Constant Degree Matrix-Driven Incomplete Multi-View Clustering via Connectivity-Structure and Embedding Tensor Learning
Authors: Zhibin Gu
Zhenhao Zhong
Xi Zhang
Bing Li
Venue: ICLR 2026 Poster (2026)
Keywords: Multi-view clustering
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: Tensor-based incomplete multi-view clustering has attracted significant research attention due to its capability to exploit high-order correlations across different views for revealing underlying cluster structures from partially observed multi-view data. However, most existing approaches construct tensors from adjacency matrices, which necessitate post-processing operations (e.g., singular value decomposition, SVD) and thereby introduce additional computational overhead and potential errors. Some approaches instead employ latent embedding tensors to avoid post-processing, but they often fail to capture the geometric structure of the underlying graph. To address these limitations, we propose **C**onst**A**nt degree **M**trix-driv**E**n incomp**L**ete multi-view clustering via connectivity-structure and embedding tensor learning (**CAMEL**). Specifically, CAMEL jointly learns view-specific latent embeddings under structured constraints and organizes them into a tensor with an ${\ell_{\delta}}$ low-rank constraint, thereby enabling coordinated optimization of graph connectivity and high-order correlations. To further mitigate the $\mathcal{O}(n^2)$ or ever higher complexity complexity associated with conventional connectivity constraints, CAMEL approximates the variable Laplacian degree matrix with a constant-degree matrix, reducing the computational cost to $\mathcal{O}(1)$. Clustering assignments are subsequently derived via $k$-means on the concatenated embeddings, eliminating the need for post-processing operations on adjacency matrices such as SVD.  Extensive experiments on nine benchmark datasets demonstrate the superior effectiveness and efficiency of CAMEL.

Published as a conference paper at ICLR 2026
CONSTANT DEGREE MATRIX-DRIVEN INCOMPLETE MULTI-VIEW CLUSTERING VIA CONNECTIVITY-
STRUCTURE AND EMBEDDING TENSOR LEARNING
Zhibin Gu Zhenhao Zhong'; Xi Zhang', Bing Li
College of Computer and Cyber Security, Hebei Normal University
School of Computer Science, China University of Labor Relations State Key Laboratory Multimodal Artificial Intelligence Systems,
Institute of Automation, Chinese Academy of Sciences guzhibinghebtu edu. on, bli@nlpr: ia. ac. en
ABSTRACT
Tensor-based incomplete multi-view clustering has attracted significant research attention due to its capability to exploit high-order correlations across different views for revealing underlying cluster structures from partially observed multi- view data. However, most existing approaches construct tensors from adjacency matrices, which necessitate post processin operations (e.g.. singular value de- composition, SVD) and thereby introduce additional computational overhead and potential errors. Some approaches instead employ latent embedding tensors to avoid post-processing, but they often fail to capture the geometric structure of the underlying graph. To address these limitations, we propose ConstAnt de- gree Mtrix-drivEn incompLete multi-view clustering via connectivity-struuuue and embedding tensor learning (CAMEL). Specifically, CAMEL jointly learns view-specific latent embeddings under structured constraints and organizes them into a tensor with an lz low-rank constraint, thereby enabling coordinated opti- mization of graph connectivity and high-order correlations. To further mitigate the O(n)) ) or ever higher complexity associated with conventional connectivity constraints, CAMEL approximates the variable Laplacian degree matrix with a constant-degree matrix. reducing the computational cost of degree matrix con- struction to (1). Clustering assignments are subsequently derived via k-means the concatenated embeddings, eliminating the need for post-processing opera- tions on adjacency matrices such as SVD. Extensive experiments on nine bench- mark datasets demonstrate the superior effectiveness and efficiency of CAMEL.
INTRODUCTION
Multi-view data widely encountered in real-world tasks, where each sample typically be represented from multiple perspectives or modalities. For example, an image can he represented from multiple perspectives, where features derived from texture, color, and local binary patterns are treated as separate views Wen et al. (2022); Gu et al. (2024). Multi-view clustering (MVC) aims to integrate information from different views to accurately uncover the intrinsic structure of the data Tang et al. (2020); Kang et al. p2 Jiang et al. (2022). In practice, however, some views may be missing; for example, certain events may be reported by some sources but absent in others, resulting in incomplete multi-view data. shihe To effectively handle such incomplete multi-view data, numer- ous incomplete multi-view clustering (IMVC) methods have been proposed, which can be broadly categorized into matrix-basee Wen et al. (2019): b); Liu et al. (2018); Li et al. (202I) and tensor- based approaches Li et al. (2024); Yu et al. (2024): Li et al. (2023b). Among them, lensor-based methods have attracted widespread attention due to their ability to effectively model high-order correlations across multiple views. For example, Gu & Feng 2024 imposes low-rank regularization on the
"These authors contributed equally.
1 "Corrsponning author.

---

Published as conference paper at ICLR 2026
View-speclfic connectivity constraint
C Constant degree matrix
Latent embelding lensor constraint
View
Graph/1
G-norm
C Concatenation of latent embedding matrices
View
Griph 2
Clustering resulls
View
Graph m
Overall framework the proposed CAMEL model, which integrates connectivity and low-rank constraints the latent embedding tensor, with constant degree matrix employed in the Laplacian to reduce computational complexity.
Figure 1:
tensor constructed from coefficient and projection matrices, thereby reducing computational com- plexity and enhancing robustness. In addition, Zhang et 2023)) employs a decomposition strategy that separates the coefficient tensor into low-rank and sparse noise components for independent opti- mization. However, these approaches require post- post-processinggJe.. SVD) of the coefficient matrices to extract the latent embedding representation for subsequent k-means clustering. The decoupling of graph learning and latent embedding learning may lead to suboptimal performance due to po- tential error propagation between the two stages. To eliminate the need for post- post-processing recent tensor-based methods such as Xu ct al. (2025): et al. (2025) construct tensors from latent embed- ding matrices, allowing k-means to be directly applied on them to obtain clustering results. Despite this improvement, major limitation remains: the absence of connectivity constraints prevents the explicit preservation of the global geometric structure within each view. While imposing connec- tivity constraints the adjacency matrix natural solution, this typically requires constructing the Laplacian matrix, which involves computing the degree matrix from the similarity matrix, re- sulting in a time complexity of (n²)) higher and significantly limiting computational efficiency. Moreover, since the latent embedding is represented as a single consensus matrix across views, ex- isting connectivity- based methods cannot naturally accommodate tensor-based constraints, making difficult to simultaneously capture both the low-rank properties of the latent embedding and the potential high-order correlations across multiple views.
To address these limitations, propose effective and novel model, CAMEL, whose overall framework detailed Figure [ The core innovation of CAMEL lies the introduction constant-degree matrix instead of variable-degree matrix in the Laplacian construction. This de- sign preserves the essential geometric structure of the graph while reducing the computational cost of munecctiviy-coostrrnnee embedding learning from O(n)) to O(1). Moreover, CAMEL inte- connectivity constraint with a low-rank constraint on the latent
grates the constant-degree-dmi embedding tensor, enabling the model learn the geometric structure of each view and construct a tensor that simultaneously captures low-rank properties and high-order correlations across views. The resulting latent embeddings can be directly used for clustering without post-processing, thereby avoiding potential errors introduced by additional steps. Compared to existing methods, our key contributions can be summarized as follows:
'The term "post-processic" free" refers to the fact that the features obtained after optimization can be di- rectly used for k-means clustering to obtain results, without requiring additional operations on these features (e.g., SVD the anchor subspace representation matrix or performing spectral clustering the similarity matrix) generate features for clustering.
2

---

Published as a conference paper at ICLR 2026
We approximate the variable Laplacian degree matrix in the connectivity constraint with a constant-degree matrix, reducing computational complexity by an order of magnitude while preserving competitive accuracy and scalability to large datasets.
We integrate the constant constant-degree on the latent embedding tensor, enabling the model learn the geometric structure of each view and construct tensor that simultaneously captures low-rank properties and high-order correlations across views.
connectivity constraint with low-rank constraint
We develop efficient algorithm for jointly solving the latent embedding matrices within an ADMM-based optimization framework and provide theoretical guarantees for its con- vergence. Extensive experiments validate the effectiveness and efficiency of CAMEL.
RELATED WORK
Tensors are well-suited for capturing high-order correlations, which makes them particularly effec- tive for imputing missing data and has led to their widespread application in incomplete multi-view clustering (IMVC). Existing tensor-based IMVC methods can generally be categorized into two groups: post- posst-pocessing-baee based approaches and post- post-processing-free free approaches.
For the peos--rocessing--. bbaeee methods, latent embeddings are typically obtained by applying SVD to the anchor subspace representation matrix or by performing spectral clustering on the similarity matrix, after which k-means is conducted
derive the final clustering results Wen et al.] (2021); Zhang et al. (2013); Zhao et al. (2023) [Lng et al. (2025 2024); Lv et al. (223) Zhang et al. (2023): Li et al. (2022b) For instance, Lonng et al. (2024) obtains per-view em- bedding features by projecting the anchor graph onto the designated matrix, and then applies a tensor low-frequency approximation the tensor constructed from these embedding feature matrices, Wen et al. (22TT) enhances representation- based graph learning by simultaneously imposing a consensus graph constraint and a tensor low-rank regularization, thereby capturing both within view and cross- view dependencies. Moreover, Lv et al. (2023) maps the original data into a low-dimensional feature subspace to achieve more compact representations, which facilitates subsequent graph construction.
the latent embeddings
As for the post-processing-ff free methods, they directly obtain the latent embedding matrix after op- timization, and clustering results are then produced by applying k-means on this matrix Li et al. 2023a); Xu et al. (2025); Liu et al. (2025); Wang et al. (2025); ) Chen et al. (2022); ; Wu et al. (2024). For example, Li et al. (2023a) introduces an orthogonal non-negative tensor factorization frame- work that extends traditional NMF to third-order tensors, enabling direct handling of multi-view data while preserving and exploiting the inherent spatial structures across views. In addition, Xu al. (2025) learns anchor-sample similarity graphs in adaptive manner, enforces structural finement in a latent subspace, and employs low-rank modeling on the latent sample embeddings. In Liu et al. (2025). the method eliminates the tensor rotation trick to prevent its interference with un- supervised clustering, and further leverages pairwise sample similarities via a linear kernel function to ensure scalability to large datasets.
PROPOSED METHOD
3.1 ANCHOR SURSPACE-BASHI TENSOR REPRESENTATION LEARNING
Let the incomplete multi-view dataset be denoted as {X" J==j, where X" E Rd.x", with d, rep- resenting the feature dimension of the v-th view, the total number of samples, and m the tal number of views. In anchor-based tensor incomplete multi-view clustering, the anchor graphs learned from different views are combined into diree-dimensional tensor, and post-processing step is performed on these anchor graphs prior to final clustering to obtain the cluster assignments. The general framework summarized as follows:
min E
R
A J
s.t. Ve, X = A'Z" + E Z" 2 (A - 1,2 2Z ..... 7
where A" o xt 7" E 1 and E" E - denote the anchor matrix, coefficient matrix, and reconstruction error matrix of the y-th view, respectively, with / representing the number of anchors.
3

