Scalable Cross-View Sample Alignment for Multi-View Clustering with View Structure Similarity
Authors: Jun Wang
Zhenglai Li
Chang Tang
Suyuan Liu
Hao Yu
Chuan Tang
Miaomiao Li
Xinwang Liu
Venue: NeurIPS 2025 spotlight ()
Keywords: Multi-view Clustering; K-means Clustering; Similarity Graph Learning; Representation Learning
Primary area: general_machine_learning

Abstract: Most existing multi-view clustering methods aim to generate a consensus partition across all views, based on the assumption that all views share the same sample arrangement. However, in real-world scenarios, the collected data across different views is often unsynchronized, making it difficult to ensure consistent sample correspondence between views. To address this issue, we propose a scalable sample-alignment-based multi-view clustering method, referred to as SSA-MVC. Specifically, we first employ a cluster-label matching (CLM) algorithm to select the view whose clustering labels best match those of the others as the benchmark view. Then, for each of the remaining views, we construct representations of non-aligned samples by computing their similarities with aligned samples. Based on these representations, we build a similarity graph between the non-aligned samples of each view and those in the benchmark view, which serves as the alignment criterion. This alignment criterion is then integrated into a late-fusion framework to enable clustering without requiring aligned samples. Notably, the learned sample alignment matrix can be used to enhance existing multi-view clustering methods in scenarios where sample correspondence is unavailable. The effectiveness of the proposed SSA-MVC algorithm is validated through extensive experiments conducted on eight real-world multi-view datasets.

Scalable Cross- Viess-View Sample Alignment for Multi-Viens Clustering with View Structure Similarity
Jun Wang'
Zhenglai Li²
Chang Tang*
Suyuan Liu'
Hao Yu!
Chuan Tang'
Xinwang Liul Natiioall University of Defense Technology, Changsha, China
Miaomiao Li
Shennzhnn Institutes of Advanced Technology, Shenzhen, China
3 Huazhong University of Science and Technology, Wuhan, China *Changsha College, Changsha, China
Abstract
Most existing multi-view clustering methods aim generate consensus partition across all views, based on the assumption that all views share the same sample arrangement. However, in real-world scenarios, the collected data across different views is often unsynchronized, making it difficult to ensure consistent sample correspondence between views. To address this issue, we propose scalable multi-view clustering method, referred to as SSA-MVC. Specifically, we first employ a cluster-label matching (CLM) algorithm select the view whose clustering labels best match those of the others as the benchmark view. Then, for each of the remaining views, we construct representations of non- aligned samples by computing their similarities with aligned samples. Based on these representations, we build similarity graph between the non-aligned samples of each view and those in the benchmark view, which serves as the alignment criterion. This alignment criterion then integrated into a late-fusion framework to enable clustering without requiring aligned samples. Notably, the learned sample alignment matrix can be used to enhance existing multi-view clustering methods in scenarios where sample correspondence is unavailable. The effectiveness of the proposed SSA-MVC algorithm is validated through extensive experiments conducted on eight real-world multi-view datasets.
pample-aligmment-beee
Introduction
Clustering aims to assign each sample to its corresponding class by leveraging the intrinsic similarities within the original data (I). With the rapid advancement of science and technology, data have become increasingly diverse in their forms of representation. The same object can often be described from multiple perspectives. For instance, video content can be represented through audio, visual, and textual modalities. Such heterogeneous but complementary data representations are collectively referred to as multi-view data 2 4 [.. To fully exploit the rich semantic information embedded in multi-view data, variety of advanced multi-view clustering algorithms have been developed in recent years 66] 9 These methods have demonstrated promising performance across wide range of real-world applications by effectively integrating complementary information from multiple views.
Despite the effectiveness of these approaches in integrating multi-view information, they typically rely on the assumption of strict one-to-one correspondence among samples across different views, which is an idealized condition in practical applications [10 In real-world scenarios, variations in
CCorreponniing Author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Featurre Ixtraction
Information Fusien
R
diag(L.M
Mutrix Mulliply
Cross Somple Similority Learning
Gross View
Similority curning
Gross Sample Similority Learning
Chaspe-- abell Monching
Vient Vie
XX
Cross Sumple Similarity
Cross View Similarity Leaming
Clustering Results
Featuree Fxttraction
R
dkag (
Figure 1: The flowchart of the proposed method. First, the baseline view is selected based the CLM criterion. Next, feature representations of the unaligned samples, denoted as [W")== are constructed. Subsequently. the cross-view similarity graphs {S") and the other views are established. Finally, these cross-vite view similarity graphs serve as alignment constraints within a late fusion multi-view clustering framework obtain a unified partition matrix F*. The final clustering results are then derived by applying k-means clustering on F"
J==1 between the baseline view
sample organization or ordering across views commonly lead to inconsistent or misaligned sample correspondences. To this end, some studies attempt to achieve sample alignment jointly with the learning of data representations. Given the effectiveness of the Hungarian algorithm in assignment problems [3]. Huang al.
[4 integrated into their clustering framework facilitate sample
alignment. However, due to semantic discrepancies between views and high intra-class similarity within views, establishing strict one-to-one alignment based solely on sample similarity limits tolerance to noise and misalignment. To address this, Yang et al. [I3] proposed alignment at the class level, reformulating it as class identification problem and introducing noise-robust contrastive loss to improve robustness. Furthermore, Ren et al. [16] leveraged sample commonality and view diversity to adaptively construct alignment matrices and designed an unsupervised data completion mechanism to handle incomplete unaligned data.
Although the aforementioned algorithms have shown promising performance in multi-view clustering with sample alignment, they still face several limitations. (1) Due to semantic discrepancies across views and the absence of supervision, establishing strict one-to-one correspondences is often difficult. In wallworldd scenarios, the sample relationships of different views are typically many-to-mann and enforcing strict matching may introduce noise and lead to sub-optimal aligmment [77] (2) While some recent methods employ joint learning frameworks that integrate alignment with feature representation enhance performance, they often fail to model explicit alignment relationships, thus limiting their scalability to other multi-view clustering methods that are not applicable in sample non-alignment
scenarios [8] [9]. (3) Clustering performance is heavily influenced by the choice benchmark view, yet selecting appropriate remains open challenge in current approaches (20).
Therefore, we propose a scalable multi-view clustering algorithm that integrates sample alignment into a unified clustering framework. To mitigate the impact of structurally noisy or disordered views, we first employ the CLM algorithm 21 to select the view that exhibits the highest structural consistency with the underlying semantic labels, designating it as the baseline. Considering that samples within same subspace can be linearly reconstructed by their peers 222 we reformulate alignment task as similarity-based reconstruction problem rather than relying rigid one-to-one index matching. Specifically, each view is structurally characterized by computing the similarity between unaligned samples and the rest of the view. Then, an alignment relationship is established by comparing these

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structural representations to that of the baseline view. Finally, the resulting alignment matrices are incorporated into a late fusion clustering framework, enabling effective alignment without the need for direct correspondence. Furthermore, the learned alignment relationship can be reused as auxiliary information enhance the performance of existing multi-view clustering methods under misaligned conditions.
Overall, the main contributions of this paper are listed as follows:
We propose to select the baseline view by measuring the similarity between sample cluster distributions and their corresponding labels within each view, effectively minimizing the impact of irrelevant noisy structural information on the alignment process.
We propose structural representation for each view based on the correlation between non- aligned and aligned samples. This representation guides cross- alignment by integrating sample-level features with intrinsic structural information.
An alternating optimization algorithm is proposed to efficiently solve the model. Its effec- tiveness is validated through extensive experiments on eight multi-view datasets.
Preliminaries
2.1 Adaptive Neighbor Graph Learning
Graph-based multi-view clustering algorithms have attracted considerable attention in recent years due to their strong capability in capturing the intrinsic structural information embedded in the original data [23 24 25 26). Based on the fact that samples within the same cluster or samples with smaller pairwise distances tend exhibit higher similarity than those from different clusters, Nie et al. [27 proposed a clustering algorithm that constructs a nearest- graph to capture local structural relationships. The objective function of this method is formulated as follows:
min
(Ixx, x, + Bs]j) s.t. s
I. 1-11-00 8t 11,
(1)
where and x, denotes the i-th and j-th samples of the original data matrix X E Raad, where is the total number of samples and is the feature dimension. The variable Sty indicates the similarity between samples x; and Xj. The parameter 3 is regularization coefficient that balances the trade-off between the similarity graph learning and the sparsity of the graph, and can be adaptively tuned during the optimization process. For a more detailed description of Eq. (T). please refer to 27
2.2 Late Fusion based Multi-view Clustering
Given multi-view datasets X E Ruxd, where de denotes the feature dimensionality of the u-th view, late fusion- based multi-view clustering methods aim to extract partition-leve clustering information from each view. A unified partition matrix is then obtained by integrating the partition information from all views. Specifically, assuming that the base partition matrices F" Bnxds are obtained from X via eigen-decomposition or other representation learning techniques, where d,, denotes the latent feature dimension, the typical mathematical formulation can be expressed as follows (88:
max 4 (F", F + (F*),
F-
where (b-)) denotes the partition fusion module, which integrates the base partitions into unified one, and (r(-) represents a regularization term designed to preserve desirable properties such as smoothness 29 sparsity (30], consistency across views [3] 32
Proposed Method
3.1 Cross Sample Similarity Learning
Late fusion-based multi-view clustering algorithms have attracted substantial attention due to their demonstrated effectiveness, and variety of advanced methods have been proposed within this
3

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However, a common assumption in these algorithms is the existence of strict one-to-one correspondence between samples across all views during the fusion of partition information. In practice, this assumption is often idealized. Due to temporal misalignment during data acquisition or storage constraints, mismatches between samples across different views frequently occur. Under such circumstances, directly fusion partition-level information without addressing the cross-view sample alignment may introduce irrelevant or inconsistent structural information, thereby degrading the quality of the unified partition and affecting the final clustering performance.
framework B3 34 35].
To overcome the limitations of traditional late fusion strategies, several algorithms have introduced a sample alignment matrix jointly with view-specific representation learning, integrating the two processes into a unified framework to enable mutual reinforcement [36] 37 [8]. In the context of unsupervised learning, sample alignment is typically inferred by exploiting the intrinsic feature similarities among data samples. However, due to the presence of substantial cross-view heterogeneity, the assumption of semantic consistency across views is often difficult to capture using rigid one-to- one matching strategies. Such hard alignment approaches fail to model potential one-to-many or many-to-one semantic relationships across views, thereby overlooking alternative and potentially meaningful alignment relationships. Moreover, although some methods attempt to embed the sample alignment process into neural network-based representation learning, they often do not explicitly model the alignment relationships themselves, which limits the scalability of these approaches.
In light of the above challenges. we propose novel strategy that utilizes the correlation between un- aligned and aligned samples within each view view-specifie structural representation. Specifically, given an unaligned multi-view dataset X" = [X;; Xž), where X I xd, and X E ma denote the aligned and unaligned samples the v-th view, respectively, and where n1 and n2 are the corresponding numbers of aligned and unaligned samples, we construct the structural representations of the unaligned samples for each view based on Eq. (T). i.c.
V ΣΣΣ
min (W") Tomi
X 16.
X 2ba
" + B s.l. W. 1 1,0 &
w/ S
(3)
where denotes the total number of views. The matrix W R2xMn represents the constructed feature representation for the e-th view in the presence of sample non-alignment. It is worth noting that the above construction of the feature representation for each view is not unique. Here we adopt the formulation given in Eq. (T) for simplicity. Nevertheless, alternative learning mechanisms could also be employed within our framework.
3.2 Cross View Similarity Learning
After obtaining the feature representations for all views, key challenge lies in constructing reliable sample alignment across views. A straightforward approach is to randomly select one view the baseline and align the remaining views it. However, due inevitable noise introduced during data collection, some views may contain structural information that does not reflect the true underlying cluster distribution. To mitigate the impact of such irrelevant or misleading information on the alignment process, we adopt the CLM algorithm to identify the most reliable baseline view. Specifically, the view that exhibits the highest consistency between its sample distribution structure and the semantic labels is selected as the baseline. The detailed selection process is defined as:
exp
1
EXEX d(x,y))
Σx 2i=1 (21, v)
H(Y, ((Y,
E Exer, d2(x, vi)
0 n 1)
exp
CLM(X) = ((x)
(5)
1+ exp (-8-H(G,X,d)
GCY G
where Y = Y2, } denotes the ground-truth -tuth cluster assignment of the dataset X, and k is the total number of clusters. Let y/ Y, denote the mean of the samples in the i-th cluster, and e c= X denote the mean of all samples. The function d.). represents the squared Fuclidean distance, and 0g = (sr e)(x E X) denotes the standard deviation of the distances between the original data samples and the global centroid. The parameter ó is pre-defined scaling factor. Based on

