SparseMVC: Probing Cross-view Sparsity Variations for Multi-view Clustering
Authors: Ruimeng Liu
Xin Zou
Chang Tang
Xiao Zheng
Xingchen Hu
Kun Sun
Xinwang Liu
Venue: NeurIPS 2025 spotlight ()
Keywords: Muti-view clustering
self-supervised learning
representation learning
autoencoder
Primary area: deep_learning
TL;DR: The paper systematically analyzes the phenomenon of varying sparsity ratios across views in multi-view learning, and proposed a targeted data-driven network architecture based on Sparse Autoencoder with Adaptive Constraints.

Abstract: Existing multi-view clustering methods employ various strategies to address data-level sparsity and view-level dynamic fusion. However, we identify a critical yet overlooked issue: varying sparsity across views.  Cross-view sparsity variations lead to encoding discrepancies, heightening sample-level semantic heterogeneity and making view-level dynamic weighting inappropriate. To tackle these challenges, we propose Adaptive Sparse Autoencoders for Multi-View Clustering (SparseMVC), a framework with three key modules. Initially, the sparse autoencoder probes the sparsity of each view and adaptively adjusts encoding formats via an entropy-matching loss term, mitigating cross-view inconsistencies. Subsequently, the correlation-informed sample reweighting module employs attention mechanisms to assign weights by capturing correlations between early-fused global and view-specific features, reducing encoding discrepancies and balancing contributions. Furthermore, the cross-view distribution alignment module aligns feature distributions during the late fusion stage, accommodating datasets with an arbitrary number of views. Extensive experiments demonstrate that SparseMVC achieves state-of-the-art clustering performance. Our framework advances the field by extending sparsity handling from the data-level to view-level and mitigating the adverse effects of encoding discrepancies through sample-level dynamic weighting. The source code is publicly available at https://github.com/cleste-pome/SparseMVC.

SparseMVC: Probing Cross-view Sparsity Variations for Multi-view Clustering
Ruimeng Liu', Xin Zou?, Chang Tang
Xiao Zheng', Xingehen Hu², Kun Sun Xinwang Liu School of Computer Science, China University of Geosciences (Wuhan)
The Hong Kong University of Science and Technology (Guangzhou)
School of Software Engineering, Huazhong University of Science and Technology "School of Computer Science, Hubei University of Technology CColleee of Systems Engineering. National University of Defense Technology
Abstract
Existing multi-view clustering methods employ various strategies to address data- level sparsity and view-level dynamic fusion. However, we identify a critical yet overlooked issue: varying sparsity across views. Cross-view sparsity variations lead to encoding discrepancies, heightening sample-level semantic heterogeneity and making view-level dynamic weighting inappropriate. To tackle these challenges, we propose Adaptive Sparse Autoencoders for Multi- Clustering (SparseMVC), framework with three key modules. Initially, the sparse autoencoder probes the sparsity of each view and adaptively adjusts encoding formats via an entropy- matching loss term, mitigating cross-view inconsistencies. Subsequently, the pirelation-infomeer sample reweighting module employs attention mechanisms assign weights by capturing correlations between early-fused global and view- specific features, reducing encoding discrepancies and balancing contributions. Fur- thermore, the cross-view distribution alignment module aligns feature distributions during the late fusion stage, accommodating datasets with an arbitrary number of views. Extensive experiments demonstrate that SparseM VC achieves state-of-the- art clustering performance. Our framework advances the field by extending sparsity handling from the data-level to view-level and mitigating the adverse effects encoding discrepancies through sample-level dynamic weighting. The source code is publicly available at htttps:/githhi: /tggttttbbbg///g/////// com/ cleste- wesste-pmmeeeeeeeeee /Sseessaaa......
Introduction
Multi-view learning has emerged as powerful paradigm for leveraging complementary information across multiple perspectives, significantly improving the performance of unsupervised learning tasks such as clustering [1, 2, 3, 4, 5]. At the same time, the sparsity of multi-view data has become pivotal focus of research, with numerous studies proposing solutions from perspectives such as activation functions [6], tensor decomposition [7, 8]. and variational autoencoders [9]. Nevertheless, while prior methods focus designing advanced approaches to address data sparsity, they often overlook a fundamental aspect- the potential variations in sparsity across different views. Given that multi view data consists of multiple views originating from distinct sources, and sparsity is pervasive characteristic in multi-view data [10, 11, 12, 13, 14]. Building upon the factual observations, natural question arises: "Does there exist a phenomenon of varying sparsity across views?"
"Corresponding author: Chang Tang (tangchange @ hust.cdu.en).
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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To quantify cross-view view sparsity variations, we define the sparsity ratio Se for the v-th view:
Sp N. _
[ 0],
(1)
where N refers to the number of samples, F refers to the feature dimension, xi, represents the i-th feature of the j-th sample in the v-th view, and the indicator function takes the value of one oi, equals zero, and zero otherwise. Zero-valued features " suggests missing dimensions or data collection errors. Our statistical and computational analysis addresses the question posed earlier and reveals that sparsity variations across views not only exist, but are widely prevalent in diverse multi-view data, as illustrated in Fig. I.
The disparity in view sparsity presents a multifaceted challenge: highly sparse views, lacking sufficient informative content, are prone to underfitting, whereas less sparse views, often bur- dened with redundant or irrelevant fea- tures, are susceptible to overfitting. Ap- plying a uniform encoder architecture or regularization strategy across such heterogeneous views compromises rep- resentational consistency and limits the model's 's capacity to extract complemen- tary cross-viee view information. To re- solve this, we adopt an adaptive and sparsity-aware encoding strategy tai- lored to individual views. This requires rethinking the autoencoder design to accommodate view heterogeneity and structural disparities. Our solution is design an autoencoder capable of adap- tively adjusting its constraints based on the sparsity ratio of each view, allowing its encoding form to evolve accordingly.
ALD-18
S1L-10
ALD0-30 3M
MSRCVI
Wiltykka
HBCA
Figure 1: Sparsity ratios across views in multi-view datasets. TOP BOX PLOTT illustrates the sparsity ratio distribution, which shows the median (orange line), interquartile range (box), and any outliers (points outside the whiskers). BOT- TOM BAR PLOT presents the sparsity ratios for each view within each dataset. To provide a more comprehensive situ- ation of sparsity variations, additional datasets are included: Digits [15]. LandUse-21 [16], RGB-D [17]. GSE [18] and 1T [19]. The cross-view sparsity ratios have been pro- cessed to improve visualization using the sigmoid function.
Due to varying sparsity across views for the same sample. encoders can introduce semantic and representational inconsistencies, affecting subsequent stages [20, 21J. As a result, it becomes essential to dynamically assess the contribution of each view based on the features extracted from different types of encoders. To address this, we design a pnrrelationiifoommat reweighting module that assigns sample-wise weights based on the correlations between global and local features, thereby balancing view contributions and discrepancies. The early fusion strategy integrates multi-view data through feature concatenation. Since our reweighting module uses the global latent representation encoded from early fusion to guide the subsequent dynamic weighted early fusion of local representations, we chose an early fusion approach that preserves the original feature values as much as possible.
In late fusion, we introduce a cross-view distribution aligmment module to align feature distributions across views, enabling robust integration and supporting datasets with arbitrary view numbers while balancing global consistency and view-specific diversity. These three core components together form the SP framework, addressing sparsity inconsistencies, encoding discrepancies, and semantic heterogeneity in a step-by-step and interdependent design.
Table I: Performance comparison of different strategies under extreme and minimal sparsity variation.
Accuracy [Ours]
NMI
Datasets
Sparsity of Different Views
[2nd]
[Ours]
[2nd]
ALO1-100 100
[0.0001. 0.0001, 0.3415, 0.6383)
82,21
14.49
77.72
92.65
11.78
90.87
MSRCVI
[0.0049, 0.0048. 0.0048, 0.3478, 0.0051. 0.0048]
97.14
15.71
91.43
94.22
f6.32
87.90
LGG Synthetic3d
[0.0040, 0.0038, 0.0078, 0.0037] [0.0017, 0.0017. 0.0017]
83.15
10.38
82.77
54.62
10.49
54.13
98.33
10.16
98.17
92.01
10.74
91.27
As presented in Table 1, we evaluate our method across range of datasets exhibiting extreme sparsity disparities. For instance, in ALO0I-100, the sparsity ratio spans from as low as 0.0001 to as high as 0.6644, revealing a substantial imbalance in information density across views. To effectively

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address such heterogeneity, our framework incorporates dynamically adaptive autoencoders that adjust both the encoding process and sparsity-aware regularization in accordance with each view's 's sparsity profile. Unlike existing methods, which generally overlook the impact of cross- view sparsity variation, our targeted design enables SparseMVC achieve consistently superior performance under severe disparity conditions. Moreover, it maintains strong competitiveness even in datasets with near- uniform sparsity variations, such as LGG and Synthetic3D, demonstrating both its responsiveness to sparsity imbalance and its robustness in more homogeneous scenarios.
To the best of our knowledge, this is the first work to explicitly identify, analyze, and define the problem of cross-view sparsity variations in multi-view data, and to propose a dedicated framework SParseMVC that offers a targeted and principled solution.
2 Related Work
For multi-view fusion, early methods [22, 23] assumed equal importance of views, ignoring view heterogeneity. Hence, dynamic weighting approaches have emerged, with attention- based meth- ods [24, 25] leading, alongside loss optimization [26, 27], kernel techniques [28, 29], and subspace methods [30]. However, uniform view-level weighting fails to address intra-view variability, high- lighting the need for sample-level dynamic weighting. Trust-based methods [31, 32, 33, 34] excel in supervised scenarios. For multi-view clustering, sample-adaptive fusion [35] has been proposed using Laplacian matrix divergence. In contrast, our framework dynamically computes sample weights correlation calculation without additional loss, while adopting decoupled design with independent global and view-specific autoencoders. More details can be found in Appendix A.2.
Sparse representation effectively captures essential features in sparse data by enforcing sparsity constraints [36] but struggles to handle the non-linear structures common in multi-view datasets [37]. Autoencoders excel at learning non-linear latent features [38, 391, yet their lack of sparsity enforce- ment limits their adaptability to varying sparsity rates across views. Sparse autoencoders combine the strengths of both sparse representation and standard autoencoders by incorporating sparsity constraints into the hidden layers [40], which have become research focus in multi-view learning [9, 41].
3 Method
Booattilly
X
Cospraasion
euphor
FA
layer
D
View-1 1: Text
Glooal
Late fusion
Connelation Informed Somple Benwighttng
Distribution
D.
X.
Aligmment
Image
Global
Locth
Ey
Dv
&+
Z
Sparse Encoders
Sparse becoders
Figure 2:
Overview of SVC framework designed to address varying sparsity across views.
This section sequentially introduces the three key submodules of SVa as illustrated in Fig. 2. begins by utilizing sparse autoencoders with adaptive constraints, which dynamically adjust the coding strategy based on the probed su, to generate latent features (Z), making the reconstructed features (X) approximate the original input features (X). ). Subsequently. the correlation between the early-fused global features (Z) and view-specific features (( (Z.))=)) >>=)) guides the computation of sample-level weights ((W,) WW )"_)) via the attention mechanism within the correlation-informeed sample reweighting module. Finally, the cross-view distribution alignment module enhances clustering performance by setting the late-fused global features as the anchor latent representation, and then simultaneously aligning the multi-view feature distribution between Y and each view-specific compressed feature ((Y, (Y,), in ) The algorithm of the framework can be found in Appendix A.

