similarity scores, allowing gradual refinement of supervision. Theoretical analysis (Prop 2.1, Thm 3.4) shows this approximates supervised contrastive learning under mild conditions. \n\n2. Versatility across weak supervision settings\nThe proposed framework adapts to both NLL (noisy labels) and PLL (ambiguous candidate labels). Results on CIFAR-10/100, CUB-200, and Clothing1M show consistent gains, especially under high noise/ambiguity.", "methods_and_evaluation_criteria": "- The proposed method builds a graph where nodes are augmented samples, and edge weights blend self-supervised similarity (from augmentations) and weakly supervised similarity (derived from labels/noise patterns).\n- The proposed method is tested on standard NLL/PLL benchmarks (CIFAR, CUB-200, Clothing1M) against 10+ baselines under varying noise/partial-label ratios.", "theoretical_claims": "I checked the proof of Thm 3.4 which seems okay. I skimmed through Appendix C (proof of Section 3) but did not check thoroughly.", "experimental_designs_or_analyses": "yes", "relation_to_broader_scientific_literature": "Weak supervision is very common in the field of machine learning literature, but contrastive learning with weak supervision has not been studied in prior works.", "essential_references_not_discussed": "All essential references are discussed.", "other_strengths_and_weaknesses": "## Strengths\n1. This paper directly addresses the \"label quality bottleneck\" in real-world data. The graph framework elegantly unifies self-supervised and weakly supervised signals.\n2. This paper proposes to connect the method to spectral graph theory, providing error bounds (Thm 3.4) and showing how label information improves clustering.\n3. Outperforms specialized methods (e.g., DivideMix for NLL, PiCO for PLL) even in high-noise regimes.\n\n## Weaknesses\n1. It seems this paper relies on uniform class distribution for theoretical guarantees—how does this hold for imbalanced data?\n2. While t-SNE plots (Fig 2) suggest better representations, deeper qualitative insights are missing.", "other_comments_or_suggestions": "None.", "questions_for_authors": "1. How does the method handle class imbalance, given the uniform class assumption in theory?\n2. What’s the computational overhead of building/maintaining the similarity graph compared to standard contrastive learning?\n3. Are there scenarios where discrete labels would still outperform continuous similarity (e.g., clean labeled data)?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uDetRbUq0h", "reviewer": "Reviewer_kufB", "summary": "The paper introduces a graph-theoretic framework for weakly supervised contrastive learning, leveraging continuous semantic similarity to better utilize ambiguous supervisory signals from imprecise class labels. This approach enhances model performance on multiple benchmark datasets in noisy and partially labeled settings.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer kufB:\n\nThank you for your valuable suggestions. We truly appreciate your review and are committed to addressing your concerns with care and attention. We sincerely look forward to engaging in a more in-depth discussion with you, as your insights are essential in helping us improve and refine our approach.\n\n> **W1 & Q1: There's a discrepancy between the caption and the content of Table 1.**\n\nWe are very sorry for this, this is a typo on our part, and we will correct the inconsistency between the caption and the table in the next revision.\n\n>**W2 & Q2: In Table 7, PiCO outperforms the proposed method with a significant performance advantage under the partial ratio setting of 0.1 on the CUB-200 dataset. The 5.89% performance advantage is remarkable. It would be beneficial to provide an explanation for why the proposed method performs poorly in this scenario.**\n\n- In addition to using contrastive learning to learn similar representations for samples from the same class, PiCO also uses class prototype-based pseudo-labeling mechanism. The synergy between the contrastive learning and the above mechanism has been proven to be very effective in several fine-grained classification tasks [1].\n- Our approach avoids the need for additional complex techniques. By incorporating our contrastive loss into the training objective, we achieve an 8.69% improvement over GFWS in this setting, highlighting the potential for even greater performance gains.\n- Although no other complex techniques are used, our method achieves the best performance in most of the remaining experiments, which also shows the effectiveness of our method in most scenarios.\n\nReference:\n\n[1] Partial label learning: Taxonomy, analysis and outlook. *Neural Networks* 161 (2023): 708-734.\n\n> **Q3: Lack of the validation on instance-dependent data on PLL.**\n\nFollowing the works in [1, 2], we have expanded our experimental results to include the settings of biased tags associated with four fine-grained instance-dependent partial label datasets. The table below demonstrates that our method achieves marginally better performance than the current state-of-the-art (SOTA) approaches, and notably, without relying on any additional tricks or techniques. In the next version of our paper, we plan to incorporate more comprehensive comparative experiments to further validate our findings.