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the above formulation, we compute a matching score that quantifies the consisteney between the structural distribution of samples and the corresponding semantic clusters in each view. The view with the highest matching score is then selected as the baseline. Accordingly, the cross-view structural similarity graph S is constructed as follows:
min S
w
willao + B(st,)² s.t. L = arg max CLM(X 1,0 si S
(6)
where S" denotes the similarity graph that captures the structural correspondence between the unaligned samples in the v-th view and those the baseline view, denoted by t. As a special case, when t = i, we define the similarity graph the identity matrix, i.e., S I.
3.3 Sample-Aligned Late Fusion Strategy
In general, a higher similarity between samples implies greater likelihood of a semantic match. Based on this intuition, we propose capture the matching criterion between unaligned samples across views by leveraging cross-view sample similarity. As discussed earlier, the ideal scenario assumes a strict one-to-one correspondence between samples across views. However, in practice, such hard 0-1 alignments are difficult to establish the absence of external supervision, due to the presence of noise and structurally irrelevant information in certain views.
To address this challenge, we reformulate the alignment problem by reconstructing unaligned samples using samples within their corresponding subspace, rather than explicitly matching index positions across views. In this way, the alignment achieved in a soft and structure-pres preseeriigg manner. By integrating this alignment strategy with the late iusion-based multi-view clustering framework, formulate the final objective function as follows:
R "M.,FMMM.R.
max MMM.0..
Tr F
On
I
M
FRR")
+ Σ
S")
offt
s.t. arg max CLM(("), F" F' F'
L,R" Rd = Σ
M
where M denotes the sample realignment matrix that maps the unaligned samples in the v-th view to those in the baseline view, while R" represents the feature rotation matrix used to align the feature space. The scalar ag indicates the weight assigned the v-th view, and is hyperparameter that controls the trade-off between feature information and structural information.
4 Optimization
4.1
Optimization Algorithms
In this section, develop iterative optimization algorithm to solve the objective function in Eq. (7) with respect to the variables R", F*, M", and ag.. The detailed optimization procedure is described as follows:
Update F*: When optimizing F* while keeping all other variables fixed, the objective function in Eq. 7 can be equivalently reformulated as:
F
max Σ FF
FİR" + F. (MTTR")) s.l. A.A.FTTR I,F"
(8)
F5
Since Fi and F independent of each other, they can be optimized separately obtain the complete solution for F". Specifically, when optimizing the variable F,, the objective function in Eq. (8) can be equivalently rewritten as:
max Σ TFFT F(R") s.t. Fi Fi ==1
5

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The optimal solution to Eq. P can be obtained by performing singular value decomposition (SVD) on the matrix FRR.. Since the optimization of Fž follows a procedure analogous to that of Fj. we omit the details here for brevity. Once the optimal solutions for both Fj and Fý are obtained, the final solution for F constructed by concatenating the two parts.
Update R": When all other variables are fixed, the optimization of R in Eq. 7 can be equivalently reformulated as:
I 0
M (1F)) F"
max Og Tr R TT F
s.t. R R" = I.
(10)
I O
Let Q" a., FVT
M F", the optimal solution for the variable R can then be obtained
like that of Fi STeeciffically by performing singular value decomposition on Q"
Update 0g: By fixing other variables, the Eq. (7) can be formulated
max
Σ To
s.t.
Σ a 11
(11)
v=1
where % C"F'F""'F'""" According to the Cauchy inequality, the optimal solution to the above optimization problem can be derived in closed form as:
7
Og
(12)
Update M": By fixing F", R", and aus M can be optimized by solving the following subproblem:
ar Tr T F FyT) + ATr s*) s.t. M M =I
(13)
Let T = a. F, F + AS". Following an optimization procedure similar to that for the variable M the optimal solution can be obtained by performing SVD on the matrix T.
In summary, the detailed procedure of the proposed method is described in the Appendix A.2
4.2 Convergence Property
In the above optimization process, each subproblem is independent, and its corresponding optimal solution can be obtained. Consequently, the proposed algorithm converges within few iterations according to Theorem [
A detailed convergence proof is provided in the Appendix A.3
Theorem 1. The proposed optimizution algorithm is guaranteed to converge to a local optimum of the SSA-MVC method.
4.3 Computational Complexity Analysis
In the proposed method, the primary computational complexity arises from three components: cross-sample similarity learning, cross-view similarity learning, and sample-aligned late fusion. Specifically, the computational cost for obtaining the feature representations (W") is O(V O(Vn, ddmar), where K denotes the number of neighbors and dmaz max (d,. d2.-.. .d d represents the maximum feature dimension across views. For the CLM algorithm. the complexity is ((dmaz), while the construction of the cross-view similarity graph requires O(V K) opera- tions. Finally, the computational cost of the late fusion step is (n²)), mainly due to the generation of the partition matrix F", Consequently, the overall computational complexity is Onn?).
R
5 Experiments
5.1 Datasets
To further validate the effectiveness of the proposed method, we conduct experiments on eight real-world multi-view datasets, including Yale, 3sources, MSRCV, 100leaves, HW, Scene, EMNIST, and Hdigit. The detailed summary of them is provided in Appendix A.4

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Table I: ACC comparison of all methods with and without Hungarian alignment on eight multi-view
datasets under a sample alignment ratio p = 50%.
Method
3sources
MSRCV
100leaves
EMNIST
Hdigit
EEOMVC
58.1820.00
59.76110.00
72.86:10.00
65.44.1..00
93.85_00.00
26.91=0.00
46.11::0.00
65.9310.00
BROMVC Hungarian
52.73-0.00
48.524-0.00
71.90++0.00
67.19:0.00
67.30+00.00
26.00-0.00
44.24+0.00
62.28+00.00
DealMVC
33.94.0.000
31.0110.00
28.1011..00
7.6910.00
47.94.10.00
2..04_00.00
45.62.1..00
65.66.10.00
DeaMMVC Hungarian
24.85-0.00
29.11+00.00
28.29+00.0
9.34+0.00
39.97-0..00
21.86-0.00
37,83++0.00
82.72--0.98
MVCAN
32.4812.50
31.1242.31
18.14.11.76
49.51.11.24
45.06.11.02
2..5
MVCAN + Hungarian
33.09=1.56
49,37444.60
51.20++2.80
40.71++1.53
47 47.08-33.56
29.75-0.54
49.56++9.49
37.37+3.76
EBMGC
39.39.0.00
38.4610.000
42.86:10.00
33.94:10.00
56.75:10.00
21.58=0.00
33.501:0.00
50.91
EBMGC + Hungarian
32.73=0.000
40.24+0.00
47.14+0.00
33.94=00.00
51 70:00.0000
26.33-0.00
41.00+00.00
59,46+0.00
Vsc_mH mH
$3.94:0.00
62.13::0000
64.29:20.00
38.56:20.000
42.50:00.00
28.03=0.00
46.472:0.00
65.1920.00
OpVuC DCMVC
53.94=0.00
57,40+0.00
30.00=00.00
53.13:=0.00
30.10=00.00
31.82=0.00
$1,09+0.00
62.90=0.00
27.15=11011
46.51144.52
45.71::2.29
48.83:10.83
19.34
26.02=0.60
59.55113600
65.74
DCMVC + Hungarian
23,88=1.19
35.15+1.10
44,57+22,46
39. 0.95
50.47=00.99
24.07=0.45
40.11+
35 35,80=0.99
LMTC
52.5813.76
48.2813.49
53.2913.15
35.58:10.94
64.9911166
0.99
41.911:..00
59.25110.30
LMTC + Hungurian
54.61:4.74
48.05+1.118
55.43443.29
35.30 A5
54.24--2.72
28.53-0.86
41.69+0.700
23,4
TMSL
24,8211.70
42.251.2.6
43.9810.97
47.47.11.0
62611066
29.13-0.49
0OM
OOM
TMSI Hungarian
68.79-2.78
56.21+-1.05
44.45+1445
47.124-1.11
53.12-0.005
27.02-0:24
00M
00M
DSTL
35.91
61.54.0.000
19.4811.811
36.87.11422
47
0..45
28.8710.40
40.9010.668
DSTI. Hungarian
37.73+2.76
59.76+0011
43.71+1443
30.57+103
43.33+0.51
19.31-0.99
30.36+0.33
50.09+0.00
Ours
64.24:3.62
64.44:::.29
83.52:1.399
70.63::1.29
96.55=00.00
35.91=0.277
77.38:3.14
71.78:1.26
Table 2: NMI comparison of all methods with and without Hungarian alignment on eight multi-view
datasets under a sample alignment ratio p = 50%.
Method
3sources
MSRCV
100kcaves
HW
Socnc
EMNIST
Hdigit
EEOMVC
62.23=00.00
39. 39.32=00.00
56.08:0.00
75.62=0.00
88.20=00.00
16.59=0.00
32.54:00.00
70.96:00.00
EEOMVC- Hungarian
57,3720.00
31.87:0.00
56.9310.00
77.03=0.00
62.65:1000
18.2610.00
29.18:10:00
53.111:.00
DeaIMVC
38.08=0.00
6.69:0.74
14.00:3.92
25 25.34:0.44
07.20=
11.89:1.09
31.39:1.59 59
3..90=1.52
DealMVC + Hungariam
23.652:0.00
7.3120048
13.08.10.17
2..34_3.63
26.08.12.49
16.311:3.23
22.80.10.48
65.46.1
MVCAN
38.43+1.87
12.87-1.67
46.19+ 1:94
69. 00.50
32.31
30.96-00899
20.14+0.13
6.689+1.68
MVCAN + Hungarian
38.53=1.22
47.72=23000 00
35.1622:.14
62.08=1.45
45,05=+5:33 03
25.7620.33
37.94=15.114
53.82+3.54
EBMGC
43.3120000
23.68:00.00
22.79100:00
22.0200000
15.99110000
11.14::.000
17.99.10:00
25.07110.00
EBMGC- Hungarian
38.18:0.00
23.98:00:00
24.79=0.00
22:0000
29.52:20.00
15.08::0.00
19:44:0.00
39,70=00.00
Vsc_ml
62.00-10.00
48.81-10.00
56.01 1:0.00
0.053
30.67.10.00
25.00
36.92.10.00
55.77.10.00
OpVaaC
55.77+0.00
36 36.86-0..0
13.63+00.00
78.00 0.00
17.74+0.00
29.69+00.00
45.94+00.00
47.51++0.00
DCMVC
26.22:22.44 31,40=11227
+:255
67.15=0.42
60.71:3..05
15.64=20.22
61.11=2.62
59.83=1..85
DCMVC + Hungarian
27.63+1.00
16.49+ .7
22.27 69
60.41-0.47
29. 29.39-+0.68
12.99+00.31
20.68++0.19 19
16.68+0.200
LMTC
57.39=2.89
40,01=44.69
33.29:+3.88 81
58.75=0.55
45.75=00555 55
23.24:20:55
24.01=0:255
47.31=00225
LMTC + Hungurian
57.791:3.67
43.85
37.60112.88
58.98
37.95
23.1610.558
24.59.0.87
47.02.12.49
TMSI.
28.68+1.225
12.02-+1.30 10
23.71 .08
69.33-0.61
48.62+0055
21.34+00336
OOM
QOM
TMSL Hungarian
68.40.1.91
31.68.0.922
25.28
6..4
18.16.10.07
19.77.0.27
0OM
OOM
DSTL.
39.59+1.18
37.00-0:00
23.41+1.9
60.51-0.55
29.05-+0550
15.54+0.33
11.64+0.22
20.40+0.115
DSTL Hungarian
41.17=1.62
40 40.46:20.38
14.3
54.13=0.51
26.08 23
14.05=00.50
15.32:20.29
42.34:20.00
Ouns
6..31
61.20.00.83
70.28
85.34_0042
92.0910.00
30.27700.20
74.84.10.4
75.50....44
5.2
Compared Methods
Ten state-of-the-art MVC methods are selected as baselines for comparison, including EEOMVC 39),
DV [4),, MVCAN [1], EBMGC [42), Vsc_mH [43], Vp [44) DCMVC 45). LMTC
46.. TMSL 47
DSTL 48
The detailed introductions of them are presented in Appendix A.5
5.3
Experiments Setup
In the experiments, four widely used evaluation metrics are employed assess the clustering performance of all compared methods: Accuracy (ACC), Normalized Mutual Information (NMI), Adjusted Rand Index (ARD,, and Flscore. For the proposed method, conducted grid search over the set (0.001, 0.01, 0.1, 1, 10, 100, 1000, 10000} to determine the best value for each dataset. The unified latent feature dimension d is set the number of clusters. Regarding the baseline methods, parameters were tuned according to the ranges provided in their respective publicly available source codes, and the best results were selected the experiments, To mitigate the influence of randomness on the experimental results, each experiment was repeated 20 times, and the mean and variance of the results are reported. All experiments are conducted on Windows 11 PC equipped with an Intel Core 17- 17-03 CPU and 64GB RAM.
5.4 Results Analysis
To facilitate a fair comparison between the proposed method and existing approaches under the sample non-alignment setting, we fix the sample alignment ratio p to 50% in the main experiments.
7