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3.1 Sparse Autoencoder with Adaptive Constraints
To handle varying view sparsity rates, we propose the sparse autoencoder with adaptive constraints (SAA), extending traditional sparse autoencoders. SAA employs adaptive loss function that integrates reconstruction and sparsity-aware entropy-matching as distinct constraints, wherein the adjustment is dynamically guided by view sparsity ratios formulated as prior knowledge.
The reconstruction loss, typically measured by mean squared error (MSE), quantifies the difference between the reconstructed output 3. and the input æ for the j-th sample:
N
Lioonn
1 N ==1
Σ
x
(2)
where N is the number of samples (batch size) for view v. Motivated by the widespread presence of sparsity variations in different views, our aim is to design function that is positively correlated with Su. allowing adaptive adjustments to both the encoder type and the strength of the sparsity constraints. We scale by (s,) outside the loss rather than tuning in Eq. (4), please refer to Appendix B.2. The
design of the adaptive weighting factor f(s.) follows a ReLU-like approach, adjusting the strength of entropy based on the probed input dataset sparsity 8, for view 2:
0,
if s, S 0
f(s.)
80-0
if 8u
0,
where the default value of is 0.01. Selecting the threshold for the sparsity ratio s, is based on the actual cross-view sparsity distribution of cach dataset. For most datasets, the sparsity ratios exhibit skewed distribution, with a significant concentration of both high and low values, as shown in Fig. and Table 1. A 9 value of 0.01 effectively captures these low-sparsity views. To enforce sparsity constraints, the antropy-matching loss is defined using Kullback-Leibhe (KL) [42] divergence, encouraging the activations ig in the hidden layer to align with a target sparsity level The matropy-matching loss and the sparsity loss derived from it are formulated follows:
H C sparse = f(Su) Centropy = f(s)) (plog
P (1 - p) log P hig
(4)
ng
where H refers to the number of units in the hidden layers of each sparse encoder, and is the target sparsity level. hgg represents the average activation of the k-th hidden unit for view which clamped lie strictly within the open interval from zero to one. Following [40], p to 0.05, which is well-validated choice that balances sparsity and the learning capacity of the autoencoder, allowing it to effectively capture key features in the data while avoiding overfitting to irrelevant features. The average activation is computed by:
hg
1 N
Σ
( a +b)
(5)
where WE is the weight matrix, by the bias term for the k-th hidden unit in the w-th view, 2 is the input feature, and at)) denotes the ReLU activation function. When se 0, Centopy is deactivated (f(s) = 0) and the sparse autoencoder degen- erates into standard autoencoder. Conversely, for input views where 8, > 8, f(s.) exhibits linear in- crease with 8.,, ensuring that the spar- sity constraint becomes more promi- nent for highly sparse inputs. Based on the above, the loss function for
Local View
Croom + f(ar)) Leatnors
Recatstruction Leas
Global View 8
D2
Creom [((e) Centrage
Sparse Representation Lose
Local
Fy
Dy
Ereen f(()) Lestnoss
Sporsity
View 1-V
Autoencoders
Constroins
Figure 3: Sparse autoencoder with adaptive constraints.

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SAA is then formulated as follows:
CSAA
(Crcoo +
Ciparse
(6)
As shown in Fig. 3, this piecewise linear design dynamically aligns the sparsity constraint with the input sparsity level and customizes the encoding strategy for each view.
3.2 Correlation-Informer Sample Reweighting
While SAA balances reconstruction and sparsity, introduces sample-specife encoding inconsisten- cies due to differences in sparsity. Additionally, while networks adjust weights through layer updates, this weighting alone cannot address the lack of communication between autoencoders. On the above bases, we drew inspiration from the concept of multi-head attention and designed the correlation- informed sample reweighting module (CSR), which leverages correlations between the early-fused global features (Z) and the view-specific local features (Z,). then computes sample-specific weights. This cascading design is devised to achieve two objectives: mitigating the encoding inconsistencies introduced by SAA and leveraging globally fused features, which preserve the relatively high-fidelity patterns of the original data structure, to supervise the computation of view correlations.
CSR adopts simplified structure inspired by multi-head attention, which captures similar effects while deviating from the standard formulation. Initially, CSR takes 2 E N and Z, E RN x as input, and projects them into the query, key, and value spaces through parallel linear transformations:
Q ZWQ, K. W. V. Wv
(7)
where Wq. WK. W. FxF are the learnable weight matrices, generating the query matrix Q NXF which encapsulates global semantic information, and the key matrix K, E N which capture view-specific features for each view. Our primary objective relationships by evaluating attention scores between queries and keys, without generating new feature representations, thus omitting the value matrix V,, To compute the correlation between and Z,, we define the correlation score C, E R.N for the w-th view based on Einstein summation convention as:
quantify inter-view
C == -5=1 Q, ())=))")))))""
(8)
VF
where VF is the scaling factor equals to the square root of the dimension of the key vector. The correlation scores are normalized via the softmax function to produce sample-specific W. E RRV as:
exp(C. (C.)
W.
V E exp(C..)
that dynamically adjust the contribution of each corresponding sample in respective autoencoders.
3.3 Cross-view Distribution Alignment
Aligning features across views is fundamental challenge in multi-view learning, as it is crucial for leveraging the complementary information provided by diverse views. The cross-view distribution Alignment module (CDA) addresses this issue by performing contrastive learning between the late- fused global features (Y) and the compressed features of individual views (Y₂), ensuring effective alignment of multi-view features within unified and shared latent space.
To mitigate the risk of dimensional collapse during alignment, potentially caused by an excessively large latent space, introduce compression layer before feeding the encoded view-specifie features into the CDA. More details are provided in Appendix B. Specifically, Y, E N is obtained from Z via the compression layer. In parallel, the global features Y E N are as follows:
Y- Σ W. Zu
(10)
where fusion function FF represents the late fusion layers and ensures that the transformed dimension matches Y.. The similarity matrix S, between Y and Y, is defined as:
S, Y (Y.))
(11)
T
5

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with denoting a temperature parameter that scales the similarity values. The sample pairs position indices in S, are defined as p and q. Positive pairs, which correspond to the same samples across views, are represented by the diagonal elements of S.. denoted as S..P. Negative pairs. which involve different samples across views, are identified using mask matrix M, E R&XN, where M = 1 if # and M = 0 otherwise. The contrastive loss for each sample is:
cXp(S
log
(12)
N q=1
(Xp M
where exp(SP-P) quantifies the similarity of positive pairs, and the denominator aggregates the exponential similarities of all pairs, weighted by the mask matrix M,. The overall CDA loss across all views is obtained by summing the individual losses for each view and averaging over all samples:
LCDA
CP
(13)
con
V=1
p=1
For contrastive learning, when samples of the same class are clustered one view, the attraction exerted by positive pairs propa- gates to other views. In contrast, although the distinction between samples does not necessarily imply class disparity, repulsion among negative pairs enables view with greater discriminative power to transmit class separations to other views. This sults in mechanism that transforms both the alignment and misalignment informa- tion at the cross-view sample level into an objective, aiming minimize intra-class while maximizing inter-class distances.
View
View,
Difficult-
Easy-to-ckassify
Easy-to-classip
Difficult-Be to classify
(a) Multi-view somple distribution in latent spoce
Global View
Regarding the role of CDA, the global view serves as an anchor, which is compared in parallel against each local view. To mini- mize the overall contrastive loss, the sam- ple distribution of the global view con- currently attracted toward all local views, thereby encouraging the features of each sample to converge more tightly in the latent space. From the perspective of the entire latent space, this process effectively facilitates overall distribution alignment. The rationale behind utilizing local views for distribution aligmment with the global view to enhance class separability lies in the ability leverage easily classified samples from one view to improve the distinguishability of harder-to-classie samples in another. The reasoning above, together with Fig. 4, illustrate the working principle of the CDA: utilizing contrastive learning as tool, through the process of aligning the distribution of sample features across views, enhancing the differentiation of leveraging easy-to- toslsso-ylsssffy samples in another, and ultimately achieving the goal of optimal alignment of features in the shared latent space.
Views
Viet,.
Easy-to-classify
(b) Gress-view distribution alignment
(c) Multi-view fusion
Figure 4: View distribution aligmment based on contrast.
to-casssiy samples in one view by
(a) GCFAgg (CVPR'23) (b) CPSPAN(CVPR'23)
(c) SCMVC (TMM²)4) (d) MVCAN (CVPR'24)
(e) SparseMVC (Ours)
Figure 5: T-SNE visualization of the features learned with recently comparative methods (a-d) and ours (c) on the ALOI- 100 dataset.

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3.4 The Overall Loss Function of SparseMVC
The total loss Luaa comprises the adaptive sparse autoencoder loss LSAA in Eq. (6), preserving data fidelity and enforcing structured sparsity via Lreen and Lentropy, and the cross-view alignment loss ECDA Eq. (13), ensuring consistent clustering across views:
Lioral
(Ciecon + f(()) - + ACR LCDA,
(14)
where AcR is the constraint ratio coefficient that controls the trade-off between CSaa and LCDA.
4 Experiments
4.1 Experimental Settings
Compared Methods. Our proposed method is compared against the following 12 state-of-the-
art multi-view clustering methods based on deep learning. DSMVC [43]. COMPLETER [44].
DCP [45], CVCL [46] and SCMVC [47] focus on dynamic contrastive learning; MFLVC [48].
GCFAgg [3], De [49] combine feature fusion and consistency; DSMVC [43] and SDMVC
[50] enhance consistency through discriminative learning; CPSPAN [51] and MVCAN [52] apply
proxy supervision and prototype alignment.
Table 2: Characteristics of kinds of multi-view datasets.
Benchmark Datasets. The selected datasets span diverse domains: Image datasets include MSRCVI [53] focusing on objects and scenes, Dermatology [54] on medical images, Out-Scene [55] on nar- ural scenes, and ALOI-100 [56] on object recognition. Image-text datasets include Wikipedia 2, which provides website cross- modal data. Omics datasets include LGG [57] focusing on brain tumor genomics and BRCA [58] on breast cancer genomics. Synthetic3d [59] supports 3D object mod- eling and recognition. Detailed properties of datasets are listed in Table 2.
Datnsets
Samples
Clusters
Views
View Dimensions
Images
MSRCVI
[1302, 48, 512, 100. 256, 210] [12, [512, 432, 256, 48)
Dermatology
Our-Scese
2,688
ALOI-100
10,300
100
[77, 13, 64, 125)
Image-Text TTxt
Wikipedio
[128, 10)
Omles
LGG BRCA
267
[2000, 2000, 383, 209]
[2000, 2000, 278,
Synthetics
Synthetic3d
600
[3. 3, 3)
Evaluation Metries. Accuracy (ACC) evaluates alignment with ground truth, normalized mutual information (NMI) measures shared information, purity (PUR) assesses cluster homogeneity, Ad- justed Rand index (ARI), measuring clustering similarity, partially utilized in experiments. For all metrics, higher values indicate better performance.
Implementation Details All experiments were conducted using Python 3.8.15 and Py PyTorch 1.13.1 on a Windows PC equipped with an AMD Ryzen 9 5900HX CPU, 32GB RAM, and Nvidia RTX 3080 GPU (16GB). Models were trained using the Adam optimizer [60], learning rate of 0.003, and a fixed seed of 50, with batch size equal to the dataset's sample count. Pre-training was performed uniformly for 300 epochs, while alignment training for 300 epochs for datasets with less than 2500 samples and 1000 epochs for larger datasets. For clustering, k-means [61] was applied with the number of clusters equal the dataset categories and 100 initializations. During pre-training, global features Z, derived from early fusion were used, while alignment training used late fusion features Y. Metrics were calculated as the average of 10 runs in the final epoch, with fine-tuning performed for specific datasets. To ensure fairness, the hyperparameters for the comparison methods were determined based on either the default global settings the configuration of the first dataset.
conducted
4.2 Comparative Results Analysis
Table 3 and 4 summarize the comparative results, leading to the following conclusions:
(1) Our method achieves state-of-the-arr performance across eight diverse multi-view datasets, along with larger-scale datasets as shown in Table 8. These results validate the versatility of our approach
7

