Confident Block Diagonal Structure-Aware Invariable Graph Completion for Incomplete Multi-view Clustering
Authors: Shuping Zhao
Yulong Chen
Jie Wen
Lunke Fei
Jinrong Cui
Tingting Chai
Venue: ICLR 2026 Poster (2026)
Keywords: Incomplete multi-view clustering
nvariable graph completion
onfident block diagonal structure learningci
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: This paper proposes a tensorized confident local structure completion method for robust incomplete multi-view clustering.

Abstract: Multi-view clustering (MVC) adopts complementary information from multiple views to reveal the underlying structure of the data. However, the conventional MVC-based methods remain a crucial challenge on the incomplete multi-view clustering (IMVC) tasks,
when some views of the multi-view data are missing. Particularly, current IMVC methods suffer from two main limitations: 1) they focused on recovering the missing data, yet often overlooked the potential inaccuracies in imputed values caused by the absence of true label
information; 2) the recovered features were learned from the complete data, neglecting the distributional discrepancy between the complete and incomplete instances. In order to tackle these issues, in this paper, a confident block diagonal structure-aware invariable
graph completion-based incomplete multi-view clustering method (CBDS_IMVC) is proposed. Specifically, we first design a confident-aware missing-view inferring strategy, where the confident block diagonal structures (CBDS) are learned to guarantee that recovered
instances of all views have the same strict invariable local structure with the constraint of CBDS. Subsequently, we proposed an invariable graph completion strategy to learn the intrinsic structure across all views. Each parts are jointly trained, complementing and promoting
each other to achieve the optimum together. Compared to other state-of-the-art methods, the proposed CBDS_IMVC demonstrates superior performance across multiple benchmark datasets.

Published as conference paper ICLR 2026
CONFIDENT BLOCK DIAGONAL STRUCTURE-AWARE IN- VARIABLE GRAPH COMPLETION FOR INCOMPLETE MULTI-
VIEW CLUSTERING
Shuping Zhao
Yulong Chen, & Jie Wen"
School of Computer Science
Shenzhen Key Laboratory of Visual Object Detection and Recognition Harbin Institute of Technology, Shenzhen
Guangdong University of Technology Guangzhou, 510006, China yb77 7774588mm 8mm edu mo
Shenzhen, 518055, China
chenyulonghit@@l... com, jiewen.pr@123 com
Lunke Fei
Jinrong Cui
School of Computer Science Guangdong University of Technology Guangzhou, 510006, China flks 12 com com
College of Mathematics and Informatics South China Agricultural University Guangzhou, 510642, China tweetyl 028002888666 63 com
Tingting Chai
Faculty of Computing
Harbin Institute of Technology
Harbin, 150001, China ttchai@hit www.hit.euucc..mmmmmmmom en
ABSTRACT
Multi-view clustering (MVC) adopts complementary information from multiple views to reveal the underlying structure of the data. However, the conventional MVC-based meth- ods remain crucial challenge on the incomplete multi-view clustering (IMVC) tasks, when some views of the multi-view data missing. Particularly, current IMVC methods suffer from two main limitations: 1) they focused on recovering the missing data, yet often overlooked the potential inaccuracies in imputed values caused by the absence of true la- bel information; 2) the recovered features were learned from the complete data, neglecting the distributional discrepancy between the complete and incomplete instances. In order to tackle these issues, in this paper, a confident block diagonal structure-aware invari- able graph completion-base based incomplete multi-view clustering method (EDDS_IVVC) is proposed. Specifically, we first design a confident-aware missing-viin view inferring strategy, where the confident block diagonal structures (CBDS) are learned to guarantee that recov- ered instances of all views have the same strict invariable local structure with the constraint of CBDS. Subsequently, we proposed an invariable graph completion strategy to learn the intrinsic structure across all views. Each parts are jointly trained, complementing and pro- moting each other to achieve the optimum together. Compared to other state-of-the-art methods, the proposed CBDSJMMVC demonstrates superior performance across multiple benchmark datasets.
"Corresponding Author: Jie Wen, e-mail: wenjic hit.edu.cn.

