Evaluate then Cooperate: Shapley-based View Cooperation Enhancement for Multi-view Clustering
Authors: Fangdi Wang
Jiaqi Jin
Jingtao Hu
Suyuan Liu
Xihong Yang
Siwei Wang
Xinwang Liu
En Zhu
Venue: NeurIPS 2024 poster (2024)
Keywords: Multi-view Clustering
Clustering Network
Primary area: learning_theory
TL;DR: This paper evaluates the contributions of views in the task of unsupervised multi-view clustering and enhances model performance through a cooperation enhancing approach.

Abstract: The fundamental goal of deep multi-view clustering is to achieve preferable task performance through inter-view cooperation. Although numerous DMVC approaches have been proposed, the collaboration role of individual views have not been well investigated in existing literature. Moreover, how to further enhance view cooperation for better fusion still needs to be explored. In this paper, we firstly consider DMVC as an unsupervised cooperative game where each view can be regarded as a participant. Then, we introduce the Shapley value and propose a novel MVC framework termed Shapley-based Cooperation Enhancing Multi-view Clustering (SCE-MVC), which evaluates view cooperation with game theory. Specially,  we employ the optimal transport distance between fused cluster distributions and single view component as the utility function for computing shapley values. Afterwards, we apply shapley values to assess the contribution of each view and utilize these contributions to promote view cooperation. Comprehensive experimental results well support the effectiveness of our framework adopting to existing DMVC frameworks, demonstrating the importance and necessity of enhancing the cooperation among views.

Evaluate then Cooperate: Shapley-based View Cooperation Enhancement for Multi-view Clustering
Fangdi Wang', Jiaqi Jin', Jingtao Hu²,
Suyuan Liu', Xihong Yang', Siwei Wang";
Xinwang Liu En Zhu
National University of Defence Technology 2 Academy of Military Science
(wangfangdi 19,, wangsiweil3, xinwangliu, enzhu @nudt.edu.cn
Abstract
The fundamental goal of deep multi-view clustering is to achieve preferable task performance through inter-view cooperation. Although numerous DMVC ap- proaches have been proposed, the collaboration role of individual views have not been well investigated in existing literature. Moreover, how to further enhance view cooperation for better fusion still needs to be explored. In this paper, we firstly consider DMVC be regarded as a participant. Then, we introduce the Shapley value and propose a novel MVC framework termed Shapley-based Cooperation Enhancing Multi-view Clustering (SCE-MVC), which evaluates view cooperation with game theory. Spe- cially, employ the optimal transport distance between fused cluster distributions and single view component as the utility function for computing shapley values. Afterwards, we apply shapley values to assess the contribution of each view and utilize these contributions to promote view cooperation. Comprehensive experi- mental results well support the effectiveness of our framework adopting to existing DMVC frameworks, demonstrating the importance and necessity of enhancing the cooperation among views.
unsupervised cooperative game where each view
Introduction
Recently, multi-view clustering has become one of the most prominent problems in unsupervised
learning [
leverages data from different views, combining them to provide richer information for clustering tasks. Generally speaking, multi-view clustering can be broadly categorized into traditional methods
2
4
[
7 and deep multi-view clustering (DMVC) [5] 9
IO
11
12 [3]
15
16. 17 [8 [9] methods. Compared to the limited feature extraction capabilities of traditional methods, deep multi-view clustering methods better preserve information from different views through flexible deep neural networks. For most DMVC methods, the key challenge is how to enhance cooperation among views to accomplish better clustering performance (20).
Numerous DMVC algorithms have already achieved significant results in downstream clustering tasks. However, there has been little research on investigating the the contribution of each view in the fusion stage. Once evaluating the view contribution, we may observe the phenomenon: One view dominates the fusion process, suppressing the collaborative contribution of other views, leading to suboptimal clustering results. We attribute this phenomenon to insufficient cooperation between views, and thus propose a fundamental assumption: More balanced contributions and more extensive cooperation among multiple views can lead to better clustering results, shown in Fig. I
"Corresponding author
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Caltech 101chh111777 datasct
UCI-digil dataset
20
0.94
0--
0.02
View Comtribution
0.55
View Costribution
0.
0.50
0.88
0.45
Fusion
0.4
Post
04-
0.88
0.35
0.64
0.82
02
00
20
60
0.50
Epochs
Brochs
View 1,2 Contribution with SCE
View 1,2.3 Cortribution with SCE
View Contrization without SCE
View 1,23 Contribution without SCE
ACC with/ without SCE
RusionCCC with/ without SCE
(b)
Figure 1: Performance evaluation view cooperation contribution and fusion ACC on (a) Caltech101- 7 (b) UCI-digit. As can be seen, proposed SCE framework clearly promotes inter-view cooperation and achieves more satisfactory performance.
To this end, we consider introducing the concept of game theory into multi-view clustering and propose the Shapley-based Cooperation Enhancing (SCE) method. Shapley value [21 22 is concept used measure the contribution of participants in game theory [3] 24]. It utilizes utility function to evaluate the different marginal contributions that each participant brings when combined with other participants, and calculates their contribution to the overall cooperation by taking the weighted average over all possible combinations of participants (2D. Upon quantifying the contribution of each view, the cooperation among views can be analyzed based on their respective contributions. By employing module that enhances view cooperation, convergence rate of views during training in accordance with their contributions. This enables the underrepresented views to receive more balanced training regimen, thereby enhancing post-fusion performance 25
can dynamically adjust the parameter
The primary contributions of this work can be summarized as follows:
Our work is the first framework evaluate the view contributions within unsupervised con- text. Leveraging the proposed View Contribution Evaluation Module, we theoretically quantify the individual contributions of views.
We propose the View Cooperation Enhancing Module based on the view contributions obtained by the View Contribution Evaluation Module, allowing suppressed views to effectively participate in the fusion process, ultimately enhancing clustering performance.
Comprehensive experiments have been conducted to showcase the versatility of our SCE framework, which can effectively evaluate the contributions of views across various MVC frameworks.
Related Work
2.1 Deep Multi-view Clustering
In deep multi-view clustering [26] 27 28 29), deep neural networks(30) 31 with multiple nonlinear
transformations are more effective at acquiring feature representations than traditional shallow models. Presently, deep multi-vies clustering methods fall into three main categories 32
33 joint
methods (34 33] 36). alignment- based methods [4 37 38 39), and other methods [00 41 Joint methods consider differences and complementarities among views, optimizing sample representations within each view's space. For example, the 35 utilizes autoencoders to generate view representations, refining them pseudo-labels for self-supervised optimization. Alignment-basee methods focus on consistency between views, mapping representations into a shared subspace. These methods often employ contrastive learning, such ProImp[43), which integrates contrastive learning