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Table 3: Clustering results small multi-views datasets. The top-ranked result is bolded, and the second-ranked result is underlined.
Synthetic3d
LGG
Dermatology
BRCA
Datasets
ACC
NM1
PUR
ACC
NMI
PUR
ACC
NMI
PUR
ACC
NMI
PUR
COMPY [CVPR'210 21] [44]
80.15
77.65
[CP [45]
97.17
87.60
97.17
59.55
44.82
73.03
72.91
77.22
80.73
57.29
39.51
60.55
[CVPR [48]
49.73
79.03
56.20
62.85
27.74
DSMVC [CVPR'22] [43]
96.83
86.64
96.83
82.77
54.13
82.77
92.74
87.82
92.74
54.52
33.53
68.84
SURE [TPAMT [62]
65.17
77.03
12.85
DealMVC [MM'23] [49]
87,50
72.07
87.50
72.28
40.55
72.28
45,53
31.13
45.53
59.55
32.79
61.56
GCFAgz [CVPR'23] [3]
61.80
88.27
79.25
CPSPAN (CVPR'23] [51]
97,83
90.15
97.83
63.30
30.53
63.30
76.26
84.63
85.20
66.83
34.48
74.12
SDMVC [TKDE'23]
7U.67
57.79
CVCL [[CCV"23] [46]
95.31
82.36
95.31
58.20
23,73
58.20
56.25
56.01
67.97
61.98
34.68
68.49
MVCAN [CVPR"24] [52]
98.17
91.27
42.57
27.18
58.38
51.58
57.79
35.70
SCMVC [TMM [47]
97.00
87.11
97.00
73.41
39.76
73.41
93.85
88. 44
93.85
50.25
30.70
60.80
SparseMVC (Ours)
83.15
95.25
89.86
95.25
70.10
20.85
and highlight its potential for wide range of downstream tasks. In comparison, other approaches such as SCMVC, MVCAN, and CPSPAN achieved relatively good results on specific datasets but failed to maintain advantage due their limited generalizability across other datasets.
(2) Our method demonstrates stability clustering performance, as evaluation metrics oscillate upward within small range and stabilize with increasing epochs, showcasing robust results. In contrast, methods like SURE and DealMVC on ALOOI- or CVCL and CPSPAN on Wikipedia fail to stabilize, with metrics either degrading significantly after peaking fluctuating dramatically without consistent improvement. During the early stages contrastive training when the embedding space's ' distribution remains uneven, potentially causing abrupt gradient fluctuations. To alleviate the instability, we adopt dynamic fusion strategy in Sec. 3.2 and a pre-training approach in which only the autoencoder is trained initially.
(3) Although our model is specifically designed to address the challenge: across across-view sparsity varia- tions, it also achieves superior performance dense datasets, such BRCA, thereby demonstrating its adaptability and broad upplicability. This is attributed to the inherent flexibility our designed sparse autoencoder, which adaptively transitions into conventional autoencoder when confronted with dense data, thereby prioritizing the reconstruction objective with greater emphasis.
(4) Compared to recent otate-of-the-art methods, our approach demonstrates superior feature rep- resentation performance by producing clearer boundaries and more compact clusters, as shown in Fig. 5. Its most notable advantage is the ability to disentangle intra-class clusters while preserving inter-class separability, which leads to better scalability and robustness when handling data with large cross-view distribution disparities.
4.3 Convergence Analysis
By analyzing the training curve in Fig. 6, we observe the following key points: (1) The evaluation metrics gener- ally exhibit oscillatory in- crease followed by stabiliza- tion, with this stability be- ing maintained as the num- ber of training epochs pro- gresses. This observation un- derscores the model's conver- gence and its robustness in maintaining stable clustering performance despite optimiza- tion challenges. (2) During the early stages of the alignment
67
04-
Valtette Metton
PUR
ARI
Trail 1an
12
02
300 330 361 300 420 210 540 shu 600
0.
300
230
420
400 480 sto
Kgesh
(a) Out-Scene
(b) MSRCVI
Figure 6: Convergence analysis of the training process. The left area of the vertical black dashed line represents the pre-training phase, while the right area stands for the view alignment training process. training phase, the evaluation metries exhibit a brief dip, which quickly followed by recovery and subsequent stabilization higher level. The fluctuations observed in the evaluation metrics

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can be attributed to LCDA in Eq. (13), which necessitates the initialization of all network parameters except the autoencoder. (3) Despite significant fluctuations in the loss for small datasets, intriguingly. these fluctuations do not propagate to the evaluation metrics, suggesting that our approach effectively mitigates the influence of instability in the optimization algorithm on the clustering structure.
Table 4: Clustering results on big (Out-Scene & ALOI--10) and small multi-view datasets.
Methods Datasets
Out-Seenc
ALOI-100
Wikipedia
MSRCVI
ACC
NMI
PUR
ACC
NMI
PUR
ACC
NMI
PUR
ACC
NMI
PUR
COMPLETER [CVPR'21] [44]
69.79
55.39
69.79
30.70
62.12
33.63
57.14
53.10
59.31
90.00
87.90
90.00
[TPAMI 23] [45]
56.03
45.59
56.32
34,01
60.28
27.14
MFLVC (CVPR 22| [48]
58.97
$1.31
58.97
33.17
73.28
33.17
40.12
27.52
41.70
63.33
66.11
64.29
DSMVC [CVPR'2] [43]
53.01
64.25
71.52
72.72
54.74
SURE MTP) [62]
60.97
48.09
60.97
10.13
34.19
11.90
50.65
39.97
54.11
91
85.84
91.43
DealMVC [MM]23] [49]
69.57
59.44
69.57
13.11
37.09
75.54
GCFAgg [CVPR'23] [3]
68.23
57.14
68.23
74
88.30
76.63
51.80
45.87
56.57
39.52
31.91
42.86
CPSPAN [CVPR'23] [51]
59.15
50.46
59.15
56.96
78.78
67.99
22.08
8.35
24.39
67.62
69.83
89.52
SDMVC [TKDE'23] [50]
56.03
46.
59.93
52.02
74.70
56.56
55.99
53.98
62.05
59.52
52.51
45.24
CVCL [ICCV'231
73.51
59.59
23.51
21.86
43.13
23.29
14.17
42,81
32.69
48.44
84.57
90.62
MVCAN [CVPR'24] [53]
70.98
58.23
49.95
67.48
83.78
56.71
59.02
55.81
67.97
71.54
60.19
71.54
SCMVC [TMM
71.54
60.19
71.54
27.72
89.42
81.05
53.54
35.59
55.84
90.95
83.92
90.95
SparseMVC (Ours)
77.49
63.34
77.49
82.21
92.65
84.19
61.04
54.79
62.91
97.14
94.22
97.14
4.4 Ablation Study
Loss Function To ensure the rigor of ablation experiment, we selected three datasets with significant
variations in view sparsity shown in Fig. 1, ensuring the functionality of the SAA. We assessed the effectiveness of individual losses in the total loss Eq. (14) of SVarseMVC, as presented in Table
Specifically, we use freson as the baseline and find that adding either Lenmopy or LCDA improves performance. Cnttropy yields substantial improve- ments on ALOI-100 and Dermatology, which have stronger variations in view sparsity. These improvements highlight the effectiveness of adap- tive encoding and view distribution align- ment as robust constraints that contribute posi- tively to the overall model training process. More- over, when all losses are activated simultaneously, the model achieves optimal performance, suggest- ing that Lentopy and LCDA complement each other synergistically. Notably, their integration does not introduce any mutual interference, further under- lining the coherence and compatibility of these objectives in driving superior learning outcomes.
Table 5: Comparison of different loss function combinations.
Loss Function
Evaluation Metrics
Datasets
Eiecon
Lenx
ACC
NM1
PUR
45.27
71.21
30.41
ALOI-100
66.35
81.65
70.62
64.11
80.23
67.33
92.65
84.19
1
70.
74.41
59.69
Dermatology
75.70
83.36
69.36
70.95
71.06
83,80
89,86
95.25
58.57
48.63
32.76
MSRCVI
70.48
65
52.08
92.38
87.62
92.38
97.14
94,22
97.14
Components Our approach focuses on view-level structural sparsity, specifically the sparsity variation across views within the same multi-view data. This differs from data-level sparsity methods, which typically apply uniform sparse encoding to all views without explicitly considering the heterogeneity of inter-view sparsity. To further validate the effectiveness of the proposed SAA module, we extend the ablation study in Table 5 by introducing two additional comparative settings: (i) uniformly sparse encoding applied to all views, which mimics methods designed for data-level sparsity; and (ii) adaptive encoding tailored each view. On top of this, we also ablate the CSR module, which reweights the local features during the late fusion stage. Regardless of whether the CSR module applied, the results in Table show that the proposed SAA, which leverages adaptive autoencoders, remains effective and consistently achieves superior performance. Results further confirm that the effectiveness arises from the synergy between adaptive encoding and sample reweighting, rather than from the use of sparse autoencoders alone. Collectively, these findings confirm that our method robust to varying sparsity across views and that the SAA and CSR modules function synergistically rather than independently.

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Table 6: Ablation study different components.
Componenis \ Dalasets
ALOI-100 100
Dermatology NMI
MSRCVI
ACC
NMI
PUR
ACC
PUR
ACC
NMI
PUR
all sparse autoencoders w/o CSR
78.56
88.92
81.12
77.37
74.38
86.03
91.90
88.56
91.90
all sparse sutocncoders
80,42
89.19
82.44
89.11
78.37
89.11
92.38
88.61
92.38
adaptive autoencoders
89.58
78.23
95.71
95.71
adaptive auioencuders (Ours)
82.21
92.65
84.19
95.25
89.86
95.25
97.14
94.22
97.14
4.5 Parameter Sensitivity Analysis
We selected the temperature parameter + in LCDA and the constraint ratio coefficient ACR. the ratio of CSAA to LCDA. the two parameters for analysis. Both coefficients were gradually increase from 0.1 to 1.9 with a step size of 0.3.
(a) Synthetic3d
(b) Out-Scene
(c) BRCA
(d) MSRCVI
Figure 7: The bar chart of clustering accuracy varying with different values of T and ACR-
In Fig. 7, we can discover that the accuracy initially increases and then decreases as T increases, remaining relatively stable within a range around 1.0. The influence of XcR on clustering performance is comparatively minor, with negligible impact when is within the range of 0.4 to 1.0. In light of Sec. 3.3, LCDA incorporates the smoothing property of the logarithmic function, which diminishes the direct effect of λcR adjustments the gradient. In contrast, changes to significantly influence clusters separability and alignment performance by modulating the nonlinear response the softmax function. A smaller enhances class separability, while larger emphasizes global consistency. Therefore, we set the default values of r and ACR to 1.0 in the loss function.
Our method is largely insensitive to hyperparameter changes. To begin with, significant performance degradation only occurs when the temperature coefficient 7 S 0.4 2 1.6. The extremely small value of 0.1 is chosen to probe the lower bound of performance degradation and is rarely used in practical applications. Furthermore, the performance fluctuation mainly occurs along the r-axis, whereas it remains relatively insensitive to changes in the constraint ratio coefficient ÀCR In practice, it is the relative weighting between loss terms that is more commonly adjusted. In addition, the accuracy-axis intentionally truncated to better highlight the differences, accentuating the visual disparity. Finally, regarding the concern that hyperparameters are not easy to tune in practice, our method maintains stable performance even under noticeable loss fluctuations, illustrated in Fig. 6.
Conclusion
This paper highlights a frequently overlooked issue in deep multi-view learning: varying sparsity ratios across views. Therefore, we systematically define, quantify, and analyze cross-view sparsity variation as fundamental characteristic of multi-view data. Our entire framework, SparseMVC, is designed to handle view-level sparsity variations with a complete data-driven and tightly integrated architecture. To tackle sparsity variation, we propose an adaptive encoding strategy that uses the sparsity ratio of each view as prior knowledge, enabling the encoder to switch between standard and sparse forms with appropriate constraint strengths. Additionally, we introduce a series of interdependent mechanisms to mitigate the side effects of representational divergence caused by non- uniform encoding. Specifically, a correlation guiiedd fusion strategy leverages global-to-local feature relationships from the early stages to guide the weighting of local features in late fusion. Moreover, distribution alignment module structurally constrains the fused representations, enhancing cross-view complementarity in the final stage. Comprehensive experiments and detailed dissections of each module validate the efficacy of SparseMVC VC. We hope this work inspires greater attention to the intrinsic characteristics of data and to the design of architectures driven by data.
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Acknowledgements
This research was supported in part by the National Natural Science Foundation of China under grants 62522604 and 62476258, the Natural Science Foundation of Hubei Province under grant 2025AFA113 13, the Interdisciplinary Research Program of HUST under grant 2025JCY3024, and the Fundamental Research Funds for National Universities of China University of Geosciences (Wuhan) (No.2024XLB7).
References
[1] Siwei Wang, Xinwang Liu,, Suyuan Liu, Jiaqi Jin, Wenxuan Tu, Xinzhong Zhu, and En Zhu. Align then fusion: Generalized scale-scale multi-view clustering with anchor matching corre- spondences. Advances in Neural Information Processing Systems, 35:5882-5895, 2022.
[2] Xin Zou, Chang Tang. Xiao Zheng, Kun Sun, Wei Zhang, and Deqiong Ding. Inclusivity induced adaptive graph learning for multi-view clustering. Knowledge-Bir Based Systems, 267: 10424, 2023.
[3] Weiqing Yan, Yuanyang Zhang, Chenlei Lv, Chang Tang, Guanghui Yue, Liang Liao, and Weisi Lin. Gefagg: Global and cross-view feature aggregation for multi-view clustering. IEEE/VVE Conference Computer Vision and Pattern Recognition, pages 19863-19872, 2023.
[4] Jing Wang and Songhe Feng. Contrastive and new-interaction structure learning for multi-view clustering. In International Joint Conference Artificial Intelligence, pages 5055-5063, 2024.
[5] Yuang Xiao, Dong Yang, Jiaxin Li, Xin Zou, Hua Zhou, and Chang Tang. Dual alignment feature embedding network for multi-omics data clustering, Knowledge -assed Systems, 309:112774, 2025.
[6] Shiping Wang, Zhaoliang Chen, Shide Du, and Zhouchen Lin. Learning deep sparse regularizers with applications to multi-view clustering and semi-supervised classification. JEEE Transactions on Pattern Analysis and Machine Intelligence, 44(9):5042-505 2021.
[7] Zhenglai Li, Chang Tang, Xinwang Liu, Xiao Zheng, Wei Zhang, and En Zhu. Consensus graph learning for multi-view clustering. JEEE Transactions Multimedia, 24:2461-2477 2021.
[8] Chao Zhang, Huaxiong Li, Wei Lv, Zizheng Huang, Yang Gao, and Chunlin Chen. Enhanced tensor low-rank and sparse representation recovery for incomplete multi-view clustering. In AAAI Conference on Artificial Intelligence, volume 37, pages 11 11174-11182, 2023.
[9] Pan Xiao, Peijie Qiu, Sungmin Ha, Abdalla Bani, Shuang Zhou, and Aristeidis Sotiras. Sc-vae: Sparse coding-based variational autoencoder with learned ista. Pattern Recognition, page 111187, 2024.
[10] Canyi Lu, Shuicheng Yan, and Zhouchen Lin. Convex sparse spectral clustering: Single-view multi-view. IEEE Transactions on Image Processing, 25(6):2833-2844 2016.
[11] Chengliang Liu, Zhihao Wu, Jie Wen, Yong Xu, and Chao Huang. Localized sparse incomplete multi-view clustering. IEEE Transactions on Multimedia, 35:5539-55511 2022.
[12] Yik Lung Pang. Changjae Oh, and Andrea Cavallaro. Sparse multi-view hand-object recon- struction for unseen environments. IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 803-810, 2024.
[13] Kaiyi Xu, Minhui Wang. Xin Zou, Chengfu Ji, Hua Zhou, and Chang Tang. Cancer-druy response prediction via feature aggregation and association graph learning. Engineering Applications of Artificial Intelligence, 10061, 2025.
[14] Xin Zou, Di Lu, Yizhou Wang, Yibo Yan, Yuanhuiyi Lyu, Xu Zheng, Linfeng Zhang and Xuming Hu. Don't i just chase "highlighted tokens" in milms: Revisiting visual holistic context retention. arXiv preprint ar 251002251. 2025.

