Unified and Efficient Multi-view Clustering from Probabilistic Perspective
Authors: Yalan Qin
Guorui Feng
Venue: ICLR 2026 Poster (2026)
Keywords: Multi-view clustering
anchor
efficiency
a unified manner
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: This paper proposes a novel method termed Unified and Efficient Multi-view Clustering from Probabilistic perspective(UEMCP).

Abstract: Multi-view clustering aims to segment the view-specific data into the corresponding clusters. There have been a large number of works for multi-view clustering in recent years. As representive methods in multi-view clustering, works built on the graph make use of a view-consistent and discriminative graph while utilizing graph partitioning for the final clustering results. Despite the achieved significant success, these methods usually construct full graphs and the efficiency is not well guaranteed for the multi-view datasets with large scales. To handle the large-scale data, multi-view clustering methods based on anchor have been developed by learning the anchor graph with smaller size. However, the existing works neglect the interpretability of multi-view clustering based on anchor from the probabilistic perspective. These methods also ignore analyzing the relationship between the input data and the final clustering results based on the assigned meaningful probability associations in a unified manner. In this work, we propose a novel method termed Unified and Efficient Multi-view Clustering from Probabilistic perspective(UEMCP). It aims to improve the explanation ability of multi-view clustering based on anchor from the probabilistic perspective in an end-to-end manner. It ensures the consistent inherent structures among these views by learning the common transition probability from data points to categories in one step. With the guidance of the common transition probability matrix from data points to categories, the soft label of data points can be achieved based on the common transition probability matrix from anchor points to categories in the learning framework. Experiments on different challenging multi-view datasets confirm the superiority of UEMCP compared with the representative ones.

Published as a conference paper at ICLR 2026
UNIFIED AND EFFICIENT MULTI-VIEW CLUSTERING FROM PROBABILISTIC PERSPECTIVE
Yalan Qin, Guorui Feng
School of Communication and Information Engineering, Shanghai University {ylg (ylqin, grfeng) shu. .ddu.
ABSTRACT
Multi-view clustering aims to segment the view-specific data into the correspond- ing clusters. There have been a large number of works for multi-view clustering in recent years. As representive methods in multi-view clustering. works built on the graph make use of view-consistent and discriminative graph while utilizing graph partitioning for the final clustering results. Despite the achieved signifi- cant success, these methods usually construct full graphs and the efficiency is not well guaranteed for the multi-view datasets with large scales. To handle the large- scale data, multi-view clustering methods based on anchor have been developed by learning the anchor graph with smaller size. However, the existing works neglect the interpretability of multi-view clustering based on anchor from the probabilis- tic perspective. These methods also ignore analyzing the relationship between the input data and the final clustering results based on the assigned meaningful prob- ability associations in unified manner. In this work, we propose a novel method termed Unified and Efficient Multi-view Clustering from Probabilistic perspec- tive(UEMCP)). It aims to improve the explanation ability of multi-view clustering based on anchor from the probabilistic perspective in an end-to-end manner. It ensures the consistent inherent structures among these views by learning the com- mon transition probability from data points to categories in one step. With the guidance of the common transition probability matrix from data points to cate- gories, the soft label of data points can be achieved based on the common transi- tion probability matrix from anchor points categories in the learning framework. Experiments on multi-view datasets confirm the superiority of UEMCP compared with the representative ones.
INTRODUCTION
As an important role in data mining and machine learning, clustering can be adopted to explore the organization and the latent structure of the data and then divide them into the corresponding clusters
Qin et al. (2021 2022
2023d
2024b 2023c;
2025)); Pu et al.
2023); Qin et al. (2025d
Qin Qian (2024): Qin et 2024c d 2025e 2023b
2025a c
2023e)
Li et 2003b c): Liu
et al. (2024 2023a 2025 Lu et al. (024); Zhang et al. 2025
et al. 2025 Liu et al. (2023b
2022b a). With the great progress in feature extraction and data generation techniques, the multi- view data has witnessed significant rise. For instance, image can be characterized with different feature representations, i.e., HOG, GIST and LBP. Multi-view clustering aims to group the data with multiple views into distinct clusters, which has been widely used in fields including object recognition and image segmentation. It is a paradigm in unsupervised learning, which is particular useful in the scenario when the explicit labels are not available.
There have been various developments for multi-view clustering domains Nie et al. 2017): Wang al. (2019; Li al. (2017): [uo al. (2018); Wang et al. [2020b 2024) in recent years, which include non-negative matrix factorization- based clustering methods Li et al. 2017), subspace-based clustering methods Luo et al. (2018) and graph-based clustering methods Wang et al. (2020b). As representive methods in multi-viww clustering, works based on the graph utilize view-conse hiw-consisteent and discriminative graph while leveraging graph partitioning for clustering. These methods construct
"Corresponding author

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Published as a conference paper at ICLR 2026
the similarity graphs and calculate the eigen-dccompose of Laplacian matrices. The computation costs of these two parts are Onn²) and O(n"), where n denotes the size of dataset. This kind of methods has gained considerable attention due to their efficacy in discovering the complex hidden structures in the data and representing inter-data relationships.