---

Published as conference paper ICLR 2026
Mi + 11l
View
View 2
Inncomplate antitiiteww data mstax
Complete dora reconstmctio
View
Confident Noek diagocal pegullazzation
Original source deta
Complete gaph inferrig
represenation
Figure 1: The flowchart of the proposed CBDS JMVC method.
INTRODUCTION
With the rapid development of multi media and sensor technologies, describing single object from different data sources or modalities has become increasingly convenient Wong et al. (2025); Fang et al. (2023), [aang jet al. (2025). For example, a web page can be described by its textual content and accompanying images;the same news might be written in different languages; facial image may be captured as a photo, infrared image, or sketch Deng et al. 2023) ; Wang et al. (201aa;; Chen et al. (2025). These diverse representations, referring the same entity, constitute what is known
-iel data Wang et al. (021b).
Each view in multi-view data corresponds to a distinct perspective, feature space, or data source. These views often exhibit both consensus (shared) and complementary (unique))informatinn Moujahid & Dornaika (2055: ; Dornaika & Hajjar (2024). The task of clustering such multi-view data into meaningful groups known as Multi- View Clustering (MVC) et al. (2022) Wang al. (2022). MVC has been extensively studied and widely applied in recent years Fang et al. (2023). For example, et al. Chen et al. (2021) developed low-rank representation-basse method enforce global structure consistency across views. Li et al. Li et al. (2021) proposed spectral consensus strategy by minimizing the disagreement among Laplacians from each view, leading more robust clustering performance. In addition, Tang et TTagg et al. 2023 introduced a subspace clustering method via adaptive graph learning and late fusion alignment.
While this approach has shown promise in some scenarios, it suffers from lacking of robustness to noisy views for it equally treatment of all available views ignores their reliability disparitiesWen al. (2021 l)
Graph-based multi-view clustering methods typically construct similarity graphs for each view using metrics like Gaussian kernels or cosine similarity. then fuse view-specific graphs through linear combination or advanced tensor operations, and perform spectral clustering on the unified graph Tang et al. (2023); Waag jet al. (2019a). However, these methods are sensitive to the choice of similarity measure. Additionally, they are prone to performance degradation when data is sparse or missing, as incomplete or noisy graphs can hinder clustering accuracy. Deep learning-based methods have also gained popularity due to their ability to learn complex nonlinear relationships and handle missing data through specialized architectures Du et al. (202T); Wang et al. (2019b); Chowdhury et al. (2025). Despite their flexibilityy. deep learning methods often require large amounts of training data and extensive hyperparameter tuning, limiting their applicability in resource-constrained scenarios.
To sum up, most IMVC methods focus on learning common graph from the available views, neglecting the latent information embedded in the missing views Cui et al. (2024). In order to tackle these issues, in this paper, confident block diagonal structure-aware invariable graph completion-based incomplete multi-view
2

---

Published as a conference paper ICLR 2026
clustering method (CBDS IMVC) is proposed. Specifically, we first propose symmetric-aware missing- view inferring strategy, where the confident block diagonal structures (CBDS) are designed to guarantee that the recovered instances of all views have the same strict invariable local structure with the constraint of CBDS. Subsequently, we proposed an invariable graph completion term to learn the intrinsic structure across all views. This ensured that the reconstructed data accurately captured the true distribution of missing instances in each view. The main contributions is summarized as follows:
1) Different from the other related papers, the proposed CBDSJIMV designed a complete representation reconstruction strategy, where the consensus complete structure across all views was learned to preserve the structure of the reconstructed data.
2) A novel confident block diagonal regularizer is proposed to guide all views toward learning invariable structure with block diagonal form. This mechanism preserves distribution of the data from each view while enforeing a consistent and strict block diagonal pattern that remains aligned across all views.
3) Extensive experiments demonstrate that the proposed CBDS IMVC can obtain the best performances compared with the other state-of-the methods.
2 THE PROPOSED METHOD
2.1 SYMMETRIC-AWARY MISSING-VIEW INFERRING
Let (x(() E gdd (n) Je=1 represent the incomplete multi-viww data, where is the total number of views and n denotes the number of subjects. Here, d(() and n(() correspond to the feature dimensionality and the number of the available samples in the v-th view, respectively. Given the affinity graph (Kr) derived from observed instances in each view, we can construct its complete graph counterpart S(r) by padding zeros for all missing instances. After obtaining the initial extended complete graph 5(v) for each view, we proposed the missing data referring strategy follows:
V
Σ v=1
-1 (00gg)))))) s.t. (o. (0) g(o)) = (X)
min MO,
Jo=1 is the reconstructed complete samples from all views, M(e) denotes the recon- structed complete data matrix for the v-th view. The operation M(v) ensures that the geometric structure of available instances is maintained, subject to the topological constraints imposed by the complete graph S. Additionally, Po, denotes the projection matrix associated with view v, which is used to record the existence of instances in the dataset X (*)) Specifically, it operates through one-hot encoding, where value in a row signifies the presence of a specific instance in view v, while a value of 0 denotes its absence.
where (J(z) E IRd
x
Specifically, Po, takes value of if the instance x exists and O if it does not exist. Mathematically, the extension of ((() be transformed as follows:
5(r) g(v)
(2)
11 10 - 112
is a Gaussian kernel function to present the symmetric relationships, G( E Ra, x serves as a projection matrix that maps the incomplete structure of the v-th view to its complete extended version. Specifically, for the j-th instance = in ((),, define ((e) as follows:
(2) where $1f e
{ 1, if a
(v) is the vth view of the ith sample
= 0, otherwise.
(3)
3

---

Published as conference paper ICLR 2026
2.2 CONFIDENT BLOCK DIAGONAL STRUCTURE PRESERVING
In order to let the referring instances preserve the data distribution of the original data, all views should maintain as similar data structure as possible. Hence, we propose the symmetric-aware missing-view inferring strategy, which adopts block diagonal structure constraint to maintain the same structure for the reconstructed data of all views as follows:
min J MO (FO)
Σ -1l - 10(0pp000)111111)11)11))111111 ||
K s.t. Po, (M,())p)))) = Po,(X)
(4)
Seeking to precisely reveal and preserve the structural distribution of the reconstructed complete data for all views, we propose the confident block diagonal structure induced regularizer || T For any affinity matrix F(l), its block diagonal regularizer
|| ||
is defined as follows:
F lle +B8 ||
F lh
(5)
where || F ||2 is close-form Z llo. which is NP-Hard problem. Here, F(t) l|g. is the proposed confident block diagonal regularization on multiple views. Specifically, F(2) captures complete structural information for both missing and available instances in each view by adopting the self-representatioon prop- erty of the data with & regularization. This approach enables selective incorporation of the most reliable information from other views to achieve structure completion. What more,the tensor low-rank constraint ll Z lle effectively captures high-order correlations across multiple views, serving a crucial component in our framework.
However, ||F'| still fails account for the latent information of missing instances in each view. To ad- dress this limitation, we propose intrinsic complete structure inference strategy, which effectively adopts complementary information across different views to recover the complete intrinsic structure of each view. Consequently, jointly optimize the referring of the complete multi-view data and the reconstruction of the confident complete graph as follows:
V
Σ 1| 1110 + X||
)a|(((())
Σ
50 Bc,) @ 1
min Jo0 M FO) u=l
#=1
(6)
8.1. Po. (M(())))))))) = Pou (8)00 Bi 11,
Bi = 1, B Σ
where À is the nonnegative penalty parameter, B E Ruxn is the linear representation matrix, 3 is the wlesent-wise multiplication to preserve the real structure of the available instances, and E = Q Qu: indicates whether both the i-th and j-th samples have instances in the e-th view. Specifically, ((+) e R xn
represents the missing-view indicator matrix, where Qi,5 = 0 means the j-th instance missing from the i-th view, while Qi, = 1 indicates its presence.
2.3 FINAL OBJECTIVE FUNCTION
The complete structure inference enables us to recover the intrinsic structure of the latent complete data for each view. To complement the main objective and enhance the clustering performance, consensus learning
4