---

[15] Li Deng. The mnist database of handwritten digit images for machine learning research [best of the web]. IEEE signal processing magazine, 2916):1 2012.
[16] Yi Yang and Shawn Newsam. Bag-of-visual-words and spatial extensions for land-use classi- fication. In SIGSPATIAL International Conference on Advances in Geographic Information Systems, pages 270 770-279,200 2010.
[17] Chen Kong, Dahua Lin, Mohit Bansal, Raquel Urtasun, and Sanja Fidler. What are you talking about? text-to-image coreference. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3558-3565, 2014.
[18] Timothy Danford, Alex Rolfe, and David Gifford. Gse: a comprehensive database system for the representation, retrieval, and analysis of microarray data. In Biocomputing, pages 539-550. World Scientific, 2008.
[19] Adam Coates, Andrew Ng, and Honglak Lee. An analysis of single-layer networks in unsu- pervised feature learning. In International Conference on Artificial Intelligence and Statistics, pages 215-223. JMLR Workshop and Conference Proceedings, 2011.
[20] Yingming Li, Ming Yang, and Zhongfei Zhang. A survey of multi-view representation learning. IEEE transactions on knowledge and data engineering, (110):8663-888)) 2018.
[21] Youwei Liang, Dong Huang, Chang-Dong Wang, and Philip S Yu. Multi-view graph learning by joint modeling of consistency and inconsistency. IEEE transactions on neural networks and learning systems, 35(2):2848-288 2862, 2022.
[22] Avrim Blum and Tom Mitchell. Combining labeled and unlabeled data with co-training. In Annual Conference Computational Learning Theory, pages 92-100, 1998.
[23] Kamalika Chaudhuri, Sham M Kakade, Karen Livescu, and Karthik Sridharan. Multi-view clustering via canonical correlation analysis. In International Conference on Machine Learning, pages 129-136, 2009.
[24] Meng Qu, Jian Tang, Jingho Shang, Xiang Ren, Ming Zhang, and Jiawei Han. An attention- based collaboration framework for multi-view network representation learning. In ACM on Conference on Information and Knowledge Management, pages 1767-1776, 2017.
[25] Kaixuan Yao, Jiye Liang, Jianqing Liang. Ming Li, and Feilong Cao. Multi-view graph convolutional networks with attention mechanism. Artificial Intelligence. 307:103708, 2022.
[26] Qianqian Wang, Zhengming Ding, Zhiqiang Tao, Quanxue Gao, and Yun Fu. Generative partial multi-view clustering with adaptive fusion and cycle consistency. IEEE Transactions on Image Processing. 30:1771-1788 2021.
[27] Jun Wang, Chang Tang, Zhiguo Wan, Wei Zhang, Kun Sun, and Albert Y Zomaya. Efficient and effective one-step multiview clustering. IEEE Transactions on Neural Networks and Learning Systems, 3509) 2023.
(28) Shudong Huang, Zhao Kang, Ivor W Tsang, and Zenglin Xu. weig-wee multi-view clustering via kernelized graph learning. Pattern Recognition, 88:174-184, 2019.
[29] Jun Wang, Zhenglai Li. Chang Tang, Suyuan Liu, Xinhang Wan, and Xinwang Liu. Multiple kernel clustering with adaptive multi-scale partition selection. IEEE Transactions on Knowledge and Data Engineering, 36(1 1::641-66655 2024.
[30] Juncheng Lv, Zhao Kang, Boyu Wang, Luping Ji, and Zenglin Xu. Multi-view subspace clustering via partition fusion. Information Sciences, 560:410-424 2021.
[31] Zongbo Han, Fan Yang. Junzhou Huang, Changqing Zhang, and Jianhua Yao. Multimodal dy- namics: Dynamical fusion for trustworthy multimodal classification. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 20707-20717 17, 2022.
[32] Xin Zou, Chang Tang, Xiao Zheng, Zhenglai Li, Xiao He, Shan An, and Xinwang Liu. Dpnet: Dynamic poly-attention network for trustworthy multi-modal classification. In Proceedings of the 3t ACM international conference on multimedia, pages 3550-3559, 2023.
12

---

[33] Xin Zou, Chang Tang, Wei Zhang, Kun Sun, and Liangxiao Jiang. Hierarchical attention learning for multimodal classification. In 2023 IEEE International Conference on Multimedia and Expo (ICME), pages 931-941. IEFE, 2023.
[34] Jian Zhu, Xin Zou, Lei Liu, Zhangmin Huang, Ying Zhang, Chang Tang, and Li-Rong Dai. Trusted mamba contrastive network for multi-view clustering. In ICASSP 2025-2025 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1 1-5, IEEE, 2025.
[35] Xiao Yu, Hui Liu, Yuxiu Lin, Nan Liu, and Shanbao Sun. Sample-level weights learning for multi-view clustering spectral rotation. Information Sciences, 619:38-51 51, 2023.
[36] Bruno A Olshausen and David J Field. Sparse coding with an overcomplete basis set: A strategy employed by vI? Vision Research, (3123333311-3223 1997.
[37] Jianchao Yang. Kai Yu, Yihong Gong, and Thomas Huang. Linear spatial pyramid matching using sparse coding for image classification. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1794-1801 1801 IEEE, 2009.
[38] Geoffrey E Hinton and Richard Zemel. Autoencoders, minimum description length and helmholtz free energy. Advances in Neural Information Processing Systems, 6, 1993.
[39] Geoffrey E Hinton and Ruslan R Salakhutdinov. Reducing the dimensionality of data with neural networks. science, 31 313(5788):504-5007 2006.
[40] Andrew Ng et al. Sparse autoencoder. CS294A Lecture Notes, 72(2011):1-19, 2011.
[41] Xi Peng, Jiashi Feng, Shijie Xiao, Wei- Yau, Joey Tianyi Zhou, and Songfan Yang. Structured autoencoders for subspace clustering. JEEE Transactions on Image Processing. 27(10):50765500 2018.
[42] Solomon Kullback and Richard A Leibler. On information and sufficiency. The Annals of Mathematical Statistics, 22(1):79-86, 1951.
[43] Huayi Tang and Yong Liu. Deep safe multi-view clustering: performance degradation caused by view increase. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 202-211, 2022.
Reducing the risk of clustering
[44] Yijie Lin, Yuanbiao Gou, Zitao Liu, Boyun Li, Jiancheng Lv, and Xi Peng. Completer: Incomplete multi-view clustering via contrastive prediction. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 11174-11188 2021.
[45] Yijie Lin, Yuanbiao Gou, Xiaotian Liu, Jinfeng Bai, Jiancheng Lv, and Xi Peng. Dual contrastive prediction for incomplete multi-view representation learning. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(4):4447-4461, 2022.
[46] Jie Chen, Hua Mao, Wai Lok Woo, and Xi Peng. Deep multiview clustering by contrasting cluster assignments. In IEEF/CVF International Conference on Computer Vision, pages 16752- 16761, 2023.
[47] Song Wu, Yan Zheng, Yazhou Ren, Jing He, Xiaorong Pu, Shudong Huang, Zhifeng Hao, and Lifang He. Self weighted contrastive fusion for deep multi-view clustering, IEEE Transactions on Multimedia, 2024.
[48] Jie Xu, Huayi Tang, Yazhou Ren, Liang Peng, Xiaofeng Zhu, and Lifang He. Multi-level feature learning for contrastive multi-view clustering. IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 16051-16061 2022.
[49] Xihong Yang, Jin Jiaqi. Siwei Wang, Ke Liang, Yue Liu, Yi Wen, Suyuan Liu, Sihang Zhou, Xinwang Liu, and En Zhu. Dealmve: Dual contrastive calibration for multi-view clustering. In ACM International Conference Multimedia, pages 337-346, 2023.
[50] Jie Xu, Yazhou Ren, Huayi Tang, Zhimeng Yang, Lili Pan, Yang Yang, Xiaorong Pu, S Yu Philip, and Lifang He. Self-supervised discriminative feature learning for deep multi-view clustering. IEEE Transactions Knowledge and Data Engineering, 35(7):7470-7482, 2023.
13

---

[51] Jiaqi Jin, Siwei Wang, Zhibin Dong, Xinwang Liu, and En Zhu. Deep incomplete multi-view clustering with cross-view partial sample and prototype alignment. IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1 16600-11609, 2023.
[52] Jie Xu, Yazhou Ren, Xiaolong Wang, Lei Feng, Zheng Zhang, Gang Niu, and Xiaofeng Zhu. Investigating and mitigating the side effects of noisy views for self-supervised clustering algorithms in practical multi-view scenarios. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 72957-22966, 2024.
[53] John Winn and Nebojsa Jojic. Locus:
Learning object classes with unsupervised segmentation. In International Conference on Computer Vision, volume 1, pages 756-763. IEEE, 2005.
[54] H Altay Güvenir, Gülşen Demiröz, and Nilsel llter. Learning differential diagnosis of erythemato-squamous diseases using voting feature intervals, Artificial Intelligence in Medicine, 13()):47-165, 1998.
[55] Aude Oliva and Antonio Torralba. Modeling the shape of the scene: A holistic representation of the spatial envelope. Proceedings of International Journal of Computer Vision, 42:145-175, 2001.
[56] Jan-Mark Geusebroek, Gertjan Burghouts, and Amoll WM Smeulders. The amsterdam library of object images. Proceedings of International Journal of Computer Vision, 61:103-112 1121011,222221111111001111211 2005.
[57] Cancer Genome Atlas Research Network. Comprehensive, integrative genomic analysis of diffuse lower-grade gliomas. New England Journal of Medicine, 372(26;2481-24998 2015.
[58] Daniel C Koboldt, Robert Fulton, Michael D McLellan, Heather Schmidt, Joelle Kalicki- Veizer, Joshua F McMichael, Lucinda L Fulton, David Dooling, Li Ding, Elaine R Mardis, et al. Tega-network, comprehensive molecular portraits of human breast tumours. Nature, 490(7418):61-70,2 2012.
[59] Abhishek Kumar, Piyush Rai, and Hal Daume. Co-regularized multi-view spectral clustering. Advances in Neural Information Processing Systems, 24, 2011.
[60] Diederik P Kingma. Adam: A method for stochastic optimization. arXiv preprint
1rXiv:1412.6980, 2014.
[61] James MacQueen Some methods for classification and analysis of multivariate observations. In Proceedings of Berkeley Symposium on Mathematical Statistics and Probability, volume 1, pages 281 Oakland, CA, USA, 1967.
[62] Mouxing Yang, Yunfan Li, Peng Hu, Jinfeng Bai, Jiancheng Lv, and Xi Peng. Robust multi-view clustering with incomplete information. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(1):1055-1069, 2022.
[63] Geping Yang. Shusen Yang. Yiyang Yang, Xiang Chen, Can Chen, Zhiguo Gong, and Zhifeng Hao. Spgmve: Multiview clustering via partitioning the signed prototype graph. IEEE Transac- tions on Neural Nenworks and Learning Systems, 2024.
[64] Kaixuan Yao, Jiye Liang, Jianqing Liang, Ming Li, and Feilong Cao. Multi-view graph convolutional networks with attention mechanism. Artificial Intelligence, 307:103708, 2022.
[65] Fangdi Wang, Jiaqi Jin, Jingtao Hu, Suyuan Liu, Xihong Yang, Siwei Wang. Xinwang Liu, and En Zhu. Evaluate then cooperate: Shapley-based view cooperation enhancement for multi-view clustering. In The Thirth-eighth Annual Conference on Neural Information Processing Systems, 2024.
[66] Zongbo Han, Changqing Zhang, Huazhu Fu, and Joey Tianyi Zhou. Trusted multi-view classification. International Conference on Learning Representations, 2021.
[67] Xin Zou, Yizhou Wang, Yibo Yan, Yuanhuiyi Lyu, Kening Zheng, Sirui Huang, Junkai Chen, Peijie Jiang, Jia Liu, Chang Tang, and Xuming Hu. Look twice before you answer: Memory- space visual retracing for hallucination mitigation in multimodal large language models. Forty- second International Conference on Machine Learning (ICML), 2025.
14