Among the existing methods based on graph, Nie al. Nie al. (2017) adaptively tune the weights of graphs across views based on their contributions to clustering by introducing a non- non-parametric weight learning strategy, resulting in the enhanced performance. Wu al. |Wu al. (2019) aim to achive superior view-consistent graph quality based on the combination of optimization function regarding the complementary information from different views and tensor nuclear norm constraints. Nie et al. Nie et al. (2018) facilitate the direct label extraction to achieve a view-consise wew-cconisttntt graph with the guidance of a Laplacian rank constraint built on the clear connected components. Despite the significant success obtained by these methods, they usually need to build x n graphs and become inefficient for the multi-view datasets with large scales.
To address the above issues, multi-view clustering methods based anchor have been presented for handling large-scale data by learning the bipartite graphs with connected components, where denotes the number of categories in the dataset. These methods usually select representative data points from the dataset as anchors, which is able to well balance the computational costs and global representation of the data. Li et al. Li et al. 2015) employ the pre-defined anchor graphs with high quality to realize multi-view clustering on large-scale datasets. Xia et al. Xia et al. (2023) effectively harness the complementary information from different views and guarantee the view consistency by using Schatten p-norm regularization on the tensor constructed by bipartite graphs Li et al. Li et al. (202)) aim to achieve clear component segregation in the bipartite graphs by imposing Laplacian rank constraints in the bipartite graph learning. You et al. You et al. (2023) introduce predefined anchor point labels and constraints on anchor points obtain discriminative cluster structures, re- sulting in superior bipartite graphs for clustering. However, these methods ignore the interpretability of anchor- based multi-vien view clustering from the probabilistic perspective, which inevitably compli- cate the acquirement of k distinct connected components. They also fail to analyze the relationship between the input data and the final clustering results by assigning meaningful probability associa- tions in unified manner.
To deal with the above issues, we propose a novel method termed Unified and Efficient Multi-view Clustering from Probabilistic (eTSpective(UEMMP) This method aims to increase the explanation ability of multi-view clustering based on anchor from the probabilistic perspective in an end-to-end manner. It is able learn the common transition probability from data points to categories shared by different views with one step, which guarantees the consistent inherent structures among these views. By introducing the common transition probability matrix from anchor points categories, we can achieve the soft labels of data points with the guidance of the common transition probability matrix from data points categories in the learning procedure. To be specific, the discrepancy between the obtained soft labels and the common transition probability from data points to categories based on the Frobenius norm is calculated as the loss function to enhance the reliability. The main contributions in our work are summarized as:
1. We propose a novel method termed Unified and Efficient Multi-view Clustering from Prob- abilistic perspective (UEMCP), which assigns the probabilistic meaning to the anchor graph and soft label of data points to increase the explanation ability of multi-view clustering model in an end-to-end manner.
2. UEMCP learns the common transition probability from data points to categories shared by multiple views with one step, which is able to ensure the consistency of inherent structures for these views. Based on the common transition probability matrix from anchor points to categories, the soft labels of data points can be achieved with the guidance of the common transition probability matrix from data points to categories in the learning framework.
3. Extensive experiments are performed on several datasets to validate the effectiveness and efficiency of the proposed UEMCP under different metrics.
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2 RELATED WORKS
2.1 MULTI-VIEW CLUSTERING
In vuew clustering process, we can obtain the final clustering outcomes by constructing the similarity graph and then run the existing clustering algorithm on this graph. Gao et al. Gao et al. (020) combine dimension reduction, feature selection and manifold structure learning multi-view clustering. It is able to reduce the effects of the existing redundancy and noise by map- ping the data with high dimension to the low-dimensional spaces. Cao al. [Cao al. (2015) use the Hilbert Schmidt Independence Criterion (HSIC) to guarantee the diversity of multiple subspace representations. The complementary among different views is further explored in this manner. Wang et al. Waan et al. (2020a) simultancously suppress noise, reduce dimension and learn the local struc- ture graph based on -nor in achieving the final clustering results. The optimal graph is able to be directly obtained and the extra processing steps are not needed in the whole procedure. Gao et al. Gao et al.] (2015) adopt a shared indicator matrix to integrate representation matrices across views, which consider the magnitudes of these representations. Sang et al. [Sang et al. (2022) learn consen- sus structure graph, consider manifold structure and reduce dimension by auto-weighted multi-view projection based on graph. The similarity graph is built on the projected data to ensure that the noisy and redundant information is removed. Wang et al. Waag et al. (2017) adopt an item related to the position for measuring the diversities among different representations and ensure that indicator shared by these representations by exploring the consistency. Wang et al. Wang et al. (2022a) analyze the multi-view dataset and learn the shared projection matrix based the diversity and consistency preserving. Wang et al. Wang et al. (2015) can better handle the noise corruption by designing angular regularizer and refining it based on the sparse decomposition. Li et al. Li eet al. (2023a) build an orthogonal projection matrix to achieve the feature representation for each view and learn the representation in an embedded space. Luo et al. [uuo al (2018) jointly explore the consistency and specificity of different views based on shared representation and particular representations. The above methods are able to achieve promising clustering results. However, these methods usually need relatively high computational complexity and space complexity, which inevitably hamper their further applications.