---

Published as a conference paper at ICLR 2026
To ensure discriminativenes. among anchors, an orthogonality constraint (A") A" I is imposed to enhance anchor diversity. Physically, Z" is constrained to be nonnegative, i... Z" 2 0. The tensor Z is constructed by concatenating and rotating {Z" Jø=1 {Z"}a= Via the operation 4. The regularization term R(-) models the tensor rank and promotes low-rank properties during optimization, while P(.) regularizes the reconstruction error enforce X AZ"
3.2 JOINT LEARNING OF CONNECTIVITY -ONDIIIITT EMBEDDING TENSOR
Although the incorporation of tensors is capable of capturing the high-order correlations across multiple views, it is still insufficient to explicitly characterize the underlying graph structure. First, since {Z" 1o=1 cannot be directly employed for clustering. post-processing step (e.g., SVD) is required to derive the cluster labels. Such reliance on post-processing not only increases the risk of obtaining suboptimal solutions but also introduces additional computational overhead. Second, Eq. (T) fails to explicitly capture the underlying graph structure, resulting in the loss of neighborhood information and reduced discriminability of the learned representations.
To address these limitations, we introduce two key enhancements. First, the connectivity constraint reformulated in view-specific manner and integrated into the subspace structure. Second, mo- tivated by recent advances Xu et al. (2025); ); Wang et al. (2025); Liu et al. (2025) : Li et al. (2023a); Feng al. (2024), impose a low-rank constraint directly the tensor constructed from la- tent embeddings, rather than on that derived from subspace representations. This design enables k-means clustering to be directly performed the latent emhedding matrices, thereby avoiding the costly SVD computation required for subspace representations. Moreover, the rank function R(-)
is approximated by the f5-norm (i.e., f=) = (1+6)
5+z Kang al. (0155), which offers smoother
and tighter surrogate than the conventional tensor nuclear norm Lu et al. (2016); Zhou et al. (2019); Lu et al. (2019) by penalizing singular values more flexible manner. In addition, the squared Frobenius norm || || is employed model the noise matrix. The overall model is thus formulated as follows:
min (A> E H
174 &
A1
E
A
Tr (H)) L'H')
(2)
s.t. X A E E"
I, H I. (H H H
where L" is the normalized Laplacian matrix, i.e., L = I- (D "'' S" R"x" is the affinity matrix, obtained by S" (Z") Z" and D E Ruxn is the degree matrix computed by summing the columns of S" The two hyperparameters A and X2 serve solely for trade-off purposes. For third-order tensor H Ia x its ls norom mathematically expressed as follows:
1
1 Th
(1 (1+))S((. 8 St
n3
In this formulation, h is defined as the minimum nı and n22, while the positive scalar serves as a parameter the &s rank function. The matrix S corresponds to the k-th frontal slice of the tensor Sy. derived from the tensor singular value decomposition (t-SVD) applied to the k-th frontal slice of H. following the model H. (V}))
3.3 CONSTANT DEGREE MATRIX- MATRIX-DRIVEN CONNECTIVITY-TENGON JOINT LEARNING
Although Eq seamlessly unifies latent representation tensor learning with connectivity constraints, the construction of the Laplacian matrix remains computationally expensive with complexity of On)²). In particular, evaluating S" (Z") Z" incurs (tn?) operations. While reordering the computational steps during optimization can reduce this cost to linear in O(n), the computation of the degree matrix D still requires column-wise summation of S" E gn which scales as O(n²). This quadratic overhead inevitably becomes a bottleneck when dealing with large-scale datasets. To address this issue, we replace the conventional variable degree matrix with constant form, D II, where 3 > controls the degree magnitude. Consequently. the Laplacian matrix is simplified to L I- (Z²) z". Based this, the overall CAMEL model can be expressed follows:
4

---

Published as a conference paper at ICLR 2026
1 Tr H -I- 1=((2") (Z"""
min (Ar "."E" 22" H
H &
A1 E F
+
22
(3)
s.t. X A'Z E" Z" E, 2
I, H -1 H² H
3.4 FURTHER DISCUSSION ON THE CAMEL MODEL
The CAMEL model introduces key innovation by efficiently integrating connectivity constraints with the low-rank learning of latent embedding tensors through a constant degree matrix approx- imation. This design not only significantly reduces computational complexity but is also justified from the following three perspectives. First, since Z" inherently contains noise, the degree matrix computed from (Z") Z is itself inaccurate. Therefore, employing a suitably approximated degree matrix does not necessarily impair clustering performance more than using (Z") z". Second, the model able to obtain relatively clean H which is the representation actually used for cluster- ing, rather than the noisier Z". As connected component partitioning is not highly fine-grained operation, once the adjacency graph captures the coarse structure, the resulting H is sufficiently reliable. Last but not least, the tensor constraint H. can further filter out noise introduced by the approximated degree matrix, reconstruction errors and missing data. By constructing the degree ma- trix through constant approximation, the computational complexity of computing D is reduced from (n²) to (1)) lowering the overall cost. This provides important insights into designing connectivity and tensor joint constraints based on latent embedding matrices.
THEORETICAL ANALYSIS
4.1 OPTIMAZITION
We employ the ADMM Lin et (2011) to optimize Eq 2 mented Lagrangian function can be formulated as follows:
Accordingly, the corresponding aug-
C({
A
#1 m In=1:
mm m J==1
IZ" m
m 1=
m
PM 1==1
W,9)
Je
m
m
||
A1
E
(H") [I
(Z") Z
W,
-9
3
(4)
m
+
H 9
Y" A
E
A 7
E
1
where y and Ç are positive parameters, 9 denotes an auxiliary tensor, and W together with Y serve as Lagrange multipliers. The complete optimization procedure is presented in the appendix.
4.2 CONVERGENCE ANALYSIS
Convergence of the optimization algorithm is theoretically ensured by Theorem 1, with the appendix offering a more comprehensive discussion of the theoretical foundations and implementation details.
Theorem The iterative sequence {J, A Zy, E Hy. Yú, 9p Wn)p produced by our op- timization scheme exhibits the following convergence properties:
The sequence {Tp)p= remains bounded;
Every accumulation point of ( is guaranteed be stationary point satisfying the
KKT conditions.
4.3 COMPLEXITY ANALYSTS
The overall computational burden our model is primarily determined by the optimization five sets of variables. These variables include A", z" E", H and g, with their respective complexities

---

Published as a conference paper at ICLR 2026
and O(mtn log(mn) + m² Aggregating these costs yields an overall computational complexity Ommtn log(n) + dmaztn), tn), with dmar indicating the largest feature dimension across the views. It indicates that our approach achieves a substantially efficient complexity.
t(d,tn+d,n), given by ddd, dd
O(ten+c²nn), dtd
EXPERIMENTS
5.1 EXPERIMENTAL SETTINGS
All experimentals are implemented in MATLAB R2023b and exe- cuted on a standard computing platform with 17-12650H CPU and 16 GB RAM. The core findings are presented in this section, while additional results and detailed analyses can be found in the Appendix.
Table 1: The datasets used in the experiments.
Datasel
Samples
Views
Datasets: We conduct experiments on nine publicly available multi-
Yales
163
view datasets. These datasets cover diverse domains including face images( (Belhumerr et al. (197)), EYaleB 10/1 1 et al. (2010)),
MSRCVI
210
EYakB10
640
an action image (Stilset(Still-DBBBB
et al. (2008))), scene im-
COII 20MV
Míeat
2000
1440
ages(Scene(Fang al. (2024)), Scene 1 Fei-FFei & Perona (2005))),
Scene
2688
Scene
4485
et al. (2022a)), COIL20MV, CI- FAR1O(Liu al. (2022))), as well as handwritten digits(Mfeat([)) ct al. (2018), Noisy MNIST). The number of samples and views for each dataset are summarized in Table
object (MSCCC
MNIST
10000
CIFÁRIO
50000
Comparison methods: Six baseline methods are selected for comparison, namely BSV (2001)(Ng
et al. (00T);; Wenn al. (2018, 2022)), Concat(Wac Wen et al. (2018 2022), PVC (2014) et al. (2014) ). IMVC-CBG
(022)(Wang et al. (2022)), PSIMVC-PG (2024) et al. (022a)), and SCSL (2024)(Liu et al. (0024)) For those methods that require hyperparameter tuning, the parameters are adjusted within the recommended ranges, whereas for our model, the search is conducted over A1 E [10- 8, 10-3), 31, Xg E [10 10 2), B [10', 10*), and [10-3 10%]
Evaluation metries: To guarantee the robustness of the evaluation, each algorithm is executed five times. The clustering results are then measured by three criteria: ACC, NMI, and PUR.
Construction of incomplete multi-view data: To simulate incomplete multi-view data, we intro- duce missing entries by randomly discarding proportion of samples each view, where the missing ratio r is selected from the set {0.1, 0.3. 0.5}. In order to avoid the situation in which a sample dis- appears from all views, one view is randomly designated to retain the corresponding data whenever such a case occurs. This procedure guarantees that every instance within the constructed incomplete multi-view dataset possesses at least one observed representation. All datasets are first processed into incomplete-view versions, and these ancomplete-view datasets remain unchanged in subsequent experiments. For example, the original Yale3 dataset is first processed into an incomplete-vier dataset Yale3_i, and all subsequent experiments conducted this fixed Yale3_i without further modifications. Unless otherwise stated (specifically in Section 5.2) the experiments are carried out with default setting of 0.5.
5.2 CLUSTERING RESULTS
Performance comparison: Table reports the comparative outcomes of different clustering meth- ods across all datasets, where the best-performing result in each case is emphasized in bold, and the runner-up result is annotated with underline. The notation "OM" denotes an out-of-memory emror. From the overall analysis of these clustering outcomes, two key insights can be drawn as follows:
(1) Our method achieves consistently superior performance across diverse datasets, including face recognition, object categorization, scene classification, handwritten digits, action recognition, and large-scale collections. In nearly all cases and under different missing rates, the proposed approach substantially outperforms baseline methods in terms of ACC, NMI, and PUR. For example, on the MSRCVI dataset with a missing rate of 0. CAMEL exceeds the second- best method by 23.05%. 29.03% %, and 23.05% in ACC, NMI, and PUR, respectively. Furthermore, on datasets such as