---

[68] Yu Kang, Erwei Liu, Kaichi Zou, Xiuyun Wang, and Huaqing Zhang. Sparse clustering algo- rithm based on multi-domain dimensionality reduction autoencoder. Mathematics, 12(10):1526. 2024.
[69] Junwei Han, Jinglin Xu, Feiping Nie, and Xuelong Li. Multi-view k-means clustering with adaptive sparse memberships and weight allocation. IEEE Transactions on Knowledge and Data Engineering. 34(2):816-88) 82-8877 2020.
[70] Qiyue Yin, Shu Wu, Ran He, and Liang Wang. Multi-view clustering via pairwise sparse subspace representation. Neurocomputing, 156: 156:12-21, 2015.
[71] Zhanxuan Hu, Feiping Nie, Wei Chang, Shuzheng Hao, Rong Wang, and Xuelong Li. Multi- view spectral clustering via sparse graph learning. Neurocomputing, 384:1-10, 2020.
[72] Jie Chen, Shengxiang Yang, Xi Peng, Dezhong Peng, and Zhu Wang. Augmented sparse representation for incomplete multiview clustering. IEEE Transactions Neural Networks and Learning Systems, 35(3):4058 1071, 2022.
[73] Claude Elwood Shannon. A mathematical theory of communication. The Bell System Technical Journal, 27(3):379-422 1948.
[74] Tianyu Hua, Wenxiao Wang, Zihui Xue, Sucheng Ren, Yue Wang, and Hang Zhao. On feature decorrelation in self-supervised learning. In /EEE/CVF International Conference on Computer Vision, pages 9598-9608, 2021.
[75] Christoph H Lampert, Hannes Nickisch, and Stefan Harmeling. Attribute- based classification for zero-shot visual object categorization. IEEE transactions on pattern analysis and machine intelligence, 36(3):453 465, 2013.
[76] Hanchuan Peng, Fuhui Long, and Chris Ding. Feature selection based on mutual information criteria of max-dependency, max-relevance, and min-redundancy. IEEE Transactions on pattern analysis and machine intelligence, 27(8):1226- 1238, 2005.
[77] David L Donoho and Michael Elad. Optimally sparse representation in general (ponorthogonal) dictionaries via él minimization. Proceedings of the National Academy of Sciences, 100(5):2197- 2202, 2003.
[78] David L Donoho. Compressed sensing. IEEE Transactions on information theory, 52(4):1289- 1306, 2006.
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NeurIPS Paper Checklist
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Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The abstract and introduction clearly articulate the core problem addressed by the paper, cross-view sparsity variations multi-view clustering, and emphasize the limitations of existing methods in handling this structural heterogeneity. The proposed solution, SparseMVC, is concisely described as incorporating sparsity-aware constraints and view-ade view-adaptive autoencoder design. These claims fully supported by both the theoretical formulation and the experimental results across diverse datasets, including those with extreme sparsity imbalance.
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3. Theory assumptions and proofs
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For completeness, we include additional discussions and experimental details in the appendices. A summary of the contents is listed below:
Appendix A
Algorithm and Comparison with Prior Work.
Appendix B
Further Experiments.
Appendix C
Visualization.
Appendix D
Computational Complexity.
Appendix E
Theoretical Analysis.
Appendix F
Limitations and Future Work.
A
Algorithm and Comparison with Prior Work
A.1 Algorithm
The training procedure for SparseMVC described in Algorithm (1).
Algorithm 1 Training Steps for SparseMVC
Input: Multi-view data { X F cluster number K, and number of training epochs Egres Evon-
Output: Late-stage fusion representation Y.
1: Initialize random seed and select Adam optimizer.
2: for epoch = 1 Epr + Ecoo do
3: Update { [Z. V by minimizing (Liccon Y and (Cintopy V utilizing Eqs. (2) and (4).
4: Update Z. formed by the concatenation of Vz utilizing Eq. (2) and Eq. (4).
5: if epoch > Eyre then
6: Update weights [W by Eq. (9).
7: Update Y by minimizing LCDA utilizing Eq. (13).
8: end if
9:
end for
10: Perform K-means clustering representation Y.
A.2 Comparison with Prior Work
To contextualize contributions, we present comparative discussion with representative dynamic weighting methods proposed in the literature. Dynamic weighting has been widely explored in multi- view learning, primarily through autention-based fusion mechanisms optimization-drris Methods such as GCFAgg [3] employ attention to emphasize discrepancies among local views, aiming refine feature alignment. Other approaches, including SPGMVC [63] and MAGCN [64], enhance feature encoding or perform view-level aggregation to improve representational quality. In parallel, techniques such as SCMVC [47], SCE-MVC [65] and TMC [66] utilize mutual information [67] or probabilistic priors to adaptively assign importance to views or losses. Furthermore, recent efforts have introduced cew-invariant representations [46] and pattotype-guide learning [52] to mitigate cross-view variability. Despite these advances, existing methods often overlook the inherent inconsistencies of sparsity between views, phenomenon that can severely degrade the effectiveness of fusion [8, 9, 68]. In contrast, our proposed approach is motivated by the need to explicitly characterize and adapt to such cross-view sparsity variations. Specifically, our framework preserves global features obtained during early fusion and integrates both global and local view representations into the fusion process. The core of our design, the correlation innecaatioonnfoomedd sample reweighting module, dynamically adjusts fusion weights based on the learned correlation between global and local views, thereby enabling fine-gres grained, sample-specific adaptation.
strategies.
While MVASM [69] addresses the challenges of ambiguous class assignments in multi-view data by introducing an adaptive sparse membership matrix, our method introduces adaptive autoencoders with view-specific encoding strategies. Moreover, unlike existing techniques [70, 71, 72] that generally apply static or view-level weighting schemes, our method performs late-stage fusion reweighting conditioned on earlier global-local interactions. This design not only enhances fusion fidelity but also
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elevates sparsity modeling from the feature level to the view level, offering principled solution to problem rarely addressed in the literature.
B Further Experiments
B.1 Dimension and Feature Compression
Fistar Disardden
Peater Disvarker
Teaen Doecaden
(a) Out-Scene
(b) MSRCVI
(c) Synthetic3d
Figure 8: The clustering evaluation metric curves with respect to feature (Z,) dimension variations.
We tested the dimensions of view-specific features Z, extracted by the autoencoder, varying from 2 to 210 in Fig 8. The experimental results demonstrated that the feature dimension increased, all evaluation metrics (ACC, NMI, PUR and ARI) initially exhibited strong oscillations but progressively improved, eventually stabilizing within small and perfooreformng range.
From the perspective of information coommation-thhorttii [73]. low-dimensional features have limited encoding capacity that hinders capturing complex data patterns, whereas high-dimensional features offer greater capacity but risk dimensional collapse [74]. The feature compression layers that we designed compress and refine Zy into Y,, mitigating the risks of overfitting and performance degradation often seen in contrastive learning. Accordingly, the model can avoid the negative effects of excessive dimensionality while leveraging increased capacity for effective feature extraction. Ultimately, we selected 64 as the dimensionality of Z, to balance computation and feature capacity.
B.2 Selection of Scaling Factor
Table Ablation study on scaling factor selection.
scaling f(su)
ihe internal coeflicient
MSRCVI
ALOI-100
Mean accuracy [Max accuracy]
I.f(s))
94,29 [95.71]
[80.39]
f(eu)
I-f(e=)
95.24 [95.24]
79.01 [80.08]
0.05
95.71 (96.67)
80.82 [81.51]
f(su)
0.05
(97.62)
[82.93]
To investigate the relative importance of the external scaling factor f(s.) and the internal ratio coefficient p in modulating sparsity within the sparse autoencoder, we conducted a set of controlled experiments in which parameter kept constant while systematically varying the other. This decoupled analysis enables a clearer understanding of how each component contributes to the overall behavior of the model. Empirical results, shown in Table 7, demonstrate that allowing f(s.) to be adaptively optimized while fixing at reasonable constant leads to superior performance in multiple evaluation metrics. The smooth variation of the constraint strength is better than directly switching between zero and one, which represents whether the sparse constraint is applied. In contrast, varying p while keeping f(*..) static yields relatively suboptimal results. These findings suggest that external view-level scaling plays a more critical role in capturing cross-view sparsity dynamics, highlighting the effectiveness of our adaptive design.
B.3 The large-seale datasets
To evaluate the effectiveness and generalization ability of our proposed method, SparseMVC. we conduct additional experiments on large-scale datasets comprising over 8,000 samples, specifically GSE [18] and Animal [75]. The GSE dataset encompasses multi-omics data across 27 categories,
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Table 8: Clustering results the large-scale multi-view datasets.
Dutasets
GSE (8,200 samples)
Animal (1 1,6333 samples)
View Sparsity Ratio
[0,877, 0.005]
[0.589, 0. 0.179, 0.355, 0.467]
Methods
ACC
NMI
PUR
ACC
NM1
PUR
MVCAN [CVPR'24]
71.40
75.41
72.29
12.06
10.25
15.48
SCMVC [TMM
62.80
74.13
65.72
16.96
15.05
20.15 10.15
SparseMVC (Ours)
capturing a broad range of biological conditions, and extensively utilized bioinformaties research. Meanwhile, the Animal dataset consists of image data featuring animals with diverse attributes, spanning 20 categories. The detailed dataset information and comparison results, presented in Table 8, highlight the performance of our approach against the latest two methods employed in our manuscript. Clustering results on large-scale multi-view datasets further emphasize the superiority of our method, demonstrating its promising applicability in other data environments and scenarios.
C
Visualization
(a)
(b)
(c)
(d)
X. features
(h) Z features
(i) Y features
Figure 9: The t-SNE visualizations the original reconstruction features (X.),, the early fusion features (Z), and the late fusion features (Y) on datasets: Dermatology (a-c), Synthetic3d (d-f), and Out-Scene (g-i).
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Figure 9 uses t-SNE for dimensionality reduction, mapping high- high-dimensional feature vectors into a wwo-dimensional space to facilitate a clearer and more insightful visualization of the distribution and structure of the data. The visualization reveals that the original reconstruction data, X,, exhibit substantial interclass entanglement, with considerable overlap between different categories. In contrast, the preliminary global features, Z., derived from autoencoder pretraining, show a notable reduction in this entanglement, suggesting an early phase of disentanglement compared to the raw features. Further refinement through multi-view alignment results in the final fused features, Y Yglobal. which exhibit the most pronounced disentanglement, yielding well-defined, separate clusters. These findings are further corroborated by the results in ablation analysis, which demonstrate that Ygiobal consistently outperforms Zu in clustering tasks, particularly in distance-based methods like K-means. This performance enhancement underscores the pivotal role of the multi-view alignment and fusion process in improving feature separability and discriminative capacity, ultimately leading to more accurate and meaningful clustering results.
D Computational Complexity
The total loss function, as defined in Eq. (14), includes two main terms: LSAA and LCVDA The term Csaa involves the reconstruction loss and the entropy-matching loss, both computed for each of the V views, each containing N samples. Since the feature dimension I' is constant and does not affect the computational scaling, the complexity of this term is determined by the number of views and samples, resulting in (V.). The term LCVDA requires pairwise similarity computations between N samples. For each pair of samples, the dot product computation has a constant cost of O(1), leading total complexity of O(N)) for computing all pairwise similarities. The contrastive loss further requires computing the numerator, denominator, and logarithmic terms for each sample, which does not increase the overall scaling beyond ((22). When combining both terms, the reconstruction and entropy-matching losses contribute D(VN), while the pairwise similarity computations dominate with (N when N is sufficiently large. Thus, the overall computational complexity of the total loss function is O(N)).
E
Theoretical Analysis
When learning from heterogeneous multi-view data, structural disparities. especially in sparsity patterns, present significant challenges to unified representation learning. Differences modality, sampling granularity, and view incompleteness lead to varying information densities across views, making fixed encoder architectures and uniform loss formulations inherently suboptimal. To address this issue, we propose the Sparse Autoencoder with Adaptive Constraints (SAA), which dynamically adjusts the strength of sparsity constraints based on sparsity ratio 8, each view. This mechanism enables the encoder to balance compression and expressiveness in view-aware manner, thereby facilitating the alignment of latent representations across structurally diverse views. In what follows, provide preliminary theoretical foundation for SAA, organized around a set of key questions that clarify its motivation, theoretical grounding, and coding formulation:
Why should the sparsity constraint adaptive?
When dealing with views of varying characteristics, uniform treatment of data may result in suboptimal representations, particularly when the model overly compresses informative in dense views overemphasizes noisy in sparse ones. Accordingly, by the principle of minimal redundancy maximum relevance (mRMR) [76], we can establish trade-offs between different views, balancing the fidelity of reconstruction and the complexity of representation.
What exactly are we coding or preserving?
Grounded in compressed sensing and optimally sparse representations [77, 177,75 78], we encode the es- sential structure sparse inputs, which, though embedded in high-dimensional space, intrinsically conform to low-dimensional semantic subspaces. Therefore, sparse activation suffices to capture their core features, with sparse coding providing a stable, efficient, and redundancy-minimitn representation aligned with this inherent geometry.
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E.1