realize
2.2 ANCHOR GRAPHI-BASED MULTI-VIEW CLUSTERING
To decrease the complexity of milti-viww clustering, the approaches based on anchor graph have been extensively presented Kang et al. (2020). The anchor graph can be obtained by constructing the relationship between m anchors and n data points, which is able to cover n data points with m anchors. Due m < the anchor graph-based multi-view clustering methods can effectively deal with datasets with large scales. For instance, Zhan et al. Zhan et al. (2088) utilize K connected components rather than rely post-processing in extracting clustering metrics and global graph can be constructed from various single view graphs. Guo et al. Guo & Ye (2019) build up the similarity relation between data points and anchors and then integrate the relations within the view as well as among views into the fused similarity matrix. Li et al. [i et al. (2022) fuse cluster graphs from different views based on a parameter-free method, which leads composite graphs by utilizing self-supervised weighting. Kang et al. Kang et al. (2020) construct smaller graphs for all views and then employ the integration mechanism to accelerate the eigen-decomposition. Zhou et al. Zhou et al. (2022) minimize nuclear norm and tensor Schatten p-norm in considering the inter- view and intra-view spatial low-rank structures of the bipartite graphs. Huang et al. |Huang et al. (2020) assign weights to individual graphs by learning and the auxiliary parameters are not needed. Yang et al. Yang et al. (2022) explore the synergistic information among different views based on the Schatten p-norm. Yu et al. Yu et al. (2022) converse the similarity learning to the consensus similarity construction.
Even though the above multi-view clustering methods have achieved preferable performance, these still neglect the interpretability from the probabilistic perspective for the anchor- based multi-view clustering, complicating the acquirement of k distinet connected components. These methods also pay few attentions to analyzing the relationship between the input data and the final clustering results in unified manner.
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3 METHODOLOGY
In this part, we present the inspiration and formulation of UEMCP, the optimization details and the analysis of computation complexity for the proposed method.
3.1 INSPIRATION AND FORMULATION
Most existing multi-view clustering methods for large-scale data usually rely on anchor graphs to reduce the algorithmic complexity. The structure of the data can be represented by choosing m anchors to reflect the entire distribution for the dataset. The relation between data points and anchors can be built based on the anchor graph and the correlation tends be stronger when a data point and anchor belong to the same category. The anchor graph can be considered as the probability transition matrix between data points and anchors due the non-negative properties and summarization being one for the row. The anchor selection and the probability transition matrix construction are separated from each other, which will inevitably affect the final performance. Different from the traditional strategy, automatically learn anchors instead of simple sampling. Besides, consensus data distribution shared by different views assumed our multi-view clustering setting. We then define the projection matrix {Po (P) to realize the consensus anchor goal with dimension being d, x as follows:
min Σ 2 PLASIEE Po.A, 1=1
s.t. P. Id, A - In S 0, 11 a=1=1
where m represents the number of anchors, X, E Rdoxn is the data of the w-th view, n indicates the number data points, denotes the unified anchor matrix and E Rmxx refers to the corre- sponding anchor graph. S 2 0 and S = 1 are adopted in the optimization function, which can be viewed as the probability transition matrix between data points and anchors.
Based on the consensus anchor A E Rdxm the transition probability matrix from anchors to cate- gories H Rexm and the transition probability matrix from data points to categories Rmxx can be calculated. The optimization function is shown as follows:
min |HS - G s.t. G 0, GT
where G E Rexm denotes the probability transition matrix between data points and categories. H E Rexm represents the probability transition matrix between anchors and categories. We then combine the above two parts and rewrite the objective function as:
min P. P..A.SHH,C
2 PLAS + *IISS G
uml
s.t. pf P, = Ia A = Im. 82 0, $- H 2
1 1, G 0, 1=1
where À > 0 is the parameter to balance different terms. Since G is adopted to achieve the labels of data points, we relax the constraints imposed G to be orthogonal and non-negative for more intuitive understanding of the category. Then each row in G just contains a non-zer zero value and the corresponding position represents the label of the data point. Therefore, the final objective function formulated as:
V
min P. R,,AHSHH.C 1=1
Σ
a - P,AST + +I|HS GIL
(4)
s.t. PT P. = La, AT A- Im S 0, 1- H 2, 111-1, Gº 1m, G 0, a -
The advantages of the above proposed method are summarized follows:
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1. Adaptive weighted:
Different from some existing multi-view clustering methods based on anchor, UEMCP adaptively learns the coefficient of each view based on their contributions to the consensus anchor graph.