---

Published as conference paper ICLR 2026
module is introduced obtain a unified low-dimensional representation shared across all views for spectral clustering. In this work, we unify these three components through a joint learning framework to achieve optimal performance. The complete objective function formulated follows:
(laloly [1 - -1())) p(o) + +111F + A - -y(o)) o 13
min MM) ((o), B,Y(
1=
(2 Ly(s s.t. (o.(M p(o)) Por(X), p00) =1, 05 B S 1, (7)
Bi =1 1, Bu, = 0, y(v) Σ
5 Ban ding( F(o)) = =0, F(w) 2 0, F(v) F(O)
1=1,i
where A and A are nonnegative penalty parameters, U E Ruxe denotes the consensus representation, and (") represents the weighting factor for balancing missing instances each view, with higher values indicating greater importance of the corresponding view. Here, Tyy denotes the Laplacian matrix of the
complete structure y()), computed as Ly(-) D(v) (M14r(()) Specifically, D(v) is diagonal matrix whose i-th diagonal element is given by D(v)i V 1
2.4 THE OPTIMAL SOLUTION
In this subsection, we employ the Alternating Direction Method of Multipliers (ADMM) to solve the opti- mization problem.
Updated ( and B: By fixing the other variables, (v) can obtained by solving the following problem:
min Ylu
(a
AiI((E(0))
-x(0)
1 + A2
K
Kesr
(v)
2
1=1
is
(8)
cdo) NO
(al
B2 1
Be (1 + B
(|((()))
1
(1(((() - 1 17.
2
1m
L=,,
12, where Kij = -M:l:1 NOO = y() EV EE {0: B and M EV 5(:) Bi,
C1
Problem 8 can be optimized by solving the following minimization problem:
(9)
in which T:(y) P(0) (A/ where p!()) (a (FtD) p(v) Kis) and A/ (a F
ΣV (a + My Ex=n, (afv)) Be, + (a
Particularly,
B can be
obtained by solving the optimization problem of ||(|))
EV 5(1) 11 which is simplex representation based optimization problem, in which B can be obtained via the accelerated projected gradient method.
min R.,/ AI. B. -0 v
Updated U: By fixing the other variables, U can be obtained by solving the following problem:
5

---

Published as conference paper ICLR 2026
((U) A2 Σ Tr(U" Uy 8.. U =I
(10)
Equation (T0) constitutes standard eigenvalue decomposition problem. The consensus representation ma- trix U can be obtained as [u1, 12, .... (te) E R where u,i = 1 are the eigenvectors corresponding to the first smallest eigenvalues of Σ 1yy
Updated ((w); By fixing the other variables, the auxiliary variable Z can be obtained by solving the following problem:
C 11 Z = min minn| + ~112 - u
((Z = || 1
+
2 111 Z(0 z('==
Σ
z
C 1| (1I)
2
where C' E R is tensor collected by all " nuclear norm minimization problem and has the following closed-form solution:
J==1- Problem 11
is typical t-SVD based tensor
u == ;
(12)
where μ np,, 2+ + A/p 2SVT is obtained by the t-SVD operation. K2 = SJ, where J E R=xixn is a diagonal tensor whose diagonal elements in the Fourier domain are expressed as Jy (i, i, = max //S// (i.
Updated ((v) and R('); By fixing the other variables, (v) and the auxiliary variable R(() can be obtained by solving the following problem:
(| - 11 + ( - c(o) 12: s.t. Po. (H( = (X).
min
(13)
J[m) (())) R(o)
where G = M() F(o) F(v) c(0)/ //.. We can respectively attain and R(e) as follows:
G(u)ag w/?
Ej G(r)as)
(14)
where G(v) the correlation with cosine distance between the ith distance and the jth distance on the wth view, [(() w(o)" (w()) y and W(v) (I(()) -
1 (uG + +A + Po, (x W),
R(O)_
(15)
A1 + 1
where (00) F(e) of
P
Updated (v) and F()); By fixing the other variables, (o) can be obtained by solving the following problem:
C(M( min M [IG ( Ar(e) 2
(16)
which ((1) = R(,) - 0(00/ /t. Afterwards, we can achieve (²) = G ((() by setting
the derivative (C(M)))/MM( = 0. Then, we can attain F(1) by solving the following minimization problem:
6