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at both sample and prototype levels. Some approaches combine joint and alignment-base based methods, like the -hare-specific method 40), which separates sample representations into shared and specific components, aligns shared representations into common space, and refines specific representations in separate spaces.
2.2 Rethinking of Cooperation in DMVC
The core concept of DMVC is to enhance clustering performance by fostering enhanced cooper- ation and utiliizigg complementary information across views. Although existing DMVC research emphasizes leveraging inter-view cooperation for improved clustering tasks, there is currently a lack of research quantifying the contribution of views during the fusion stage to enhance cooperation. Multi views and multi modalities [44] are two closely intertwined domains. Within the realm of multimodal fusion, studies have surfaced highlighting the issue of subpar collaboration among modalities, prompting endeavors to alleviate the imbalance in modality cooperation. Wang et al. [45] uncovered varying convergence rates among diverse modalities, leading scenario where jointly trained multimodal models struggle to match or outperform their unimodal counterparts. Furthermore, Peng et al. [25] demonstrated that in the context of audio-visual learning, superior performance in the audio modality hampers the optimization of the video modality.
Drawing inspiration from prior studies, research aims to evaluate the individual contributions of each view mulli-view clustering fusion. Nevertheless, evaluating view contribution based solely on labels in unsupervised environments is infeasible. Moreover, employing weights measure for view contribution may not align with all DMVC structures, especially those based on contrastive learning frameworks. Consequently, quantifying view contributions in fusion and fostering their cooperation pose significant research challenges.
2.3 Shapley Value: a Method for Evaluating Contribution.
In order to reasonably and effectively evaluate the contributions of each view in the fusion process, we refer the theoretical framework of game theory. The Shapley value [46] within game theory proves instrumental in quantifying participants' contributions to cooperative coalitions. Denote X' x= (x, } as the alliances of n participants. For a participant x;, the Shapley value considers 2; contribution in every set that includes him. Let S, {S C X' E S} represent the set of all subsets of that include the participant 2i, then, the overall contribution of member Fú, i.e. the Shapley value, can be expressed as
Shapley, = Σ (Is) - 1)1(n - |s|) (u(s) - v(s((t))).
(1)
n!
SES;
where |s| signifies the cardinality of set v(-) represents the utility function; [u(s) - v(s{{it)
quantifies the marginal contribution of x; to set 8; and (lsl-1)((n1el)) denotes the weight assigned to this marginal contribution, determined by the probability of set 8 occurring.
nd
Given the advantageous characteristics of Shapley values-efficiencyy symmetry, dummy, and ad- ditivity- have increasingly employed them for explaining machine learning models. For instance, Lundberg al. [47] proposed method to interpret predictions by computing feature importance based on Shapley values. Hu et al.
[48] applied Shapley values to evaluate modality contributions in supervised tasks, while Wei et al. [49 explored modality assessment using Shapley values. However, these studies focus mainly on supervised scenarios, making the definition of utility functions for Shapley values in unsupervised settings challenging task yet to be resolved.
3 Method
3.1 Problem Statement
Given a multi-view dataset X' = (X (x,,,,,,XXV)) where V is the number of views, and (( = 1, ..., V) denotes the original feature space of the e-th view. Consider the clustering problem on the dataset, each cluster is represented by clustering center u.)" .j I,.,, K, where K is the number of clusters. To achieve better clustering performance, we map the original fea- ture space (v) to latent embedded feature space (()) through non-linear mapping function
3

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Basehine MVC Modill
E-Shaple--baeee based View Concribution Evaluation Modille af
Stuples -eapl Vew Costriluaaco
Decideer
Decidder
Shaplen boaed hertri
Data Distribution
Vient Vana B Vier Vinws
Baanie Macck
KARAK
6.6.
RARAN
6.666
Figure 2: The framework of our proposed SCE model. SCE framework can be applied to most mainstream DMVC methods. After obtaining the cluster distribution from the selected method, SCE iteratively using the following two modules: View Contribution Evaluation Module and View Cooperation Enhancing Module. View Contribution Evaluation Module computes the marginal contribution views in each combination(Shaple value), in which the optimal transport distance between the view cluster distribution and the strengthened cluster distribution after fusion serves the utility function of Shapley formula. View Cooperation Enhancing Module controls the convergence speed ratio of different views through the calculated view contribution.
(1)
(²)
((),, where ((+) represents the learnable parameters of mapping function for the
g(o)
u-th view.
3.2 View Contribution Evaluation Module
Modified Shapley Value for Multi Views
To assess the contributions of each view MVC tasks, we treat each view as player and introduce the Shapley value referring Eq. (1) Let X (x(0)) J be the set of all views, S, {S C
(*) S} represent the set of all subsets of that include the y-th view (()), Define an internal metric of clustering E independently of labels to serve as a utility function, which will be elaborated
on in the next subsection. Then the Shapley value of the w-th view can be calculated as
([s) 1)((V |s|)
Shapley. =
Σ
E(s) (eb)-E(sS-S ((()(X((1))
(2)
VI
SES.
where |s| denotes the number of views in the view-set
Utility Function in Shapley Value
In Shapley value, the utility function is used to evaluate the benefit created by coalition. In DMVC. where each view acts as a participant, the utility function needs to be designed to reflect the contribution of any combination of views in fusion, without the need for label assistance. In this scenario, we constructed a novel function E using the following method.
Inspired by the DEC model [50]. we assume that the data from each view follows a Student's 's t- distribution. Following the approach outlined by (5T]. we utilize the t-distribution as kernel function to evaluate the distance between the sample embeddings and the cluster centers μ4, within view:
(1 ||, 1, 11,11//a-- ديج (1 ||, M4 13/a)
di
where a represents the degrees of freedom in the 1-distribution. qy denotes the probability of assigning the feature 2; to the cluster center pjj

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Fix the degrees of freedom a to 1, the cluster distribution of multi-view data can be considered as combination of single-view cluster distribution based on weight matrix π.:
Σ. (0) (p)
1 1 119)-1
q: Σ 1' (1+
(4)
11))
where the weight matrix RRxV represents the importance of the cluster centers μ4 in different views. This weight matrix π can be obtained by performing row normalization on an unconstrained trainable matrix W:
(0)
(5)
For any given combination of m-view union U weight matrix n² by selecting the corresponding columns in the matrix W. By applying Eq. (3), we derive the data distribution for the fused view, which is denoted q²t
(x)),)
(
(Mm))), obtain the
In particular, uf 0, which means no view is involved in the clustering process, we consider q to be uniform distribution. This means that the probability of any embedding belonging to any cluster μi is equal and given by 1/K, where K is the total cluster numbers. this case, we denote the resulting data distribution qº as
1NNKS q --= K
1
(6)
where 1NNK N a matrix in which all elements are equal to 1.
According to the principle of self-distillation in DEC, we sharpen the fused distribution q from all views to obtain more powerful target distribution p:
P.y (g:))) E9
Since consider the distribution p the trend of the fusion process, the closer the fusion distri- bution q² (combined with U = (ux) ((2), ,..., ( (4m) }) participation view set in the fusion process. Therefore, we can use the optimal transport distance from distribution q" to distribution p measure of confidence for distribution q²é Additionally, based on our desire for the metric to possess monotonicity and boundedness, we define the utility function based on distribution transportation as:
is to the distribution p. the higher the
B(U) = + OT q")
where OT(:)) is calculated as wasserstein distance and E(-) E (0, 1]. The closer the data distribution to the pseudo-labels p. the larger the value of the metric E(.).
In the process of computing the optimal transport OT(:,), we consider the distribution q as probability set where K cluster centers are assigned by N samples. The probability of each cluster center is 1 K.. For example, when calculating OT (T(q(x()) .q (x(())) the distribution of views can be represented as [1/K, 1/ K,.., 1. IKK", and the distance matrix (Di, Pigre kxk can be calculated as
Dú, =I E (1) (2) 12,
where the j-th cluster center for the w-th view can be represented as c(0) = } JNNI.
Shapley-based View Contribution: Formal Representation
In the above subsections, we have already discussed the integration of cooperative game theory with multi-view clustering tasks. We have also defined utility function E that reflects the accuracy of clustering. Now, combining the Eq.
2 and (8), we present the formal representation of view-
contribution through Shapley values:
Shapley, Σ
|s
1[((V
|s|)!
1 1 OT(p.')) + OT(p, q((001))))))) (10)
1
V!
SES.
5