2. Probabilistic: UEMCP assigns the probabilistic meaning the anchor graph and soft label data points to increase the explanation ability of multi-view clustering model.
3. Consistent: Despite the differences the distributions for different views, the intrinsic structure of the data among these views is able to be consistent with the guidance of the introduced common transition probability matrix from anchor points to categories.
3.2 OPTIMIZATION
To deal with the optimization problem in Eq. (4), we give an alternate optimization algorithm in finding the solution to each variable with the others being fixed.
S-subproblem:
When the other variables are fixed, the objective function regarding S is written as
min
a - P.AS + SIISS - GI s.t. 12.220, 0, 111
(5)
We transform the above optimization problem as the Quadratic Programming (QP) in the following:
min St, S. ff s.t.
1h 1=1=1, S20,
(6)
where W 2(E of A)I, ft - P.A 2A H. We then solve the QP
problem for each row in the optimization process.
A subprobbem:: When the other variables are fixed, the objective function regarding A is written as
min L P.AS s.l. AT AT
We rewrite the optimization problem of A in the above as follows:
maxTr(A"O), s.t. A=
(8)
where C EV=1 o X.SS. Assuming the singular value decomposition (SVD) of
VA'AVA we achieve the optimal A by calculating UAV.T.
P. P.-subbroblem: When the other variables are fixed, the objective function regarding P, is written
as
min
a X,
PLAS s.t. P.P. Pa I.
(9)
We then transform the above problem as follows:
Tr(P. Bo), s.l. Pd P. I,
(10)
P.
where B, X,,ST AT, Supposing the SVD of P. is Up Ep, Vž the optimal P, can be achieved by calculating Up, Vi.
H -uspproobe:m: When the other variables are fixed, the objective function regarding H is written as
min |HS GlE s.t. H 2 0, 1=1
(11)
The above objective function can formulated by the minimization problem as follows:
min H G.y s.t. H 2 0, =11
(12)
H
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The optimal row of the above formulation is
arg min ||G., S.,il?
(13)
Then the optimal cluster assignment is achieved by minimizing the above optimization problem. G-subproblem: When the other variables fixed, the objective function regarding G is written as min G
HS - Glle, s.t. GO G Im, G 2 0.
(14)
Likewise, the minimization of G is formulated as:
maxTT((" J), s.t. GT Ly G 0, where HS. Therefore, the optimal UJV with being UYE.
(15)
a.-subproblem: With the other variables being fixed, the objective function regarding a, is written as
I
P. s.t. 1=1=1
(16)
Then the optimal a, achieved according to Cauchy- Cauchy- Bww.akkwssy--Shwaaaaae inequlllll
inequality as follows:
X P AS
Qgy
(17)
Due to the convex property in each sub problem, the objective function monotonically decreases in each iteration until convergence. The detailed process in solving the proposed UEMCP is shown in Algorithm 1.
Algorithm 1: Algorithm of UEMCP
Input: Duraset (Xx Ve number of clasters &.
Output: Probabistt tranission between data points and categoriis paz.
Initialize Po. [==]p=
Update
Updiste {Po)p p-1 by solving Fq. (10)s
Update sulving (8):
Update G solving
Update by solving Eg (13):
Update a by solving Eq. (17):
until convergence:
(a) Caltech20 (b) Scene15
(c) SUNRGBD
(d) Reuters (c)
(f) AWA
(g) NMNIST
NUSWIDE
Figure 1: Parameter study of all datasets.
3.3 COMPLEXITY ANALYSIS
The total computational complexity of the proposed UEMCP contains the optimization cost of all variables. It needs (dd,d²) to conduct SVD and (d,,kk?) to achieve the optimal P. by performing matrix multiplication, where ΣV= dy Likewise, the time cost of updating A includes O (dm²) for SVD and (ddmk²) in matrix multiplication. Besides, solving S consumes (nm²² for all data vectors. It costs Omk?² and (mh)3) for H in SVD and matrix multiplication, respectively. For updating G, the corresponding time cost is (mnnk). It just needs ( in calculating a,, Therefore, the computational complexity of UEMCP is (n)) regarding the number of data points.

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Table Clustering results based on ACC (%) on all datasets. "N/A denotes out of memory.