---

Published as conference paper ICLR 2026
[(F(o)) min A1 ((((((o))(())))))) o 1 + (ich M p(0) " 1 1
(17)
F(o)
where =Z))) z(0) +. By setting the derivative (((E(t))/OFtt = 0, we can obtain Fle)
(uM(t)) G4 G ( ( M + pl) + A) E
Updated p(v); By fixing the other variables, p(v) can be obtained by solving the following problem:
c(o) L(P = 1 (11(00) p(() + - 11e + Allp6o11|
(18)
Then, we can obtain the optimal solution of p(v) as follows:
p(o) = U2 F(o) + clu)
(19)
where Q presents the shrinkage operator.
Let wo()) denote we can achieve a( as alo) = (wlo))//(a-))/ Ev ( (w))))///11))
addition, the
Lagrange multipliers c( clu), c("), and cl can be updated 011) c1 + (((()) M F())),
ch cla¹ + ((F(t) 2()),, C = cfal + ((((t) Ei=,,e4 5(()) Br,)) ,nd Ct = -c + +
p(o)), respectively. Particularly, can be updated l min(pps, Mmax), where and P'maw tWO constants.
2.5 COMPUTATIONAL COMPLEXITY ANALYSIS
This paper employs an iterative optimization scheme to solve each variable in the objective function. The update procedures for (+) and B involve only simple element-wis wise operations, rendering their computa- tional cost negligible. The most expensive step is updating which requires an eigenvalue decomposition of Occ²²) with the first minimum eigenvalues. In the step of Z((), the majority of the time cost is consumed by incorporating t-SVD, FFT, and inverse FFT operations, with respective computational complexities of O(V2n2) and O(Vn^log (n)) Speciffcally, the computation of J(v), R(*), M F(v), and p(e) only con- tain some simple operations, where the computational complexity can be ignored. Finally, the total time complexity can be calculated as O(Ven² + 22n² Vn))..(n)).
3 EXPERIMENTS AND ANALYSIS
3.1 DATASETS DESCRIPTION AND INCOMPETED DATA CONSTRUCTION
BBCSport constitutes document database comprising 737 news articles pertaining to five sports: athletics, cricket, football, rugby, and tennis, sourced from the BBC Sport website between 2004 and 2005. For the purpose of evaluating various algorithms grounded in Integrated Multi-view Clustering (IMC), a subset from the BBC sport multi-view datasets, encompassing four distinct views, was utilized. This specific subset contains 1 116 samples, where each of the four views possesses a feature dimensionality
BBCSpor[] T
7

---

Published as a conference paper ICLR 2026
ACC (%)
NMI (%)
Purity (%)
Data
Methods
0.3
0.5
0.3
0.5
0.3
0,5
MVLJI
72.5
53.45=1,49
59.09:5.05
38.85
80.75=1.32
02.93:1.49
AWIMVC
78.33+22.25
64.75=2.69
69.51=2.17
49,48+2.03
85,42=0.97
73.55+1.20
BBCS
UEAF
68.6245..87 62.93=5.69
58.43:13.57
46.51=4.52
78.28=3.85 71.03+33.88
AGCJMMV
80.17:13.19 70.86:6.14
67.79:14.88
52.41-15.22
83.79-3.83
76.03:14.54
LBIMVC
78.79:13.02 74.66+3..9
70.31::199
65.82-42.72
85.17=0.94
84.27+42.70
HLSCG
82.17::1.11 73.68+0.28
70.22:1:..2
62.66-+1.00
86.14-1811
83.09:10.46
Proposed
84.77:1.117 75.48=0.15
71.23:1042
66.41
86.74-2.35
84.79:22.36
MVL.IV
48.544.166 52.43+
62.81:0.99
63.28+0.85
52.78=1.60
56.53++1.21
AWIMVC
46.55+0.71 33.49+1.77
50.71+1.90
54.30+1.32
41.70=1.61
13.5
C0
UEAF
47.22466.20 36.04+3.68
52.31+:5.6
44.46+66.14
48.82-4.91
8..8
AGC IMVC
83.5412.41 76.1813.96
82.22:1:.04
80.59155.55
86.94=141
79.1713.77
LBIMVC
81.08:12.55 76.4011..58
82.35:12.01
79.6911341
84.39=1.50
80.46:12.21
HLSCG
82.27122.06 78.85
84.6311.33
1.,60
85.50=2.44
76.90112.08
Proposed
84.05+2.46 80.65=3.40
85.64+3.10
80.06+1.79
87.72-2.57
1..64+1.72
MVL.IV
48.54.11.64 52.43.1..15
62.81
63.2810.85
52.78-1.60
5..53111221
AWIMVC
46.55.10.71 33.4911.77
50.71
44.30.1.32
11.70-1.61
1..5
Caltech-7
UEAF
42.71 36.32+4.22
31.07+
2..02+1..7
78.26-2.12
76.29+ 1,93
AGCJMVC
57.31+2.13 55.10=2.66
59.47=1.28
59.37+2.36
61.59=2.01
60.22++2.57
LBIMVC
56.5610.46 56.25=1.15
57.69+0.83
59.66+2.03
62.55=1.07
59.68
HLSCG
65.13:11.52 63.45--2.21
64.774-2.31
61.21141.75
79.50-1.29
77.36
Proposed
66.73412.35 64.38
67.06:12.31
63.49
80.46 2.29
7.65
MLV JIV
36.81 31.11=0.27
66.8312.53
62.63=20114
38.16=1.01
32.74::0.41
AWIMVC
41.02+1.67 33.27=0.68
62.09:1.05
61.22++0.93
43.68=0.48
36.49::0.74
UEAF
30.59+3..7 26.74-9.66
60.46+7.94
59.1443.38
32.22-4.41 28.07++3.97
BUAA
AGC IMVC
26.7440..77 25.48+2.70
53.87+0.61
57.51+5525
28.01=1.37
26.67++3.72
LBIMVC
46.49+42.63 39.65+1.60
58.56+0.50
54.33+3.75
39.45-2.73
37.60+2.001
HLSCG
45.33:12.16 38.35
63.8711.06
61.68:1114
47.51=1.21
1..38
Proposed
49.35112.79
42.98112.47
68.48:0..9
64.25::2.29
50.68-241
46.55.1.32
IV
40.02+2.58 43.75+3.55
51.83+1.78
52.28+2.39
41.37-2.65
48.98-44.32
AWIMVC
41.5711145 30.6813.69
41.72.12.50
52.3313.31
32.40-2.55
41
Noisy COIL-20
UEAF
40.50.14.14 32.0513.35
47.68.14.79
36.66.14.51
40.03-2.59
33.62.15.70
AGCLMVC
72.55.1..17 71.
75.35.13.58
71.91
79.96-2.74
70.0712.667
LBIMVC
73.66++5.22 71.32+2.75
73.6744.45
71.2543.77
75.55=2.17
71.79+5.53
HLSCG
71.12+22.77 69.82=3.11
1..55+1120
70.46+2.55
81.04=1.54
3..31
Proposed
75.2443.31
73.88=2.70
78.33#2.711
7..34+1180
83.25-4.64
74.55+22.36
Table ACC (%), NMI (%), and Purity (%) of different methods on the BBCSport, COIL-20, Caltech-7, and BUAA datasets, respectively.
1991, 2063, 2113, and 2158, respectively. The Columbia Object Image Library (COIL-20) Library(CCOIII-100 comprises a total of 1,440 images spread across 20 different classes. Simultancously. 10% Gaussian noise was imposed into the original samples to generate the Noisy COIL-20 database. To create the multi-view dataset, we extracted three feature types from each image: deep features using the VGG-F Aiddi al. (2024), Local Binary Patterns (LBP) features Zeng et al. (2023), and vectored pixel features. Mcanwhile, the Caltech10 database encompasses 101 object categories, with each class having between 40 800 images Fei-Fei et al. (2004). For the comparison experiments conducted in this paper, subset known as Caltech-7 was chosen, which includes 1474 images from 7 classes. Specifically, the selected multi-view dataset for Caltech-7 consists of two views: GIST and LBP features Li et al. (2015). Referring to Zhao et al. 20016)
8