\n\n|         | WSC(Our)  |  CEL[1]   | DIRK[2] | NEPLL[3] | IDGP[4] |\n| :-----: | :-------: | :-------: | :-----: | :------: | :-----: |\n| CUB200  | **69.10** |   68.60   |  66.60  |  62.88   |  58.16  |\n| CARS196 | **87.96** |   86.22   |  85.31  |  85.05   |  79.56  |\n| DOGS120 | **79.65** |   78.18   |  75.97  |  74.84   |  74.86  |\n| FGVC100 |   77.94   | **78.36** |  76.86  |  75.36   |  72.48  |\n\nThe experimental details are shown in the table below ($\\mathrm{resnet34}^{*}$ denote resnet34 pre-trained on Imagenet)：\n\n| Hyper-parameter | CUB200 | CARS196 | DOGS120 |FGVC100|\n| :---: | :---: | :---: | :---: | :---: |\n| Model | $\\mathrm{resnet34}^{*}$ | $\\mathrm{resnet34}^{*}$  | $\\mathrm{resnet34}^{*}$ |$\\mathrm{resnet34}^{*}$|\n| Image Size | 224 | 224 | 224 |224|\n| Batch Size | 256 | 256 | 256 |256|\n| Learning Rate | 0.1 | 0.1 | 0.1 |0.1|\n| Weight Decay | 1e-4 | 1e-4 | 1e-4 |1e-4|\n| LR Scheduler | Cosine | Cosine | Cosine |Cosine|\n| Training Epochs | 500 | 500 | 500 |500|\n| Classes | 200 |196 | 120 |100\t|\n|$\\alpha$|2|2|2|2|\n|$\\beta$|0-400|0-400|0-240|0-200|\n\n> **Q4: Only one state-of-the-art (SOTA) method published in 2024 is adopted for comparison, while the others were published in 2022 or earlier.**\n\n- Recent advances in partial label learning have primarily concentrated on instance-dependent partial label problems, exemplified by several notable methodologies: CEL [1], a class-wise embedding-guided disambiguation approach; DIRK [2], a self-distillation-based label disambiguation framework; NEPLL [3], which employs normalized entropy for sample selection; and IDGP [4], a generative method modeling candidate label generation processes. \n- While these methods provide thorough investigations into label disambiguation mechanisms for instance-dependent scenarios, they notably neglect the critical exploration of leveraging ambiguous information for representation learning—a research gap addressed by our work. Comprehensive comparative experiments under instance-dependent partial label settings will be included in our subsequent manuscript version, accompanied by detailed discussions in related work sections.\n\nReference:\n\n[1] Mixed Blessing: Class-Wise Embedding guided Instance-Dependent Partial Label Learning (KDD'2025)\n\n[2] Distilling Reliable Knowledge for Instance-Dependent Partial Label Learning. (AAAI'2024)\n\n[3] Candidate-aware Selective Disam-biguation Based On Normalized Entropy for Instance-dependent Partial-label Learning. (ICCV'23)\n\n[4] Decompositional Generation Process for Instance-Dependent Partial Label Learning. (ICLR'2023)", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "I have gone through the theorem and found no obvious errors", "experimental_designs_or_analyses": "The paper experimentally evaluates its weakly-supervised contrastive learning framework through extensive testing on noisy and partial label datasets (e.g., CIFAR-10 and CIFAR-100), demonstrating performance improvements over existing methods via quantitative metrics and t-SNE visualizations of learned representations. \n\nHowever, for PLL, it is better to add the experiments on instance-dependent PLL data.", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:\n1. The paper introduces a graph-theoretic framework for weakly supervised contrastive learning named WSC.\n2. Extensive experiments demonstrate that WSC consistently outperforms existing baselines across diverse noise levels.\n3. The manuscript is clearly written and well-structured, ensuring ease of readability.\n\nWeaknesses:\n1. It looks like there's a discrepancy between the caption and the content of Table 1. The caption mentions that both the mean and standard deviation values are reported, but only the mean values are shown.\n2. In Table 7, PiCO outperforms the proposed method with a significant performance advantage under the partial ratio setting of 0.1 on the CUB-200 dataset. The 5.89% performance advantage is remarkable. It would be beneficial to provide an explanation for why the proposed method performs poorly in this scenario.\n3. For PLL, the authors lack of the validation on instance-dependent data on PLL.", "other_comments_or_suggestions": "Please refer to the weakness", "questions_for_authors": "1. It looks like there's a discrepancy between the caption and the content of Table 1. The caption mentions that both the mean and standard deviation values are reported, but only the mean values are shown.\n2. In Table 7, PiCO outperforms the proposed method with a significant performance advantage under the partial ratio setting of 0.1 on the CUB-200 dataset. The 5.89% performance advantage is remarkable. It would be beneficial to provide an explanation for why the proposed method performs poorly in this scenario.\n3. For PLL, the authors lack of the validation on instance-dependent data on PLL. \n4. Only one state-of-the-art (SOTA) method published in 2024 is adopted for comparison, while the others were published in 2022 or earlier. It would be better to adopt more SOTA methods as comparison methods to further validate the effectiveness of the proposed method.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7zSwThxAfM", "reviewer": "Reviewer_LHXf", "summary": "This paper proposes a graph-theoretic framework for contrastive learning with weakly-supervised information. This framework is recognized as effective according to the superior results in noisy label learning and partial label learning by introducing the continuous semantics similarity to define positives and negatives. Theoretical analysis shows that the framework could approximate supervised contrastive learning under mild conditions.