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The sum of all view contributions is then calculated as:
Σ Shapley, = + (T( 1 OT qd)
(11)
As p and qr evolve throughout training, the of Shapley values for all views also fluctuate accordingly. To obtain the contribution proportions of each view to the fused view, let
Shapley.
Pv E Shopley,
(12)
where You is the normalized Shapley value, determining the respective contribution of different views.
3.3 Theoretical Analysis
To further demonstrate the generalization performance of the View Contribution Evaluation Module, we apply it to the alignment-basse method and the joint method, respectively. The relevant theoretical derivations are given below.
Theorem 1: For alignment- methods, considering the representation (z.") and { I on two views, with the infoNCE contrastive learning loss:
exp(z: 2
4
log
(13)
exp(z) (1)
2j /n) exp(z!
(1)
23 4/1//m)
N 2N
con (1,2)
(14)
==1
where N is the sample number and 0 scalar temperature hyperparameter. lcon (1, 2) brings the representations between views closer, leads to the average of view contributions, i.e., (01 62 - 0.
Theorem 2: For joint methods, considering the representation ( = gge
= (x ((1) and (
f62 (2), ((2)) on two views, with a simple view-level weight fusion during the fusion process:
1i Z (1) + Z (2) There are situations where view 2 is dominated by view I and cannot make its contribution, i.e.. w1 W2-
Remark: Theorem indicates that alignment-based methods tend to align the contributions of different views. Under our assumption, the cooperation between views is already sufficient, optimizing the model based on contributions can only lead limited improvements. Theorem 2 suggests that in joint methods, one view may suppress another, highlighting insufficient cooperation between views. In such cases, enhancing cooperation between views theoretically improves model performance. We will elaborate on the View Cooperation Enhancing Module in Section 3.4 and validate these conclusions through experiments. The proofs for Theorems and can be found in Appendix.
3.4 View Cooperation Enhancing Module
After acquiring contributions of different views, seek to enhance their cooperation by dynamically regulating views' training speeds according to these contributions. This proportional control guaran- tees active involvement of all views in the integration process. For this purpose, we introduce the convergence speed ratio, labeled as k, derived from the contributions of each view.
To ensure the applicability of our Shapley-bard based Cooperation Enbancing(SCE) method to most multi-view clustering tasks, we consider a typical multi-view clustering framework: firstly, obtain embeddings for each view by utilizing autoencoders; then, fuse the embeddings from different views using global optimization goal. Let ((o)
(X( (()),(()) represent the encoder model for the v-th view, g denote the fusion method for multiple views, and represent the loss function for computing the fusion loss. The overall fusion loss L be represented as
L(,,) (2 22,0,))))
fac
(2)
(VV)
Jo (x (()))()))))
(15)

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In practice, the parameters for the e-th view is updated as
0!!)) geo L(
(16)
With the normalized Shapley values calculated by Eq. (1) to dynamically monitor the contribution differences between different views, we are able to adaptively modulate the gradients via an empirical formula:
kg e
(17)
where r a temperature hyperparameter to control the degree of modulation. Thus k, is not greater than 1, and the larger the contribution of the view, the smaller the kn. We integrate the coefficient k; into the Adam optimization method, and the update of 0(4) at iteration t follows:
0 B!!)
ky 1l
al
0g 0f.
(18)
0g 0f. o0(")
where K2 is used as a relative convergence speed ratio for the v-th view in gradient descent.
Theorem 3: For joint methods that use view-level weight fusion Z. w/ (, + wz where
( f00 (x()) () and (2) = (22) (x g(2)), gradient modulation in Eq. (8 allows the two
views to contribute more evenly, i.e.
Theorem indicates that through dynamic gradient adjustments, we can harmonize the learning progress among different views, attain a more average distribution of contributions, and enhance the cooperation among views. The proof for Theorem can be found in Appendix.
Remark:
Algorithm 1: Shapley-based Cooperation Enhancing Multi-view Clusterig(SCE-MMCC)
Input:
Multi-view dataset [X(*))= temperature hyperparameter
Output: cluster labels 9
// Warm-up training
Obtain initial cluster assigmments through a selected algorithm;
// Optimization with SCE module while the fusion loss has not converge do
// View Contribution Evaluation
Obtain shapley value for each view with Eq.
and 10):
Normalize the shapley value view contribution with Eq. (12):
// View Cooperation Enhancing
Calculate the convergence speed ratio k, for each view with Eq. 17
Update the parameters of each view relatively with Eq. (88)
return;
3.5 Implementation
This subsection outlines the implementation our SCE module, detailed Algorithm obtaining initial cluster center assignments, we iteratively employ the following two modules:
[ After
View Contribution Evaluation. First, we calculate the cluster distribution for each individual view as well as the combined view. Next, we compute an enhanced distribution p. After utilizing the optimal transport distance between distributions as the utility function, we can calculate view contributions.
View Cooperation Enhancing. We utilize the calculated contribution values to influence the training process of each view using a convergence ratio k. The larger the contribution value of a view, the less the convergence ratio & will be. This will coordinate the training process of the views, enabling better cooperation among views, and allowing them to collaborate more effectively.
4 Experiments
In this section, we implement experiments on alignment- method and joint method to verify the effectiveness of the proposed thcorems and SCE module by addressing the following questions:
7

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RQ1: For alignment-basee based MVC methods, do the contributions of views calculated by the View Contribution Evaluation Module exhibit relative uniformity. thereby validating Theorem 1?
RQ2: In the case of uniform view contributions, can model performance be further enhanced by the View Cooperation Enhancing Module?
RQ3: For joint methods, are extreme view contributions present, thereby validating Theorem
RQ4: Can model performance be enhanced by harmonizing view contributions through View Cooperation Enhancing Module, thereby validating Theorem 3?
RQ5: How do the
impact the performance of SCE-MVC?
Table Dataset summary.
Dataset
Views
Samples
Clusters
Dataset
Views
Samples
Clusters
CUB
2
600
10
UCI-digit
3
2000
10
Caltech101-7
2
1474
7
STLI0
4
13000
10
HandWritten
3
2000
10
Reuters
5
1200
6
Table 2: Analyzing view contribution under InfoNCE=Kmmenns and Prolmp frameworks on CUB and Caltech101-7 datasets.
View Contribution
Metries NMI
Dataset
Method
Φ1
d2
ACC
ARI
InfoNCE+K Kmenns Kmeans
0.491
0.509
0.715
0.748
0.626
InfoNCE-Kmeanstrr
0.493
0.507
0.717
0.747
0.626
CUB
433
0,556
0.443
0.825
0.756
0.671
ProIMP+SCE
0.484
0.516
0.832
0.762
0.678
MHloNCE+Kmeang
0.484
0.516
0.351
0,486
0.272
InfoNCE+ KMC
0.496
0.504
0.364
0.485
0.281
Caltech101-7
ProlMP +3
0.489
0.511
0.382
0,468
0.281
ProIMP+SCE
0.499
0.501
0.382
0,470
0.279
4.1 Datasets, Metrics and Experimental settings
Our experiments utilized six multi-view datasets, including CUB Caltech101-7 HandWritter UCI-digi STL1 and Reuter? The detailed information these datasets is listed in the Table Meanwhile, three widely used metrics are adopted in our experiment, including clustering accuracy (ACC), normalized mutual information (NMI) and adjusted rand index(ARI).
For fairness, We conduct all experiments on PyTorch platform using the NVIDIA 2060 GPU. Besides, ten state-of-the-art MVC methods are introduced: DEMVC 40MVC[]4), SiMVC SDSNE[15) 16) SDMVC[[7], DSMVC APADC 52). DMIC and Prolmp 43
4.2 Evaluate Alignment-based Methods(RQ1 & RQ2)
In this section, we apply the SCE methods to two alignment baasmeet--baeddd MVC methods:
Utilizing infoNCE contrastive loss and employing Kmeans after concatenation.
Incorporating contrastive learning both the sample and prototype levels as ProIMP
We applied these two approaches to the CUB and Caltech datasets, each comprising two views. Detailed experimental results presented the Table 2 The results lead following conclusions:
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Collection