---

Published as conference paper ICLR 2026
ACC (%)
100
ACC (%)
100
90
50
ACC(%)
100
50
ACC(%)
0
80
g
10 5
10
10
108
10
10
60
10.
10
10
10.57
70
10
10
X2
A1
A2
A1
À
(a)
(b)
Figure 2: Clustering accuracies versus different combinations of parameters with 30% missing rate on the (a) BBCSport and (b) COII--20databaee respectively.
subset the BUAA-visnir face database was chosen to evaluate the proposed method, which comprises two views, i.e., 90 visual images and 90 near-infrared images, all belonging the first 10 classes.
Incomplete multi-view data construction: this paper, the incomplete multi- dataset was created by randomly removing 30% and 50% instances in each view. Then, every algorithm method was carried out on each dataset 10 times, and the average value was reported as the final result. The evaluation metrics used were clustering accuracy (ACC), normalized mutual information (NMI), and purity ?. All experiments conducted in this study were run on MATLAB R2020a, utilizing hardware setup with 16.0 GB of RAM and 3.40 GHz CPU.
3.2 COMPARISON WITH TTATE-OF-THE--
In the comparative experiments, the proposed method was evaluated alongside a range of state-of-the-art IMC methods., i.e., AWIMVC et al. (2024), and HLSCG Zhao et al. (2025). Table I reports the experimental results on the BBCSport, COLI-20. Caltech-7, 7, and BUAA databases, respectively. From Table I it can be observed that: 1) The proposed method can always obtainsthe best clustering performances on different incomplete multi-view databases. shows that graph-based methods have clear advantages compared to the other related methods. 2) Because the proposed method is capable of uncovering higher-order correlations among various views, it outperforms the other methods in terms of performance.
al. (2020), UEAF et al. (2019), AGC.IMVC LBIMVC Wen
3.3 PARAMETER SENSITIVITY ANALYSIS
Algorithm requires tuning of three key parameters, i.e., Å1 λ2, and X. To determine the optimal param- eter combinations for each dataset, we conducted extensive experiments on the BBCSport and COIL-20 datasets. As shown in Fig. - our analysis reveals that the proposed method always demonstrates remarkable stability with respect to Å₁ across the range [10-5 10a] and Xa E [10-5, 101,,002 10-2 as well as À in the range of [10- 10-2].
3.4 CONVERGENCY ANALYSIS
We empirically validate the convergence of the proposed method through comprehensive experiments on the BBCSport and BUAA datasets. Figure illustrates the relationship between iteration counts versus both clustering accuracy and objective function values across multiple datasets with 30% missing view rates. The results clearly demonstrate that clustering accuracies stabilize within few iterations while objective values decrease monotonically convergence, thereby confirming the convergence properties of method.
9