Dutsset
ETLMSC
SEMC
BMVC
LMVSC
FMONOF
FPMVS
MSC-BG
EDMC
Ours
Calsech101-20 20
08.95:0.200
59. 09,501:0.05
16.95:10.00
29:20:1:0:00
49.50:10.00
66.20:10.00
61.80=0.20
65.10:00.05
68.00:10.20
Scene15
20.50=0005
22.08-0001
29,00+0000
307550000
24.20:00.00
40.50=00.00
45.40=00.00
48.00=0.50
50.27 1:115
SUNRGBD
10:20:00.00
11.58-0002
15.69
13:20:10005
Reuters
11.50
12.00::0000
15.90:1..20
16.20:100000
12.00:10.25
20.58:10.00
23.65-0.10
29.60:10.00 25.27=0000
NUS NUSWIDEOBJ
N/A
00005
13722+0001
15:20:0000
12.00+00001
19.54=0.00
19.75=0.15
21.10=0.00
23.56-1-0.05
AWA
NIA
3.95-:0.05
8.724:0.01
7.40::0.55
8.25
9.00100.00
8.80:10.00
9.15::0.05
10,62 0.000
MNisy
5.29:100000
10.28:10.00
Table Clustering results based on NMI (%) on all datasets. "N/A denotes out of memory.
Dutoset
ETLMSC
SFMC
BMVC
LMVSC
FMONOF
FPMVS
MSC-BG
EDMC
Ours
Caloech
4..26:20.05
43.15::0.00
16.46:1:005
41:00:100000
31.47:200000
63.26:20.01
51.05-0.00
$2.00=00.05
53.60110.00
Scome15
32.50:00.00
30.181:0.10
13.00:0.000
3025::0.10
22.64::0000
45.70::0.15
39,46=0.00
40.00=0.05
42.70::0
SUNRGBD
18.75:0.005
23.00=0:01
Reuters
12,90:0.000
2.58:10.00
18:20:1:0000
20.30:1:000
9.80:10000
25.20::0000
24.37=0.05
28.00=0.05
29.95:10.00
NUS NUSWIDEOBI
NLA
0,95--0.005
+-005--0.01
12.75+00.15
5.22+0001
13.27-00.01
13.15=0.00
14.70=0.02
16.20 -0.00
ARA
NIA
0.321:0.01
13.56.1:0.00
8.621.0.01
7.95-10.00
10.46.10.11
9.82:10.01
0..15:00000
12.00.10.10
MNIST
1.0.10.011
2.59:10.05
2.80.20.00
3.05
3.60::0.00
(a)
(b) 1cene
(c) SUNRGBD (d) Reuters (e)
AWA
(g)
NoisyM-
Caltech101-20
NUSWIDE
NIST
Figure 2: Sensity investigation on all datasets.
4 EXPERIMENTS
We validate the proposed UEMCP seven widely adopted multi-view datasets by performing tensive experiments based on the clustering performance and computational efficiency in this sec- tion. The memory of the adopted device for running experiment is 8G.
4.1 DATASETS
The proposed UEMCP is evaluated on seven multi-view datasets and show the details of these datasets on Table 1. 1altcch101-20 consists of 2386 data points from 20 categories, which is a subset of the collection Caltech101. Scene15 contains total three views and 4485 instances in 15 categories. SUNRGBD has 45 classes and 10335 indoor scene images from two views. Reuters consists of 18758 data points and five views over six classes. NUSWIDEOBJ the object image dataset, which includes 30000 images from 31 classes. AwA is an animal dataset with attributes, composed of 50 kinds of animals in six features. NoisyNMIST has 50000 samples and views in 10 categories.
4.2 COMPARED METHODS AND EXPERIMENTAL SETTINGS
The proposed UEMCP is compared with the eight multi-view clustering works shown in the follow- ing. BMVC Zhang et al. (2019) reduces the computational complexity by concurrently learning the consensus clustering result and multi-view binary representation. LMVSC Kang et al. (2020) constructs a consistent bipartite graph based on a multi-graph fusion strategy and determines the cluster labels by spectral clustering. SFMC et al. (2022) uses bipartite graph learning strategy with Laplace rank constraints to build view-consistent bipartite graphs. ETLMSC Wu et al. (2019) applies spectral clustering to the constructed probability matrix from a tensor, resulting in the final results. FMCNOF Yang ef al. (2021) adopts the anchor selection and non-negative matrix factor- ization based fast clustering method. FPMVS Wang al. (2022b) achieves joint optimization by combining graph construction and anchor learning in a framework. MSC-BG Yang et al. (2022) constrains the bipartite graph with the Scatten p-norm effectively capture the complementary in-

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Table 3: Clustering results based on Purity (%) on all datasets. "N/A denotes of memory.
Dutseet
ETLMSC
SFMC
BMNC
LMVSC
FMONOF
FPMVS
MSC-BG
EDMC
Calsech101-2 20
71.20:0.005
65,00:00.20
42.50
12.50:1:0055
65.78.10.00
52.35:10.10
66.20:10.00
59.57-0.16
61.58:00.00
69.79 0..9:::0.10
Scene15
00.00=0000
42.85-00.00
45.0040000
49:0000000
41.35=20.50
48.20=00.00
46.40=0.00
47.89=00:0
52.50-00.00
SUNRGBD
Reuters
11,46:0.00
9.82100.15
15:20:1:0000
18000110000
10.35:10.05
23.19::0.00
22.50=0.00
26.06:0.00
28.00:10.00
NUS NUSWIDEOBJ
N/A
15.20+00.50
21.75
22.00+0..10
24.00+00.00
23.57=00055
26.28=0.00
27.00=0.00
29.58 -0000
AWA
NIA
3.40
8.424:0.00
1:.00:0000
11.85.10.15
13.59.10.000
0.00
0..00
18.35
MoisyMNIST
0.95
4.10::0.00
Table 4: Clustering results based Fl-score (%) on all datasets. "N/A denotes out of memory.