Why should sparsity constraints be adaptive?
From Max- Dexendeney to mRMR and our view-wise coding. Let S= { feature set and c be the target. Peng et al. [76] formulate max-dependency as maximizing the mutual information between the joint selected features and the target:
- be selected
max 8
()):c),
c(S;c) = ((S;c) = ( Tmm c).
(15)
Because directly estimating (S; is difficult when m is not tiny, they introduce a first-order surro- gate: max-relevance (ra [S] Ex,s [( c), and control redundancy by mainreduudancy (z), then combine them as the mRMR criterion max m (Se c)
R(S) ?S?
R(S)) (difference form) [76]. In our setting, for each view v. the encoder produces a hidden code h. = (hi,..., 1/ S and the input as the "target" с (unsupervised but still information (ioormattontthhorttte)) we use a weighted difference form to allow view-wise calibration, and the view-wise analogue of mRMR becomes:
for input random variable X,, Interpreting hidden units as the "selected features"
H LE h()
max
Σ x.) - Bu H i=
(16)
max-relovance
min-redundancy
where B, 2 0 is the redundanccy/constraint strength for view U.
Why 8, depends on view sparsity s, redundancy scales with the effective degrees of freedom. SparseMVC explicitly measures the sparsity ratio s, of view e as a prior statistic in Eq. (I). A larger s, indicates a higher fraction of zero (or missing/default) entries and thus lower effective information density in X In this regime, the first term in Eq. (16) maximizing relevance alone encourage multiple hidden units to focus the same limited set of informative patterns, which increases dependence among {hg} and inflates the redundancy term.
To relate redundancy to the sparsity regularizer used by sparse autoencoders, we follow the standard sparse autoencoder interpretation [40] and treat each hidden unit as a Bernoulli activation variable when discussing entropic quantities. Then, for any pair (i,
[hh; bb)) Hh,)) (h) h2) S H(h)),
(17)
where (-) denotes Shannon entropy of the corresponding Bernoulli variable. This inequality shows that suppressing the entropies (and co-activations) of hidden units controls an upper bound on pairwise dependence, and thus provides principled route to reducing redundancy when the view is highly sparse.
SAA realizes adaptive mRMR via entropy-matching. SVarseMVC instantiates the adaptive redundancy coefficient as a deterministic function of s, in Eq. (3) and implements redundancy control via the For numerical stability in KL computation, the estimated mean activation is clamped to remain strictly within (0, 1) in implementation. This design matches the mRMR logic at the view level: large views precisely those where redundancy inflation is most likely, f(..) increases the effective redundancy penalty; s.-small views receive weak or no sparsity constraint (e.g., f(sx) = 0 below f), preventing unnecessary relevance loss.
KL penalty (sparse autoencoder prior) on the mean activation hy in Eq. (4).
Conclusion. According to the mRMR principle [76], effective representations should maximize relevance while minimizing redundancy. SparseMVC operationalizes this principle with view- dependent redundancy strength, using the sparsity ratio 8, to adaptively scale the KL--based sparsity penalty. As result, sparse views benefit from stronger redundancy suppression, whereas dense views retain greater expressive capacity, yielding a principled, data-driven trade-off between information preservation and redundancy reduction across heterogeneous views.
E.2
What exactly are coding preserving?
This question concerns the rationale and effectiveness of using sparsity constraints and sparse autoencoders to represent sparse data. Sparse data are characterized by low information density:
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only small fraction of entries are nonzero and carry informative structure, while many coordinates may correspond to missing/default states. This suggests that an effective representation should focus capacity on the informative components, rather than distributing it uniformly across all dimensions. Consequently, representations of sparse inputs are often encouraged to be sparse as well, activating only small subset of latent units that capture the dominant patterns supported by the nonzero entries, thereby reducing redundancy.
Sparse inputs and informative support. Formally, let the j-th sample from view v be denoted by 23 R"y, with sparsity level || k < Ne. Define the (sample-dependent) support set S = [i [i # 0}. In sparse views where zeros primarily represent missing/default entries, the informative variation is mainly concentrated on the restricted subvector "j,0y, while the complement " (An.. contributes little signal. Thus, allocating excessive modeling capacity to the zero-valued region (e))) may introduce unnecessary degrees of freedom and increase redundancy.
Reconstruction error and support-aware desideratum. Consider linear decoder dictionary Dr E IR n. X and a latent code oz E "R"".. Let Day denote the row submatrix of D indexed by 22j
and let D Dayy denote the row submatrix indexed by the complement of Q7. Then the reconstruction error decomposes as:
| D,a2 = 1 2i + PPaa. Paap.
(18)
The second term measures spurious energy produced on coordinates that zero in the input. A natural desideratum for sparse-view reconstruction is therefore to keep |Pjayy- [ small. Impor- tantly, sparsity of oz alone does not guarantee this term is exactly zero; rather, sparsity regularization reduces the degrees of freedom available fit (D))c and encourages the decoder to rely on a small number of atoms, which empirically helps suppress spurious energy outside the informative support.
Sparse coding and recoverable representations. Under this support-aware viewpoint, sparse coding seeks representation of the form:
F= arg min |y D s.t. ||allo S,
(19)
with 8 m.. Classical results in sparse approximation characterize uniqueness in the noiseless exact-representation case, e.g., if x", ". = oi and 8 < 1 (D then the sparsest solution is unique [77]. Compressed sensing theory further establishes stable recovery guarantees under Restricted Isometry Property (RIP)-type conditions for standard ( -baeed decoders [78]. These guarantees justify sparse coding as a canonical representational prior for signals whose structure is concentrated on low-dimensional support.
Sparse autoencoder as amortized sparse recovery. A sparse autoencoder implements the above sparse recovery principle in an amortized and differentiable manner. For view v. the encoder produces hidden activations whose empirical mean is given by:
1 N N
N
F
h
Σ o (WE 1=1
bb)
W
+ 88),
(20)
1=1
i=1
where (-)) is chosen so that the quantity used in the KL term can be interpreted as a probability, and hy is clamped to lie strictly within (0, 1) in implementation before computing the KL divergence. Enforcing hg to match a small target sparsity level via KL-divergence sparsity penalty imposes an lo-like constraint in expectation on the latent representation. As a result, for each sparse input 2.. only a small subset of hidden units tends to be active, which qualitatively mimics sparse coding behavior without explicitly solving a combinatorial optimization problem. Eq. (20) also makes explicit that sparsity is enforced through an empirical average (over samples or minibatches) rather than per-sample hard sparsity constraints.
Conclusion. Sparse coding theory and compressed sensing results [77, 78] provide principled rationale for representing sparse-view signals with sparse codes under standard recoverability as- sumptions. A sparse autoencoder operationalizes this idea by combining reconstruction with Kl.-based sparsity penalty, yielding compact representations that suppress redundant co-activations and emphasize patterns supported by the informative entries of the input.
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Limitations and Future Work
While effective in addressing cross-view sparsity variations, the current approach has several limi- tations that suggest directions for further improvement. To begin with, SparseMVC demonstrates strong performance across a variety of multiview data, but its effectiveness real-world applica- tions with noisy or incomplete scenarios remains to be fully explored. Additionally, the use of contrastive learning inherently introduces computational overhead, making it unlikely to rank among the fastest available approaches. Moreover, the framework assumes well-aligned views, whereas slight inter-view misalignment may occur in real-world scenarios.
A potential direction for future work is to incorporate structural information across views, which can be modeled through priors such as graph connectivity, inter-view relational graphs. or mutual information
that have been extensively explored in prior multi-view repre- sentation learning research. In contrast, the present study primarily focuses on feature-level sparsity and its adaptive regularization, rather than modeling explicit structural dependencies. Incorporating structural priors may further enhance representation quality, particularly in scenarios where inter-view relationships are semantically meaningful.
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Reviews:
[{"review_id": "e7nKp6E6Uc", "reviewer": "Reviewer_p2jA", "summary": "This paper addresses the varying sparsity problem in multi-view data by proposing an adaptive sparse autoencoder for MVC. To mitigate encoding disparities across views, an attention mechanism is incorporated based on inter-view correlations. Furthermore, to enhance representation consistency, a contrastive loss is introduced to align view-specific representations with a unified global representation.", "questions": "1. How is multi-view data fused in the early fusion strategy?\n2. Why does the sparsity of the ALOI-100 dataset differ between Figure 1 and Table 1? There appears to be an inconsistency in the reported sparsity of the ALOI-100 dataset between Figure 1 and Table 1.", "limitations": "YES", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\nThis paper tackles cross-view sparsity variation in multi-view learning and conducts experiments on datasets with pronounced sparsity differences across views, such as ALOI-100.\nWeakness:\n1.\tThe method addresses sparsity variation by applying an adaptive weight based on an existing entropy-matching loss. This solution is relatively simplistic; additional theoretical justification is recommended to support the claimed contribution.\n2.\tMost of the evaluated datasets are small-scale, typically containing only a few hundred samples, which may limit the robustness of the conclusions.\n3.\tThe method lacks comparisons with MVC approaches specifically designed to handle data-level sparsity. An ablation study of the adaptive weight component is also necessary to validate the effectiveness of the proposed SSA. As currently presented, the results do not convincingly argue the method’s ability to handle varying sparsity.", "was_revised": true, "final_justification": "After carefully reading the rebuttal, I found that most of my concerns have been fixed, including adding more large-scale datasets which also exhibit SOTA performances, and more experiments w.r.t the effectiveness of the SAA module. So I would like to raise the score to acceptance.", "rebuttal": "Thanks for your constructive reviews and valuable suggestions. Below, we will address your questions and concerns one by one.\n\n---\n\n> ***Question 1**:* How is multi-view data fused in the early fusion strategy?\n\nThanks. The early fusion strategy integrates the data from each view through **feature concatenation**. Since our method uses the global latent representation encoded from early fusion to guide the subsequent dynamically weighted late fusion of local representations, we chose an early fusion approach that preserves the original feature values as much as possible.\n\n---\n\n> ***Question 2**:* Why does the sparsity of the ALOI-100 dataset differ between Figure 1 and Table 1? There appears to be an inconsistency in the reported sparsity of the ALOI-100 dataset between Figure 1 and Table 1.\n\nThanks, and I apologize for any misunderstanding or confusion. As explained in the figure caption, the sparsity ratios in **Figure 1** are transformed using **the sigmoid function**, which shifts the baseline from 0 (the bottom of the image) to 0.5 (the middle of the image) for better visualization, as many views have very small sparsity values (less than 0.01) that would be hard to see with the original scale.\n\n---\n\n> ***Question 3**:* The method addresses sparsity variation by applying an adaptive weight based on an existing entropy-matching loss. This solution is relatively simplistic; additional theoretical justification is recommended to support the claimed contribution.\n\nThanks. The entropy-matching loss serves merely as one component for regulating the sparsity of representations, and our contribution is by no means limited to this design alone. Rather, by systematically defining, quantifying, and analyzing cross-view sparsity variation as a fundamental characteristic of multi-view data, we construct an integrated, data-driven framework that is specifically motivated by this widely existing yet previously underexplored property. To address sparsity variations across views within a dataset, we propose not only an adaptive encoding strategy that enables the encoder to automatically transition between standard and varying degrees of sparse encoding, but also a series of interdependent mechanisms to mitigate the side effects of representational divergence caused by such differentiated encoding. Specifically, we incorporate a correlation-guided fusion strategy that leverages global-to-local feature relationships established in the early stage to guide the weighting of local features during late fusion. Moreover, a distribution alignment module further constrains the fused representations structurally, thereby enhancing cross-view complementarity in the final stage.\n\nRegarding the theoretical analysis, we have discussed the necessity of adaptive sparsity constraints in a summarized manner in Appendix E, where we explain how uniform encoder architectures and fixed loss functions are often ineffective in handling sparsity variations across different views. We also outline how our method balances compression and expressiveness by adjusting the strength of sparsity constraints, which in turn enhances the alignment of latent representations. We hope this partially addresses your concerns, and, in line with your suggestion, we will further expand this section to more comprehensively substantiate the contributions of our method.\n\n---\n\n> ***Question 4**:* Most of the evaluated datasets are small-scale, typically containing only a few hundred samples, which may limit the robustness of the conclusions.