---

Published as conference paper ICLR 2026
20
Objective funstion value
Chjoctive lunction value Churteingg scouray
Oljective fundion valua
Objective function
18
8
ACC (%)
Custaring accuracy
ACC (%)
70
80
5
10
15
20
25
30
10
15
20
25
30
Iterations
Iterations
(a)
(b)
Figure 3: Objective function value and clustering accuracy of the proposed method versus the number of iterations on the (a) BBCSport and (b) BUAA databases, respectively.
3.5 ABLATION ANALYSIS
In this sub-section, experiments on the COIL-20 and noisy COIL-20 databses with 10%, 30%, and 50% view missing were conducted, where the proposed model with the high-confidens block diagonal regularizer and the low-rank constraint were compared. Fig. 4 shows the comparative results of the proposed method, where the proposed method signiffcantly outperforms the low-rank constraint model.
Law-ank constrant
Low rank constrant
95
proprsed block disporal repulerizer
78
The proposed block diagonal regulerize bser.
76
74
ACC
85
72
70
75
68
70
20
30
40
50
06
10
20
30 40
50
Rate of Views
Missing Rate
(a)
(b)
Figure 4: Clustering accuracies (ACC, %) of the proposed method and the conventional low-rank constraint model on the COIL-20 and noisy COIL-20 databases with different misng rates, respectively.
4 CONCLUSIONS
This paper proposed novel and efficient CBDS IMVC method for IMVC taks, which introduced complete representation reconstruction strategy, where the consenus complete structure across all views was learned preserve the structure of the reconstructed data. Specifically, a confident block diagonal regularizer is pro- posed to guide all views toward learning an invariable structure with block diagonal form. This mechanism preserves distribution of the data from each view while enforcing consistent and strict block diagonal pat- tern that remains aligned across all fiews.Extensive experiments demonstrate superior missing data recovery and IMVC performance. Future work will extend this framework to broader application domains.
10

---

Published as conference paper ICLR 2026
ACKNOWLEDGMENTS
This work was supported in part by the Natural Science Foundation of Guangdong under Grant 1647,, in part by Shenzhen Science and Technology Program under Grant JCY1202400131051155077 and in part by the National Natural Science Foundation of China under Grant 62576112 and 62576140.
REFERENCES
Oussama Aiadi, Belal Khaldi, Aicha Korichi, Mourad Chaa, Mohamed Ben Bezziane, and Ibrahim Omara. Fusion of deep and local gradient-based features for multimodal finger knuckle print identification. Clus- ter Computing, 27(6):7541-7557, 2024.
Jie Chen, Shengxiang Yang, Hua Mao, and Conor Fahy. Multiview subspace clustering using low-rank representation. IEEE Transactions Cybernetics, 52(11):1234412237 2021.
Jie Chen, Hua Mao, Wai Lok Woo, Chuanbin Liu, Zhu Wang, and Xi Peng. One-step adaptive graph learning for incomplete multiview subspace clustering. IEEE Transactions on Knowledge and Data Engineering, 2025.
Anal Roy Chowdhury, Avisek Gupta, and Swagatam Das. Deep multi-view clustering: A comprehensive survey of the contemporary techniques. Information Fusion, pp. 103012, 2025.
Jinrong Cui, Yuting Li, Han Huang, and Jie Wen. Dual contrast-driven deep multi-view clustering. IEEE Transactions on Image Processing, 2024.
Shijie Deng, Jie Wen, Chengliang Liu, Ke Yan, Gehui Xu, and Yong Xu. Projective incomplete multi-view clustering. IEEE Transactions on Neural Networks and Learning Systems, 2023.
Wanyu Deng, Lixia Liu, Jiangiang Li, and Yijun Lin. Auto-weighted incomplete multi-view clustering. IEEE Access, 8: 8:138752- 11118722211872,, 2020.
Fadi Dornaika and Sally El Hajjar. Towards unified framework for graph-based multi-view clustering. Neural Networks, 173:106197 2024.
Guowang Du, Lihua Zhou, Yudi Yang, Kevin Lü, and Lizhen Wang. Deep multiple auto-encoderrbasse multi-view clustering. Data Science and Engineering, ((3):323-338, 2021.
Uno Fang, Man Li, Jianxin Li, Longxiang Gao, Tao Jia, and Yanchun Zhang A comprehensive survey on multi-view clustering. IEEE Transactions on Knowledge and Data Engineering, 32(12):12350--2368 2023.
Li Fei-Fei, Rob Fergus, and Pietro Perona. Learning generative visual models from few training examples: An incremental bayesian approach tested on 101 object categories. In 2004 conference on computer vision and pattern recognition workshop, pp. 178-178. IEEE, 2004.
Yeqing Li, Feiping Nie, Heng Huang, and Junzhou Huang. Large-scale multi-view spectral clustering via bipartite graph. In Proceedings of the AAAI conference on artificial intelligence, number 1, 2015.
Zhenglai Li, Chang Tang, Xinwang Liu, Xiao Zheng, Wei Zhang, and En Zhu. Consensus graph learning for multi-view clustering. IEEE Trunsuctions on Multimedia, 14:2461-24722 2021.
Abdelmalik Moujahid and Fadi Doraiik.. Advanced unsupervised learning: a comprehensive overview of multi-view clustering techniques. Artificial Intelligence Review, 58(8):1-52, 2025.
11