ETLMSC
SFMC
LMVSC
FMONOF
MSC-BG
Caleech101-20
0..30::..05
52.85:1:0000
48. 50:1000000
37.69 1.000100.00
49:20:10.00
18.58:10.15
18.69-0.10
38:46:00.00
62.49:1.0.00
Scame15 15
25.20:10.00
23.50:10.10
27.00:100000
28:48:::000
29:00:00.06
27.18:20.00
28.65-0.05
30.00=0.002
32.45::0.00
SUNRGRDD
16. 06.0020.000
15:202:0:05
18:53:0000
12.46:20.10
14.0::0005
1..8420.00
17.98=0.00
20.40=0.00
23.00:10.05
12.00:10.00
1146610.10
NUSWIDEOBI
N/A
11:0::00000
1048:::0000
9:96::015
8.80:00000
9.80:00000
11.50=0.10
14.27=0.00
16.70:20.00
AWA
N/A
4,37::0.00
3.50::0000
5.5240000
4 4:90:00000
3.78:0000
5.90
6.27
7.40:11.00
NoisyMNIST
NIA
NIA
5.5.10001
6.881:0000
8.10:10.19
12.58:20.00
13.90=0.00
14,20:00000
14.75:10.00
formation and spatial structure among the views. EDMC Qin et al. (2024a) learns anchors from instance and cluster level in the row and column space of the feature representations across views.
To eliminate the randomness in initialization, we run K-means for 50 times in achieving the clus- tering results. For fair comparison, the parameters in the compared methods are set to the same as the ones in their original literatures. We then evaluate the clustering performance by four widely adopted metries, i.e., ACC, NMI, Purity and Fi-score. To ensure the precision for the experimental results, we report the mean and variance from 20 iterations as the final results.
(a)
(b)
(c)
(d)
Figure 3: Ablation study all datasets.
4.3 INFLUENCE PARAMETER À
To study the influence of parameter À on the final clustering performance, we perform experiments for investigation the range of (0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1). According to Fig. 1, we find that choosing the proper parameter is crucial in improving the final clustering results. The parameter with too large or small values not helpful in achieving the desired performance. We can find that the stable performance be obtained when À is in the range of (0.1, Besides, the best performance achieved when À = 0.5 for the proposed UEMCP.
4.4 EXPERIMENTAL RESULTS AND ANALYSIS
We present the clustering results of the proposed UEMCP and compared methods on seven bench- mark datasets. 'N/A' is adopted to show that the method encounters the out-of out-of-memory issue on the dataset. According to Tables 2-5, we can draw conclusions as follows:
Compared with the existing multi-view clustering methods based on subspace, the anchor works are more proper for the dataset with large scales and can achieve better performance on most cases, demonstrating the necessarity of adopting anchor graph
Some methods employ two-stage clustering manner, i..., ETLMSC, which will lead the memory issues for datasets with large seales. In contrast, the proposed UEMCP performs anchor selection to mitigate such issues, especially for the large-scale clustering tasks.
8

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Published as a conference paper at ICLR 2026
Figure 4: Logarithm of running time different datasets.
(a) Caltech101-20
(b) Scene
Reuters
(d) NUSWIDEOB/
Figure 5: Convergence curves on different datasets.
UEMCP shows obvious advantages over other compared methods under different metrics. For example, UEMCP improves 2.70% over EDMC on Scene15 in terms of NMI, which can be explained by that assigning the probabilistic meaning to the anchor graph and soft label of data points for increasing the explanation ability of multi-view clustering end- to-end manner is necessary.
4.5 SENSITY INVESTIGATION
In this part, we study the influence of the anchor number for the final clustering results. We vary the anchor number in the range of {2k, 4k, 5k: 6k) and investigate the sensity in terms of different metries, where lt is the number of clusters in the dataset. According Fig. 2, we observe that more anchor number can help lead to better performance and too large anchor number is not necessary the final results, i.e., the better clustering performance can be achieved when anchor number equal to 5k on NUSWIDEOBJ.
4.6 ABLATION STUDY
To study the significance of the regularization term in the final objective function, we perform ab- lation study by removing the second term in validation. According to Fig. 3, we find that the performance significantly drops without considering the probability transition matrix between data points and categories. is because the soft label of data points can be achieved with the guidance of the common transition probability matrix from data points to categories in the learning procedure.