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Table 3: Miew clustering performance on six benchmark datasets. The optimal results are marked in bold, and the suboptimal values are underlined. O/M denotes out-of of-omoormooyy error encountered during the training process.
Calecatu1
Meties
NMI
NMI
DENIV
0.30
0.88
6377
D308
0275
0.900
0077
CoxvE 14
0.30
0.355
0152
0355
0.50
0228
0365
0.39
0.55
0.30
0.355
0.37
02255
0.163
0055
0.307
0,003
0.189
S.NVC 141
13523
0.258
0136
0.32
0435
0252
0285
0.166
0335%
0.35
0.21
0.359
0.190
0083
0090
03356
0.304
0.99
SO8NE 13
0.368
0.508
0.361
0.775
0.787
0.70
0846
0.891
0.516
0.766
0.878
0.819
00
0M
00
0233
0.200
0.077
MILVV 15
1.001
0323
026
0203
0322
0.89
0920
0333
0.70
0302
0.835
0.557
1211
0.24
0137
0.399
0,300
0207
SESVV
17
0.366
0.335
0233
0.62
0.64
0.517
0630
0.643
0.001
0.495
0.519
0.390
0283
0.362
0156
0.77
0.047
0113
DSMVC
0.454
0.92
0.000
0855
0.807
0911
0.540
031
0275
0.102
0.22
521
0.305
0.71
0278
DMICC13
Viewi
Viewz
View3
View4
w/o SCE
W. SCE
w/o SCE
0.5
Caltechios
W/ SCB
01b
UCI-digit
8
0/b
ST1.00
w/o SCE
0.69
SCB
0.3
06
View Contribution
Caltechton UCT-digit
StLao
(a)
(b)
Figure 3: (a) View contribution is more balanced with SCE. (b) Fusion performs better with SCE.
Assessed with the View Contribution Evaluation Module, the contributions of the two views in alignment-bass based methods are remarkably similar, validating the assertion of Theorem
As the contributions of the views tend to equalize, the utilization of the View Cooperation Enhancing
Module approach only yields subtle improvements the alignment-based methods.
4.3 Evaluate then Cooperate Joint Methods(RQ3 & RQ4)
In this subsection, we apply the SCE model to a joint MVC methods We have carried comprehensive experiments
six datasets with varying numbers of views, as depicted in Table and Fig. [) In Table 3] we showcase the performance of SCE-MVC in comparison to ten benchmark methods. Fig. illlustrates the ablation study of DMJC with and without SCE. Based on the aforementioned experimental results, we draw the following conclusions:
Fig. 3(a) illustrates the changes in view contributions with and without the SCE module across three datasets. Prior to the use of the SCE module, there a scenario where certain views dominate while others suppressed, a pattern particularly evident in the Caltech101-7 dataset, aligning well with the conclusions of Theorem 2.
After introducing the View Contribution Enhancing Module, the contributions of the views become more balanced, leading to a significant improvement in the model's performance. Specifically, the ACC, NMI, and ARI metries the Caltech101-7 dataset all witness increase of over 10 percentage points, thus validating the conclusion of Thcorem 3. More detailed contribution comparison data can be found in the Appendix.
Table 4: Sensitive analysis on UCI-digit and STL10 datasets. The optimal results are marked in bold.
0.5
1.0
1.5
Datasets UCI-digit
ACC
NMI
ARI
ACC
NMI
ARI
ACC
NMI
ARI
0,871
0.825
0.767
0.882
0.830
0.777
0.886
0.834
0.788
STL10
0.303
0,257
0.152
0,330
0.256
0.162
0.320
0,259
0.159
2.0
2.5
3.0
Datasets
ACC
NMI
ARI
ACC
NMI
ARI
ACC
NMI
ART
UCI-digit
0.903
0.842
0.805
0.927
0.865
0.847
0.930
0.860
0.842
STL
0.317
0.263
0.160
0.319
0.260
0.160
0.317
0.259
0.160

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4.4 Sensitive Analysis(RQ5)
In SCE-MVC, the singular hyperparameter essential for fine-tuning the convergence equilibrium among distinct views according to their contributions. In sensitive analysis, we varied T within {0.5, 1.5, 2, 2.5,3), as illustrated in Table 4 As increases, the changes in clustering results metrics become less pronounced, which can be attributed to the characteristics of the exponential functions in Eq. Specifically, as increases, the convergence ratio ky tends to be closer zero.
Conclusion
This paper explores the roles of views in multi-view clustering through the lens of game theory. We introduce novel contribution evaluation module founded on the Shapley value. Our experiments showcase the adaptability of this module across mainstream MVC frameworks, accurately assessing each view's impact. By scrutinizing the functions of diverse views during fusion, we devise cooperation enhancing module to bolster cooperation among views. Extensive dataset validations underscore the effectiveness and robust applicability of our approach and theorems.
Acknowledgments and Disclosure of Funding
This work is supported by National Natural Science Foundation of China under Grant No. 62276271, 62325604, 62406329, 62476280, 62476281 I and National Science and Technology Innovation 2030 Major Project under Grant No. 20222D0209103.
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A Proof of Theorems
A.1 Proof of Theorem
Theorem 1: For alignment-based methods, considering the representation {z ) and (2 )
)i=1
on two views, with the infoNCE contrastive learning loss:
exp(z; (1) 7 (T)
(2)
e
log
(19)
j= exp(z (1)
(11) /m) exp(z (1) 1.00 2//0/1))
Loon (1,2)
(20)
2N
where N is the sample number and 0 scalar temperature hyperparameter. loon (1, 2) brings the representations between views closer, leads to the average of view contributions, i.e. (01 d2 +
Proof: Referring to Definition of Perfect Alignment in [5], we say the two views are perfectly
aligned if x x!(² i 2, ..... N. Assuming that with the training of contrastive learning, the representations of the two views tend to align perfectly. Then, there exists limiting representation
Z* such that lim 2 - lim [| Z- O.
Due to the convergence of sample representations from both views, the centroids and sample dis- tributions of the views also tend converge, i.e. lim [n(1) م lim | μ and lim ( - q''l lim ( q'll = O, where μ³ and q" represent the clustering centroids and sample distributions respectively when the views are perfectly aligned.
Therefore, the utility function based on distribution transportation tends to converge towards a same value, namely
E((i)) 1+ (p. qr) i {1,2}.
(21)
According to the definition Shapley values, the same utility function values yield same Shapley values. Therefore, the normalized Shapley value 101 62l O.
A.2 Proof of Theorem 2
Theorem 2: For joint methods, considering the representation z = (x()) and (
fgr X (2) 0(2)) on two views, with simple view-level weight fusion during the fusion process:
Z. = z + w Z (2) There are situations where view 2 is dominated by view I and cannot make its contribution, Le., wh >> W2-
Proof: Suppose the representation of the two views:
z = (1)
g0) ((1)),
(22)
2) = 5
((2)),
and view-level weight fusion is adopted in the fusion process:
Z = w Z (1) + w (2)
(23)
where wi and W2 represent the weights of View and View 2 respectively, satisfying + w2 Conduct Kmeans the fused representation the overall optimization goal is
1.
K
E
z
p4: 12
(24)
2
1=1
zEC.
where K the number of clusters, C, is the sample set of the i-th cluster, and μ is the centroid of C,, which can be represented as
#4
1
Σ z,
(25)
IC
ZEC.
14