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Table 3: ARI comparison of all methods with and without Hungarian alignment on eight multi-view
datasets under a sample alignment ratio p = 50%.
Method
3sources
MSRCV
10lleaves
HWW
Scene
EMNIST
Hdiggt
BEOMVC
38.19100.00
29.7910.00
46.02.10.00
37.47-0.00
56.99_0..00
7.76:10.00
18.37.1..00
53.61200.00
BROMVC Hungarian
32.46-0.00
21.51+0..0
45.98+0.00
36.53-0.00
32.12+0.00
6.93+0.00
16.33+00.00
42.8++00000
DeaIMVC
11.75=0..0
2511=0.44
4.49:1..2
1,89=0.25
21.77=1.06
6.57=0.91
23,83+11.50 59
38.45+1.65
DeaMMVC+ + Hungarian
3.35
0.242:061
3.4910.21
2.6120049
18.7741.58
8.9842.30
15.57.20499
65.76.1181
MVCAN
10.54=1.71
4.92=1.55
35.98+2.09
30.37=1.53
22.24=0058
16.37:0060
16,49:0.330 38
50.04+2.04
MVCAN + Hungarian
11.10=11.07
29.57-14.45
24.14:13.45
20.57=1.66
30.14:15.39
12.52:0.100
29.21
42.3011..52
EBMGC
15.43=00.00
12.29=0.00
14.78+0.00
15. 15.36-0.00
31.06++0.00
5,75+00.00
12.37+00.0
22.04+00.00
EBMGC Hungarian
9.89+0.00
13.21=00.00
17.44=00.0
15.36=0.00
24.43-0.000
8.60=0.00
15.06:20.00
36.00+0.000
Vsc mH
37,27+0..00
36.47-0.00
45.47+00.0
23. 23.39-0.00
20.92++0.00
13.26+00.00
24.93+00.00
46.54+0.000
OpVoC DCMVC
30,40=0.00
33 33.68=00.00
5,49:+0.00
40.64-0.0 0.00
10.90=00.00 00
15.39:00.00
34.32=+0.00
38 0000000000
4.69111055
19.77.15.08
19.55
29.23=00.1
53.01:2.79
8.8110.24
48.53:13.72
49.4212.03
DCMVC + Hungarian
2.01+00844
4.26++1.13
14,93++ 1.54
0.750-0.73
24.12-40.33
6.88-+0224
13.51=00577
12.12:00.2
LMTC
52,32=3.88
28. 28.15:24.27
24.59+3.8 88
17 17.24=0.81
39.20=0097
11.48+0.43
15:42=0.64
40, 40,59+00.35
TMTC + Hungarian
32.71+4.911
30.32-22.02
27.45+3..0
17.58- 1.45
19.07-+.117
11,26++0.43
15.09-0.78
37.54+22.25
TMSL
2,64=1:005
8.57=1.75
15.62:0..83
31.41=1.155
37.67=00777
10.85=0.31
OOM
COM
TMSL Hangarian
47.94:13.13
32.77=11.34
16.48:0.77
31 31.38-100 1,02
23.93-10.06
9,9910.21
OOM
OOM
DSTL
1.24
24.19=00:00
1-2022
18.49-0.77
20.12:4.37
6.11+0.20
7.25-:0:22
15.17
DSTL Hungarian
13.70122.01
37,03-10.36
14.86110.97
0.71
18.22.1.336
5.4410.25
9.31
29.22.10.00
Ours
48.91-2.68
45.98-1.61
66.34 0.66
59.97 0.95
92.49-0.00
17.15--0.32
68.52-2.12
65.53+00443
Table 4: Flscore comparison of all methods with and without Hungarian alignment on eight multi- view datasets under a sample alignment ratio p = 50%.
Method
Yale
Aspurces
MSRCV
100leaves
HW
Scene
EMNIST
Hdigit
BROMVC
42.10+-0.00
44.77-00.0
53.93-0.00
18.17-0.00
88.29-0000
15.02+0.000
28.53+00.00
58.97+00.00
BBOMVC - Hungarian
36,75=0..00
36 36.99=0.00
53,83=0.00
37.28=0.00
41.02=00.00
14.41=0.00
26.36220.00
49.21=00000
DeaIMVC
25.781.0.0
31.29-1.10
26.27:1..57
6.6810.15
12.01:0.87
14.8810.9
3..84
4..7311.50
DeaIMVC++ + Hungarian
23.89=0.00
30.27=2.35
25.19:1.200
7.66:21.06
30.95=1194 1.4
18.8611.72
27.34:20044
69.41:1.61
MVCAN
25.09111.61
29.14
47.00..1.67
1..49-1.29
34.95.0.45
22.78....4
25.18.10.26
1.93
MVCAN à Hungarian
25.35+1.43
37.97+2..44
28.31-1885
39.76+3.43
19.98+0.15
37.12-11.11
49.72+3.79
EBMGC
20.59=0.00
29.02=00.0
26 26.60=0.00
16. 16.16=0.00
37.92:20.00
12.13=0.00
21.13=20:00
29.84:20.00
EHMGC Hungarian
15.38-0.00
29.77=0000
28. 28.90=0..0
16, 1600.00
31.95+0..0 00
14.80+00000
23.55+00.00
42.40+00.00
Vsc_mH
41.70=0.00
52.20=00.00
53.91=0.00
24.41=0.00
29.42:20.00
20.55::0.00
32.83:10.00
$1.99200.00
OpVaC
34.92:10.00
46.62-10.00
20.44110.00
41.36.00.00 0.00
20.65.10.00
21.5911..00
41.09110.00
44.991.0.00
DCMVC
19.48-0.98
41.10+2.31
35.18+1.60 1.69
35.08-0.61
59.95-43.01
15.57-00.23
58.15-3.28
56.02+2.455
DCMVC + Hungarian
17.08=0.81
36.92=11.97
30.19::.32
25 0..58806655
33.17=0062
13.94:0.119
25.51=0033
21.22+0.25
LMTC
36.59+3.61
42.40+33.28 28
35, 35.28++3.33
18.07-0811
45.29+0087
17.68+00.41
24.48++0.42
46.62+0031
LMTC Hungarian
36.95:44.57
13.81=1.57
37.70=2.554
18.41=1433
29.32:20.89
17.4320.40
24.34:0:42
43.942:2044
TMSL
9.04.10.97
32.77-12.01
27.43:10.74
32.09 1.14
44.06.10.68
17.4610.34
OOM
OOM
TMSI. Hungarian
51,19+2,91
47.76-+1.00 1.06
28.15+00.77
32.07
1.01
31.54+0.06
16.37+0,20
OOM
0OM
DSTL 17.50.11.09
47.67-10.00
25.27
19.35 0.75
12.6510.22
16.81.1.116
23.70110.07
DSTI. Hungarian
19.25+1.87
51.00+0.25 28
26.87+0.44
12.79-0.71
26.72+0.34 34
12.06+0.224
18.65+00.22
37.49+0.00
Ours
52,17=2,43
1..59:1134
7..03:0.57
60.37=0.94
93,24:20.00
22.94::0.29
69.12::1.85
69.16:2037
Due to space constraints, results under other alignment ratios are provided in the Appendix and can be found in Tables 8 for reference. Notably, some baselines are not directly applicable to the non-aligned scenario. For fair evaluation, we apply the Hungarian algorithm to align the data before using these methods. Clustering results under the four evaluation metrics are shown in Tables | 4 with the best and the second results highlighted in bold and underlined respectively. Methods that encounter memory overflow are marked as OOM. Based on the results reported in the Tables, several key observations can be obtained:
(1) The proposed algorithm consistently outperforms most baseline methods, including those using Hungarian- brasaddi--baedddeee sample alignment. For example, the EMNIST and MSRCV datasets, it achieves ACC improvements of 10.66% and 17.83% over the second-best methods, EEOMVC and DCMVC, respectively. Similar gains are observed across other datasets, highlighting the method's effectiveness in capturing true cross-view sample correspondences and enhancing clustering performance.
(2) Our method is superior to existing methods such as Vsc_mi and OpVuC, which are designed for non non-aligned sample clustering. These two kind of methods rely on mining alignment relationships directly from raw features without explicitly modeling the structural hierarchy within each view. Moreover, they employ a hard matching strategy, determining class correspondences based on pairwise sample similarity. Due to high intra-class similarity, this often results in unstable alignment matrices, adversely affecting algorithm convergence and leading to performance fluctuations.
(3) Compared with deep clustering methods such as DealMVC, MVCAN, and DCMVC, the pro- posed method demonstrates notable advantages. Although deep neural networks possess strong representation capabilities, they often rely on the assumption of consistent semantic information
8

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x x
18 22
234
Objectts Valte
Objerthe Vallee
Objectthe Toltee
2213
2214
Namber of Inerailo
Number of Inerations
Namber Ierraiions
Namber of Inrraiom
(a) Yale
(b) 3sources
(c) MSRCV
(d) 100leaves
Obpeetive
Okpestive 2
Obgoetive
11 16 26 36 41
11
14
23 26 36 61
16 11
23 28 33
36 41
11 14 31 26
51
HW
Scene
EMNIST
(h) Hdigit
Figure 2: The objective function values of the proposed method across iterations.
-0-3001
-0-301
Chineter
6GH M1
20
1800 1888
690 M1
20 1801 1000
65% M1
1900
MI
M1
80
1900 1900
(a) Yale
(b) 3sources
MSRCV
(d) 100leaves
XM0 ---ARI -
-M-VMI -e-AR -e-iuue
Panmertr Yaluo
Parimattr Yaluo
Parsmtrtrr Yaluo
Parmmttrr Valoo
(e) HW
(i)
Scene
(g) EMNIST
(h) Hdigit
Figure 3: Clustering performance of the proposed method with varying values of the parameter X.
across views. This assumption breaks down the presence of sample misalignment, resulting in inconsistent feature learning and diminished clustering performance. Moreover, the use of the Hungarian algorithm for late fusion alignment does not consistently lead to performance gains and can even degrade results. This may be due incorrect alignments introducing noisy or misleading information, ultimately impairing the effectiveness of the model.
5.5 Convergence and Parameter Sensitivity Analysis
In the previous section, we theoretically established that the proposed algorithm converges within finite number of iterations. In this section, further verify the convergence behavior empirically. The corresponding experimental results are illustrated in Fig. 2 As shown the figure, the proposed method typically converges within approximately 10 iterations across all datasets, which empirically confirms its favorable convergence properties.
The results of our proposed method across varying À values are presented in Fig. Overall, the method demonstrates strong robustness to X, with stable performance on most datasets. Notably, fluctuations on datasets like 3sources suggest higher sensitivity, may be can attributed to significant semantic divergence among views, which underscores the importance of appropriately weighting structural similarity.

---

5.6 Ablation Studies
We conduct ablation studies assess the contribution of the proposed cross-view structural similarity module to clustering performance. Specifically, we denote the model without this module as SSA- MVC w/o CVS. The results, shown in Fig. 4 indicate that incorporating the module consistently improves sample alignment and clustering performance across most datasets. These findings highlight the effectiveness of the module and its integral role in the overall framework.
BMAAAAYT BISA AVE
SMLLAVC BILA NVE
EWLLAWV BALL-MY MVT WW
SMALANTE BRSA AYC-
Verturtutte
Chaserrag Perfurttnnee
Chonering Pesfurtanee
Chioeiig Vefurrtaare
Chieerigg
ACC NHI ARI Flseure
ACC
N30 ARI Flooore
ACC
NO0 ART Plooce
ACC
NHI ARI Flseare
Erelaation Metric
Evaluntion Metric
Evalantion Metric
Eraluntion Metric
(a) Yale
(b) 3sources
(c) MSRCV
(d) 100lcaves
Pefummare
Chasactieg Performaace
Chastering Perfemmtare
XMI
SMI
SMI
NMI
Evaluation Menic
Exaluation Metric
Evaluation Metric
Evaluation Menic
(c) HW
Scene
EMNIST
(h) Hdigit
Figure Clustering performance of the proposed method and its variant on eight multi-view datasets.
5.7 Effectiveness of the Alignment Strategy
To evaluate the scalability of our proposed method, we assess its effectiveness on the clustering algorithms that do not inherently handle sample misalignment. Specifically, under an alignment ratio of p 50%, we use M to realign the originally non-aligned 100leaves multi-view data and compare the performance of several bascline algorithms on both the original and the realigned 100leaves. As shown in Table 5 our method can benefit the clustering performance of these algorithms in the non-aligned setting, demonstrating its effectiveness and potential for generalization to other methods.
Table Results of competitors on the 100leaves under a sample alignment ratio of p = 50%.
Sctting
Metrie
DealMVC
MVCAN
EBMGC
DCMVC
LMTC
TMSL
DSTL
Unaligned
7.69:10.00
49.51::1.24
33.94:0.00
48.83:0.83
35.58-0.91
47.47:1.40
36.87=1.42
ACC
Aligned-Ours
12.42+00.52
48.81+1.28
43.06-0.00
53.75+0.889
40.82 1.31
48.18+1.37
35.60-0.90
Unaligned
25.34:0.44
69.50=0.95
58.22=0.00
67.15=0.44
58.75=0.55
69.33=0.61
60.53=0.55
NMI
Alugned-Ours 38.01
69.95-0.59
64.52-0.00
72.10
64.58 0.80
70.54=0.57
61.09 0.53
ARI
Aligned-Ours
Unaligned
5.16:10.12
189910.25
3..37+1.53
3.958:0.99
24.62=0.00
15.36=0.00
3.850
29.23+0.91
17.24=081
1.24
1..4
32.67:1.17
18.49=0.77
19.38=0.66
Unaligned
6.68:10.15
17.4921.29
16.16=0.00
35.08:0.61
18.07=0.81
32.09=1.14
19.35=0.75
Fiscore
Aligned-Ours
11.47:10.14
37.74
2..33:0.00
41.38::0.85
25.21 1.23
33.34:1.15
20.23=0.67
6 Conclusion
This paper proposes a scalable multi-view clustering algorithm to tackle sample non-alignment. By selecting baseline view via the CLM algorithm and leveraging structural similarities between aligned and non-aligned samples, the method guides cross-view alignment and integrates the resulting alignment matrix into late fusion clustering framework. Experiments on eight benchmark datasets validate the effectiveness of the proposed method.
10