4.7 RUNNING TIME ANALYSIS AND CONVERGENCE ANALYSIS
To validate the computational efficiency, we list the running time of our UEMCP on different datasets. As shown Fig. 4, we observe that UEMCP needs relatively shorter running time on different datasets, which demonstrates the computational efficiency of our method. As previously demonstrated, the time complexity of UEMCP is nearly to (n). Though less running time is needed by BMVC, the simple procedure ignores to fully explore the information from multiple views, leading to relatively poor performance. We perform experiments to demonstrate the conver- gence of the proposed UEMCP benchmark multi-view datasets. According to Fig. 5, find that the objective values of UEMCP gradually decrease for each iteration and converge in about 20 iterations, which clearly show the convergence of the proposed UEMCP.

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Published as a conference paper at ICLR 2026
5 CONCLUSION
We propose a novel method termed Unified and Efficient Multi-view Clustering from Probabilistic perspective(UEMMC) in this paper, which can improve the explanation ability of multi-view cluster- ing based on anchor from the probabilistic perspective in an end-to-end manner. UEMCP learns the common transition probability from data points categories in one step, ensuring the consistency of inherent structures for these views. The alternate minimizing algorithm is given to solve the for- mulated problem. Extensive experiments illustrate the effectiveness and efficiency of UEMCP on seven multi-view datasets under four metrics.
ACKNOWLEDGMENTS
This work was supported by Eastern Talent Plan Leading Project under Grant BJKJ2024011 and National Natural Science Foundation of China (62402303).
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Reviews:
[{"review_id": "VCR5kB6AYG", "reviewer": "Reviewer_baAD", "summary": "This paper shows a new method termed Unified and Efficient Multi-view Clustering from Probabilistic perspective (UEMCP), which assigns the probabilistic meaning to the anchor graph and soft label of data points to increase the explanation ability of multi-view clustering model in an end-to-end manner. UEMCP is able to learn the common transition probability from data points to categories shared by multiple views with one step, which ensures the consistency of inherent structures for these views.", "questions": "See the Weakness box.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The authors propose a novel method termed Unified and Efficient Multi-view Clustering from Probabilistic perspective for increasing the explanation ability of multi-view clustering based on anchor from the probabilistic perspective. Besides, the expression is very clear with satisfied writing and novelty. With the guidance of the common transition probability matrix from anchor points to categories in the learning framework, the soft labels of data points are able to be achieved.", "weaknesses": "1. The authors can give the brief operations in Optimization part for Section 3.2. Then the main idea for optimization is more clear for readers.\n\n2. To study the influence of parameter λ on the final clustering performance, the authors perform experiments for investigation in the range of {0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1}. The authors should explain why they choose such range in parameter selection.\n\n3. The authors can give more details for running time analysis part in this work, i.e., the reason why the proposed UEMCP needs relatively less time in the experiment.\n\n4. The authors should confirm the typo error and check the whole paper to avoid such issues for the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PIHEbh1Rjz", "reviewer": "Reviewer_zh1j", "summary": "This paper introduces a novel anchor-based multi-view clustering method, UEMCP. The primary goal is to improve the interpretability of anchor-based clustering by framing it from a probabilistic perspective within an end-to-end learning model. The method aims to ensure structural consistency across views by learning a common transition probability matrix from data points to cluster categories in a single step.", "questions": "Could the authors provide a more detailed analysis of the running time results shown in Figure 4? Specifically, please elaborate on the comparative efficiency of UEMCP against the baselines.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper proposes an end-to-end framework to enhance the explanation ability of anchor-based multi-view clustering, which is a valuable goal. The method is designed to ensure the consistency of inherent structures across views. A key aspect is the use of a common transition probability matrix from anchors to categories, which guides the soft label assignment for data points within the learning framework. The experimental results on several datasets are strong and show a clear improvement over existing methods.", "weaknesses": "1.  The methodology is plagued by confusing and inconsistent notation. For example, in Eq.1, $P_v$ should be $d_v \\times d$, but its constraint is $P_v^T P_v = I_m$ (it should be $I_d$). The variable $S$ is defined with different dimensions in Eq.1 ($m \\times n$) and Eq.2 ($l \\times n$). Similarly, $H$ has conflicting dimensions in Eq.2 ($c \\times l$) and Eq.3 (implying $c \\times m$). The dimension $l$ is introduced in Eq.2 without definition. These issues severely affect readability.\n2. The paper presents a running time analysis, but the memory used for the experiments are not provided. Besides, the discussion of the running time is very brief. A more detailed comparison and analysis against the other methods are needed.\n3. The formatting of the references is inconsistent and should be carefully checked throughout the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UV7Q3kx9yl", "reviewer": "Reviewer_ixUY", "summary": "This paper proposes a method termed Unified and Efficient Multi-view Clustering from Probabilistic perspective (UEMCP), which assigns the probabilistic meaning to the anchor graph and soft label of data points to increase the explanation ability of multi-view clustering model in an end-to-end manner. UEMCP learns the common transition probability from data points to categories shared by multiple views with one step, which is able to ensure the consistency of inherent structures for these views. Experiments are performed on several datasets to validate the effectiveness and efficiency of the proposed UEMCP under different metrics", "questions": "1.The authors are expected to give detailed reason why choose $ \\{0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1\\} $ as the range in performing experiments for investigation.