---

Published as a conference paper at ICLR 2026
Table 2: Comparison of clustering results across varying levels of missing data.
Metrook
ACL
FUX
1X01
PCR
TCR
BSV
35
Cooat
11.76-1160 65
$4.30-1135
35
1344455
30 30.04-001
23
20 20443-4.99
24.08-4400
Yake
PSIMVC-PG
IMVC-CRG
57.736000
664880.31
0535-00.0
4500-0333
03
55-00.00
56.744000
33 81
10 10:0-0000
19 19.87-0029
33.52-037
21 21:85-000
SUSI
01011-0000
0301100000
1
33.94-0000
BSV
13.33210045
7.61
74.29 39
$4.29
Sa 56.563541
24 10:0255
44,862278
PVC
02.2 62.78:5.77
6408223
50,2529 20
1.85
56.7583477
MSKCV1
PSINYC-PP PG
4500750000
41/56e4075
50013000
30500 26
182900
19:32-8043
SCSL
14.56-505 00
64 0.30
74 24.2000mm
10000000
5o 00:00.0000 D0
66 10.19200 00
02 1112000
70.45:0000
CAMEL
97.81 -.43
95 Al -0.5%
97.84
$485
97.0 25
07.50 20.36
BSV
IVC
21 0300
2252
12.52-000
SAI-DD
ISIMYC-IG
IMVC-BBG
20:00:0000
100
28 25.22000
20.440010
814-000
8991000
31,2811000
21:0825118
CAMEL
58.66 1.58
-141
S7 18.30
53,47 -9.39
63.45
20.30
74.5% 90.4
06.1 $440
T4LTH -146
BSV
11:00110 70
27
10 13
Cooatt
18.94155
K11-2.96
1006-205
540- 1.5
17:35-1735
BYAKBID
PSIMVC-PG
IMVO-ERG
91M4-6-11
9000 0.000-00
33.38 1.
1647-0111
5479-615
00
5406-0
53 1110111
18601000
1355-014
1200-0m
1567015
14
17:81
131-000
1315-009
1319:0m
222-
2000 00
20-78-000
BSV
33.08:3555 56
3.33
300 2000
+7478100 m0
50 02 24
51.321 72
57.67:5344
4535-1099
41,06:3.09
PVC
600 0.00
5:2 0m
=3 0000
5.39 0.00
5165 000
0000 2000
5,55 = 0000
COLLDIVV
PSINYC-PG PG
56:51=1.30
05555
5
4350 3
+0 0000000
5331
364442050
SCSL
-581-500 00
28.5 050
50008000
S251500 00
53
3355000 00
CAMEL
66.38 1.27
7017 2058
71.22
18
72 2105
BSV
6:211:335
67
41482313
PVC
05.90 181
30
Mirat
PSIMYC-PG
IMPC-CBG
53 51000-0000
00:000000
05.0411000
100
5350:00
33 5S-312000
21342010
192200
25:0-10000
CAMEL
99.65 13.0
9904 20.00
99.65 20/0
96.99 26.16
$7.99
-2.94
95.73
96.35 500
09.29 44.52
BSV
73 11-111
Concat
4.07
JA75+120
0611117 71
2095-201
17-54-188
Scese
PSIMVC-IG
IMVC-ETGG
43:37
3187 0.00
39 0.50
31 10700-00 820
15 15-0--000 01
26 01
10.00
38 1000
1310100 0m
1101000
2008 D:00
$1.34 1104--000
21 0.00
16.78 16:0000000
1:4 0.50
1700
1957-000
21.50
BSV
Cooaa
31 31 0.33
540055033
27.25 20
31 50
$1.388000
10.75 0.08
24.54 =363
IVC
30.59 0.61
13:0144
64
25.1 340
51 1100011 19
25
Sccse15
IMVC-CBB
SCSE.
37:11= 0.00
25.5
26.7:0000
00:00 60
51 11110:00
17.486013
20
17557 16 00
55
1250 000
-5540.00
LOC40
13:00:0000
CAMEL
73:33 2
25.21 25:21-111
42-30-1-33
57.94-1.67
BSV
23.24
23.05
59.45
1:70
0100
1.3
47.15 21.59
PVC
93.00 4.00
2134
03.19 25500
22.27
$3.35 11.41
91.33
$4.55 2 5.55
05.59 24.10
54 24 5.5
CIBARIO
PSIMYC-PO
IMVC-CBG
98.12 = 8.00
25.50
05 60 0.00
98
03.55
08 08322000
95.91 20000
06.14
05.11
9205 20000
92.52220
20100
92/06:10000
CAMEL
$9.99 00 0.00
99.95:00000
99 99.99.9.00
99:95
2 0.80
99.82
0.00
99.95 0.00
97.20
97.20:33.6
95,89:5546
15:41 2131
10 11.1
11.25 TOX
1176
TTAT - 1176
ME11 1787
Crncal
35.93
1:01
31 0.57
43:19 0.47
90:33
34.33
1.19
19.55
31:43 1.04
0.77
51:00 0.00
Novay MNIST
PSIMYC-IG
39:09
0.00
0/00
41.38
0.00
1.00
12/00
810
30.57
0.00
33.44 1100
(1:0
25
1.09
IMVCSCING
0.00
32:21
0.20
46.47
1100
31:27
1100
34.65
8000
31.45
0.00
33.75 8.00
15.52 (1.0
26 0.00
SUSI.
21:1
800
GA8
0150
33.50
0.00
18.98
11:00
5.16
0.00
14.5 8:00
0.00
16 (1 0.000
MSRCVI, Mfeat, Scene, and CIFARIO, most results achieved by CAMEL surpass 95%. Even as the missing rate increases, CAMEL also maintains relatively stable performance.
(2) Compared with both traditional methods (BSV, Concat, and PVC) and recently proposed matrix- based approaches (IMVC-CBG, PSIMVC-PG, and SCSL), CAMEL consistently delivers more competitive results. This superiority stems from the effective integration of connectivity and tensor constraints, which enables a comprehensive modeling of global graph structures, precise learning of latent embeddings, and accurate characterization of high-order correlations across multiple views. In addition, by eliminating the need for post-processing, our approach avoids error accumulation, thereby ensuring robust and high-quality clustering outcomes.
presents a comparison of clustering performance and computational time with and without the use of constant degree matrix. Specifically, CAMEL-vI employs learnable degree matrix, whereas CAMEL adopts a constant degree matrix, with all other compo- nents kept identical. A comparative analysis between CAMEL and CAMEL-v1 yields the following two observations:
Runtime comparison:
(1) In several experiments, CAMEL achieves higher values in ACC, NMI, and PUR compared with CAMEL-vI. Even in cases where its performance is slightly lower, the gap remains negligible. This phenomenon can be attributed to the inherent noise in graph structures, which introduces noise into
7