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Acknowledgments
This work was supported by the National Natural Science Foundation of China (NO. 62325604, 62276271, 62441618).
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[49] Yijie Lin, Yuanbiao Gou, Zitao Liu, Boyun Li, Jiancheng Lv, and Xi Peng. Completer: Incomplete multi-vies clustering via contrastive prediction. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pages 11174-11188 1133, 2021.
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A Appendix
A.1 Limitations
As revealed by the construction process of the view-specifie structural representations, the proposed method exhibits certain limitations when applied to scenarios where samples across views are completely unaligned. In such cases, a feasible approach is explore the cross-view structural correlations by computing similarities between all samples in the to-be-aligned view and those in designated baseline view. This cross-view relational information can then be integrated into the overall clustering framework after the main convergence process. Accordingly, future research will focus on designing multi-view clustering algorithms that are specifically tailored handle fully unaligned sample scenarios.
A.2
The Pseudo Code of the Proposed Method
The detailed algorithm optimization processes presented in the following.
Algorithm 1 The Algorithm of SSA-MVC.
1: Input: Unaligned multi-view data {X")
Jem1, the number of clusters k, the unified feature dimension d, and the hyper-parameter A.
2: Construct the cross-view similarity graph (S*) Je=1 via Eqs. 3
3: Initialize [R*) (M' J {av)e=1.
4: while not converge do
5: Update F via Eq. (8)
6: Update [R"V Via Eq.
10
7: Update {ap, / via Eq.
8: Update {M") Tom1 via Eq.
13
9: end while
Conduct k-means clustering algorithm the consensus partition F".
11: Output: Clustering results Y.
A.3 Proof of Theorem
Proof. The proof can be divided into two parts, i.e., the objective function is upper bounded, and monotonically increasing.
The objective function upper bounded.
Given that M = I, the overall objective function in Eq. 7 can be simplified as follows:
Σ T +
a. CFFRR")
T Sr)
R MM,,,, aa,
(14)
s.t. C
FF -IR R
> a L =I
0 M
For any two distinct views v and v', where v + v', the following inequality holds:
Tr (a FFFF
$ Th (CCFF"")
(15)
$ 2 Tr (CCFR") + TI " (C''''"")
15

---

Based on the above inequality, we can further derive that:
Tr
F
ag CFF"R
< F*) Ir
1
Fr(F"FF")
a'.
F
Rd
(16)
2
$ (d dn²)
where is the unified feature dimension, and is the number of samples. Moreover, for the regularization term, we have:
A ATMMTT"") (Tr(M (M) Tr(S s*)) = (nz + (T(SSTT"))
(17)
where ni a number unaligned samples. Since Tr(S= is constant and À is fixed, the term XIr(M 5t) upper bounded.
Therefore, the overall objective function in Eq. is guaranteed to be upper bounded.
2) The objective function is monotonically increasing.
In the aforementioned optimization procedure, it is apparent that the sub-problems involving the vari- ables F*, R", and M respectively, reduce to classical Orthogonal Procrustes Problems. Throughout the iterative solution process, the objective function values associated with these variables exhibit monotonically non-decreasing trend. Additionally, the optimization with respect to the variable constitutes standard linear objective maximization problem under quadratic constraints, which simi- larly guarantces monotonically non-decreasing progression of the objective value during updates. Let ((**, M R", a,) represent a simplified form of the objective function defined in Eq. (7). Thus, the following inequalities hold:
O(FF)) (P), (M)) (R))()) (()) z (((F')(ot)) (MM))(+)) (R"))tt {av (ott))
(18)
where the superscript (p) and (p + 1) denote the number of iterations.
Therefore, based on the aforementioned properties, we can conclude that the proposed algorithm is guaranteed converge during the optimization process.
A.4 Datasets Description
In our experiments, eight multi-view benchmark datasets are used to verify the effectiveness of proposed method, including Yale 3sources5) MSRCV 100leave HW Scene [49], EMNIST and Hdigi In the following, will give detailed introduction to them.
Yale: This dataset comprises 165 samples distributed across distinct classes. Each sample
is characterized by three heterogeneous feature sets: 4096-dimensional Intensity descriptor, 404-diimnniionl Local Binary Pattern (LBP) descriptor, and a 6750-dimensional Gabor descriptor.
3sources: It comprises 169 samples collected from three distinct news media sources: BBC, Reuters, and The Guardian. Each sample is represented by three views, corresponding the textual content extracted from each respective source. The feature dimensions for these views are 3560. 3631, and 3068 for BBC, Reuters, and The Guardian, respectively. All samples are annotated with one six semantic classes
"https://visio. www.riiiioo.uusddco wwt./cooteet/yllllfffaaaa facll--f-cc-dddaassss
"http://mlg. www...w.dd..eeeee /e//ataasets//soources.ss. html
"https////ldte dta. htt.dddtt...aaaaaiiieeeeeee /
ics uci.
hhutps:/archivv..cc. www.eiiii...i...........
https://wwww.m http:://wwww.i.... wwt.:/iiii..oii///oottttoomoomoe soviil/ppddass--ddervvveeeeemeie
'httss//css.u httpss//ssnnuueedee. edu.edu/ roweis/data.htm
16

---

Table 6: Summary of eight benchmark multi-view datasets.
Datasets
CCaasses
#Samples
# Views
# #Feature Dimensionalities
Yale
15
165
3
4096; 3304; 6750
3sources
169
3
3560; 3631; I; 3068
MSRCV
7
210
5
24; 576; 5 512; 256; 254
100leaves
100
1600
3
64; 64; 64;
HW
10
2000
6
216; 76; 64; 6; 240; 47
Scene
15
4485
3
20: 59; 40
EMNIST
10
10000
4
576; 944; 512; 640
Hdigit
10
10000
2
784; 256
MSRCV: It comprises 210 image samples, each labeled with one of seven semantic classes. For each sample, five-view feature representation is provided to capture diverse visual characteristics. Specifically, the dataset includes the following feature descriptors: 24-dimensional Color Moments (CM), 576-dimensional Hístogram of Oriented Gradients (HOG), 5 512-dimensional GIST, 256- dimensional Local Binary Patterns (LBP), and 254-dimensional Gabor Energy-based Texture (GENT) features.
The dataset comprises 1600 samples distributed across 100 distinct leaf species. Each sample is characterized by three complementary feature views: 64-dimensional Texture Histogram (TH), a 64-dimensional Fourier Shape-based Metric (FSM), and a 64-dimensional Statistical Descrip- tor (SD). These multi-view features encapsulate diverse morphological and structural characteristics of the leaves, rendering the dataset highly suitable for the evaluation of multi-view learning and clustering algorithms.
100leaves:
HW: The dataset comprises 2000 samples, each annotated with of ten distinct class labels. encompasses six heterogeneous views, each representing diverse feature modalities extracted from the same set of samples. Specifically, these views include: 76-dimensional FOU features, 2 216-dimensional FAC features, 64-dimensional KAR features, 240-dimensional PIX features, 47- dimensional ZER features, and 6-dimensional MOR features.
Scene: The dataset comprises 4485 samples distributed across 15 distinct scene categories. Each sample is deseribed by three complementary visual modalities: 1 ci0-ddmensional GIST deserip- tor capturing the global spatial layout, 1100-dimensional PHOG feature encoding local shape information, and 2240-dimensional LBP representation characterizing texture patterns.
The dataset comprises 10000 samples distributed across 10 distinet classes. Each sample characterized by four heterogencous views, each providing complementary information derived from different feature sets. Specifically, the dimensionalities of the features corresponding to the four views are 576, 944, 512, and 640, respectively.
EMNIST:
Hdigit: The dataset comprises 5000 handwritten digit images, representing the ten classes from to 9. These samples are drawn from two distinct sources: the MNIST and USPS digit datasets. By integrating variations in handwriting styles and image resolutions inherent to both domains, the dataset offers comprehensive and challenging benchmark for evaluating digit recognition algorithms.
A.5 Compared Methods Introduction
In this section, the specific introduction of ten state-of-the-at multi-view clustering methods is illustrated in the following.
EEOMYC (TNNLS 23) [39] This method efficiently performs one-step multi-view clustering by constructing anchor-based similarity graphs to learn unified latent partition representations, enabling direct extraction of discrete clustering labels. By integrating latent information fusion and clustering into a joint framework, it significantly reduces computational complexity while improving clustering accuracy large-scale datasets.
CeACM MM 23) 40]: The proposed method addresses the limitation of existing multi-view clustering models by aligning similar yet distinct samples across views through dual contrastive
17

---

Table 7: Clustering performance of all compared methods on eight multi-view datasets under sample alignment ratio of p = 25%.
Dausee
Metri
BEOMYC
DealNvC
MVCAN
EBMGC
Vic_iH
OPANCC
DCMVC
LMTC'
TMSL
DSTL
Quss
ACC
47 17.88-0000
10.3010.00
26.182194
27.2720000
43.402000
24.242000
22.121104
47.5523.35
232121-13
30.27:1445
63.78:2.15
NMI
520822000
39910000
33.8522677
33,22-000
43022000
27.262000
274221.111
26.001000
35.3221.33
45.35-12.21
Flacor
156050.00
50,996150
13.27-000
1000
$4340,00
166180.0
31.44:1.4
7.149063
13.72:1.06
48.75-2.06
AEC:
M 45
20/941 76
53.18-000
5538 1L0
46 00300000
53.14399
45A06-188
16.00-1744
45-26-0279
3sornces
N641
ART
20.03
00416.55
47628.17
9.24:2.28
13,54:0000
35
0.00
1516-009
25.1922000
154342.18
5.10-135
33-76:5.62
5.84-9:73
1.041094
20.33:0..5
10.00
53.6510.11
41.20 18.69
Flacor
91
27.3161447
2300
12:0.0
34.0933.03
26.7764.29
30.57=1-17
53.246-9.59
ACC
0195 19,00
76.77 -000
55.24-0000
12. 12.290060
18 00
MISRCV
N6I
ARI
$1.27 23039
13
6.019.05
30 39.58.6.76
19,520000
5.7 0.0
40.09:0000
1X00-100 100
7.5918.00
12110.00
50.3621.89
15 12
33.35u2.00
2145-119 78
18
11.72207
40440.31
18 18.0
06.4510.944
68-49 HURU
Flicuss
30 11:30
23
23:55-11119
40,0215.335
16.780000
42
38 14:
51 16
32.4761.46
18:52:0.28
22:52:1223
06.87739.77
100feaves
XMI
ART
50.0010000
19 1000
0.1910.06
1366610112
19.6-00.00
58.251065
50,0000000
624-000
04.50:000
1159--000
6671000
57,0040.59
7.01 0.8
26.30
50.03:050
10.29-10.91
83.1129.27
Flaoss
109225002
3.8760.55
21.1550677
7:12:0000
12.000000
1.336000
21.530.54
30.00-00.90
55.74299.90
MII
AUC
27 34-10344
38
8:00:0.05
55440.03
0.750000
00000-0000
10.270000
4441000
7206:0-88
03 AT
2000:100
3.19-2--0 10
96-35--1180
21.59:00000
02
15.5640,02
11,99:0000
17
53:000
44,00
ACC
37
26 26.7:00377
14,25:0000
559966555
25 15:0 TV
Sccne
ART
16.45100000
7,2410.00
3.830.15
4540.00
14.99...15
276600000
144
20
13440/00
62610.00
247--116
5:64::026
10A8-0-10
22.7750330
18
1000:10:22
10046115
27.79-00.32
Flaour
20
8.12:0.00
20056:0000
40:0000
9.22-014
16.5750344
15.110054
10.52:0116
ACC
16 10
43 15
26
17.662000
10.06:1100
00M
19 15:0511115
75.434-4255
EMNIST
XMI
ARI
11 111000000
00
50 50.613319
25
5:79:0.10
1.91-0112
3.00000000
30.73:0000
78 31-0000
4104:000
10.1150.77
0084
DOM
4.3320.07
3.7844.05
70.66:1.87
61A77:3.56
Flicose
21.17:0000
32 A3
13.7430.09
11,70:000
34.59
25.0500000
55.2013.29
16.32:0.88
0004
12:00:0.07
65.6533.13
Hdiglt
XMI
ARI
192144122
35.5311346
43
0.00
201--100
48.48:0.00
14:1100))
6.2210.00
18940.00
55.374097
4273340.81
45.5632.18
34.99-1.4 A2
00M
00M
11.77:0.05
802-40.03
63.35--3300
74887-1.00
Flaoor
50.55-0.00 0000
21.9421.09
S1.0424.18
12,63:0.00
45.36::000
13.7
50.6921005
41.5621.29
0.044
17.26:0.03
07. 07.28:22.59
calibration losses at both global and local levels. This approach effectively integrates cross-view feature similarity and reliable class information, enhancing clustering performance and robustness.
MVCAN (CVPR 24) [41]: MVCAN is theoretically grounded deep multi-view clustering method designed to mitigate the impact of noisy views by allowing unshared parameters and inconsis- tent clustering predictions across views. It employs a two-level iterative optimization enhance representation learning, achieving multi-view consistency, complementarity, and robustness to noise.
EBMGC (TPAMI 24) [42]:
This method effectively leverages consistent neighbor information across multiple views through a novel Cross-view Good Neighbors Voting module, while a balanced regularization term based on the p-power function adapts clustering to diverse data distributions. By incorporating graph coarsening and an accelerated coordinate descent algorithm, this method achieves superior clustering performance with high efficiency.
Vse_ Vsc_mH (Neural Networks 24) 43:
This method effectively addresses the View-shuffled Problem by simultaneously establishing cross-view correspondences through a global alignment and modified Hungarian algorithm, and performing clustering via matrix factorization. This integrated approach enables robust clustering on shuffled multi-view data with varying alignment ratios, supported by both thcoretical convergence guarantees and strong empirical performance.
VuC (TMM 24) 44): This method simultancously addresses instance alignment and clustering within a unified framework, effectively handling fully unaligned multi-view data without relying on any pre-aligned samples. By leveraging a novel global-local aligmment strategy grounded in geometric invariance and relaxed k-means clustering approach, OpVuC robustly processes data at any alignment level, demonstrating superior performance across benchmark datasets.
DCMVC (TIP 24) 45 : This paper introduces a deep multi-view clustering network with a dual contrastive mechanism that simultaneously enhances inter-cluster separation and within-cluster compactness learn seatterinnn-iinndy representations. By integrating dynamic cluster diffusion and neighbor-guided positive alignment losses, it effectively fuses multi-view features into discriminative consensus representations, achieving superior clustering performance.
LMTC (CVPR 25) [46]: This method removes the tensor rotation trick to avoid inadvertent label information and introduces a large-scale multi-view tensor clustering approach that incorporates pair-wise similarities via an implicit linear kernel. This results in an efficient, linear-come complexty algorithm that effectively improves clustering performance without relying on sequential data order.
TMSL (KBS 25) 47 This method enhances traditional based multi-view clustering by leveraging tensor low-rank representation to capture the intrinsic data structure, resulting in more
18