---

where [C,] represent the number of samples in the sample set C,.
For arbitrary sample 2 CRyz the corresponding centroid is
M4g = wi (1) (2)
W2 w2
(26)
The loss corresponding to the sample 2 is
B 2 2j - pk; l1
2 (wiz)" 11) =wzz) w2zz
(w M4 w2
(1)
(2)
11
(27)
w1
23 (1) (2) + w2 (
(2)
Mkx 113
(2)
W1 (2) mk ) + (1 - w1) (z)²) Mk; 11
2
Derive with respect to w1 for Ej.
ØEj ((2h) - -14, (2)² 1k; ))
(1)
2) (2)
W1 (z))) pk;
(1) + u2
(2)2) (2)
[22 14k
-14;;
Owi
= w1
||2 (1) 1 w2 l12)
(2) kg
11 (wi - w2) (z) (2) pl
(1) T
(2)
(28)
The above only considered the individual sample zj to the total loss. When considering all samples
K |C,1
E
(29)
then derive with respect to wi for E,
K |C 0E,
aE
(30)
Ou'
Ow1
k=1j=1
If N represents the total number of samples, taking the expectation Avg(-) of both sides of the equation yields
OE Owl
K
Caal
Avg
Avg
Owci
k=1
(EE
N
Avg
Own
N
w1
Avg
[1z)) (1)
mby 11111 Avg ((x1) 14: (1()) (2(2)
(1)
(2)
W2
Avg
(1z)
(2) p (13) Avg ((20)- -1
(1) T
( (z)²) pi;
(2)
(31)
When OE ow:
0, the analytical solution for wi can be calculated, and at this point
11 (2)
1kg 12 (2()
(2)
(1) T 1b; -
(2)
Mk;
W1
(32)
U12
||z 1k, 11 (z(1) pky
(1)
(2)
u 4k;
where expression Avg(12 (
1kg (2) reflects the level of difficulty in clustering for view
i (i {1,2}); the clearer the clustering structure, the smaller the Avg(llz)) = (12)
15

---

And (z()
dot product similarity, reflecting the correlation between the two views; the smaller the correlation between views, the closer (1)
Mk, 2 Mk, be
(1)
(2)
(z(")- Pky
(2) is to O.
Assuming significant differences in clustering structures between two views, with view be- (2)
(1) ing the dominant view over view 2, where Avg(Izz) Mig (13) Avg(llz) - Mky (1) and
Avg( (21)) M R, (z.)) m )²") =o O, then when the loss of multi-view clustering converges, the weights the two views
(2) W1 Avg 2 kk
1,
(33)
212 Avg
Pk
which means in the fusion of views, view takes the lead, while view 2 can only play limited role. However, in unsupervised tasks, the clarity of the clustering structure is unrelated to the quality of the clustering results. Even though the clustering structure of view 2 may not be clear, the clustering results of view 2 could still be accurate. In this scenario, > can be considered that the cooperation between the two views suboptimal.
A.3 Proof of Theorem 3
Theorem For joint methods that use view-level weight fusion uy ()) + Zz where ( / ( and ( = S (x(2); ((2)), gradient modulation in Eq. (8) allows the two views to contribute more evenly, i.e. we
Introducing the suppression factor & (where k 1) suppresses the convergence speed of the dominant view's 's parameters. In this case, the parameter updates become:
Proof:
{ ((1) = ((1) k ag)) 0(2) = g(2) -7 09(2)
(34)
dE
where the overall optimization objective for multi-view learning is given by
F
11l12
(35)
zEC.
Due to the equivalence ||z- -m112 W1 (1) the Cauchy-Schward inequality, we obtain:
pb:
(1)
w2 2 1k 1|²
(2)
in accordance with
||z mil12 - (w) + wz) (1
(1)
(2) 23 Mkg
(2)
M4
+
(36)
2
Therefore, the upper bound on the overall loss E' is jointly determined by the individual view losses and the view weights. As both ||z)" #k,
(1) |l3 and ||z, u4: xx!2)113 ||2 can be seen the Kmeans loss for
single view, during one iteration, if suppress the gradient descent for the dominant view (view
1). [lz) 1
(1) l2 would increase compared to not suppressing it, while ||z (2) 4 [|3 remains
(2)
unaffected. The upper bound on the overall optimization objective shifts, leading to differences in the single-view parameters and weights convergence.
2i + 1. This implies that the As the loss converges, ratio of weights between the two views, weight the non-dominant view increases, allowing more fully engage the fusion process. Consequently, there is improved cooperation between the views, enhancing the fusion process.
16

---

B Further Experiments
B.1 View Contribution Comparison w/o SCE
In this subsection, we conducted additional ablation experiments on the SCE module based on the DMJC method across six datasets. Table[Sjilustratee the variations in view contributions before and after using SCE. The experimental findings further corroborate our conclusions:
There are cases where certain views in some datasets do not effectively contribute, such as the 2-nd view of C011 the 2-nd view of CUB, the 3-rd and 4-th views STL10, and the 5-th view of Reuters, all with contributions below 0.1.
Through the SCE method, the contributions of suppressed views experienced significant growth, with the contribution of the 2-nd view of Caltech 101-7 increasing from 0.032 to 0.433, and the contribution of the 5-th view Reuters increasing from 0.107 0.232.
Table View Contribution Comparison wo SCE
Dataset
Method
d2
03
44
DMIC 0.968
0.032
Caltech101-7
DMIC+SCE
0.567
0.433
DMJC
0.974
0.026
\
CUB
DMIC+SCE
0.892
0.108
DMJC
0.671
0.191
0.138
UCI-digit
DMIC+SCE
0.636
0.197
0.167
DMIC
0.661
0.192
0.147
HandWritten
DMIC+SCE
0.645
0.178
0.177
DMJC
0.479
0.425
0.062
0.044
STL10
DMIC+SCE
0,432
0.409
0.083
0.076
DMJC
0.301
0.248
0.23
0.114
0.107
Reuters
DMIC+SCE
0.177
0.235
0.138
0.218
0.232
Table 6: Three views training on the UCI-digit dataset and fuse the results of different views. The optimal results are marked in bold, and the suboptimal values are underlined.
Fusion Views
ACC
View
View2
View3
without SCE
with SCE
0.871
0.927
0.860
0.909
0.801
0.849
0.648
0.659
0.857
0.902
0.873
0.921
0.856
0.866
Table 7: Three views are used for training in the first row, and then selected two of them each time for training, and compared the fusion results.
Training Views (without SCE)
nim-view
ACC
View
View2
View3
3
0.873
4
X
0.861
2
0.856
x
0.707
B.2 Detailed Discussions of View Roles
In this subsection, we conducted further exploration on the UCI-digit dataset to clarify the roles played by different views in cooperation and the effectiveness of cooperation enhancing module. In
17