---

Published as a conference paper at ICLR 2026
Table Performance and runtime comparison between CAMEL-VI and CAMEL.
Dita
PUX 100
ACL
7UK
JAL
PER
HEDD
CAMTI
KUN9 1100
1331 -515
03.00 -2.5
CAMEE
0030
1030
05W0 18.35
18.36
PARIT VI
LOILTRNY
CARDE-I
CANIL
7500:340
12:1
1.31
Mici
CAMIRE
9609
90.99 -3.01
99.33
1.48
Sime
CANPY
100
CAMEL
2002
38.00.00
2521
03.51.1155
16.11
CAMEL-V
LK
134
US
0M
0M
00
0M
UN
0M
0M
AG5
AG0
124
129
107
124
12"
10t
10r
100
(a) Yale3
(h) Scene
(c) CIFARIO
Figure 2: Analysis of Hyperparameter Sensitivity in the CAMEL Model
81-
Yale3
(b) MSRCVI
(c) EYaleB10
Figure 3: Effect of anchor quantity on the CAMEL model.
the computed degree matrix. Approximating the degree matrix with suitable constant allows the model to substantially reduce computational complexity without sacrificing clustering accuracy.
(2) On small-scale datasets, the runtime difference between the two strategies is negligible. How- ever, on large-scale datasets, CAMEL runs significantly faster than CAMEL-vI confirming its lower computational complexity and validating the effectiveness of the constant degree matrix.
5.3 PARAMETERS ANALYSIS
Hyperparameters analysis:We search for the optimal hyperparameters within the ranges \ [10 8 10-3) and \a 10-2). As summarized in Figure 2 the model exhibits slight fluc- tuations with respect to Ay; the best value of x2 is 10-3, which clearly outperforms the cases of Ag = 1 and A2 = 10-4
Impact of anchor quantity: We vary the number of anchors within the range [c, 5c], where denotes the number of classes. As illustrated in Figure 3, the clustering performance exhibits fluc- tuations as the anchor count changes. Increasing the number of anchors does not necessarily lead to better results, since additional anchors may introduce more noise while also incurring higher computational cost. Therefore, it is prudent select fewer anchors long satisfactory per-
8

---

Published as a conference paper at ICLR 2026
formance be maintained. In practice, setting the anchor number 2c or 3e provides desirable trade-off between performance and efficiency.
The parameter 3 is associated with the degree matrix and controls the scaling of node degrees. We search within the range [10, 10*). As shown in Figure 4 performance also fluctuates with varying 3 values, which can be attributed instability caused by approximating the degree matrix with a constant. Nevertheless, experimental results suggest that 10 and 10a generally reliable choices.
Impact of the degree parameter 3:
33
(a) Yale3
(b) MSRCV1
(c) Scene15
Figure Effect of parameter on the CAMEL model.
Impact of the parameter d: The parameter belongs to the éo-morm rank function used in the tensor regularization term. We conduct experiments with å ranging from 10-3 to 1. As presented in Figure 5 clustering accuracy undergoes significant fluctuations as varies, highlighting the crucial role of tensor regularization in filtering noise. Since the performance consistently poor when 10 we recommend searching for the optimal value within the range [10-2, 1].
0.4
60n
Con
360
(a) Yale3
(b) MSRCVI
(c) EYaleB10
Figure 5: Effect of parameter on the CAMEL model.
5.4
CONVERGENCE
(a) Yale3
(b) EYaleB10
(c) CIFARI0
Figure 6: Training convergence trends of the CAMEL model.
To evaluate convergence, we employ two indicators: the reconstruction error (RE), formulated as min X A EeI boo and the match error (ME), defined as H 91 As depicted in

---

Published as a conference paper at ICLR 2026
Figure both RE and ME decrease rapidly and approach zero within the first 25 iterations, thereby demonstrating strong convergence behavior.
5.3 ABLATION STUDY
To investigate the synergistic effect of the connectivity constraint and the tensor constraint, we con-
ducted ablation studies. Specifically, CAMEL-C denotes
Σ Tr((H)) "
H'"",, while CAMEL-T
t=1
represents [ leg. 4 that using only CAMEL-C or CAMEL-T yields worse per- formance than the full model, and the significant performance improvement achieved by combining both components confirms their strong complementarity. Specifically, the connectivity constraint (CAMEL-C) serves connected component constraint that explicitly enforces the formation of well-connected clusters in the latent space, effectively preventing the fragmentation of clusters and ensuring that data points from the same category form connected subgraphs. Meanwhile, the ten- sor constraint (CAMEL-T) operates across multiple views by leveraging the high-order correlations among them, effectively aligning the complementary information from different views into co- herent tensor representation. This dual mechanism allows CAMEL-C to guarantee the structural connectivity of clusters within each view's embedding space, while CAMEL-T captures the global consensus and shared patterns across views, creating synergistic effect where intra-view connec- tivity and cross-view consistency mutually enhance each other. Together, they provide more robust and comprehensive representation learning framework that neither component could achieve alone.
Table 4: Evaluation of the CAMEL model ablation Yale3, MSRCVI I and EYaleB 10 datasets.
Components
Datasets
Yale3
MSRCVI
EYaleB10
CAMEL-C
CAMEL-T
ACC
NMI
PUR
ACC
NMI
PUR
ACC
NMI
PUR
28.
28.9
30.79
41.81
29.80
42.86
25.47
16.30
26.88
48.36
52.49
50.67
57.43
46.76
60.29
26.56
17.33
27.16
61.82
65.17
63.03
97.90
96.03
97.90
55,72
51.21
55.78
CONCLUSION
This paper proposes a unified framework that integrates connectivity constraints with latent embed- ding tensor constraints. In this framework, view-specific connectivity constraints generate multiple latent embedding matrices, which are jointly optimized in tensor form to capture high-order cor- relations across views. Cluster labels directly obtained by applying k-means the learned embeddings, thereby avoiding additional post-processing. To reduce the computational complex- ity from (nn²) ((1), introduce constant-degree matrix into the Laplacian construction of the connectivity constraint. Extensive experiments demonstrate the effectiveness and efficiency of CAMEL, while relatively high standard deviations on certain datasets reveal limitation of the constant-degree approximation.
ACKNOWLEDGEMMENTS
This work was supported in part by the National Natural Science Foundation of China (No. 62506116 16, No. U24A20331), in part by the Hebei Natural Science Foundation (No. F2025205002), in part by the Science Research Project of the Hebei Education Department (No. BJ2026004), and part by the Beijing Natural Science Foundation (No. 1.251005).
REFERENCES
Robert Bartle and Donald Sherbert. Introduction real analysis. John Wiley & Sons, Inc., 2000.
Peter N. Belhumeur, Joao P Hespanha, and David J. Kriegman. Eigenfaces fisherfaces: Recog- nition using class specific linear projection. IEEE Transactions on pattern analysis and machine intelligence (TPAMI), 19(7):711-72) 1997.
10

---

Published as a conference paper at ICLR 2026
Jie Chen, Zhu Wang, Hua Mao, and Xi Peng. Low-rank tensor learning for incomplete multi- view clustering. IEEE Transactions on Knowledge and Data Engineering(TKDE), 35(11):11556 11569, 2022.
Yongyong Chen, Shuqin Wang, Xiaolin Xiao, Youfa Liu, Zhongyun Hua, and Yicong Zhou. Self- paced enhanced low-rank tensor kernelized multi-view subspace clustering. IEEE Transactions on Mullimedia(TMMM 24:4054-4066, 2021.
Si-Guo Fang, Dong Huang, Xiao-Sha Cai, Chang-Dong Wang, Chaobo He, and Yong Tang. Efficient multi-view clustering via unified and discrete bipartite graph learning. IEEE Transactions on Neural Networks and Learning Systems (TNNLS), 35(8):11436111 11447 2024.
L. Fei-Fei and P. Perona. A bayesian hierarchical model for learning natural scene categories. In 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), volume 2, pp. 524-531, 2005.
Wei Feng, Dongyuan Wei, Qianqian Wang, Bo Dong, and Quanxue Gao. Multi-view clustering based on deep non-negative tensor factorization. In Proceedings of the 32nd ACM ACM Interna- tional Conference on AMM MM), pp. 1130-1138, 2024.
Donald Geman and Chengda Yang. Nonlinear image recovery with half-quadratic regularization. IEEE Transactions on Image Processing(TIP), 4(7):932-946, 1995.
Zhibin Gu and Songhe Feng. From dictionary to tensor: A scalable multi-view subspace clustering framework with triple information enhancement. Advances in Neural Information Processing Systems(NIPS), 37:103555-103577 2024.
Zhibin Gu, Zhendong Li, and Songhe Feng. Topology-driven multi-view clustering via tensorial refined sigmoid rank minimization. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD), pp. 920 931, 2024.
Nazli Ikizler, R. Gokberk Cinbis, Selen Pehlivan, and Pinar Duygulu. Recognizing actions from still images. In 2008 19th International Conference on Pattern Recognition(ICPRR) pp. 1-4, 2008
Guangqi Jiang, Jinjia Peng, Huibing Wang. Zetian Mi, and Xianping Fu. Tensorial multi-view clustering low-rank constrained high-order graph learning. IEEE Transactions on Circuits 2022.
and Systems for Video Technology/TCSVT). 320)
Zhao Kang, Chong Peng, and Qiang Cheng. Robust pea via nonconvex rank approximation. In 2015
IEEE International Conference on Data Mining (ICDM), pp. 211-22112000 2015.
Zhao Kang, Wangtao Zhou, Zhitong Zhao, Junming Shao, Meng Han, and Zenglin Xu. Large-scale multi-view subspace clustering in linear time. In Proceedings of the AAAI conference on artificial intelligence(AAAT) volume 34, pp. 4412-4419, 2020.
Adrian S Lewis and Hristo S Sendov. Nonsmooth analysis of singular values, part i: Theory, Set-
Valued Analysis, 13:213-241, 2005.
Jing Li, Quanxue Gao, Qianqian Wang, Ming Yang, and Wei Xia. Orthogonal non-negative ten- sor factorization based multi-view clustering. Advances in Neural Information Processing Sys- tems(NIPS), 36:181::1111 111188188882211 2023a.
Lusi Li, Zhiqiang Wan, and Haibo He. Incomplete multi-view clustering with joint partition and graph learning. IEEE Transactions Knowledge and Data Engineering/TKDE), 35(1):589-60) 2021.
Miaomiao Li, Siwei Wang, Xinwang Liu, and Suyuan Liu. Parameter-free and scalable incomplete multiview clustering with prototype graph. IEÉE Transactions on Neural Networks and Learning Systems (TNNLS), 35(1):300-310, 2022a.
Shao- Yuan Li, Yuan Jiang, and Zhi-Hua Zhou. Partial multi-vies view clustering, In Proceedings of the AAAI Conference Artificial Intelligence(AAAI), pp. 1-7, 2014.
11