---

Table 8: Clustering performance of all compared methods on eight multi-view datasets under sample alignment ratio of p = 75%.
Dauset
Metri
BEOMVC
DeIMVC
MVCAN
BBMOC
VacnH
Optac
DCNVC
LMTCC
TMSL
DSTL
Ours
ACC
58.7220000
35.7650:00
47.27-0-77
42.58-00.0
56 56.35-10.0
47.2740000
41.0823-72
60003-415
33,9732322
0.30022.22
03-9122.89
01:40:0000
41.6750000
51.80:1008
51.58-0.00
450771446
07:38:1.77
10.251000
42.49:0.00
306230.00
31.90:0062
2000
36
254454000
32.11-105
4339
51,02-2.12
AEC:
65:09110000
90 34
41.99:155
42.00-10 1:00
633300000
53 41.540000
6111-191
50.3 64
52.002 17
05:3610.26
N611
00
5.32
28.13.3.299
28.48-0000
55
53.48-19.00
38.55100000
44.27:4.344
46.072271
30882110 0L
39.6240.00
61,18:9.10
061-0 0.90
546510.0
10.0 10
10.36
Flacur
05.2660000
40006
11.00
4640000
55.51=3.4-
46.54::2.00
4319340.00
58.5119.25
ACC
16.57 10,00
06.52-1.2
66 100
10 49:4000
21624000
4560-008
15.29
16-275
MISRCV
ARI
56.66
18.00
11.18 34,3621300
33-7610.66
1.79
40.27:00.0
41.00 0.00
0.00
59.3 50.3922000
48:57-0000
00
9,9720.00
08 0.00
42.62:109
26 1.000-00
58.63-803
$2.90 8.48
42.5 29
30.4421.15
10 122211399
70085 0.54
75.06:99.00
Flicoss
39.2030.57
33.13=1.400
00:2100000
56.35:10:00
1900640,00
40036-1.00
59.15:2.99
41.0242.17
30.581105
74.25:9946
11:00-144
100leaves
XMI
ARI
46 00
32.6612.35
81.15:0611
$2.67-133 1.56
60.00
12 15:15-1 010
75.2710000
10 10.201000
S1ST4000
$1.38:034
1588
101
73.1430.1
Flaous
47 57.500.00
9 9,9
18.45
12.70:0000
32.
01
03.7020000
34,7750.00
59.17-8066
104
#413:1152
57.112112
07.3621.40
AUC
61.9946.35 18
34:40.34
76 114
141001100
34554000
09375-333
57 578812281
MMI
51.7360.10
55.46:0.52
01:10:000 0000
24.5:10.00
149010.00
3:26:0055
120
5429202
0124:0445
53,0950,34
57.57=0.51
62:53-0000
23.01:00:00
5:38
$6.902200
ACC
27 42
30:35:20100
30
35 1982008
32 20 140
33 33002001
25 25
ST.SE-OOTT
Sccre
ARI
27.7110000
17:9740000
22.36.0.19
7816667
34547:855
19:40 9:46
25:00:00000
15 11:
2894110000
14:75-1000
259410.00
14:340/0
29:52:1106
27.54:076 076
245420077
1010-0000
21 2212022
Flacur
30.15::0:00
22.56630.4
26.77:9.52
21.10:00.0
21.27::.00
2031,000
24.44:0.55
20.69
30.50:0007
15.9610.20
23.92:8.49
ACC
50 50.45:10:00
16
10015:11000
47 47.38:0000
47.7440000
5613622.7
5281-200
356346.79
W6-M633.20
EMNIST
XM3
ARI
50.6510.00
11:55-10100
42.5120.45
34.4400.22
4:2
51.38:00.00
47:33-1UR11
38.722:0000
76. 26:3644000
42.6410.00
90 00
59 19:26-1455
3..67:0.32
78 1K:0--10
DOM
DOM
21 21.3300.00
1345-40.0
78-66-1-3866
77.94:1.29
Flicoss
41.32:0000
42.5 42.58.0.29
45.48:00.3
$3.05=0000
3407:1000
37 37.5510.000
$3.66:1.97
36.11:0044
DOM
22.43:0.65
73.48-12.69
07.76:40:00
Hdigl
ARI
50.194000
63.6413.11
02.5615.12
67.74:5.71
0255--63
30
93.35-0.00
94.78-11 110
52 03.8600000
43 434700000
39.4710.00
294940000
6676-4411
74.01=2.32
91.71-057
58.01-0311
54.7130.33
26-6660.99
R167--001
70-76-0081
Flaons
62.9920.00
67.8334.48
67.27
96.30-00.0
40.25:0000
36.3130.00
73.23-3.77
56.64:0050
DOM
36.0130.53
76.50-00.11
reliable and robust multi-subsppac representation. Integrated into unified framework solved via the augmented Lagrangian algorithm, TMSL. can also serve as a versatile post- post-processing strategy to improve the performance of various existing TMVC methods.
DSTL (TMM 25) [S]: This method efficiently captures high-order correlations among multi-view latent semantic representations while disentangling comantic-related and unrelated components to reduce feature redundancy. By aligning semantic-related features across views through a consensus indicator, DSTL achieves scalable and robust multi-view clustering without relying affinity graphs.
A.6 Experimental Results with Varying Sample Alignment Ratios
To further assess the effectiveness of the proposed algorithm under different sample alignment rates, experiments were also conducted at 25% and 75% alignment rates, with detailed results shown in Tables 7 The findings confirm that our proposed method consistently maintains superior performance compared to other approaches. These results collectively validate the robustness and effectiveness of our algorithm in handling sample misalignment scenarios.
19