\n\nThanks for the suggestion. We conduct further experiments on the large-scale datasets GSE [1] and Animal [2]. Below are the larger dataset details and comparison results using the latest methods employed in our manuscript:\n\n| **Model \\ Dataset** |    GSE    | Samples 8,200 | View Sparsity Ratio [0.877, 0.005] |  Animal   | Samples 11,673 | View Sparsity Ratio [0.589, 0.179, 0.355, 0.467] |\n| :-----------------: | :-------: | :-----------: | :--------------------------------: | :-------: | :------------: | :----------------------------------------------: |\n|                     |    ACC    |      NMI      |                PUR                 |    ACC    |      NMI       |                       PUR                        |\n|   MVCAN [CVPR’24]   |   71.40   |     75.41     |               72.29                |   12.06   |     10.25      |                      15.48                       |\n|   SCMVC [TMM’24]    |   62.80   |     74.13     |               65.72                |   16.96   |     15.05      |                      20.15                       |\n|  SparseMVC (ours)   | **73.18** |   **78.42**   |             **74.26**              | **19.90** |   **17.92**    |                    **24.13**                     |\n\n---\n\n> ***Question 5**:* The method lacks comparisons with MVC approaches specifically designed to handle data-level sparsity. An ablation study of the adaptive weight component is also necessary to validate the effectiveness of the proposed SSA. As currently presented, the results do not convincingly argue the method’s ability to handle varying sparsity.\n\nThanks. Our approach focuses on view-level structural sparsity, specifically the sparsity variation across views within the same multi-view data. This differs from data-level sparsity methods, which typically apply uniform sparse encoding to all views without explicitly considering the heterogeneity of inter-view sparsity. To further **validate the effectiveness of the proposed SAA module**, we extend the ablation study in Table 5 by introducing two additional comparative settings: **(i) *uniformly sparse encoding applied to all views, which mimics methods designed for data-level sparsity***; and **(ii)** ***adaptive encoding tailored to each view*.** On top of this, **we also ablate the CSR module**, which reweights the local features during the late fusion stage. Regardless of whether the CSR module is applied, the results in the bottom table show that the proposed SAA, which leverages adaptive autoencoders, remains effective and consistently achieves superior performance.  Collectively, these findings confirm that our method is robust to varying sparsity across views and that the SAA and CSR modules function synergistically rather than independently.\n\n|   Method \\ Dataset    |           | ALOI-100  |           |           | Dermatology |           |           |  MSRCV1   |           |\n| :-------------------: | :-------: | :-------: | :-------: | :-------: | :---------: | :-------: | :-------: | :-------: | :-------: |\n|                       |    ACC    |    NMI    |    PUR    |    ACC    |     NMI     |    PUR    |    ACC    |    NMI    |    PUR    |\n| all sparse ae w/o CSR |   78.56   |   88.92   |   81.12   |   77.37   |    74.38    |   86.03   |   91.90   |   88.56   |   91.90   |\n|     all sparse ae     |   80.42   |   89.19   |   82.44   |   89.11   |    78.37    |   89.11   |   92.38   |   88.61   |   92.38   |\n|  adaptive ae w/o CSR  |   81.04   |   89.58   |   83.17   |   88.83   |    78.23    |   88.83   |   95.71   |   92.69   |   95.71   |\n|  adaptive ae (ours)   | **82.21** | **92.65** | **84.19** | **95.25** |  **89.86**  | **95.25** | **97.14** | **94.22** | **97.14** |\n\n**Reference**\n\n[1] Danford T, Rolfe A, Gifford D. GSE: a comprehensive database system for the representation, retrieval, and analysis of microarray data[C]//Pacific Symposium on Biocomputing. Pacific Symposium on Biocomputing. 2008: 539.\n\n[2] Lampert C H, Nickisch H, Harmeling S. Attribute-based classification for zero-shot visual object categorization[J]. IEEE transactions on pattern analysis and machine intelligence, 2013, 36(3): 453-465."}, {"review_id": "Kz32Y2a7CA", "reviewer": "Reviewer_ar36", "summary": "This paper proposes a multi-view clustering method focusing on the sparsity difference across views. Thanks to the design of the sparse autoencoder, the correlation-informed sample reweighting module, and the cross-view distribution alignment module, the proposed method achieves the state-of-the-art performance on eight datasets.", "questions": "I would like the authors to address my concerns raised in the weaknesses section.", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "- Strengths\n\t1. The proposed method is evaluated on eight datasets and achieves promising clustering performance.\n- Weaknesses\n\t1. The motivation of this work is questionable. Why must sparsity correspond to less informative features? The information about zeros depends on its physical meaning behind. For example, true zero values also represent the characteristics of the sample. \n\t2. While this work claims that sparsity harms the multi-view clustering performance, the method seems to be designed for handling low-quality features in general. In other words, only a few components in the proposed method are specifically designed to handle data sparsity. In fact, the most related design, sparse auto-encoder, has already been proposed in previous works. As a result, the contribution of this work seems limited.\n\t3. The datasets used for evaluations are too small. Methods evaluated and tuned on datasets with only hundreds of points might not generalize well to other datasets and scenarios.\n\t4. The proposed method is also sensitive to hyperparameters, which are not easy to tune in practice, given the unsupervised nature of clustering.", "was_revised": true, "final_justification": "Thanks for the responses. With the authors' explanations and clarifications, my concerns have been addressed, so I would like to raise my score.", "rebuttal": "We sincerely appreciate your constructive comments and suggestions. Below, we will address each of your questions and concerns.\n\n---\n\n> ***Question 1**:* The motivation of this work is questionable. Why must sparsity correspond to less informative features? The information about zeros depends on its physical meaning behind. For example, true zero values also represent the characteristics of the sample.\n\n**(1) Why must sparsity correspond to less informative features?** Thanks. According to sparse coding and compressed sensing theory [1-2], sparse features lack sufficient capacity to preserve underlying structure and thus carry less informative for downstream tasks. While individual zeros may sometimes be meaningful, our concern lies in the overall representational capacity of a view. For example, in a 1024-dimensional embedding, if only a few dimensions are activated, the representation is likely information-poor. This phenomenon aligns with the theoretical insight that excessively sparse or under‑activated representations—especially when unevenly distributed across views—struggle to capture the full semantics of the data.\n\n**(2) We focus on statistically meaningful patterns of structural sparsity, rather than simplistically equating zero values with a lack of semantic significance.** In our work, sparsity refers to the structural variation across views within the same multi‑view dataset, where some encoded views exhibit extremely limited activation and markedly low information density  [2]. This form of structural sparsity, which frequently produces high-dimensional but low-rank features, limits expressive capacity and complicates alignment and clustering. Therefore, while individual zeros can be meaningful, our study emphasizes statistically significant sparsity patterns across views.\n\n**(3) In multi-omics data, zeros often arise from sequencing dropout or detection limits rather than representing meaningful biological signals [3–4]**. For instance, methylomic and proteomic profiles frequently contain such zeros, while transcriptomic data are typically more complete. In these cases, a zero simply indicates that no signal was detected in that dimension. \n\n**(4) Motivation: addressing view-wise imbalance caused by cross-view sparsity variations, rather than sparsity itself or data‑level sparsity.** In summary, we neither assume that zeros are inherently uninformative nor disregard them. Instead, our framework targets cross-view sparsity variations within the same dataset, an objective statistical property that does not depend on the semantic meaning of zeros. Consequently, we adopt an adaptive and view‑specific encoding strategy tailored to view sparsity differences.\n\n---\n\n> ***Question 2**:* While this work claims that sparsity harms the multi-view clustering performance, the method seems to be designed for handling low-quality features in general. In other words, only a few components in the proposed method are specifically designed to handle data sparsity. In fact, the most related design, sparse auto-encoder, has already been proposed in previous works. As a result, the contribution of this work seems limited.\n\nWe sincerely thank the reviewer for raising this important and insightful concern.\n\n(1) To begin with, we would like to clarify that the primary focus of our work is **not on general data sparsity** or low-quality features, but rather on a more specific and previously underexplored issue: **the sparsity variations across views within the same multi-view data**.\n\n(2) Moreover, our entire framework is specifically designed to handle view-level sparsity variations, a prevalent yet underexplored characteristic of multi-view data, through a complete and tightly integrated data-driven architecture, where **the sparse autoencoder serves only as one means of enforcing sparse encoding**. To address the sparsity variations across views, **we design an adaptive encoding mechanism that leverages the sparsity ratio of each view as prior knowledge, allowing the encoder to transition between standard and sparse modes with appropriate constraint strengths**. To mitigate the enlarged representation gap caused by heterogeneous encoding, we further introduce a correlation-guided reweighting module. It adjusts each view’s contribution during late fusion based on **correlations between late-fused local and early-fused global features**. In addition, a distribution alignment module reinforces inter-view complementarity through structural constraints.\n\n(3) To the best of our knowledge, we are **the first to systematically conduct statistical analysis of view-wise sparsity ratios and formally define the problem of cross-view sparsity variation in multi-view clustering field**. Our contribution lies not only in architectural innovation, but also in the discovery and analysis of fundamental multi-view data characteristics.\n\n(4) Finally, in further ablation summarized in the bottom table, we evaluate two variants: **(i)** all views use sparse autoencoders, and **(ii)** adaptive encoding is applied, both with the CSR reweighting module considered. Compared with above settings, our view-adaptive encoder, which adjusts the encoding strategy based on the sparsity level of each view, consistently achieves superior performance. Results further confirm that the effectiveness arises from the synergy between adaptive encoding and sample reweighting, rather than from the use of sparse autoencoders alone.\n\n|   Method \\ Dataset    |           | ALOI-100  |           |           | Dermatology |           |           |  MSRCV1   |           |\n| :-------------------: | :-------: | :-------: | :-------: | :-------: | :---------: | :-------: | :-------: | :-------: | :-------: |\n|                       |    ACC    |    NMI    |    PUR    |    ACC    |     NMI     |    PUR    |    ACC    |    NMI    |    PUR    |\n| all sparse ae w/o CSR |   78.56   |   88.92   |   81.12   |   77.37   |    74.38    |   86.03   |   91.90   |   88.56   |   91.90   |\n|     all sparse ae     |   80.42   |   89.19   |   82.44   |   89.11   |    78.37    |   89.11   |   92.38   |   88.61   |   92.38   |\n|  adaptive ae w/o CSR  |   81.04   |   89.58   |   83.17   |   88.83   |    78.23    |   88.83   |   95.71   |   92.69   |   95.71   |\n|  adaptive ae (ours)   | **82.21** | **92.65** | **84.19** | **95.25** |  **89.86**  | **95.25** | **97.14** | **94.22** | **97.14** |\n\nWe hope this work inspires greater attention to the intrinsic characteristics of data and to the design of architectures driven by data.\n\n---\n\n> ***Question 3**:* The datasets used for evaluations are too small. Methods evaluated and tuned on datasets with only hundreds of points might not generalize well to other datasets and scenarios.\n\nThanks for the suggestion. We conduct additional experiments on the large-scale datasets: GSE and Animal (please refer to the reference cited for Reviewer p2jA above). Below are the larger dataset details and comparison results using the latest methods employed in our manuscript:\n\n| **Model \\ Dataset** |    GSE    | Samples 8,200 | View Sparsity Ratio [0.877, 0.005] |  Animal   | Samples 11,673 | View Sparsity Ratio [0.589, 0.179, 0.355, 0.467] |\n| :-----------------: | :-------: | :-----------: | :--------------------------------: | :-------: | :------------: | :----------------------------------------------: |\n|                     |    ACC    |      NMI      |                PUR                 |    ACC    |      NMI       |                       PUR                        |\n|   MVCAN [CVPR’24]   |   71.40   |     75.41     |               72.29                |   12.06   |     10.25      |                      15.48                       |\n|   SCMVC [TMM’24]    |   62.80   |     74.13     |               65.72                |   16.96   |     15.05      |                      20.15                       |\n|  SparseMVC (ours)   | **73.18** |   **78.42**   |             **74.26**              | **19.90** |   **17.92**    |                    **24.13**                     |\n\n---\n\n> ***Question 4**:* The proposed method is also sensitive to hyperparameters, which are not easy to tune in practice, given the unsupervised nature of clustering.\n\nThanks. To begin with, a noticeable performance drop occurs only when the *temperature coefficient $\\tau$* ≤ 0.4 or ≥ 1.6. The extremely small value of 0.1 is chosen to probe the lower bound of performance degradation and is rarely used in practical applications. Furthermore, the performance fluctuation mainly occurs along the $\\tau$ direction, whereas it remains relatively insensitive to changes in the constraint ratio coefficient $\\lambda_{\\text{CR}}$. In practice, it is the relative weighting between loss terms that is more commonly adjusted. In addition, the y‑axis was intentionally truncated to better observe the differences, which in turn accentuates the visual disparity. Finally, regarding the concern that hyperparameters are not easy to tune in practice, our method maintains stable performance even under noticeable loss fluctuations, as illustrated in Fig. 6.