---

Published as a conference paper at ICLR 2026
Xiang-Long Li, Man-Sheng Chen, Chang-Done Bongg-Dog Wang, and Jian-Huang Lai. Refining graph struc- ture for incomplete multi-view clustering. IEEE Transactions on Neural Networks and Learning Systems(TNNLLS) 35(2):2300-233 2022b.
Xingfeng Li, Yinghui Sun, Quansen Sun, Jia Dai, and Zhenwen Ren. Distribution consistency based fast anchor imputation for incomplete multi-view clustering. In Proceedings of the 31st ACM International Conference on Multimedia(AAM MM), pp. 368-376, 2023b.
Xingfeng Li, Yuangang Pan, Yuan Sun, Quansen Sun, Yinghui Sun, Ivor Tsang, and Zhenwen Ren. Incomplete multi-view clustering with paired and balanced dynamic anchor learning. IEEE Transactions on Multimediaa TMM), 2024.
Zhouchen Lin, Risheng Liu, and Zhixun Su. Linearized alternating direction method with adaptive penalty for low-rank representation. Advances in Neural Information Processing Systems (NIPS), pp. 1-9, 2011.
Guangcan Liu, Zhouchen Lin, and Yong Yu. Robust subspace segmentation by low-rank represen- tation. In Proceedings of the 27th International Conference on Machine Learning (ICML), pp. 663-670, 2010.
Jiyuan Liu, Xinwang Liu, Chuankun Li, Xinhang Wan, Hao Tan, Yi Zhang, Weixuan Liang, Qian Qu, Yu Feng, Renxiang Guan, et al. Large-scale multi-vies view tensor clustering with implicit linear kernels. In Proceedings of the Computer Vision and Pattern Recognition Conference( pp-
20727-20736, 2025.
Suyuan Liu, Xinwang Liu, Siwei Wang, Xin Niu, and En Zhu. Fast incomplete multi-viww clus- tering with view-independent anchors. IEEE Transactions on Neural Networks and Learning Systems(TNNLSS) 35(6):7740-777) 2022.
Suyuan Liu, Junpu Zhang. Yi Wen, Xihong Yang, Siwei Wang, Yi Zhang, En Zhu, Chang Tang, Long Zhao, and Xinwang Liu. Sample-level cross-view similarity learning for incomplete multi- view clustering. In Proceedings of the AAAI Conference on Artificial Intelligence(AAAI)), pp- 14017-14025, 2024.
Xinwang Liu, Xinzhong Zhu, Miaomiao Li, Lei Wang, Chang Tang, Jianping Yin, Dinggang Shen, Huaimin Wang, and Wen Gao. Late fusion incomplete multi-view clustering. JEEE Transactions on Pattern Analysis and Machine Intelligence(TPAMM)) 41(10):241-22433 2018.
Zhen Long, Qiyuan Wang, Yazhou Ren, Yipeng Liu, and Ce Zhu. S2mvte: a simple yet efficient scalable multi- tensor clustering. In Proceedings of the Computer Vision and Pattern Recog- nition Conference( CVPR), pp. 26213-201 2-2222,, 2024.
Zhen Long, Qiyuan Wang, Yazhou Ren, Yipeng Liu, and Ce Zhu. Tirlf4mve: Tensor low-rank and low-frequency for scalable multi-view clustering, IEEE Transactions on Pattern Analysis and Machine (nteliggenc(TPAAMI), 2025.
Canyi Lu, Jiashi Feng, Yudong Chen, Wei Liu, Zhouchen Lin, and Shuicheng Yan. Tensor robust principal component analysis: Exact recovery of corrupted low-rank tensors via convex optimiza- tion. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition (CVPR), pp. 2249-5257, 2016.
Canyi Lu, Jiashi Feng, Yudong Chen, Wei Liu, Zhouchen Lin, and Shuicheng Yan. Tensor robust principal component analysis with a new tensor nuclear norm. IEEE Transactions on Pattern Analysis and Machine IntelligenceTPAAMI), 42(4):925 9388 2019.
Wei Lv, Chao Zhang, Huaxiong Li, Xiuyi Jia, and Chunlin Chen. Joint projection learning and tensor decomposition- based incomplete multiview clustering. IEEE Transactions on Neural Networks and Learning Systems(TNNL.S) 2023.
Andrew Ng. Michael Jordan, and Yair Weiss. On spectral clustering: Analysis and an algorithm.
Advances in Neural Information Processing Systems (NIPS), pp. 1-8, 2001.
12

---

Published as a conference paper at ICLR 2026
Feiping Nie, Rui Zhang, and Xuelong Li. A generalized power iteration method for solving quadratic problem on the stiefel manifold. Science China Information Sciences(SCIS), 60(1 1):11110,,, 2017.
Feiping Nie, Lai Tian, and Xuelong Li. Multiview clustering via adaptively weighted procrustes. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), pp. 2022-2030, 2018.
Chang Tang, Xinwang Liu, Xinzhong Zhu, En Zhu, Zhigang Luo, Lizhe Wang, and Wen Gao. Cgd: Multi-view clustering via cross- view graph diffusion. In Proceedings of the AAAI conference artificial intelligence(AAAI),, volume 34, pp. 5924-5931, 2020.
Pham Dinh Tao and LT Hoai An. Convex analysis approach to de programming: theory, algorithms and applications. Acta mathematica vietnamica, 22(1):289 -55, 1997.
Siwei Wang, Xinwang Liu, Li Liu, Wenxuan Tu, Xinzhong Zhu, Jiyuan Liu, Sihang Zhou, and En Zhu. Highly-efficient incomplete large-scale multi-view clustering with consensus bipartite graph. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 9776-9785, 2022.
Xinxin Wang, Yongshan Zhang, Jie Zhang, and Yicong Zhou. Incomplete multiview clustering using discriminative feature recovery and tensorized matrix factorization. IEEE Transactions on Circuits and Systems for Video Technology(TCSV7), 2025.
Jie Wen, Yong Xu, and Hong Liu. Incomplete multiview spectral clustering with adaptive graph learning, IEEE Transactions on Cybernetics (TCYB), 50(4):1418-142) 2018
Jie Wen, Zheng Zhang, Yong Xu, Bob Zhang, Lunke Fei, and Hong Liu. Unified embedding align- ment with missing views inferring for incomplete multi-view clustering. In Proceedings of the AAAI Conference on Artificial Intelligence(AAAT), volume 33, pp. 5393-5400, 2019.
Jie Wen, Zheng Zhang, Zhao Zhang, Lei Zhu, Lunke Fei, Bob Zhang, and Yong Xu. Unified tensor framework for incomplete multi-view clustering and missing-views inferring. In Proceedings of the AAAI Conference on Artificial Intelligence(AAAT), volume 35, pp. 10273-10281, 2021.
Jie Wen, Zheng Zhang, Lunke Fei, Bob Zhang, Yong Xu, Zhao Zhang, and Jinxing Li. A survey on incomplete multiview clustering. IEEE Transactions on Systems, Man, and Cybernetics: Systems (TSMC), 53(2): 100::1136-1144 1149, 2022.
Tingting Wu, Songhe Feng, and Jiazheng Yuan. Low-rank kernel tensor learning for incomplete multi-view clustering. In Proceedings of the AAAI Conference on Artificial Intelligemce(AAAl), volume 38, pp. 15952-15960, 2024.
Wei Xia, Quanxue Gao, Qianqian Wang, Xinbo Gao, Chris Ding, and Dacheng Tao. Tensorized bipartite graph learning for multi-view clustering. IEEE Transactions Pattern Analysis and Machine Intelligence (TPAMI), 45(4)551875520,, 2022a.
Wei Xia, Quanxue Gao, Qianqian Wang, Xinbo Gao, Chris Ding, and Dacheng Tao. Tensorized bipartite graph learning for multi-view clustering. IEEE Transactions on Pattern Analysis and Machine Inteltggence(PPMMI, 45(4):5187-52D0) 2022b.
Yuan Xie, Dacheng Tao, Wensheng Zhang, Yan Liu, Lei Zhang, and Yanyun Qu. On unifying multi- view selfrrepresentatiion for clustering by tensor multi-rank minimization. International Journal 2018.
of Computer Vision(IJCV),
Deng Xu, Chao Zhang, Cong Guo, Chunlin Chen, and Huaxiong Li. Fast incomplete multi-view clustering with adaptive similarity completion and reconstruction. In Proceedings of the AAAI Conference on Artificial Intelligence(AAAll) volume 39, pp. 21734-21742, 2025.
Shengju Yu, Siwei Wang, Pei Zhang, Miao Wang, Ziming Wang, Zhe Liu, Liming Fang, En Zhu, and Xinwang Liu. Dvsai: Diverse vheweshared anchors based incomplete multi-view clustering. In Proceedings of the AAAI Conference on Artificial Intelligence(AAAl),, volume 38, pp. 16568- 16577, 2024.
13

---

Published as a conference paper at ICLR 2026
Changqing Zhang, Huazhu Fu, Si Liu, Guangean Liu, and Xiaochun Cao. Low-rank tensor strained multiview subspace clustering. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), December 2015.
Chao Zhang, Huaxiong Li, Wei Lv, Zizheng Huang, Yang Gao, and Chunlin Chen. Enhanced tensor low-rank and sparse representation recovery for incomplete multi-view clustering. In Proceedings of the AAAI Conference on Artificial Intelligence(AAAl), volume 37, pp. 11174-11182, 2023.
Shuping Zhao, Jie Wen, Lunke Fei, and Bob Zhang. Tensorized incomplete multi-vies clustering with intrinsic graph completion. In Proceedings of the AAAI Conference on Artificial Intelli- gence(AAAT), volume 37, pp. 11327-11335, 2023.
Pan Zhou, Canyi Lu, Jiashi Feng. Zhouchen Lin, and Shuicheng Yan. Tensor low-rank represen- tation for data recovery and clustering. IEEE Transactions Pattern Analysis and Machine Imellizence(TPAMI))4335 378 2019.
14