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Justification:
The detailed proof of Theorem can refer to Appendix A.3
Guidelines:
The answer NA means that the paper does not include theoretical results.
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Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Answer: [Yes]
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Reviews:
[{"review_id": "qwdTVeb9mu", "reviewer": "Reviewer_PvMp", "summary": "This paper proposes a scalable sample-alignment-based multi-view clustering method. The proposed approach first employs a cluster-label matching algorithm to select the view, and then constructs representations of non-aligned samples by computing their similarities with aligned samples. Extensive experiments demonstrate the effectiveness of the proposed method.", "questions": "(1) The title of this paper is 'Scalable Cross-View Sample Alignment for Multi-View Clustering with View Structure Similarity'. However, it lacks the most important experiments with fully misaligned views. This raises a question: Is the proposed method unable to address the issue of fully misaligned views?\n\n(2) This paper considers the issue of view misalignment. Therefore, performance validation should be compared more with the SOTA methods of view misalignment instead of traditional multi-view clustering methods.\n\n(3) The proposed method uses a feature rotation matrix to align the feature space. Is the rotation matrix effective for the sample, i.e., can the learned rotation matrix be used to output aligned samples?\n\n(4) The presentation of the paper can be further improved. For example, the roles of each component in Eqs. (4), (5) and (7) should be specifically explained.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n(1) This paper proposes a novel alignment strategy to address the issue with partially aligned views. Extensive experiments demonstrate the effectiveness of the proposed method.\n\nWeaknesses:\n(1) The title of this paper is 'Scalable Cross-View Sample Alignment for Multi-View Clustering with View Structure Similarity'. However, it lacks the most important experiments with fully misaligned views. This raises a question: Is the proposed method unable to address the issue of fully misaligned views?\n\n(2) This paper considers the issue of view misalignment. Therefore, performance validation should be compared more with the SOTA methods of view misalignment instead of traditional multi-view clustering methods.\n\n(3) The proposed method uses a feature rotation matrix to align the feature space. Is the rotation matrix effective for the sample, i.e., can the learned rotation matrix be used to output aligned samples?\n\n(4) The writing of this paper can be further improved. For example, the roles of each component in Eqs. (4), (5) and (7) should be specifically explained.", "was_revised": true, "final_justification": "After reading the other reviewers' comments and the authors' response, most of my concers have been addressed. So I raise the rating from 3 to 4.", "rebuttal": "**Q1: Is the proposed method unable to address the issue of fully misaligned views?**  \n**A1**: Thanks for your valuable comment. Although our method is primarily designed for partially aligned multi-view scenarios, it is also theoretically and practically applicable to fully unaligned views. Theoretically, our framework is built on the assumption that all views share a common underlying clustering structure. Even in the absence of explicit sample correspondence, samples belonging to the same cluster across different views exhibit implicit structural consistency. To capture this, we employ an orthogonality-constrained alignment matrix $\\mathbf{M}$ that aligns samples based on their structural behavior in the latent space, rather than their correspondence. In practice, under fully unaligned scenarios, i.e., alignment ratio $\\rho=0$, we adapt our strategy by discarding the alignment-based similarity construction. Instead, we compute cross-view similarity based on the affinity between partition matrices $\\mathbf{F}^v$ of different views. This allows us to preserve and exploit high-level clustering consistency without relying on direct sample-wise alignment. The learned alignment relationship thus enables indirect alignment by capturing common clustering structures. Furthermore, to demonstrate the effectiveness of our proposed method on fully unaligned scenarios, we compare it with Vsc_mH and OpVuC, which are among the few existing methods capable of handling fully unaligned multi-view data. According to the experimental results, we can find that our method outperforms the competitors in most of the datasets. Thus, these results confirm the adaptability of our method in addressing full sample misalignment by leveraging shared structural information at the partition level.  \nClustering results of compared methods under fully unaligned views (alignment ratio $\\rho=0$)\n|  Metric | Methods |    Yale    |  3sources  |    MSRCV   |  100leaves |     HW     |    Scene   |\n|:-------:|:-------:|:----------:|:----------:|:----------:|:----------:|:----------:|:----------:|\n|         |  Vsc_mH | 36.36±0.00 | 40.24±0.00 | 39.05±0.00 | 13.94±0.00 | 13.50±0.00 | 29.45±0.00 |\n|   ACC   |  OpVuC  | 24.24±0.00 | 24.26±0.00 | 24.29±0.00 |  8.75±0.00 | 12.90±0.00 | 10.90±0.00 |\n|         |   Ours  | 59.36±2.40 | 55.77±0.42 | 68.31±0.66 | 54.93±1.37 | 19.25±0.95 | 19.88±0.65 |\n|         |  Vsc_mH | 41.79±0.00 | 11.27±0.00 | 21.78±0.00 | 44.50±0.00 |  3.42±0.00 | 27.85±0.00 |\n|   NMI   |  OpVuC  | 28.96±0.00 |  4.53±0.00 |  5.97±0.00 | 32.91±0.00 |  0.93±0.00 |  3.52±0.00 |\n|         |   Ours  | 63.97±1.26 | 50.21±0.97 | 49.65±0.93 | 73.11±0.86 |  5.11±0.36 |  9.98±0.63 |\n|         |  Vsc_mH | 14.61±0.00 | 11.07±0.00 | 12.71±0.00 |  4.82±0.00 |  0.10±0.00 | 14.39±0.00 |\n|   ARI   |  OpVuC  |  2.57±0.00 |  0.27±0.00 |  1.59±0.00 |  1.08±0.00 |  0.02±0.00 |  0.56±0.00 |\n|         |   Ours  | 40.97±1.94 | 37.35±0.57 | 43.51±1.02 | 39.29±1.28 |  2.25±0.23 |  4.78±0.34 |\n|         |  Vsc_mH | 20.49±0.00 | 31.99±0.00 | 25.05±0.00 |  6.09±0.00 | 15.99±0.00 | 21.10±0.00 |\n| F1score |  OpVuC  |  8.74±0.00 | 19.24±0.00 | 17.57±0.00 |  2.53±0.00 | 10.97±0.00 |  9.06±0.00 |\n|         |   Ours  | 44.75±1.80 | 49.59±0.46 | 51.44±0.88 | 39.90±1.26 | 12.06±0.21 | 11.34±0.32 |\n\n**Q2: Should be compared more with the SOTA methods of view misalignment instead of traditional multi-view clustering methods.**  \n**A2**: Thank you for your valuable suggestion. To better evaluate the effectiveness of our method in handling view misalignment, we have included tow representative state-of-the-art methods specifically designed for unaligned multi-view clustering, i.e., TUMCR[1] and DAGF[2], and the corresponding experimental results are presented in the following, where we report ACC under two different alignment ratios: 0\\%(fully unaligned) and 50\\% (partially aligned). As shown, our method consistently achieves superior performance across six benchmark datasets in both settings. For example, under the fully unaligned scenario, our method outperforms TUMCR and DAGF by large margins on datasets such as MSRCV and 100leaves. Similar trends can be observed under the other datasets. Therefore, these results demonstrate the adaptability of our method in dealing with both fully and partially misaligned scenarios.  \nACC comparison with view misalignment multi-view clustering methods under different alignment ratios  \n| Alignment Ratios | Methods |    Yale    |  3sources  |    MSRCV   |  100leaves |     HW     |    Scene   |\n|:----------------:|:-------:|:----------:|:----------:|:----------:|:----------:|:----------:|:----------:|\n|                  |  TUMCR  | 29.09±0.00 | 36.09±0.00 | 26.67±0.00 | 47.44±0.00 | 15.70±0.00 |  9.59±0.00 |\n|        0%        |   DAGF  | 17.33±0.24 | 33.31±0.00 | 26.05±1.45 |  8.61±0.08 | 18.02±2.34 | 11.83±0.56 |\n|                  |   Ours  | 59.36±2.40 | 55.77±0.42 | 68.31±0.66 | 54.93±1.37 | 19.25±0.95 | 19.88±0.65 |\n|                  |  TUMCR  | 35.76±0.00 | 42.60±0.00 | 37.14±0.00 | 41.75±0.00 | 41.35±0.00 | 14.51±0.00 |\n|        50%       |   DAGF  | 18.36±0.36 | 33.90±0.01 | 27.90±1.37 |  9.46±0.11 | 22.37±2.53 | 13.44±0.57 |\n|                  |   Ours  | 64.24±3.62 | 64.44±1.29 | 83.52±0.39 | 70.63±1.29 | 96.55±0.00 | 35.91±0.27 |  \n  \n[1] Ji J, Feng S, Li Y. Tensorized unaligned multi-view clustering with multi-scale representation learning[C]// KDD. 2024.  \n[2] Jiang H, Tao H, Jiang Z, et al. Unaligned multi-view clustering via diversified anchor graph fusion[J]. Pattern Recognition, 2025.  \n  \n**Q3: Can the learned rotation matrix be used to output aligned samples?**  \n**A3**: Thanks for your constructive comments. In our method, the feature rotation matrix $\\mathbf{R}$ is primarily learned in the latent feature space to align the view-specific representations with the unified representation, rather than to directly realign the original samples. Specifically, $\\mathbf{M}$ operates on the extracted features of samples, mapping them into a shared feature space. Therefore, while the learned rotation matrix can produce aligned feature representations, it does not yield aligned samples.  \n  \n**Q4: The roles of each component in Eqs. (4), (5), and (7) should be specifically explained.**  \n**A4**: Thanks for your advice. The metric in Eq.(4) acts as an enhanced separation-compactness ratio that evaluates the structural quality of a clustering $\\mathbf{Y}$ with respect to the entire dataset $\\mathbf{X}$. Specifically, H integrates scale-invariant normalization through $\\sigma_{d^2}$, exponential smoothing to mitigate the effects of data shifts, and a balanced weighting of inter-cluster and intra-cluster distances. As a result, H provides a robust and comparable measure of cluster quality across different views, even when the data distributions and cluster cardinalities vary. On the basis of Eq.(4), Eq.(5) defines the final clustering quality measure, i.e., it averages pairwise cluster scores and applies a logistic transformation to normalize the range to 0-1. This design not only ensures class-cardinality invariance but also provides a directly interpretable and bounded quality score. Therefore, with Eqs. (4)-(5), we can quantify both intra-view compactness and inter-view separability in a scale-free manner and consistently identify the view that fully preserves the intrinsic structure of the data, regardless of sample size, feature scales, or the number of clusters. In this way, we can avoid biases caused by poorly aligned or low-quality views for subsequent cross-view alignment.  \nAfter selecting the base view according to the Eqs.(4)-(5), the final clustering objective function can be formulated, as presented in Eq.(7). The first part of Eq.(7) is a sample alignment-based late fusion clustering framework, where the sample alignment matrices $\\mathbf{M}^v$ and the feature rotation matrices $\\mathbf{R}^v$ are jointly learned to map each view-specific clustering result $\\mathbf{F}^v$ into a shared space, enabling the consensus partition $\\mathbf{F}^*$ to integrate multi-view information effectively. The second part introduces a structure-guided alignment term, where $\\mathbf{S}^v$ encodes the similarity between the non-aligned samples of each view and the baseline view. This regularization term drives $\\mathbf{M}^v$ to capture the correct correspondence by preserving cross-view structural relationships."}, {"review_id": "vabiLkdvrE", "reviewer": "Reviewer_gq64", "summary": "The proposed method addresses unalignment samples in multi-view clustering by aligning views through a benchmark-based cluster-label matching strategy and constructing cross-view similarity graphs, enabling effective clustering without requiring sample correspondence.", "questions": "1. Can the proposed method be effectively applied to cases where samples are entirely unaligned across views?\n\n2. The same approach is used for both unaligned sample representation and cross-view similarity—are these the only feasible choices, or can alternative methods be employed?\n\n3. As alignment performance depends on the benchmark view, how is its reliability ensured?\n\n4. While the datasets are standard in multi-view clustering, could the authors clarify how they were preprocessed to simulate sample misalignment?\n\n5. The proposed method belongs to MVC methods with shallow models, so it is recommended to compare more SOTA shallow MVC methods like tensor-based or subspace-based methods.", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The strengths of this manuscript are listed as follows:\n\nThe proposed method effectively handles unalignment multi-view data by eliminating the need for strict sample correspondence across views; The use of cluster-label matching to select a benchmark view ensures more reliable and consistent alignment across different views; By integrating the alignment criterion into a late-fusion framework, SSA-MVC maintains scalability and flexibility across diverse datasets.\n\nThe weaknesses of this manuscript are listed as follows:\n\nCan the proposed method be effectively applied to cases where samples are entirely unaligned across views; the same approach is used for both unaligned sample representation and cross-view similarity—are these the only feasible choices, or can alternative methods be employed; as alignment performance depends on the benchmark view, how is its reliability ensured; while the datasets are standard in multi-view clustering, could the authors clarify how they were preprocessed to simulate sample misalignment?", "was_revised": true, "final_justification": "Thanks for authors' response. This paper proposed an interesting method with promising results, and I have no further questions.", "rebuttal": "**Q1: Can the proposed method be effectively applied to cases where samples are entirely unaligned across views?**  \n**A1**: Thanks for your valuable comment. Our method remains effective even in the absence of any sample-level correspondence across views. Rather than depending on explicit alignment, it captures the shared clustering structure implicitly present across different modalities. This is achieved by projecting data into a latent space where structural patterns are preserved through orthogonality constraints. These constraints ensure that the alignment process respects the intrinsic geometry of each view, enabling the model to discover coherent groupings without relying on direct sample matching. In fully unaligned scenarios (i.e., alignment ratio $\\rho$=0), we adapt our strategy by discarding alignment-based similarity measures and instead computing cross-view relationships via the affinity between view-specific partition matrices. This allows the model to emphasize clustering-level consistency, which serves as a reliable proxy for alignment under severe view discrepancy. To assess the practical effectiveness of our method in such settings, we conduct comparisons with two representative models designed for unaligned multi-view clustering: VSC_mH and OpVuC. Extensive experiments on six benchmark datasets demonstrate that our method consistently achieves higher performance across most metrics. These results confirm that even in the most challenging case of complete sample misalignment, our framework successfully exploits latent structural information to produce robust clustering results.  \nClustering results under fully unaligned views\n|  Metric | Methods |    Yale    |  3sources  |    MSRCV   |  100leaves |     HW     |    Scene   |\n|:-------:|:-------:|:----------:|:----------:|:----------:|:----------:|:----------:|:----------:|\n|         |  Vsc_mH | 36.36±0.00 | 40.24±0.00 | 39.05±0.00 | 13.94±0.00 | 13.50±0.00 | 29.45±0.00 |\n|   ACC   |  OpVuC  | 24.24±0.00 | 24.26±0.00 | 24.29±0.00 |  8.75±0.00 | 12.90±0.00 | 10.90±0.00 |\n|         |   Ours  | 59.36±2.40 | 55.77±0.42 | 68.31±0.66 | 54.93±1.37 | 19.25±0.95 | 19.88±0.65 |\n|         |  Vsc_mH | 41.79±0.00 | 11.27±0.00 | 21.78±0.00 | 44.50±0.00 |  3.42±0.00 | 27.85±0.00 |\n|   NMI   |  OpVuC  | 28.96±0.00 |  4.53±0.00 |  5.97±0.00 | 32.91±0.00 |  0.93±0.00 |  3.52±0.00 |\n|         |   Ours  | 63.97±1.26 | 50.21±0.97 | 49.65±0.93 | 73.11±0.86 |  5.11±0.36 |  9.98±0.63 |\n|         |  Vsc_mH | 14.61±0.00 | 11.07±0.00 | 12.71±0.00 |  4.82±0.00 |  0.10±0.00 | 14.39±0.00 |\n|   ARI   |  OpVuC  |  2.57±0.00 |  0.27±0.00 |  1.59±0.00 |  1.08±0.00 |  0.02±0.00 |  0.56±0.00 |\n|         |   Ours  | 40.97±1.94 | 37.35±0.57 | 43.51±1.02 | 39.29±1.28 |  2.25±0.23 |  4.78±0.34 |\n|         |  Vsc_mH | 20.49±0.00 | 31.99±0.00 | 25.05±0.00 |  6.09±0.00 | 15.99±0.00 | 21.10±0.00 |\n| F1score |  OpVuC  |  8.74±0.00 | 19.24±0.00 | 17.57±0.00 |  2.53±0.00 | 10.97±0.00 |  9.06±0.00 |\n|         |   Ours  | 44.75±1.80 | 49.59±0.46 | 51.44±0.88 | 39.90±1.26 | 12.06±0.21 | 11.34±0.32 |  \n  \n**Q2: The same approach is used for both unaligned sample representation and cross-view similarity—are these the only feasible choices, or can alternative methods be employed?**  \n**A2**: Thank you for your insightful comment. In our method, we adopt the adaptive neighbor graph learning for both unaligned sample representation learning and cross-view similarity learning. This approach provides a unified and effective way to capture local geometric structures and inter-view relationships. Of course, we acknowledge that this is not the only feasible solution. Alternative strategies, such as metric learning techniques, multiple kernel learning, could also be employed for them. We chose the current method due to its flexibility and effectiveness in our preliminary studies. Therefore, future work could explore the impact of these alternative methods.  \n  \n**Q3: As alignment performance depends on the benchmark view, how is its reliability ensured?**  \n**A3**: Thanks for your constructive comments. In the existing alignment-based multi-view clustering methods, the baseline view is either randomly selected or chosen heuristically without considering the intrinsic clustering quality of each view. This randomness can lead to sub-optimal alignment, especially when the selected view exhibits poor cluster structure or noise. To address this, we introduced a baseline view selection mechanism based on the CLM algorithm, and the detailed introductions can refer to Section 3.2. As presented in Eqs. (4)-(5), the measures assess the separation and compactness of cluster structures in a scale-invariant, shift-invariant, and class-size-invariant manner, enabling a fair comparison of clustering quality across unaligned views. Specifically, we compute the CLM score for each view and select the one with the highest score as the baseline view, thereby ensuring that alignment is guided by the most reliable structural information. Therefore, compared with the existing baseline view selection methods, the proposed method mitigates the limitations of random selection and enhances the stability and performance of the alignment process.  \n  \n**Q4: While the datasets are standard in multi-view clustering, could the authors clarify how they were preprocessed to simulate sample misalignment?**  \n**A4**: Thank you for your valuable comments. As existing multi-view datasets are originally aligned across views, we simulate the sample misalignment scenario by artificially disrupting the arrangement of the samples. Specifically, we first apply the CLM algorithm to determine the baseline view. Then, we select a fixed proportion of samples with the same indices across views to serve as aligned samples for each view. The remaining samples in each view are considered as unaligned by randomly shuffling their order. Additionally, in the baseline view, we preserve the original sample order to maintain consistency with the ground-truth labels. Through this process, we construct multi-view datasets with varying degrees of sample misalignment, which allow us to effectively evaluate the robustness of our proposed method under sample misalignment scenarios.  \n  \n**Q5: It is recommended to compare more SOTA shallow MVC methods like tensor-based or subspace-based methods.**  \n**A5**: Thank you for your insightful suggestion. In our experimental comparison, we have added two representative shallow multi-view clustering methods, i.e., TPCH [1] and ESTMC [2], and we report results under a 50% sample alignment ratio, with and without Hungarian alignment. The results demonstrate that our proposed method consistently outperforms them across all datasets in four clustering metrics. Notably, applying Hungarian alignment to these baselines does not necessarily lead to performance gains and sometimes even degrades the results, indicating that rigid post-processing alignment may be unreliable when sample correspondences are uncertain. In contrast, our method achieves significantly better and more stable performance by jointly modeling soft sample alignment and cross-view structural consistency, further demonstrating its effectiveness in handling partially aligned multi-view data.  \nClustering performance under a sample alignment ratio $\\rho$=50%  \n| Metrics |     Methods     |    Yale    |  3sources  |    MSRCV   |  100leaves |     HW     |    Scene   |\n|:-------:|:---------------:|:----------:|:----------:|:----------:|:----------:|:----------:|:----------:|\n|         |       TPCH      | 27.88±0.00 | 30.77±0.00 | 33.81±0.00 | 32.38±0.00 | 25.20±0.00 | 19.15±0.00 |\n|         |  TPCH+Hungarian | 33.94±0.00 | 30.18±0.00 | 27.62±0.00 | 16.25±0.00 | 17.95±0.00 | 15.43±0.00 |\n|   ACC   |      ESTMC      | 51.42±3.74 | 49.59±0.91 | 54.95±2.22 | 35.53±1.04 | 69.63±4.64 | 23.81±0.92 |\n|         | ESTMC+Hungarian | 32.94±1.71 | 36.92±0.56 | 45.21±1.96 | 35.54±1.18 | 45.94±0.81 | 20.77±0.50 |\n|         |       Ours      | 64.24±3.62 | 64.44±1.29 | 83.52±0.39 | 70.63±1.29 | 96.55±0.00 | 35.91±0.27 |\n|         |       TPCH      | 32.65±0.00 |  7.03±0.00 | 14.31±0.00 | 56.35±0.00 |  9.51±0.00 | 12.65±0.00 |\n|         |  TPCH+Hungarian | 35.95±0.00 |  8.31±0.00 |  8.18±0.00 | 45.32±0.00 |  3.17±0.00 |  5.27±0.00 |\n|   NMI   |      ESTMC      | 56.44±3.05 | 34.70±1.56 | 45.57±2.96 | 58.76±0.63 | 61.21±2.06 | 17.37±0.78 |\n|         | ESTMC+Hungarian | 35.61±1.59 | 19.24±0.73 | 23.58±1.52 | 58.84±0.73 | 23.72±0.55 | 14.11±0.31 |\n|         |       Ours      | 69.31±1.32 | 61.20±0.83 | 70.28±0.55 | 85.34±0.42 | 92.09±0.00 | 30.27±0.20 |\n|         |       TPCH      |  5.33±0.00 |  2.54±0.00 |  6.72±0.00 | 12.62±0.00 |  5.14±0.00 |  5.78±0.00 |\n|         |  TPCH+Hungarian |  8.84±0.00 |  3.41±0.00 |  2.08±0.00 |  3.10±0.00 |  1.30±0.00 |  2.93±0.00 |\n|   ARI   |      ESTMC      | 30.82±4.14 | 25.26±1.08 | 32.04±2.42 | 17.15±0.91 | 52.12±3.56 |  7.97±0.41 |\n|         | ESTMC+Hungarian |  8.19±1.26 |  9.98±1.00 | 15.50±1.07 | 17.24±1.03 | 18.60±0.48 |  6.49±0.23 |\n|         |       Ours      | 48.91±2.65 | 45.98±1.61 | 66.34±0.66 | 59.97±0.95 | 92.49±0.00 | 17.15±0.32 |\n|         |       TPCH      | 11.39±0.00 | 22.24±0.00 | 19.87±0.00 | 13.66±0.00 | 14.74±0.00 | 12.33±0.00 |\n|         |  TPCH+Hungarian | 14.60±0.00 | 21.95±0.00 | 16.32±0.00 |  4.12±0.00 | 12.73±0.00 | 10.53±0.00 |\n| F1score |      ESTMC      | 35.18±3.89 | 39.76±0.87 | 41.53±2.05 | 17.98±0.90 | 56.94±3.18 | 14.34±0.39 |\n|         | ESTMC+Hungarian | 13.96±1.18 | 27.77±0.65 | 27.33±0.86 | 18.08±1.02 | 26.85±0.44 | 12.91±0.22 |\n|         |       Ours      | 52.17±2.43 | 56.59±1.34 | 71.03±0.57 | 60.37±0.94 | 93.24±0.00 | 22.94±0.29 |  \n  \n[1] Wang Z, Li X, Sun Y, et al. TPCH: tensor-interacted projection and cooperative hashing for multi-view clustering, AAAI. 2025.  \n[2] Ji J, Feng S. Anchors crash tensor: efficient and scalable tensorial multi-view subspace clustering, IEEE TPAMI, 2025."}, {"review_id": "wD5mpyxowU", "reviewer": "Reviewer_oX5B", "summary": "A novel multi-view clustering method based on cross-view sample alignment is proposed in this manuscript, and its effectiveness is demonstrated through experiments on eight multi-view datasets.", "questions": "While the proposed late-fusion strategy elegantly avoids the need for early alignment, it raises an interesting question: to what extent does this design truly exploit the structural and semantic complementarities between views? One might wonder whether certain types of cross-view information—particularly those that only emerge through joint modeling—are inadvertently underutilized in the alignment process.\n\nAnother intriguing aspect is the reliance on cluster-label matching for selecting a benchmark view. This appears to work well in practice, yet it remains somewhat unclear how robust this step is to imperfections in the initial clustering. Given that early-stage clustering often involves noise or instability, especially in high-dimensional or heterogeneous views, it would be worth exploring how such variance might influence the view selection outcome—and, by extension, the final alignment quality.", "limitations": "Yes.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:  \n(1) The paper addresses the challenge of multi-view clustering with unaligned samples in a principled and practical manner. The proposed framework is conceptually clear and yields competitive results across multiple benchmarks.  \n(2) The use of cluster-label matching (CLM) for benchmark view selection is a noteworthy design choice. This mechanism appears to improve alignment stability, particularly in the presence of inconsistent or noisy views.  \n(3) The introduction of an alignment graph to connect aligned and unaligned samples enhances representational flexibility and contributes to performance gains under partial correspondence settings.  \n\nPoints for Consideration:  \n(1) It would be helpful for the authors to clarify the specific motivations behind adopting the CLM mechanism for benchmark view selection. A brief discussion comparing it to alternative strategies (e.g., view consensus or uncertainty-based methods) could further strengthen the justification.  \n(2) As the alignment process is anchored on a single benchmark view, there is a potential risk that misrepresentative or noisy data within this view could influence the quality of alignment across other views. Some discussion on the mitigation of such risks would be valuable.  \n(3) Constructing similarity graphs between non-aligned and aligned samples is central to the proposed method. However, the paper could benefit from further analysis on how initial clustering inaccuracies might affect the propagation of alignment errors.  \n(4) While the alignment matrix plays a key role in sample correspondence, its robustness under noisy benchmark views remains unclear. Additional empirical or theoretical insight into the stability of alignment in such cases would enhance the completeness of the work.", "was_revised": true, "final_justification": "Most of my concerns have been resolved. I would like to recommend acceptance.", "rebuttal": "**Q1: It would be helpful for the authors to clarify the specific motivations behind adopting the CLM mechanism for benchmark view selection. A brief discussion comparing it to alternative strategies could further strengthen the justification.**  \n**A1**: Thanks for your valuable suggestion. The motivation for adopting the CLM-based mechanism lies in its ability to provide a view-independent evaluation of clustering quality. Unlike many existing methods that either randomly select a baseline view or rely on view consensus, our approach aims to directly assess the intrinsic structural information of each view's clustering. Consensus-based strategies typically identify a baseline view by measuring its agreement with other views. However, such methods may favor views that are consistent yet collectively poor in quality, leading to confirmation bias across views. Similarly, uncertainty-based methods rely on the distributional confidence, but they often lack explicit consideration of clustering structure, especially in the absence of ground truth labels. In contrast, the CLM mechanism jointly evaluates intra-cluster compactness and inter-cluster separability in a scale, shift, and class cardinality-invariant manner. This allows for fair and consistent comparison across heterogeneous views. By selecting the view with the highest CLM score, the most structurally reliable view is used in the proposed method, thereby enhancing the overall robustness of cross-view alignment.  \n  \n**Q2: As the alignment process is anchored on a single benchmark view, there is a potential risk that misrepresentative or noisy data within this view could influence the quality of alignment across other views.**  \n**A2**: Thanks for your insightful comments. We fully agree that anchoring the alignment process on a misrepresentation or noisy baseline view could propagate errors to other views and negatively affect overall performance. To mitigate this risk, our framework employs a carefully designed selection mechanism based on the CLM score, which evaluates each view's clustering structure through a combination of compactness and robustness to data scale, shift, and class imbalance. By computing the CLM score for each view, we explicitly avoid relying on random selection. Instead, we select the view with the highest structural quality as the baseline view, ensuring that it reflects well-formed clusters and minimizes the chance of incorporating noise from poorly separated structures. Furthermore, since CLM is constructed to be scale and data cardinality-invariant, it is less sensitive to outliers or imbalanced cluster sizes, making it more robust in practice. Additionally, according to the extensive experimental results in Section 5, we can observe that it consistently achieves satisfactory alignment performance across diverse multi-view datasets.  \n  \n**Q3: The paper could benefit from further analysis on how initial clustering inaccuracies might affect the propagation of alignment errors.