---

Table 8: Three additional datasets summary.
Dataset
Views
Samples
Clusters
WikipediaArticles
2
693
10
SCENE15
2
1474
7
NoisyMNNSST
2
70000
10
Table 9: Multi-view clustering performance on three additional 2-view benchmark datasets. The optimal results are marked in bold, and the suboptimal values are underlined. O/M denotes out-of- memory error encountered during the training process.
Methods
Wikipedia
SCENE15
Metrics
ACC
NMI
ARI
ACC
NMI
ARI
ACC
NMI
ARI
DEMVC(2020 IS)
0.322
0.253
0.136
0.301
0.282
0.126
0.629
0.583
0.488
CoMVC(2021 CVPR)
0.261
0.176
0.158
0.239
0.287
0.118
0.278
0.265
0.095
SiMVC(2021 CVPR)
0.205
0.074
0.121
0.227
0.266
0.114
0.236
0.242
0.049
SDSNE(2022 AAAT)
0.586
0.525
0.363
0.412
0.440
0.238
O/M
O/M
O/M
MFLVC(2022 CVPR)
0.429
0.346
0.288
0.389
0.409
0.217
0.990
0.975
0.979
SDMVC12022 TKDE)
0.274
0.183
0.164
0.316
0.312
0.141
O/M
O/M
O/M
DSMVC12022 CVPR)
0.629
0.569
0.481
0.286
0.202
0.102
0.447
0.369
0.273
APADC(2023 TIP)
0.491
0.391
0.338
0.445
0,449
0.268
0.694
0.801
0.682
DMIC(2020 TKDE)
0.623
0.594
0.478
0.294
0.328
0.183
0.776
0.743
0.647
DMIC+SCE
0.648
0.607
0.508
0.344
0.392
0.223
0.824
0.800
0.703
ProIMP(2023 IJCAI)
0.570
0.505
0.416
0.442
0.460
0.279
0.992
0.976
0.983
ProIMP+SCE
0.576
0.508
0.420
0.444
0.462
0.280
0.993
0.978
0.985
T we illustrate the ACC of all views, single view, and paired views after training through three views of UCI-digit. Before we used the view cooperation enhancing module, the fusion results of ViewI and View3 were superior to those all views, although the ACC of View3 as a single view was worst amongst all. Therefore, we consider that views have shortcomings in cooperation. View dominates the fusion process, resulting in View2 and View3 not playing a better role. To this end, we used the view cooperation enhancing module to enhance the contributions of View2 and View3.
Furthermore, we compared the results in Table when only two views participated in training. Although the fusion of ViewI I and View3 showed superior performance when three views participated in training, when training with only View1 and View3, the performance of fusion is significantly lower than that of training with three views. Besides, the fusion performance of paired views is not as good as that of three views, proving that each view in the dataset can play a positive role in cooperation, and this effect can be further expanded by enhancing cooperation.
B.3 Experiments Additional Datasets
In this subsection, we conducted experiments on three additional datasets with two views: WikipediaaAtticesSSEAl
(555) and NoisyMNIST The detailed information about the datasets is shown in Table Considering the scale of the NoisyMNIST dataset, all experiments were conducted on PyTorch platform using the NVIDIA 3090 GPU. The experimental results on these three datasets are presented in Table The experimental results confirm proposed theory and demonstrate the validity our View Contribution Evaluation Module as well as the effectiveness of our View Cooperation Enhancing Module.
C Limitations and Future Work
In this section, we discussed the limitations of the SCE model and potential future work:
On some extreme datasets, certain view contributions obtained by the View Contribution Evaluation Module may be negative, which contradicts the non-negativity definition of Shapley values. A
http::/yaann.D lecun. www..a/.....//e////miiisstttttt.
18