---

Published as a conference paper at ICLR 2026
A APPENDIX
The detailed optimization derivations, the proof of convergence, and the remaining experimental results are provided in this appendix.
A.1
LLM USAGE
We employ LLMs solely to refine the language and wording for improved fluency and precision, with no further application beyond this purpose.
A.2 DETAILED OPTIMIZATION PROCESS
The augmented Lagrangian function the optimization problem is given as:
(((A) I==1: [E")
Je=1:
(Z"
Je=
[H")
J==1:
Ju=1:
(Y")
JE=1 [B") W,
Tr(( 1 ( Z²)"" Z") + (W, H 9)
=11 A
(5)
<
+
H
Y",X" A'Z" - E") + X A A - E
We can decompose Eq. 5 into several subproblems and solve them sequentially.
A "-Suupooblemm ing optimization:
By fixing the other variables, the update of A can be obtained from the follow-
arg max Tr((A'))M*)) (Av)T A"=I
(6)
where M (Y" , (Z")" The optimal solution of A UA VA where UA- and VA v are the left and right singular matrix M
H" Subproblem Treating the other variables as constants, the subproblem with respect to H is formulated as:
HH) H 227 ( + Tr((W") (H" - GG") + H -
This subproblem can be equivalently reformulated as:
min (Ho) H==I
Tr((H") (P(HH""TTTAL))))EE) (A²L")H") + (r((H)) (W -G)")
To make the problem more tractable, further relax it into the following form:
max (H") H TH( ((H)HLLHH)) L'HH) + TT((H") TC)
where = that L" is positive definite, and it is fixed at 0.001 throughout all experiments. clear that this becomes quadratic optimization problem the Stiefel manifold Nie al. (207)), which can be efficiently solved by Algorithm Let D BI. In this case, have D
al C (G" -- W". The constant parameter a is introduced ensure
Consequently, L can be expressed L of A2 (a - A2) À (Z*) Z". Algorithm the adjacency matrix TZ E Ruxn is multiplied by H E R To improve efficiency, the computation is reordered as (Z") (Z"H"), which reduces the complexity to (tcn) and avoids the (n?)) cost of explicitly forming (Z)) Z
v
E SS-uproobolem By fixing the other variables, the subproblem with respect to E can be formu- lated as:
mén A E + (Y", A"Z" - E²) + - A'Z" E E
(10)
Minimizing this function with respect to E" yields the following optimal solution:
1
E"
Y + p(X" A'Z")]
(11)
2
15

---

Published as a conference paper at ICLR 2026
Algorithm 1 Optimization Algorithm of HD -L--uubooo
Input: X X"HH'
Output: H
1: ensure 'H'
( = I:L = al A2L a positive definite matrix.
2: while not converge do
3: Update F 2L" C 2(a H (Z'H"
4: Compute the SVD of matrix FUUVV =
5: Update H UV
6: end while
7:
Output H
Z" D'-Subproblem When focusing on Z" while treating the remaining variables as fixed, the opti- mization problem can be expressed as:
P 1 1 A E
min 2 20
- A E
(12)
The above formulation imposes nonnegativity constraint, making Z" essentially a nonnegative representation matrix to optimized. Accordingly, the update rule for Z" is:
z" = max(0, [AA) (" ((AA))(X -E))
(13)
G-Subproblem
Keeping other variables fixed, g can be derived by solving
g = arg min "10lle, + 1 = (7
W
Ç )
(14)
g
By applying the following theorem, the optimal solution of can be determined:
Theorem Suppose D E R
Xn2Xna is a tensor, and tensor singular value decomposition (t-SVD) be S V Then the minimization problem under the Es Es-norm can be formulated as follows:
1
min Cllg
DI
(15)
2
The closed-form optimal solution Eq. 150 can be expressed as:
g" = (0)
T
(16)
Here, iff (Sy), 113) represents tensor with diagonal frontal slices. For each slice, the diag- onal entry Q2 f ( ((, is determined by solving the following subproblem:
(4,c(S) = min #20
(= ()--))(.))) Cf(x)
(17)
2
where the parameter satisfies Ç O, and f(x) represents the rank function of the Es-norm, namely
f()
(1+6)p
6+2
Since Eq. 17 involves a mixture of convex and concave components, can be solved using the framework of tifference-of-comee programming Tao & An (1977), resulting in the closed-form
expression below:
Öf
iter++1
(t,
(18)
where (5, and iter refers to the iteration step.
16

---

Published as a conference paper at ICLR 2026
Lagrange multipliers and penalty parameters The penalty parameters and Ç together with the Lagrange multipliers W and Y" are updated according to:
Y =Y + μ( A"Z" E")
W = w + C(H 99)
(19)
# Mutti μr min(H,, Mmaz)
C= =mii(S Smaz)
Impute X
X (A "A" ),miss
(20)
This step performs imputation for the missing entries in the original data, where the subscript mis denotes the indices of the missing samples. A complete description of the optimization process for CAMEL summarized in Algorithm where the subscript ava refers the indices of the available data points.
Algorithm Optimization Algorithm of CAMEL
Input: Incomplete multi-view dataset X") pm anchor number t. Output: Clustering labels
JE= cluster count balancing parameters Å1. d2 and
Initialize: Vu, (Z") Jims = 1, (Z")
(2"))aav ),ve 0,H 0, E Y 0.9
0, W- 0, -= 11-10------0 10-1, 7/p 1k 1.2, /max Smax 1012 10-2,--2 10-5
2: while not converge do
3: Update {A 11 1m
by Eq. 6
4: Update E" J==1 by Eq.
5: Update 1m by Eq.
7: Update Y mm Tomm [m by Algorithm
6:
Update [H"
1==1 W. μr. by Eq. 19
8: Update 9 Te=1 by Eq. (16)
9: Update [X" [===== by Eq. 20
10: Check the convergence conditions: X A"Z" E"Iloc < and H g <e
11:
end while
12: The final clustering assignments are generated via k-means applied to the concatenated latent
embedding matrix [ H .... H
A.3 CONVERGENCE ANALYSIS
This subsection provides a detailed proof showing that Theorem in Section 4.2 establishes the theoretical guarantee CAMEL's convergence. first introduce a necessary lemma:
Lemma Lewis & Sendov (2055) The expression F(X) f(v(X)) defines a mapping F (o) (X),... ..... .,.,)) represents the collection of singular values
Rmxn + R, where u(X) of X E mux, with min n). The singular value decomposition of X written as X Udiag(v(X))) symmetric and difserentiable at the point v(X). With these conditions in place, the subdifferential of F(X) at X can be expressed as
Moreover, assume that the function IR" + R is both absolutely
OF(X) OX
Udiig((f((())))V
(21)
where (f(((X)) (8f(ar()) (X))
afia, (fia 0x
Proof that the sequence 10 is bounded: Consider the (p + 1)-th iteration. Solving for E: d.p+1 columnwise yields:
(xp(X A 22 + y
(22)
qptt1 2A1 + 14
17

---

Published as a conference paper at ICLR 2026
Moreover, the multiplier Y" update is given by:
Y p+1 Y + 4p(X" - A 2+1 Ep+1)
(23)
The q-th column y 1.p+1 of the multiplier Y is updated as:
y = Ng. + Mp(x" A 2 1
(24)
Substituting Eq 22 into the Eq 24 gives:
2
9. + 211 (xx
y p.p+1
A 2
(25)
2\1 + Mp
2 H4p
Taking the 4--nomm of both sides yields the recursive inequality:
2A1
2/1/4p
1 112 S
2A1 up 1y + 2 + Up
|
A A 11111l22
(26)
where fp= "p > are bounded constants since A is constant and Up is bounded. Finally, given that the initial multiplier yo the penalty parameters {/p], and the data term x A are bounded, the recursive inequality implies sup, Iyy < co, and thus the sequence (Y;) is bounded.
At iteration (p + 1), the first-order optimality condition regarding gp+1 is characterized by:
0 = p1 + (p(9 - Hpp+1) Wp
(27)
Meanwhile, the variable W is updated as:
W P + SpHHpt gp+1)
(28)
By combining these two relations, we arrive at:
a11gttlle. =Wp+1
(29)
The tensor G can be factorized via the SVD as g u * S VT. Applying Lemma
|
we obtain:
|alls p+ llea
u * im (S,), (1.3) *
113
(30)
ny min(n m2))
1 (0f(5)
Σ
Σ
(i,
As a consequence, the Frobenius measure of 11 lle is controlled by finite constant, which ensures its boundedness. Combining this fact with Eq. (29). it follows that the sequence (W
p 1,0
uniformly bounded. In addition, making use of the update scheme outlined in Algorithm 2 one
can establish the inequality below:
L(A p+1 p+1: p+1 p 9 9pt1, Y Wp.pps Cp)
V E Z
S LLA E E. pr pt 9p p. Y W p. Up. Sp)
H
p
L(AY pr E. pr
H 9 p p-1 W 1, 14p 11- Sp-1)
Y
So + Sp-1
Wp p
W w ||
(31)
2
m
Up + Hp-1
Y Y p-
2p D=l
Iterating the inequality over p = 1, 1,2,..., n yields:
L(A 1 1 1 H 9 p+1 Y Wp Pop Sp)
C(Ai, Ey, Zi H 91, Yo Wa, po, Co)
Sp + Sp-1
w Wp Wp-1l1
2
(32)
Hp 1p-1
Σ Y v
+
2
18