---

Published as a conference paper ICLR 2026
Chuan Tang, Kun Sun, Chang Tang, Xiao Zheng, Xinwang Liu, Jun-Jie Huang, and Wei Zhang. Multi- view subspace clustering via adaptive graph learning and late fusion alignment. Neural Networks, 165: 333-343, 2023.
Hao Wang, Yan Yang, and Bing Liu. Gme: Graph-based multi-view clustering. IEEE Transactions on Knowledge and Data Engineering, 92(6):1116-112) 1129, 2019a.
Hao Wang, Yan Yang, Bing Liu, and Hamido Fujita. A study of graph-based system for multi-view cluster- ing. Knowledge-Born -aaedg Systems, 163:1009-1019, 2019b.
Qianqian Wang, Zhiqiang Tao, Wei Xia, Quanxue Gao, Xiaochun Cao, and Licheng Jiao. Adversarial multiview clustering networks with adaptive fusion. IEEE transactions on neural networks and learning systems, 34(10):7635-764) 2022.
Siwei Wang, Xinwang Liu, Li Liu, Sihang Zhou, and En Zhu. Late fusion multiple kernel clustering with proxy graph refinement. IEEE Transactions on Neural Networks and Learning Systems, 34(8):4359-4370, 2021a.
Siwei Wang, Xinwang Liu, Xinzhong Zhu, Pei Zhang, Yi Zhang, Feng Gao, and En Zhu. Fast parameter-free multi-view subspace clustering with consensus anchor guidance. IEEE Transactions on Image Processing, 31:556-568, 2021b.
Xinxin Wang. Yongshan Zhang, and Yicong Zhou Highly efficient rotation-invariant spectral embedding for scalable incomplete multi-view clustering. In Proceedings of the AAAI Conference on Artificial Intel- ligence, volume 39, pp. 21312-21320, 2025.
Jie Wen, Zheng Zhang, Yong Xu, Bob Zhang, Lunke Fei, and Hong Liu. Unified embedding alignment with missing views inferring for incomplete multi-view clustering. In Proceedings of the AAAI conference on artificial intelligence, volume 33, pp. 5393-5400, 2019.
Jie Wen, Ke Yan, Zheng Zhang, Yong Xu, Junqian Wang, Lunke Fei, and Bob Zhang. Adaptive graph completion based incomplete multi-view clustering. IEEE Transactions on Multimedia, 23:2493-2504, 2021.
Jie Wen, Gehui Xu, Chengliang Liu, Lunke Fei, Chao Huang, Wei Wang, and Yong Xu. Localized and balanced efficient incomplete multi-view clustering. In Proceedings of the 31st ACM international con- ference on multimedia, pp. 2927-2935, 2024.
Wai Keung Wong, Lusi Li, Lunke Fei, Bob Zhang, Anne Toomey, and Jie Wen. Confident local structure- aware incomplete multiview spectral clustering. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2025.
Wei Xia, Quanxue Gao, Qianqian Wang, Xinbo Gao, Chris Ding, and Dacheng Tao. Tensorized bipartite graph learning for multi-view clustering. IEEE Transactions Paltern Analysis and Machine Intelli- gence, 45(4):5187-5202, 2022.
Xiaomei Zeng, Xingcong Zhao, Xinyue Zhong, and Guangyuan Liu. A survey of micro-expression recogni- tion methods based on lbp, optical flow and deep learning. Neural Processing Letters, 5565);5995-6026, 2023.
Handong Zhao, Hongfu Liu, and Yun Fu. Incomplete 2392-2398, 2016.
visual data grouping. In IJCAI, pp.
Shuping Zhao, Lunke Fei, Qi Lai, Jie Wen, Jinrong Cui, and Tingting Chai. High-confident local structure guided consensus graph learning for incomplete multi-view clustering. In Proceedings of the Thirty- Fourth International Joint Conference on Artificial Intelligence, UCAI '25 2025. ISBN 978-1-956792 06-5. doi: 10. 00.24963/jjcai.222577779 URL http://///././/////o//oo/.././..
2496 http://0.222401//1.......2.
12