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negative Shapley value indicates that the corresponding view has performed poorly in the fusion process, raising the question of whether these views should be discarded, which could be potential area of future research.
The View Cooperation Enhancing Module narrows the contributions between views through gradient modulation, but it cannot extend the contributions between views. Whether there exists more flexible method control the contribution relationship between views an intriguing question.
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Reviews:
[{"review_id": "PuinxfyhQm", "reviewer": "Reviewer_zda3", "summary": "This paper firstly considered DMVC as an unsupervised cooperative game where each view can be regarded as a participant. Then, the authors introduced the shapley value and propose a novel MVC framework termed Shapley-based Cooperation Enhancing Multi-view Clustering (SCE-MVC), which evaluates view cooperation with game theory. In summary, this paper was well written with obvious superiority.", "questions": "I have the following questions:\n-- What will happen if the view contribution is push away from each other? \n-- Are there scenarios where narrowing the contribution between views fails to enhance the effectiveness of multi-view clustering?", "limitations": "The authors have adequately addressed the limitations and potential negative societal impact of their work.", "rating": 6, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "-- A MVC framework was designed that utilizeD game theory and Shapley values to evaluate and elevate inter-view cooperation. \n-- The experiments were sufficient, and the analysis of the experimental results was adequate.", "weaknesses": "-- In this paper, why utilize $\\phi_i$ to measure the contribution of views instead of the view weight $w_i$? The article's explanation on this is not clear enough, and there is a lack of experiments to demonstrate the relationship between $\\phi_i$ and $w_i$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. The relationship between $\\phi$ and $w$:**\n\nThanks. Evaluating the view contribution using $\\phi$ calculated by shapley value, instead of relying solely on pre-set weights $w$, stems from a systemic perspective on the process of multi-view clustering. In multi-view of representation learning and fusion, conventional static weights ($w$) overlook the intrinsic interactions among views. Contrarily, Shapley values’ distinctive marginal analysis capability allows for precise quantification of each view’s average incremental contribution to the system’s overall performance. Therefore, $\\phi$ reveals not only the fundamental worth of each view, but also showcasing the additional value they generate through dynamic interactions.\n\n**2. Pushing away contributions:**\n\nThanks. Pushing away contributions among different views can be seen as the antithesis of our SCE module. In this context, the fusion of views collapses onto one or a few views, failing to leverage the complementary information of multi-view information.\n\n**3. Counterexamples:**\n\nThanks. There are situations where the contributions of views are brought closer together without enhancing the clustering performance. The fundamental assumption of DMVC is: Each view contains complementary information beneficial for downstream tasks, better view cooperation leads to better model performance. However, if a dataset obeys this assumption that a certain view contains a significant amount of noise and erroneous information, detrimental to the clustering task, increasing the contribution of that view will result in an overall decrease in clustering performance.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gQ7xYB7dex", "reviewer": "Reviewer_TWtF", "summary": "This research merges game theory with multi-view clustering by introducing the Shapley-based Cooperation Enhancing (SCE) approach. It features a module to systematically evaluate each view's contribution. The approach promotes view cooperation by adjusting the training convergence rate of view parameters based on their contributions. Extensive experiments on various datasets demonstrate the method's effectiveness when applied to different MVC frameworks.", "questions": "1) Figure 3(a) indicates that the method does not equalize the contribution value of each view. Why do the contribution values become identical after adding the SCE module to the comparison-based method in Table 2? Please provide a detailed discussion.\n2) What criteria were used to classify DMVC?\n3) Is DMJC representative of joint methods? Have other joint methods employed similar frameworks?", "limitations": "Yes", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "1) The paper integrates the Shapley value from game theory into DMVC, allowing for precise assessment of each view's contribution.\n2) Theoretical analysis is thorough, with clear and intuitive figures.\n3) The manuscript is well-organized and clearly written.", "weaknesses": "The article categorizes DMVC into alignment-based and joint methods. What criteria were used for this classification? Furthermore, only one DMJC method is used as a representative for joint methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. Different characteristics of joint method (Figure 3(a)) and alignment-based method(Table 2):**\n\nThanks. Figure 3(a) illustrates the change of view contributions of DMJC (a joint method) with/without SCE. In the joint methods, views' representations are optimized in their respective spaces, leading to uneven contributions in the fusion process. Even with the utilization of the SCE module, it can only alleviate the imbalance in view contributions without guaranteeing complete consistency. This is an inherent characteristic of the joint method framework.\n\n|    Dataset   |   Method   |   $\\phi_1$   |   $\\phi_2$   | ACC |\n|--------------|------------|-------|-------|-------|\n|      CUB     | DMJC       |   0.974  |   0.026  |  0.758  |\n|      CUB     | DMJC+SCE   |   0.892  |   0.108  |  0.797  |\n| Caltech101-7 | DMJC       |   0.968  |   0.032  |  0.469  |\n| Caltech101-7 | DMJC+SCE   |   0.567  |   0.433  |  0.583  |\n\nOn the other hand, the two alignment-based methods in Table 2 inherently bring views' representations closer, leading to more consistent contributions between views. With the application of the SCE module, it is possible to achieve essentially consistent view contributions. These experimental results validate the theoretical framework proposed in our study.\n\n|    Dataset   |   Method   |   $\\phi_1$   |   $\\phi_2$   | ACC |\n|--------------|------------|-------|-------|-------|\n|      CUB     | ProIMP     |   0.556  |   0.443  |  0.825  |\n|      CUB     | ProIMP+SCE |   0.484  |   0.516  |  0.832  |\n| Caltech101-7 | ProIMP     |   0.489  |   0.511  |  0.382  |\n| Caltech101-7 | ProIMP+SCE |   0.499  |   0.501  |  0.382  |\n\n**2. Criteria of classifying DMVC:**\n\nThanks. Our classification of DMVC methods is inspired by [1], which categorizes multi-view representation learning into (1) joint methods, (2) alignment methods, and (3) shared and specific methods, based on whether the representations of views are brought closer in the same space. In shared and specific methods, shared representations are brought closer to each other in the same space, while specific representations are individually optimized in different spaces, which can be seen as a combination of (1) and (2). Similar classification methods are also found in [2], [3], [4]. Drawing on this classification approach, our work divides DMVC into joint methods, alignment-based methods, and other methods.\n\n**3. Similar frameworks to DMJC:**\n\nThanks. The methods of DMJC have found numerous applications in MVC research in recent years. Approaches such as SURER [5], SGDMC [6], and DFP-GNN [7] have also employed similar strategies, namely optimizing view representations in respective spaces and using the sharpening of fused distributions as self-supervised signals to guide training.\n\n**Reference**\n\n[1] Jia X, et al. Human collective intelligence inspired multi-view representation learning—Enabling view communication by simulating human communication mechanism[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022, 45(6): 7412-7429.\n\n[2] Baltrušaitis T, Ahuja C, Morency L P. Multimodal machine learning: A survey and taxonomy[J]. IEEE transactions on pattern analysis and machine intelligence, 2018, 41(2): 423-443.\n\n[3] Li Y, Yang M, Zhang Z. A survey of multi-view representation learning[J]. IEEE transactions on knowledge and data engineering, 2018, 31(10): 1863-1883.\n\n[4] Jia X, et al. Semi-supervised multi-view deep discriminant representation learning[J]. IEEE transactions on pattern analysis and machine intelligence, 2020, 43(7): 2496-2509.\n\n[5] Wang J, Feng S, Lyu G, et al. SURER: Structure-Adaptive Unified Graph Neural Network for Multi-View Clustering[C]//Proceedings of the AAAI Conference on Artificial Intelligence. 2024, 38(14): 15520-15527.\n\n[6] Huang Z, Ren Y, Pu X, et al. Self-supervised graph attention networks for deep weighted multi-view clustering[C]//Proceedings of the AAAI Conference on Artificial Intelligence. 2023, 37(7): 7936-7943.\n\n[7] Xiao S, Du S, Chen Z, et al. Dual fusion-propagation graph neural network for multi-view clustering[J]. IEEE Transactions on Multimedia, 2023, 25: 9203-9215.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "x4fCuSG5ex", "reviewer": "Reviewer_URj2", "summary": "The study centers on improving task performance via deep multi-view clustering (DMVC) and fostering cooperation among different views. Specifically, the study evaluates view contributions, emphasizing the significance of strengthening cooperation among views.", "questions": "In the unsupervised multi-view scenario, what is the physical meaning of the contribution value of each view proposed in this paper? What is the relationship between the quantitative value of the view's contribution and the clustering performance of a single view ?", "limitations": "The authors have adequately addressed the limitations and potential negative societal impact of their work.", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Considering multi-view tasks from a collaborative standpoint represents a novel approach, with the paper's motivation being notably fresh. Moreover, the paper elucidates potential contribution imbalances in the joint method and addresses them through the SCE method, thereby enhancing cooperation among views.", "weaknesses": "When dealing with datasets comprising more than two views, such as three views, how can one assess whether the contribution of the views has become more evenly distributed after employing SCE? While the paper visually presents the contributions of the views, could a quantitative method be provided for this evaluation?