\n\n**Reference**\n\n[1] Donoho D L. Compressed sensing[J]. IEEE Transactions on information theory, 2006, 52(4): 1289-1306.\n\n[2] Donoho D L, Elad M. Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization[J]. Proceedings of the National Academy of Sciences, 2003, 100(5): 2197-2202.\n\n[3] Cao Z J, Gao G. Multi-omics single-cell data integration and regulatory inference with graph-linked embedding[J]. Nature Biotechnology, 2022, 40(10): 1458-1466.\n\n[4] Baysoy A, Bai Z, Satija R, et al. The technological landscape and applications of single-cell multi-omics[J]. Nature Reviews Molecular Cell Biology, 2023, 24(10): 695-713."}, {"review_id": "o1OYpkuprD", "reviewer": "Reviewer_bBMF", "summary": "The paper rigorously quantifies and identifies variations in view-level sparsity ratios across a wide range of multi-view data, presenting an innovative adaptive, unsupervised representation learning network. While prior research on sparsity in multi-view clustering has predominantly focused on addressing sparsity at the dataset level, this work makes a significant contribution by targeting internal view sparsity discrepancies within a dataset. Specifically, it introduces SparseMVC, a framework built upon a sparse autoencoder with adaptive constraints, further enhanced by a correlation-informed sample reweighting module designed to alleviate the impact of encoding disparities and constraint-induced inconsistencies.", "questions": "- CDA module is designed to align feature distributions across views. Could you provide a more detailed technical explanation of how this alignment is accomplished? Are there any potential risks of overfitting when this alignment is applied too aggressively across views?\n\n- Could the correlation computation between global and view-specific features in the CSR module be compromised by noise introduced from highly sparse views? How is the reliability of the resulting attention weights ensured under such conditions?\n\n- Could you provide a more detailed explanation for the choice of Sv in Eq. (3)?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- Quality: This paper demonstrates a commendable level of quality, with a methodologically sound and well-executed design that reflects careful engineering and a solid understanding of the multi-view clustering problem. The experimental design is rigorous, and the empirical results are convincingly presented, showing clear improvements over existing baselines.\nClarity: In terms of clarity, the paper is written with precision and coherence. The problem is articulated clearly from the outset, and the architectural design is logically motivated and systematically explained. The visualizations are informative and enhance the reader's understanding of the proposed framework.\n\n- Significance: Regarding significance, the paper tackles a meaningful yet underexplored issue in the field—the cross-view sparsity problem—which is both practically relevant and theoretically challenging. By directing attention toward data-level issues rather than solely on architectural refinements, the work offers a compelling shift in perspective that could inspire future research directions.\n\n- Originality: With respect to originality, the proposed SparseMVC framework exhibits an impressive level of conceptual novelty. The integration of adaptive sparsity modeling, sample reweighting based on correlation, and distribution alignment constitutes a multifaceted and coherent strategy that differs from conventional multi-view learning paradigms. Each module contributes distinctly to the overarching objective, and their unified deployment reflects thoughtful innovation.\n\nWeaknesses: \n\n- Although the framework is reasonably efficient, the use of contrastive learning inherently imposes computational overhead, making it unlikely to rank among the fastest available approaches. \n\n- The assumption of well-aligned views may limit its robustness in practical scenarios where slight inter-view misalignments are often unavoidable.", "was_revised": true, "final_justification": "Thank the author for their rebuttal. All my concerns have been resolved, and I will maintain the score.", "rebuttal": "Thanks for your acknowledgment of our work. Below, we will address each of your questions.\n\n---\n\n> ***Question 1**:* **CDA module** is designed to align feature distributions across views. Could you provide a more detailed technical explanation of **how this alignment is accomplished**? Are there any **potential risks of overfitting** when this alignment is applied too aggressively across views?\n\nThanks. The CDA module aligns feature distributions by drawing together positive pairs between the late-fused global view and each local view and pushing apart negative pairs in the latent space. Technically, this is achieved by minimizing the cosine similarity of diagonal elements while maximizing that of the off-diagonal ones. Based on the convergence analysis , as shown in **Table 6**, our method effectively mitigates the risk of overfitting caused by excessively aligned feature distributions. Notably, the evaluation metrics remain relatively stable, even though the loss fluctuates on small datasets, as the number of alignment iterations increases.\n\n---\n\n> ***Question 2**:* Could the correlation computation between global and view-specific features in the CSR module be compromised by noise introduced from highly sparse views? How is the reliability of the resulting attention weights ensured under such conditions?\n\nThanks. First, as shown in **Figure 8** provided in **Appendix B.1**, a 64-dimensional parameter space is sufficient to effectively encode high-dimensional data views, even when the original dimensions (see **Table 2**) differ by up to two orders of magnitude. Moreover, the sparsity ratio in most views remains below 0.9 (less than one order of magnitude), suggesting that the influence of invalid features is largely suppressed during the sparse encoding process. Second, the correlation computation is used to assign sample-level weights, which helps to counteract the impact of sample-wise dimensional collapse. Finally, the attention weights in CSR are computed independently of other modules, ensuring that encoding and alignment constraints do not interfere with correlation estimation.\n\n---\n\n> ***Question 3**:* Could you provide a more detailed explanation for **the choice of $s_v$** in Eq. (3)?\n\nThanks. As shown in **Table 1**, we have evaluated several datasets with extreme sparsity $s_v$ variations, such as ALOI-100, where the low sparsity $s_v$ view is 0.0001 and the high sparsity $s_v$ view is 0.6644. Our method includes dynamic autoencoders that adjust encoding and loss constraints based on view sparsity. In datasets with minimal sparsity differences, such as BRCA and Synthetic3d, performance remained at SOTA levels, showing no noticeable degradation. Subsequently, to evaluate whether to select the external scaling factor $f(s_v)$ or the internal ratio coefficient $\\rho$, we conducted experiments by keeping one constant and varying the other. The results provided in **Appendix B.2** indicate that optimizing the external scaling factor, while maintaining a fixed internal ratio coefficient, yields the best performance."}, {"review_id": "NoRYPMXdpv", "reviewer": "Reviewer_h93U", "summary": "This manuscript explicitly introduces and analyzes the phenomenon of cross-view sparsity variations in multi-view data, a challenge that has been largely overlooked by existing clustering methods. In response to this newly identified issue, the authors propose SparseMVC as a targeted and systematic solution. SparseMVC employs an adaptive sparse autoencoder that dynamically adjusts encoding constraints for each view according to its sparsity level, reweights sample contributions based on correlations between local and global features to mitigate inconsistencies, and aligns feature distributions across views to enhance clustering robustness and thereby improve generalization.", "questions": "(1) How does the correlation-informed sample reweighting module differ from attention-based or other dynamic weighting methods in multi-view clustering? What limitations of these prior approaches does it address?\n(2) Why does the SparseMVC framework apply an additional transformation to the encoded latent features (Z), producing compressed representations (Y)? What motivates this design choice?\n(3) What criteria guided the selection of the sparsity threshold (τ)?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: The manuscript introduces a novel, data-driven approach to addressing cross-view sparsity variations, a challenge that has been largely overlooked in prior research. By focusing on sparsity at the view level, the authors offer a refined and effective solution, SparseMVC, which enhances both the accuracy and robustness of clustering results. SparseMVC is meticulously designed, incorporating three key components that collectively tackle the problem from multiple angles. Each module is distinct yet seamlessly integrated, ensuring a comprehensive approach to the complexities posed by varying sparsity across views. Furthermore, the paper is exceptionally clear and well-structured, beginning with a concise problem statement and thoroughly explaining the rationale behind the proposed solution, making the complex ideas both accessible and academically rigorous. Lastly, the significance of this work lies in its emphasis on addressing inherent data characteristics, rather than merely refining existing models. By shifting the focus toward data-driven model design, this paper opens up new avenues for future research in multi-view learning.\n\nWeaknesses: While the method demonstrates strong performance across a variety of datasets, its effectiveness in real-world applications with noisy or incomplete data remains to be fully explored. Further analysis is needed to evaluate how well SparseMVC handles extreme cases of sparsity or data imperfections in practical scenarios. Additionally, while the model shows strong performance in the downstream clustering task, its applicability to other tasks remains unexplored. The figures, especially the diagrams illustrating the modules, could be improved, particularly in terms of the consistency of symbols and the definitions of certain input variables.", "was_revised": true, "final_justification": "Thanks for the author rebuttal. I am satisfied with the author response and would like to accept this work.", "rebuttal": "Thanks for your acknowledgment of our method. We will address your questions one by one.\n\n------\n\n> ***Question 1**:* How does **the correlation-informed sample reweighting module differ from attention-based or other dynamic weighting methods** in multi-view clustering? What **limitations of these prior approaches** does it address?\n\nThanks. The key differences between our correlation-informed sample reweighting (CSR) module and existing dynamic weighting methods can be analyzed from two perspectives: design foundation and intended purpose. \n\n(1) For the foundation: Our method preserves the global features extracted during early fusion and incorporates both global and local features as inputs. Unlike attention-based fusion methods such as GCFAgg (CVPR'23) that emphasize disparities among local views, our CSR module performs dynamic view weighting based on the correlation between global and local features. It also differs from other strategies that depend on view-invariant representations (e.g., CVCL, ICCV'23), loss-based optimization (e.g., SCMVC, TMM'24), or prototype-driven fusion (e.g., MVCAN, CVPR'24).\n\n(2) For the purpose: Prior attention-based fusion methods aim to enhance feature encoding (e.g., SPGMVC, TKDE'21) or view-level aggregation (e.g., MAGCN, AIJ'22), and others like SCMVC (TMM'24) and TMC (TPAMI'22) emphasize view or loss weighting via mutual information or Dirichlet priors. In contrast, our method applies sample-specific weighting in the late fusion stage, guided by the correlation between global and local views established during the early fusion phase.\n\nAddressed limitations: The limitations of existing methods stem from the absence of an autoencoder capable of dynamically adjusting based on the sparse differences between views, as well as the inability to effectively leverage the relationship between globally fused features and local view features from earlier fusion stages. The SAA and CSR modules are specifically designed to address these two key shortcomings.\n\n------\n\n> ***Question 2**:* Why does the SparseMVC framework **apply an additional transformation to the encoded latent features** $Z$, producing compressed representations $Y$? What **motivates** this design choice?\n\nThanks. As shown in **Table 8 (Appendix B.1)**, larger latent feature dimensions in the autoencoder tend to improve overall performance. At the same time, excessively high-dimensional representations in contrastive learning can lead to dimensional collapse and unstable clustering. To address this trade-off, the compression layer applies dimensionality reduction to balance the performance of the CSR and CDA modules.\n\n------\n\n> ***Question 3**:* What criteria guided **the selection of the sparsity threshold**?\n\nThanks. Selecting the threshold value for the sparsity ratio $s_v$ is based on the actual cross-view sparsity distribution of each dataset. For most datasets, the sparsity ratios exhibit a skewed distribution, with a significant concentration of both high and low values, as shown in **Table 1**. A sparsity threshold value of 0.01 effectively captures these low-sparsity views."}]

Author rebuttal:

Meta-review: The paper studies an underexplored problem (i.e., cross-view sparsity variation) and proposes SparseMVC, a coherent three-part solution (adaptive sparse autoencoder, correlation-informed sample reweighting, and cross-view distribution alignment). Reviewers found the problem meaningful, the design well-motivated, and the empirical results convincing (especially after the authors’ rebuttal added larger-scale experiments (GSE, Animal) and stronger ablations), leading to a consensus recommendation to accept. Remaining concerns to address in the final version include stronger theoretical justification for the entropy-matching/adaptive weighting choices, clearer delineation of novelty versus reused components, more practical guidance on hyperparameter sensitivity and computational costs, and improved figure/notation clarity; the authors should also release code and reproduction scripts. Overall, the work could advance multi-view clustering by reframing a data-driven challenge and delivering an effective, empirically validated approach.
Decision: Accept (spotlight)