---

Published as a conference paper at ICLR 2026
It follows directly that:
Go Gp-1 < 00, 2C
Up + pp-1 00
(33)
pm1 2
Since the initial value of the objective function C(A). Ej, Zi, Hi, 91, Yū Wu. No; (o) finite, and the sequences {Yp)p= (Wp)p= together with the sums Σp=1 2
is
Sptcp-1
and Ep=1 Pet are uniformly bounded, it follows that the augmented Lagrangian 2n
C(A p+ E p+l Z p+1
H 1+11 9pt1, Yp, W Mps Sp) is bounded throughout the entire iterative pro-
cess.
We next make of the following identity:
C(A p+1
E
1+
H
p+1 9pt1, Y, Wp, Ppp Sp)
= -11p+tllle + A
E p+1
11?
+ A2
Tr p+1 P (4p+1) T TT H + (Wp. Hpt1 -pt1)
n=l
+
So 1Hpt1 - 1pt1l12 + Σ
Y x X",
- A Z A p+l p+l Ep+1 1) + Hp X
E p+
11
2
2
u=l
(34)
All quantities on the right-hand side of Eq. 34 are finite. Among them, the term || 1pp+1 Iles of
particular importance: its boundedness ensures that the corresponding singular values S (2, 2) do not diverge. This observation leads directly to the relation below:
min(ni, m2)
| IFF
_ n13
p. 11
11
(1,0)
(35)
n1 k=1
This guarantees that the sequence {g 120 remains bounded. In addition, the update rules im- ppppp=1 and {E pp=1 bounded sequences well. {
mediately show that {Ap]p= 1= {Z Consequently, the combined sequence {Jp A Ep D pr Y Gp. W p is confined to bounded set.
On the convergence of accumulation points towards stationary KKT solutions: According to the Weierstrass-Bolzan theorem Bartle & Sherbert (2000), any bounded sequence admits a con- vergent subsequence. Hence, {Jp)p= has at least one accumulation point, denoted by J.. Conse- quently, obtain:
lim
Z H Y 9p. WP = E. H. S...W)
(36)
From Eq. 199 the following relations can be derived:
Y
Y
W p+1 Wp
X
A Zp+1
Ep+1
p+1
Z p+1 9p41
(37)
Pp
So
100 Since the sequences (Y)), p=d and (Wp following conditions:
p=1 remain bounded, the accumulation point satisfies the
X A. E. = 0, H. 1.. =
(38)
Moreover, the satisfaction of the first-order optimality conditions by E and gp+1 leads to:
Y = O E W. 0 G.le,
(39)
Thus, we deduce that the accumulation point J. complies with both stationarity and primal feasi- bility. Accordingly, any subsequential limit of (J,),= constitutes stationary point obeying the KKT conditions of the proposed optimization model.
19

---

Published as a conference paper at ICLR 2026
A.4 SUPPLEMENTARY EXPERIMENTS
This subsection supplements the experimental results that are not included in the main text. Ta- bles 5 and present the remaining ablation studies, which further validate the synergistic effect of the connectivity constraint and the tensor constraint in CAMEL. Figures 7 g 9 and [T] illustrate the influence of different parameters on other datasets, namely hyperparameters, the number of an- chors t. the degree matrix parameter B, and the rank function parameter 8, [igure[iilllsteern convergence behavior on the remaining datasets, again demonstrating the favorable convergence of CAMEL.
the
400
109
169
121
72"
10
(a) Still-DB
(b) Scene
(c) MSRCVI
05
05
A00
12"
137
100
12
160
12
107
(d) Mfeat
(c) EYaleB10
(f) COIL
Figure 7: Analysis of Hyperparameter Sensitivity in the CAMEL Model on the rest datasets.
(a) Still-DB
(b) 1cene15
(c) Scene
620
82-
(d) Mfeet
(c) COIL20MV
(I) CIFARIO
Figure Effect of anchor quantity on the CAMEL model on the rest datasets.
20

---

Published as a conference paper at ICLR 2026
03
200
(a) Still-DB
Scene
(c) Mfeat
(d) EYaleB10
(e) COIL20MV
(I) CIFARIO
Figure 9: Effect of parameter f on the CAMEL model on the rest datasets.
60
2oo
360
(a) Still-DB
(b) Scene
(c) Scene
4.
3.-
5471
(d) Mfeat
(c) COIL
(f) CIFARIO
Figure 10: Effect of parameter on the CAMEL model on the rest datasets.
Table 5: Evaluation of the CAMEL model ablation on Still-DB, COIL20MV and Mfeat datasets.
Compunenis
Datasets
Still-DB
COIL20MV
Mfcat
CAMFLCC
CAMEI-1
ACC.
NMI
PUR
ACC
NMI
PUR
ACC
NMI
PUR
27.15
8.20
29.55
43.32
50.35
45.83
52.96
46.90
54.36
28.52
32.03
44.49
54.80
46.85
40.75
46.05
74.78
66.61
74.78
70.32
77.31
72.41
96.35
96.79
97.29
Table 6: Evaluation of the CAMEL model ablation on Scene, Scene 15 and CIFAR10 datasets.
Compunents CAMEL-C
Datasets
Scene NMI
Scene15 NMT
CIFAR10
CAMEL-T
ACC
PUR
ACC
ACC
NMT
PUR
31.67
19.34
32.92
20.31
13.64
22.02
82.54
75.57
83.47
42.05
31.35
43.97
21.82
21.41
25.65
85.65
92.74
89.54
95.42
90.31
95.42
$7.04
47.33
57.94
93.96
97.20
95.89
21

---

Published as a conference paper at ICLR 2026
10.0
200
10
18
(a) Still-DB
(b) Scene
(c) Seene
200
10
300
10
122
(d) MSRCVI
(e) Mfeat
(f) COIL20MV
Figure 11: Training convergence trends of the CAMEL model on the rest datasets.
A.5 THE IMPACT OF DIFFERENT NORMS ON CAMEL
We adopt the fo-mo Kang et al. (2015) to address the theoretical limitations of the traditional tensor nuclear norm (TNN) Xie et al. (2018). As convex relaxation, TNN imposes uniform penalties on all singular values, which tends to overly suppress principal information components and proves inadequate for noise removal. In contrast, the non-convex f--norm employed in this work offers adaptive penalization: it imposes stronger penalties on smaller singular values and weaker penalties on larger ones, thereby filtering more noise during low-rank approximation while better preserving essential information. We also acknowledge that numerous studies have proposed various other types of non-convex surrogates (e.g.. Geman Geman & Yangl (1995), Laplace Chen al. (202I), Schatten 4p Xia et al. (2022b), etc.). Fundamentally, these non-conve, surrogates follow a similar underlying principle- assigning differentiated penalties to singular values achieve more refined characterization of the low-rank structure. Our choice of the ég-norm is motivated by its empirically verified performance: across multiple datasets, consistently delivers competitive results.
To further assess the effectiveness of the fo-norm, we replaced it in our model with TNN and several representative non-convex surrogates (including Geman, Laplace and Schatten fp), and carefully tuned the parameters for each alternative. The comparative results are summarized in the table (th missing rate is 0.5).
Dalasets
Yale3
MSRCVI
SHII-DB
EValeB10
COILZOMV
Mical
Scene
Scene 15
CIFARIO
TNN
X0.6+1
21.622893
21.80=1
15,8640.40
10.36=0
12.
44.40:0000
9.82=0.26
33,23+1 1,50
Geman
58,30+2.82 82
16.
72.55=0.96
53.59+1.37
67. 67.25522.25
99. 99.20+0.00
44.50=0.00
15.01=1.16
94.92+7.07
Lapace
33.21+3.87
29.42+1.17
1.28
46.28=3399
18.23+1.16
90.1342.19
Schatten& f
75.7646.64
93.9921.65
87.1124.99 95
57.0945 96
81.78:12.71
98.6140.13
70.4520.00
76.2324.11 10
33.44z1.19
61.8242.27
97.90+0.26
74.78+0.46
55.72+5.42
70.32+1.33
96.3546.62
95.42+0.00
57.04+3.30 30
93,96+8.10
Table 7: Experimental results on different norms.
The experimental results clearly show that, compared with the traditional TNN, employing any convex surrogate within the CAMEL framework leads to improved clustering performance. This demonstrates that non-convex penalties, by adaptively treating singular values, effectively preserve informative components while suppressing noise. Moreover, although the performance of differ- ent non-convex functions varies across datasets, the proposed f--norm consistently exhibits strong competitiveness: it achieves the best performance on MSRCV1 and Scene, and ranks second on datasets such as Yale3, Still-DB, EYaleB COIL20MV, 1cene15, and CIFARIO. These findings confirm that adopting the fo-mr for tensor low-rank modeling is both reasonable and effective. We would also like to clarify that the primary contribution our work does not lie designing
22