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer LHXf:\n\nThanks for your valuable suggestions, we will try to address your concerns and we are eager to engage in a more detailed discussion with you.\n\n> **W1: Lack of in-depth analyses of experiment results...**\n\n- We analyze the effectiveness of the proposed method in detail in the theoretical understanding of Section 3. It can be roughly understood in the following order:\n  - **Supervised information improves representation learning.** Theorem 3.4 describes how much the quality of learned features can be improved when using accurate supervised information. It can be seen that the features learned by the graph constructed with supervised information usually have a smaller error upper bound than those learned by only self-connection.\n  - **Weakly supervised information can approximate supervised information.** Through Theorem 3.7, we try to show that the features estimated using weakly supervised information can approximate the features obtained by supervised information to a certain extent. This theorem also indirectly illustrates how the proposed method works in weakly supervised scenario.\n- Your proposal made us realize that there is a lack of some supplementary explanations between our theoretical analysis and experimental verification. We will emphasize the reasons why the model behind the theoretical results is valid in the experimental part when the revised version is uploaded.\n\n> **W2: Some typos.**\n\nWe are very sorry for this. We will check typos carefully and correct them in the next revision.\n\n> **Q1: Question on hyperparameter settings.**\n\nThe setting of the hyperparameters $\\alpha$ and $\\beta$ is related to the number of categories in the dataset. Denote $C$ is the number of class, setting the ratio of $\\beta$ to $\\alpha$ at $2C$ can achieve a promising performance in general.", "claims_and_evidence": "Yes, this paper provides sufficient theorems and outstanding experimental results to support the convincing claims.", "methods_and_evaluation_criteria": "Yes, the proposed method adopts a novel view of contrastive learning by framing it as a graph spectral clustering problem, and the edge weights are constructed based on self-supervised connectivity and weakly-supervised information, thereby learning expressive weakly-supervised representations.", "theoretical_claims": "I have checked all the theoretical claims and proofs, which are correct in my angle. However, there are still some typos in the proofs.", "experimental_designs_or_analyses": "The experimental designs adopt two weakly-supervised learning paradigms and several well-known datasets to validate the superiority of the model. However, the analyses are not in-depth enough.", "relation_to_broader_scientific_literature": "This paper is relative to the field of weakly-supervised learning. The excellent experiment results and rigorous theorems may contribute to the broader scientific literature.", "essential_references_not_discussed": "The key references are provided to follow the main idea of the paper, and I have no essential references to provide.", "other_strengths_and_weaknesses": "Strengths\n- The technique is novel. This paper proposes a graph-theoretic framework for weakly-supervised contrastive learning, which mines the relationships between sample pairs by integrating the self-supervised and weakly-supervised information into the augmentation graph.\n- The theoretical analysis is sound. This paper provides sound theorems to explain how to realize weakly-supervised contrastive learning, and offers a rigorous performance guarantee to show that the proposed framework can approximate supervised learning.\n- The model performance is outstanding. Many SOTA models are adopted to compare with the proposed WSC, and the experiment results show the superiority of WSC in both noisy label learning and partial label learning.\n\nWeaknesses\n- Lack of in-depth analyses of experiment results. The analyses in Section 4.1 and Section 4.2 just focus on the performance improvements, but do not explain why the model has wonderful performance, e.g., how the model works in the weakly-supervised scenario. So more in-depth analyses should be provided.\n- Some typos. There are some typos in the paper, for example, $\\mathcal{D} and \\mathcal{D}_{\\mathcal{Q}} $ in line 625.", "other_comments_or_suggestions": "No, I have no other comments or suggestions.", "questions_for_authors": "- Question on hyperparameter settings. I find that there exist significant differences in the numerical settings of $alpha$ and $beta$, so I want to know how to balance the self-supervised and supervised information in Eq. (5), and find ideal $alpha$ and $beta$ jointly.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes WSC (Weakly-Supervised Contrastive Learning), a graph-theoretic framework that replaces rigid binary positive/negative sampling in contrastive learning with continuous semantic similarity. By blending self-supervised augmentations with weak supervision signals (e.g., noisy/partial labels), WSC achieves state-of-the-art performance in noisy label learning (NLL) and partial label learning (PLL). Theoretical analysis shows WSC approximates supervised contrastive learning under mild conditions.

WSC makes a methodological contribution by bridging graph theory and weak supervision, with strong empirical/theoretical validation. The rebuttal strengthened the paper’s rigor and scope, addressing all major critiques.
Decision: Accept (spotlight poster)