\n\n2.The difference from existing jobs, especially those related to probability transitions, is not very prominent, It is recommended to supplement relevant work, highlight the differences of the paper", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Most existing multi-view clustering methods for large-scale data usually rely on anchor graphs to reduce the algorithmic complexity. The structure of the data can be represented by choosing $ m $ anchors to reflect the entire distribution for the dataset. The relation between data points and anchors can be built based on the anchor graph and the correlation tends to be stronger when a data point and anchor belong to the same category. Besides, a consensus data distribution shared by different views is assumed in our multi-view clustering setting. The anchor graph can be considered as the probability transition matrix between data points and anchors due to the non-negative properties and summarization being one for the row. The anchor selection and the probability transition matrix construction are separated from each other, which will inevitably affect the final performance. Different from the traditional strategy, the authors automatically learn anchors instead of simple sampling", "weaknesses": ". To study the influence of parameter $ \\lambda $ on the final clustering performance, the authors perform experiments for investigation in the range of $ \\{0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1\\} $. According to Fig. 1, the authors find that choosing the proper parameter is crucial in improving the final clustering results. The parameter $ \\lambda $ with too large or small values is not helpful in achieving the desired performance. However, the authors do not give detailed reason why choose $ \\{0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1\\} $ as the range in performing experiments for investigation.\n2. The authors present the clustering results of the proposed UEMCP and compared methods on seven benchmark datasets. 'N/A' is adopted to show that the method encounters the out-of-memory issue on the dataset. According to Tables 2-5, we can draw conclusions. However, the authors do not bold the best clustering results in Tables 2-5 for the experiment part. Therefore, the authors are expected to bold the best clustering results in Tables 2-5 for the experiment part.\n3. To validate the computational efficiency, we list the running time of our UEMCP on different datasets. As shown in Fig. 4, we observe that UEMCP needs relatively shorter running time on different datasets, which demonstrates the computational efficiency of our method. Though less running time is needed by BMVC, the simple procedure ignores to fully explore the information from multiple views, leading to relatively poor performance. Considering that the running time analysis is given in this paper as shown in the above, the authors are expected to list the memory of the adopted device in performing the experiment.\n4. In the reference part, some publication name is called for short, i.e., Inf. Sci., and some publication is called for the whole name, i.e., Proceedings of the AAAI Conference on Artificial Intelligence. It is observed that these manners of publication names are not consistent. Then the authors should correct this issue and ensure that the forms of the references in this paper are consistent.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "AXa7c9hNPQ", "reviewer": "Reviewer_1Zmv", "summary": "This paper proposes a novel method termed Unified and Efficient Multi-view Clustering from Probabilistic perspective (UEMCP). It aims to improve the explanation ability of multi-view clustering based on anchor from the probabilistic perspective in an end-to-end manner. It ensures the consistent inherent structures among these views by learning the common transition probability from data points to categories in one step. With the guidance of the common transition probability matrix from data points to categories, the soft label of data points can be achieved based on the common transition probability matrix from anchor points to categories in the learning framework.", "questions": "The authors are expected to give the reason why they choose {2k, 3k, 4k, 5k, 6k} as the range in the experiment for the proposed method.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The proposed method, which proposes a novel method termed Unified and Efficient Multi-view Clustering from Probabilistic perspective. It aims to improve the explanation ability of multi-view clustering based on anchor from the probabilistic perspective in an end-to-end manner. The soft label of data points can be achieved based on the common transition probability matrix from anchor points to categories in the learning framework.", "weaknesses": "1. The authors summarize the advantages of the proposed method based on the Adaptive weighted, Probabilistic and Consistent property. However, there is no further explanation for Probabilistic property in illustrating the advantages of the proposed UEMCP. The authors should add the related further explanation for Probabilistic property in the paper.\n2. The second best results in Tables for the experiment part can be highlighted to make the authors more obviously grasp the clustering performance gains.\n3. The authors do not give detailed analysis regarding the convergence study of the proposed UEMCP. More analysis can be given for the convergence study part in the experiment, i.e., the iteration speed for the proposed UEMCP in the paper.\n4. The authors study the influence of the anchor number for the final clustering results in the experiment. They vary the anchor number in the range of {2k, 3k, 4k, 5k, 6k} and investigate the sensitivity in terms of different metrics, where k is the number of clusters in the dataset. The authors are expected to give the reason why they choose {2k, 3k, 4k, 5k, 6k} as the range in the experiment.\n5. The authors are expected to adjust the size of the presentation for Algorithm 1 of the proposed method in the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)