---

Published as a conference paper at ICLR 2026
particular non-convex surrogate. Rather, the main novelty is constant-degree driven connec- tivity constraint integrated with a low-rank constraint on the latent embedding tensor. The tensor rank surrogate itself is modular within framework, allowing different functions to be seamlessly incorporated without altering the overall architecture. The experimental evidence demonstrates the extensibility and flexibility of method: it accommodates various tensor low-rank surrogates while maintaining linear computational complexity, consistently delivering robust and competitive performance.
A.6 THE ASPECTS IN WHICH H PROVIDES ADVANTAGES OVER
First, in terms of representation quality, each row of H E R corresponds to sample's em- bedding in a e-dimensional latent space, where с denotes the number of clusters. Each dimension directly reflects the association strength of the sample to specifie cluster, providing clear seman- tie interpretabilityy. and can thus be regarded as an effective cluster indicator matrix. In contrast, E R represents the coordinates of samples in subspace spanned by anchor points. Since t is typically much larger than c, Z lacks direct characterization of cluster structure and exhibits seman- tic ambiguity. Second, in terms of clustering performance, since H inherently contains cluster-level structural information, applying k-means directly to it yields high-quality clustering results. This eliminates the need for post-processing steps(e.g.. SVD) required by methods based on extract implicit structures, thereby avoiding associated errors and computational overhead, and enhancing the method's efficiency and stability.
To further validate the advantage of H over Z in the CAMEL framework, we conducted additional clustering experiments using (referred to as CAMEL-Z) and compared them with the original CAMEL model. As shown in Table & CAMEL -Z consistently underperforms CAMEL across all datasets, further demonstrating the superiority of using H.
Table Comparison of clustering results of CAMEL-Z and CAMEL across varying levels of miss-
ing data.
Metheds
RMI
PUR
ACT
PUR
Yale3
CAMET-Z
CAMEL
SL-4-1777
74.904400
79.25+3.19
ST.6T1
75.643341
7261+500
7621+415
459FF11
6182-2225
W877585
65.17-253
1772 16
63.05-2544
16 MANSSTI
MSRCVI
CAMEL-Z
CAMEL
97.110033
71312419
95.61-0.58
97.81=0043
97. 07.1430005
94.5411.99
314122
97. 04.045
40.76:425
27.00-026
96.63-0.59
31.8:288
44.99338
07.90-226
Stil DB
CAMEL2
CAMEL
393621.88
45.34
10.0
61.8811.0
67.4510.00
53.4740.99
67 67.45.0.30
7A.7820 46
66.61:8440
74.7820446
10.751102
EYbiel110
CAMEL
CAMEL
519721778
22.521149
51.0910.88
13.8EE 1.36
52.44155
21000 10
21 11.41110
48 18.5914773
48 44.944.49
15.725544
1.75
42
51.1125455
15.7825335
19:28-143
CO0 20MV
CAMEL-Z
CAMEL
68.38:2.27
79.17:0.56
T1 11.7322311
71.22-1.44
05.11 05.185551 19
34
69 2917777777
71.0710.05
70811155
70.3221.33
00S1 97
1.33
17.31-0.70
72.41:0055
421991133
Miaа
CAMBI AZ
CAMEL
00.0300.00
71.30-1-00 178
09:00130.00
65 08
71.47-1 58
0000
96.971616
55 118811000
97.99222.81
97.7244.45
55 15
96.35
1255-115 51
96.79-6.62
19.97-994
97.2914.32
10 1533--84
Scose
CAMELE Z
CAMEL
00.29-0.00
19.3310:1
30,76:10:22
03.50:1..00
00029-0000
31
96 00.9010000
91.64:0000
29.0030111
41
93.991000
313553037
25.4250000
50.31-0013
19:30-1002
45-42-000
Scone15
CAMELIZ
CAMEL
22:23-2002
26, 20115
69 10:10:99
75:21
31.44100
61 6111999997
54.0081.00
62:11:18
52.8
21 30
47:35-1:15
10:18:11355
579961677
2138
06:20-00:00
25.46-13.73
10-5,4
65-19-3.9
2.
99:05-00:00
98.09-0000
09.05+00.00
00.87:00.00
05.05-0.00
99.86-8
23


Reviews:
[{"review_id": "zGgZGW1v4u", "reviewer": "Reviewer_bSK1", "summary": "The manuscript develops a novel tensor-based approach for incomplete multi-view clustering tasks. The main ideas of the work are:\n* Employ view-specific connectivity constraints and tensor learning to capture high-order cross-view correlations;\n* Perform clustering on the learned embeddings to eliminate additional post-processing;\n* Introduce a approximation of Laplacian matrix for efficiency.", "questions": "Please refer to the weaknesses.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The work improves clustering performance and meanwhile enhances algorithm efficiency;\n* Detailed theoretical analyses are provided, and experimental results are sufficient;\n* The paper is well-written.", "weaknesses": "* Why the authors adopt the norm of $\\ell_\\delta$-norm (i.e., $f(x) = \\frac{(1+\\delta)x}{\\delta + x}$) ? What its specific advangates?\n* Some details require further explanation. For example, what is the clear definition of the \"post-processing-free\" methods? It seems that some post-processing-free methods still requires an extra clustering procedure on the learned features.\n* The ablation study requires a deeper analyses, and current discussion cannot well illustrate the contribution of each component.\n* Some typos should be corrected.", "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": "Da7Z5HZF8D", "reviewer": "Reviewer_MEf5", "summary": "This paper proposes a novel incomplete multi-view clustering method that seamlessly integrates the low-rank tensor learning, latent learning, anchor learning as one unified framework. However, this paper is with limited contributions, since the proposed method is the combination of existing methods such as \"Tensorized multi-view subspace representation learning\", \"Generalized latent multi-view subspace clustering\", and so on.", "questions": "In Equation ( 3), the authors have added the \"Tr[(Hv )⊤(I − 1β (Zv )⊤Zv )Hv\" into the objective of the proposed model. The author states that the proposed model could reduce the running times. However, I think the introduction of  \"Tr[(Hv )⊤(I − 1β (Zv )⊤Zv )Hv\"  could yield a heavy computational burden, rather than reduce the running times.", "limitations": "", "rating": 2, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "The paper is well-written.\nThe authors also illustrate the motivations, details, and optimization of the proposed method.", "weaknesses": "1、In Figure 1, what is the function of “Constant degree matrix”? How to communicate with other modules of the proposed method?\n2、Many incomplete and complete multi-view clustering methods assume that there exists some sparse noise such as deadline. The authors select the ||.||_F as the regularizer for noise removal. How to deal with the datasets with sparse noise?\n3、How to extend the l_\\sigma norm from the matrix format into the tensor format? How to define the l_\\sigma tensor norm? There are many nonconvex surrogates such as l_p norm, Schatten l_p norm, and so on, for nonconvex low-rank tensor learning. The authors failed to compare the proposed method with other representative nonconvex surrogates for high-order modeling. \n 4、Small-scale testing datasets. The testing datasets are relatively small-scale! I want to see the performance of the proposed method on the large-scale dataset.", "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": "bLA9ZlNayz", "reviewer": "Reviewer_q5uJ", "summary": "This paper proposes a tensor-based method for incomplete multi-view clustering named CAMEL. CAMEL jointly learns view-specific latent embeddings under structured constraints and organizes them into a low-rank tensor, enabling coordinated optimization of graph connectivity and high-order correlations while avoiding costly post-processing (e.g., SVD). Extensive experiments on nine benchmark datasets demonstrate its superior effectiveness and efficiency compared to existing methods.", "questions": "See Weakness section.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.The proposed method learns view-specific embeddings under structured constraints and organizes them into a low-rank tensor, effectively overcoming the limitation of traditional methods that cannot simultaneously capture graph connectivity and high-order correlations.\n\n2.The manuscript presents extensive experiments that demonstrate the effectiveness and efficiency of the proposed method.\n\n3.The manuscript provides detailed theoretical derivations and proofs, which theoretically validate the effectiveness of the proposed model.", "weaknesses": "1. The hyperparameters $\\lambda_1$ and $\\lambda_2$ in Equations (1)--(3) are not defined, leaving their values and roles unclear.\n\n2. The manuscript lacks experimental comparisons between the $\\ell_\\delta$-norm and the conventional tensor nuclear norm in terms of performance.\n\n3. The authors state that Equation (2) involves quadratic computational complexity; however, since Equation (2) is based on an anchor method, it should already have low complexity. It is unclear where the quadratic complexity arises. \n\n4. Lines 259--262 contain a repeated statement, which affects the manuscript’s clarity.", "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": "JxZZUd4alQ", "reviewer": "Reviewer_Q7LQ", "summary": "To effectively model the rich information in multi-view data while improving computational efficiency, this paper develops a constant-degree matrix-based framework for incomplete multi-view clustering. View-specific latent embeddings are learned under structured constraints and organized into a tensor with an $\\ell_\\delta$ low-rank constraint, enabling joint optimization of graph connectivity and high-order correlations. CAMEL further reduces the $O(n^2)$ complexity of traditional connectivity constraints by approximating the variable Laplacian degree matrix with a constant-degree matrix. Theoretical analysis and extensive experiments validate the method’s effectiveness.", "questions": "Please refer to the weaknesses mentioned above.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-written and presents the work in a clear and rigorous manner.\n2. By approximating the variable Laplacian degree matrix with a constant-degree matrix, the proposed method avoids the $O(n^2)$ or higher computational complexity associated with traditional connectivity constraints, thereby improving overall efficiency. The authors also validate the approach on large-scale datasets, demonstrating its practical applicability.\n3. The CAMEL consistently achieves superior clustering performance compared to baseline methods across multi-view datasets of different types.", "weaknesses": "1. One of the core contributions of this work is the use of the latent representation H instead of the subspace representation Z for clustering. It would be helpful if the authors could clarify in which aspects H provides advantages over Z, such as representation quality, or clustering performance.\n2. The paper replaces the traditional Laplacian matrix $L_v = I - (D_v)^{-1/2} S_v (D_v)^{-1/2}$ with a constant form $L_v = I - \\beta (Z_v)^\\top Z_v$, but the underlying mechanism and justification for this approximation are insufficiently discussed.\n3. The parameter sensitivity analysis uses different ranges for $\\lambda_1$ and $\\lambda_2$, but the rationale for selecting these ranges is not clearly justified.\n4. The construction of incomplete multi-view data is insufficiently described. It is unclear whether the missing elements are consistent across different datasets and compared methods, and whether the randomly removed elements are the same across experimental runs.\n5. The paper refers to the dataset as \"Still2\", while other references consistently use \"Still-DB\". Could the authors clarify whether these names refer to the same dataset or different ones?", "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)