Reviews:
[{"review_id": "8u6llnarFG", "reviewer": "Reviewer_o5ae", "summary": "In this paper, a Confident Block-Diagonal Structure (CBDS)-based framework is presented for incomplete multi-view clustering. The key assumption is that the recovered instances across all views should exhibit a consistent and strictly invariant local structure, enforced through the CBDS constraint. To further enhance robustness, an invariable graph completion mechanism is introduced to infer the intrinsic cross-view structure while accommodating missing views. All components are jointly optimized in a unified learning framework, enabling mutual reinforcement between structure recovery and clustering. The experiments conducted on multiple benchmark datasets demonstrate that the proposed approach consistently outperforms baseline methods.", "questions": "1. The transition from Eq. (6) to Eq. (7) feels a bit unclear and somewhat repetitive. Could this part be streamlined or explained more smoothly to make the flow easier for readers to follow?\n2. Could you clarify why the element-wise Hadamard product is used in Eq. (6)? A bit more explanation on its purpose (e.g., handling observed entries or enforcing consistency) would help readers understand the motivation behind this choice.\n3. Section 2.4 has a lot of detailed math, which may distract from the main ideas. Would it be possible to simplify this section or move some of the detailed derivations to the appendix so readers can focus more on the key concepts?\n4. There are a few typos and formatting issues. Please proofread the paper to improve clarity.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "1. The method adopts a complete representation reconstruction mechanism to infer missing views, ensuring that the recovered multi-view representations preserve the underlying data structure across views. \n2. By explicitly learning a consensus and complete structure shared by all views, the framework enhances global structure alignment and reinforces clustering consistency. This cross-view agreement helps achieve more stable and reliable clustering outcomes.\n3. The introduction of a confident block-diagonal regularizer encourages each view to maintain its intrinsic distribution while enforcing a strict and invariant block-diagonal structure across views.", "weaknesses": "1. While the method handles missing views and distribution discrepancies, it does not explicitly address robustness to noise or outliers. It is better to include additional experiments with synthetic or real noise to demonstrate the method's resilience in practical scenarios.\n2. The current Figure 1 provides only a high-level conceptual overview, and some steps in the pipeline remain unclear from the illustration. Please enhance the figure with clearer modules, detailed annotations, and flow directions.\n3. The optimization part, especially Section 2.4, is a little long and detailed. It would be helpful to simplify this section or move some of the detailed derivations to the appendix.", "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": "Y8alcL9AwG", "reviewer": "Reviewer_uJCn", "summary": "Incomplete multiview clustering is a significant and challenging sub-field within the research direction of multi-view clustering. This paper proposed a block diagonal structure completion method to guide the learning of the missing-instances of different views. Moreover, the re-constructed complete structures from different views can be adaptively aligned. The experimental results proved its positive performance in IMVC tasks.", "questions": "See Weaknesses.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The block-diagonal matrix represents a crucial property of data. It can clearly reveal the relationships within clusters and between clusters. This paper consider the IMVC problems from a unique perspective, where the property of the block diagonal structure of data is utilized to recover missing data from different views. \n\n2. The proposed method designs the model simultaneously from the perspectives of missing data prediction and complete structure derivation. The joint learning strategy of this method enables the missing predicted data to maintain the structure of the original data on the one hand, and the complete predicted data to promote the derivation of the complete structure on the other hand.", "weaknesses": "1. The comparative experiments lack several state-of-the-art incomplete multi-view clustering (IMVC) methods. Including these would provide a more convincing evaluation of the proposed approach.\n\n2. Could the authors provide further analysis or experimental evidence to justify the necessity of the proposed block diagonal constraint, particularly in comparison with self-representation or low-rank regularization techniques?\n\n3. The paper would benefit from a more detailed explanation of the symbol “Pov” used in Equation (1), which would help readers better understand the proposed method.\n\n4. It would be beneficial for the authors to incorporate additional visual evaluation approaches. Moreover, how did the authors set the hyperparameters? Please provide a complete set of basis and methods for setting them.", "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": "BFTmANT3Cf", "reviewer": "Reviewer_vwi6", "summary": "This paper proposed a confident block diagonal structure-aware invariable graph completion-based incomplete multi-view clustering method, which designed a confident-aware missing-view inferring strategy to guarantee that recovered instances of all views have the same strict invariable local structure. This mechanism preserves distribution of the data from each view while enforcing a consistent and strict block diagonal pattern that remains aligned across all views.", "questions": "1.Compared with the other block diagonal regularizers, what are the obvious advantages of the proposed model?\n\n2.Could the authors provide a more detailed explanation of Eq. (4)?\n\n3.Could  the authors provide a more detailed explanation for the function of each part in Eq. (6)?\n\n4.The mathematical formulas and explanations are too complex. Can the mathematical solution process be simplified, and add more explanations of the motivation and contributions of the proposed method?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1.This paper is well written with clear organization and significant contributions.\n\n2.Different from the other related papers, this paper designed a complete representation reconstruction strategy. This approach learns a unified, consensus structure that is invariant across all views, effectively maintaining the intrinsic structure of the recovered data.\n\n3.A novel confident block diagonal regularizer was proposed to guide all views toward learning an invariable structure with block diagonal form.", "weaknesses": "1.All equations presented in the manuscript must include clear definitions of their parameters. \n\n2.Can the authors further clarify the importance of block diagonal structure learning compared with the other related methods.\n\n3.The conclusion should clearly summarize the main findings.", "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": "ur7X16kKVF", "reviewer": "Reviewer_FAVd", "summary": "This paper proposes an incomplete multi-view clustering method named CBDS_IMVC. By introducing confidence-aware block diagonal structure perception and invariant graph completion strategies, it aims to address the issue that existing methods overlook structural consistency and distribution differences during missing view recovery. The method jointly learns the complete representations of multiple views and employs block diagonal regularization to ensure all views share a consistent intrinsic clustering structure, thereby enhancing clustering performance. Experimental results demonstrate that this method outperforms state-of-the-art approaches on multiple benchmark datasets.", "questions": "1. How does the time complexity of the proposed method compare with other methods? Please supplement the analysis of time complexity.\n\n2. Although the experimental results have demonstrated the superiority of this method, could you further explain the differences and connections between it and other block diagonal methods?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. A confidence-aware block diagonal structure regularization is proposed to ensure that all views maintain a consistent local structure after recovery.\n\n2. A joint learning framework is adopted to simultaneously optimize view recovery, graph completion, and clustering representation, thereby improving overall performance.\n\n3. Multi-view complementary information is introduced to effectively leverage cross-view information for recovering the structure of missing instances.", "weaknesses": "1. While the empirical results are promising, a more in-depth theoretical analysis of why CBDS and invariable graph completion lead to better clustering performance would strengthen the paper. \n\n2. Please double-check the definitions of variables, typos, and citations overall in the manuscript.\n\n3. The complex mathematical solution process takes up too much space, resulting in insufficient necessary experiments, such as ablation experiments.", "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)