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. Quantification of the equilibrium level of view contributions:**\n\nThanks. Due to the normalized characteristic of view contributions, the cooperation level among views with/without SCE can be compared by calculating the variance of view contributions, denoted as $D(\\phi)$. A smaller variance indicates a more consistent contribution among views, reflecting improved cooperation between views.\nThe table below illustrates the values of $D(\\phi)$ for DMJC with/without SCE. It is evident that using SCE could result in a lower $D(\\phi)$ for the views, indicating a better cooperation among them.\n\n|  $D(\\phi)$ | Caltech101-7 |  CUB  | UCI-digit | HandWritten | STL10 | Reuters |\n|--------|:------------:|:-----:|:---------:|:-----------:|:-----:|:-------:|\n|   DMJC   |     0.219    | 0.225 |   0.057   |    0.054    | 0.040 |  0.006  |\n| DMJC+SCE |     0.004    | 0.154 |   0.046   |    0.049    | 0.029 |  0.001  |\n\n**2. Physical meaning of view contributions:**\n\nThanks. In unsupervised multi-view scenario, the physical meaning of view contributions lies in their influence on fusion. The greater the view's contribution, the more significant its influence on the fusion progress.\n\n**3. Relationship between view contribution and view performance:**\n\nThanks. There is no link between the contribution of a view and its performance. Based on the fundamental assumption of multi-view learning that each view contains complementary information beneficial for downstream tasks, examining the performance of a single view independently is deemed meaningless. Furthermore, it is infeasible to evaluate the quality of a single view under unsupervised conditions. Hence, the aim is to enhance the cooperation among views by ensuring a more balanced view contribution.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9Y3UiFwJHL", "reviewer": "Reviewer_v5TM", "summary": "The author introduces a Shapley-based cooperation enhancement framework aimed at fostering collaboration among different views. The SCE-MVC method incorporates cooperative game theory, considering each view as a participant in the model and assessing their contributions using the Shapley Value.", "questions": "The alignment-based method proposed in Theorem 1 will make the contribution values of several views the same. Combined with the experimental results in Table 2, does this mean that the end point of the view contribution optimization proposed in this paper is contrastive learning? If not, please explain in detail the difference between the method in this paper and the contrastive learning method?", "limitations": "The authors have adequately addressed the limitations and potential negative societal impact of their work.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "Viewing each view as an individual player within game theory represents a fresh perspective in multi-view clustering. Also, enhancing clustering performance through balancing view contribution is both well-founded and innovative.", "weaknesses": "1. Using the SCE module in an alignment-based framework only provides a marginal improvement to the model. Does this imply that the SCE module is ineffective in the alignment-based framework? \n\n2. The view contributions of alignment-based method is much balanced than view contributions of joint methods. Does this imply that the alignment-based method is much better than the joint method? It's not reasonable since the clustering performance of alignment-based methods may not necessarily be better than that of joint methods.\n\n3. Is the complexity of computing Shapley values truly O(n!)? When dealing with a larger number of views, can this evaluation framework still be utilized for computation?\n\n4. Are the loss functions L in Eqs (15) and (16) on page 6 the same? If so, there is a problem of inconsistent dependent variables. In addition, $D_ij$ in Eq. (9) is a scalar and should not be bolded.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. the use of SCE module on alignment-based methods:**\n\nThanks. The SCE method consists of two modules: the View Contribution Evaluation Module and the View Cooperation Enhancing Module. For alignment-based methods, the View Contribution Evaluation Module obtains the contributions of views, where these contributions are remarkably close in the alignment-based frameworks, agreeing with the results deduced from our theory. \n\n|    Dataset   |     Method     |  $\\phi_1$ |  $\\phi_2$ |\n|------------|--------------|-----|-----|\n|      CUB     | InfoNCE+Kmeans | 0.491 | 0.509 |\n|      CUB     | ProIMP         | 0.556 | 0.443 |\n| Caltech101-7 | InfoNCE+Kmeans | 0.484 | 0.516 |\n| Caltech101-7 | ProIMP         | 0.489 | 0.511 |\n\nBuilding upon the consistent view contributions, the limited enhancements brought by the View Cooperation Enhancing Module can be elucidated. Our experimentation and analysis of alignment-based methods with/without SCE in the paper are not aimed at showcasing the enhancement brought by SCE to the model, but rather at validating the integrity of our theory and the soundness of the View Contribution Evaluation Module.\n\n**2. Is the alignment-based method superior to the joint method?**\n\nThanks. Given a DMVC framework, the SCE module can evaluate the cooperation among views and make optimizations based on the assessment. If the cooperation among views is insufficient, it indicates potential for improvement upon the original clustering performance, while it does not imply that the original clustering results are poor. For example, on the Caltech101-7 dataset, while DMJC may not match ProImp in view cooperation, it exhibits superior clustering performance. Moreover, the inconsistent view cooperation of DMJC suggests that using the SCE module can yield remarkable enhancements to the model. Our work, from the perspective of view contributions, highlights potential limitations of joint methods, and does not assert that alignment-based methods are superior than joint methods in clustering performance.\n\n| Dataset      | Method | $\\phi_1$  | $\\phi_2$  | ACC   |\n|--------------|--------|-------|-------|-------|\n| Caltech101-7 | ProIMP | 0.489 | 0.511 | 0.382 |\n| Caltech101-7 | DMJC   | 0.968 | 0.032 | 0.469 |\n\n**3. Algorithmic complexity of SCE module:**\n\nThanks. SCE module involves computing the fusion distribution for any combination of views. With V views, the complexity of SCE amounts to $O(2^V)$. While the $O(2^V)$ complexity is manageable for a small number of views, it can become cumbersome with a larger number of views. To mitigate the computational burden of calculating Shapley values for numerous views, approximate algorithms can be employed. For instance, the TreeSHAP [1][2] algorithm can compute Shapley values **in polynomial time**, offering a more efficient approach.\n\n**4. Representation of symbols in the formula:** \n\nThanks. The $L$ in Equation 15 is an abbreviation for $L(\\theta^{(v)}_t)$ in Equation 16. The representation of the formulas will be unified in the final version.\n\n**Reference**\n\n[1] Yang J. Fast treeshap: Accelerating shap value computation for trees[J]. arxiv preprint arxiv:2109.09847, 2021.\n\n[2] Muschalik M, Fumagalli F, Hammer B, et al. Beyond TreeSHAP: Efficient Computation of Any-Order Shapley Interactions for Tree Ensembles[C]//Proceedings of the AAAI Conference on Artificial Intelligence. 2024, 38(13): 14388-14396.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mDV9kCmzAS", "reviewer": "Reviewer_CDST", "summary": "This paper studies multi-view clustering and seeks to investigate the view cooperation issue. The authors consider DMVC as an unsupervised cooperative game and regard each view as a participant. Compared with the existing methods, this consideration is new and interesting. Based on the novel idea, the authors proposed SCE-MVC, a novel shapley-based cooperation enhancing multi-view clustering method. The paper is well-organized. The experiments are convincing.", "questions": "The article involves rich theoretical and mathematical knowledge. I have a question to the designation of the model: Which design is the key to improving model performance?", "limitations": "NA", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. The paper proposes a new point also an interesting point for multi-view clustering tasks, i.e., considering the multi-view collaboration as a cooperative game. \n\n2. The experiments are sufficient and convincing. The authors validate the method from many aspects. The proposed SCE-MVC obtains much better performance on six diverse datasets.", "weaknesses": "1. Figure 2 is confusing. The specific structure of View Cooperation Enhancing Module is not clearly presented.\n\n2. There are many formulas and symbols. It is suggested to add a notation table.\n\n3. Although the authors try to explain model (1), it is still difficult to understand Shapley Value from the model. In addition, many variables are not clearly explained. The authors should present more information about the model and explain all variables used in this model, such as S_i, {i}, s\\{i}， etc.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. Illustration in Figure 2:**\n\nThanks. Improvements have made to the View Cooperation Enhancing Module in Figure 2 in the Figure 1 of global reponse, presenting the details of gradient modulation within this module.\n\n**2. Notification table:**\n\nThanks for your suggestions. A notification table will be given in the final version.\n\n**3. Key to improving model performance:**\n\nThanks. Previous DMVC methods did not adequately consider the aspects of **view cooperation**, especially the joint methods, which may lead to the collapse of fusion results onto one or a few views. Building upon the fundamental assumption of multi-view learning that each view contains complementary information beneficial for downstream tasks, fully leveraging the role of each perspective and enhancing cooperation leads to better model performance.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank the SAC, AC, and PCs for their efforts and constructive comments, which are helpful in further improving the quality of our manuscript. We respond to your questions carefully one by one carefully, and we hope our responses can address your concerns.

Note that there are five tables and one figure in the attached PDF, corresponding to RQ1 for Reviewer TWtF, RQ1 for Reviewer URj2, RQ1 and RQ2 for Reviewer v5TM, and RQ1 for Reviewer CDST.

Meta-review: This paper treats DMVC as an unsupervised cooperative game, with each view acting as a participant. The authors introduce the Shapley value and propose a novel MVC framework that assesses view cooperation using game theory. Extensive experiments validate the effectiveness of the proposed framework. While some potential issues remain, all reviewers acknowledge the contributions of this paper and have recommended it for acceptance.
Decision: Accept (poster)