**  \n**A3**: We thank your insightful comments. We agree that inaccuracies in the initial clustering stages may potentially affect the construction of similarity graphs between aligned and misaligned samples, thereby influencing the propagation of alignment errors. To mitigate this, we integrate both feature-level information and structural-level information to construct the consensus representation. This dual-perspective design ensures that the model does not rely solely on any single level of information, and enables correction and compensation for localized clustering inaccuracies during the optimization of the consensus representation. Furthermore, the late fusion clustering framework is introduced to decouple the final clustering decision from the initial alignment assumptions, providing an additional layer of robustness. Essentially, even in the presence of some local misalignment, the global fusion step can improve the clustering results by weighted integration of diverse information from multiple views, without being overly affected by any single misaligned view.  \n  \n**Q4: While the alignment matrix plays a key role in sample correspondence, its robustness under noisy benchmark views remains unclear.**  \n**A4**: We appreciate your valuable comment regarding the robustness of the learned alignment matrix under noisy baseline views. From a theoretical perspective, our alignment matrix learning framework is designed to incorporate global structural information from multiple views rather than relying solely on local pairwise similarities. This global view enables the alignment matrix to be less sensitive to localized noise or misalignment in any single baseline view. Furthermore, the alignment objective jointly optimizes for consistency across views, implicitly enforcing robustness through structural constraints and regularization. These design choices ensure that the learned correspondences capture the intrinsic relations among samples, improving stability despite noise. Empirically, we validate this theoretical robustness by conducting experiments where the alignment matrix learned by our method is applied to align clustering results from several different algorithms. Compared to using the classical Hungarian algorithm, which is primarily a local matching approach, our alignment matrix consistently achieves better clustering accuracy across diverse datasets and algorithms. The detailed experimental results are presented in the following, and the detailed introductions can refer to Section 5.7. This demonstrates that our approach can generalize beyond the specific baseline view used during alignment learning and effectively mitigate the influence of noise views. In a word, the theoretical design and empirical evidence strongly support the conclusion that our learned alignment matrix is stable and robust under the noise baseline views.  \nResults of competitors on the 100leaves under a sample alignment ratio of $\\rho$=50%  \n| Metrics |    Setting   |   DealMVC  |    MVCAN   |    EBMGC   |    DCMVC   |    LMTC    |    TMSL    |    DSTL    |\n|:-------:|:------------:|:----------:|:----------:|:----------:|:----------:|:----------:|:----------:|:----------:|\n|   ACC   |   Unaligned  |  7.69±0.00 | 49.51±1.24 | 33.94±0.00 | 48.83±0.83 | 35.58±0.94 | 47.47±1.40 | 36.87±1.42 |\n|         | Aligned+Ours | 12.42±0.52 | 48.81±1.28 | 43.06±0.00 | 53.75±0.89 | 40.82±1.31 | 48.18±1.37 | 35.60±0.90 |\n|   NMI   |   Unaligned  | 25.34±0.44 | 69.50±0.95 | 58.22±0.00 | 67.15±0.42 | 58.75±0.55 | 69.33±0.61 | 60.53±0.55 |\n|         | Aligned+Ours | 38.01±0.46 | 69.95±0.59 | 64.52±0.00 | 72.10±0.49 | 64.58±0.80 | 70.54±0.57 | 61.09±0.53 |\n|   ARI   |   Unaligned  |  1.89±0.25 | 30.37±1.53 | 15.36±0.00 | 29.23±0.91 | 17.24±0.81 | 31.41±1.15 | 18.49±0.77 |\n|         | Aligned+Ours |  5.16±0.12 | 30.58±0.99 | 24.62±0.00 | 36.50±0.87 | 24.46±1.24 | 32.67±1.17 | 19.38±0.68 |\n| F1score |   Unaligned  |  6.68±0.15 | 37.49±1.29 | 16.16±0.00 | 35.08±0.61 | 18.07±0.81 | 32.09±1.14 | 19.35±0.75 |\n|         | Aligned+Ours | 11.47±0.14 | 37.74±0.93 | 25.33±0.00 | 41.38±0.85 | 25.21±1.23 | 33.34±1.15 | 20.23±0.67 |  \n  \n**Q5: To what extent does this design truly exploit the structural and semantic complementarities between views?**  \n**A5**: Thanks for your constructive comments. Indeed, it is critical to ensure that the late-fusion strategy fully leverages the structural and semantic information among views. To address this, we designed our framework with the following considerations. Firstly, while the fusion process is conducted at a later stage, we do not rely solely on independent view-wise clustering results. Instead, the alignment phase incorporates both feature-level information and graph-structural similarities, which are used to learn a unified latent representation space. This design allows the alignment process to indirectly capture inter-view dependencies before fusion. Secondly, to address the potential underutilization of cross-view information that only emerges through joint modeling, we introduce a weighted consensus mechanism in the fusion step. Specifically, each view contributes to the final clustering based on its alignment consistency and structural reliability, allowing high-quality views to reinforce low-quality ones and flexibly preserving multi-view complementary information. In summary, although our method does not perform explicit joint modeling in the traditional sense, it captures structural and semantic information through a combination of unified representation learning and adaptive weighted fusion, and the experimental results demonstrate the effectiveness of our proposed method.  \n  \n**Q6: Another intriguing aspect is the reliance on cluster-label matching for selecting a benchmark view.**  \n**A6**: We sincerely appreciate your insightful comments. Unlike methods that assess each view independently, CLM measures the relative consistency of each view concerning all other views through a pairwise, normalized label-matching function. By aggregating over all $\\binom{k}{2}$ cluster pairs, CLM captures global structural similarity in a redundant and smoothed manner, which effectively reduces the impact of local noise or instability in individual views. Furthermore, this strategy enables CLM to select the view that is most structurally aligned with the consensus of the remaining views, even when all views contain some level of noise. As such, the view selection process becomes inherently more robust than strategies that either randomly choose a baseline view or rely on the absolute clustering performance of a single view. Furthermore, as observed in Section 5.7, using our alignment matrix to align the clustering results of other algorithms achieves superior performance compared to the traditional Hungarian matching approach, which demonstrates the robustness and generalization of the selected baseline view."}, {"review_id": "qJfxDdO44F", "reviewer": "Reviewer_HmvP", "summary": "A scalable multi-view clustering algorithm based on sample alignment is proposed in this paper, where an alignment matrix is learned and incorporated into a late-fusion framework to facilitate clustering in the absence of aligned samples.", "questions": "a. Without explicit correspondence, what principle underlies the computation of cross-view similarity between non-aligned samples?\n\nb. What is the rationale for enforcing orthogonality on the alignment matrix M? A clear justification is currently lacking and should be provided.", "limitations": "Yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "a. The manuscript is well-written and introduces a novel perspective on addressing sample misalignment in multi-view clustering.\n\nb. Experimental results on eight benchmark datasets demonstrate the effectiveness of the proposed method.\n\nc. It remains unclear whether cross-view similarity between unaligned representations can reliably capture semantic correspondence under strong view heterogeneity.\n\nd. The scalability of the method to large-scale data is not thoroughly discussed, particularly regarding the computational implications and potential optimizations.\n\ne. While the use of a k-nearest neighbor graph is intuitive, the sensitivity of the model to the choice of k warrants further analysis.", "was_revised": true, "final_justification": "The authors have well solved my concerns and I decide to raise the rating.", "rebuttal": "**Q1: It remains unclear whether cross-view similarity between unaligned representations can reliably capture semantic correspondence under strong view heterogeneity.**  \n**A1**: Thanks for your careful review. We would like to clarify that in our framework, the cross-view similarity between unaligned representations is not intended to directly capture semantic correspondence or high-level semantic consistency across views. Instead, it mainly estimates sample-level correspondence between views under unaligned scenarios. Specifically, our approach constructs cross-view similarity graphs by leveraging the view-specific graph structures, i.e., the similarity between the aligned samples and the unaligned samples for each view. These similarity measures are designed to identify samples that are likely to be counterparts across views, even in the presence of view heterogeneity. They are not responsible for learning semantic representations, but rather provide a relational criterion to guide the sample alignment process.  \n  \n**Q2: The scalability of the method to large-scale data is not thoroughly discussed, particularly regarding the computational implications and potential optimizations.**  \n**A2**: Thanks for your constructive comments. As detailed in Section 4.3, the primary sources of computational cost in our proposed method include cross-sample similarity learning, cross-view similarity construction, and the late fusion stage. Among these, the largest computational complexity is the generation of the partition matrix $\\mathbf{F}^v$ during the late fusion process, which leads to an overall complexity of $\\mathcal{O}(n^2)$. Fortunately, this cost can be significantly reduced in practice. In particular, when more efficient or approximate strategies are adopted for generating the feature matrix $\\mathbf{F}^v$, such as sparse graph construction, anchor-based clustering, or randomized low-rank approximations, the computational complexity can be reduced to nearly linear in the number of samples. Moreover, other components of the proposed method, such as CLM-based baseline view selection and cross-view similarity learning, scale linearly or near-linearly concerning the number of samples. In our experimental results, the proposed method has already demonstrated the ability to handle relatively large datasets within a reasonable time.  \n  \n**Q3: While the use of a k-nearest neighbor graph is intuitive, the sensitivity of the model to the choice of k warrants further analysis.**  \n**A3**: Thank you for your valuable comment. We agree that the choice of the $k$-nearest neighbors parameter may impact the performance of the proposed method, and thereby we conduct additional experiments to analyze the proposed method's sensitivity to different values of $k$ under 50% sample alignment. As shown in the following experimental results, we evaluated clustering performance in terms of ACC, NMI, ARI, and F1score across a range of $k$ values ($k$=5, 9, 15, 19, 25) on six benchmark datasets. The results demonstrate that while minor performance fluctuations exist across different $k$ values, the overall performance remains relatively stable. For instance, on the HW and 100leaves datasets, the ACC and NMI remain consistently high regardless of the choice of $k$, suggesting that our method is robust to the $k$ setting. Although some datasets exhibit moderate variations, these changes do not significantly affect the general trend of the ranking of results. Moreover, the best performance does not concentrate on a single $k$ value, indicating that the model is not sensitive to a specific value. In a word, these observations confirm that our method maintains a stable and competitive performance across a reasonable range of $k$ values, suggesting that it is not highly sensitive to the neighborhood size. This robustness enhances the practicality of our method, especially when the optimal $k$ is not known a priori.  \nEffect of $k$-nearest neighbors on ACC under 50% sample alignment.  \n| Different $k$-nn |    Yale    |  3sources  |    MSRCV   |  100leaves |     HW     |    Scene   |\n|:----------------:|:----------:|:----------:|:----------:|:----------:|:----------:|:----------:|\n|       $k$=5      | 63.33±2.99 | 62.49±0.95 | 85.76±0.15 | 70.85±1.29 | 95.29±3.48 | 35.63±1.49 |\n|       $k$=9      | 64.79±2.48 | 61.39±0.50 | 84.45±0.32 | 70.65±1.36 | 96.07±0.04 | 36.17±0.58 |\n|      $k$=15      | 64.24±3.62 | 64.44±1.29 | 83.52±0.39 | 70.63±1.29 | 96.55±0.00 | 35.91±0.27 |\n|      $k$=19      | 64.52±3.09 | 62.57±0.33 | 83.38±0.60 | 70.76±0.98 | 95.84±2.94 | 36.60±0.75 |\n|      $k$=25      | 64.09±2.40 | 64.56±0.18 | 83.31±0.57 | 70.68±0.91 | 95.28±3.34 | 36.11±1.03 |  \n  \nEffect of $k$-nearest neighbors on NMI under 50% sample alignment.  \n| Different $k$-nn |    Yale    |  3sources  |    MSRCV   |  100leaves |     HW     |    Scene   |\n|:----------------:|:----------:|:----------:|:----------:|:----------:|:----------:|:----------:|\n|       $k$=5      | 68.18±1.41 | 57.97±0.55 | 72.96±0.32 | 84.55±0.33 | 91.43±1.33 | 30.81±0.44 |\n|       $k$=9      | 69.34±1.16 | 57.67±0.13 | 71.66±0.45 | 84.65±0.52 | 91.42±0.07 | 29.85±0.55 |\n|      $k$=15      | 69.31±1.32 | 61.20±0.83 | 70.28±0.55 | 85.34±0.42 | 92.09±0.00 | 30.27±0.20 |\n|      $k$=19      | 69.26±1.82 | 58.81±0.31 | 70.93±0.37 | 84.50±0.41 | 91.75±1.09 | 29.53±0.68 |\n|      $k$=25      | 68.91±1.51 | 62.05±0.06 | 70.87±0.38 | 84.61±0.38 | 91.41±1.15 | 30.16±0.29 |  \n  \nEffect of $k$-nearest neighbors on ARI under 50% sample alignment.  \n| Different $k$-nn |    Yale    |  3sources  |    MSRCV   |  100leaves |     HW     |    Scene   |\n|:----------------:|:----------:|:----------:|:----------:|:----------:|:----------:|:----------:|\n|       $k$=5      | 47.26±2.43 | 43.06±1.24 | 70.11±0.31 | 58.95±0.98 | 91.12±3.36 | 17.45±0.71 |\n|       $k$=9      | 49.23±1.67 | 41.67±0.65 | 67.97±0.58 | 59.34±1.49 | 91.48±0.08 | 16.78±0.50 |\n|      $k$=15      | 48.91±2.65 | 45.98±1.61 | 66.34±0.66 | 59.97±0.95 | 92.49±0.00 | 17.15±0.32 |\n|      $k$=19      | 49.24±2.82 | 43.88±0.32 | 66.45±0.80 | 58.94±1.14 | 91.80±2.68 | 16.91±0.63 |\n|      $k$=25      | 48.46±2.11 | 47.10±0.18 | 66.34±0.78 | 59.09±1.00 | 91.10±3.06 | 17.66±0.56 |  \n  \nEffect of $k$-nearest neighbors on F1score under 50% sample alignment.  \n| Different $k$-nn |    Yale    |  3sources  |    MSRCV   |  100leaves |     HW     |    Scene   |\n|:----------------:|:----------:|:----------:|:----------:|:----------:|:----------:|:----------:|\n|       $k$=5      | 50.59±2.25 | 54.20±1.03 | 74.27±0.27 | 59.36±0.96 | 92.02±2.98 | 23.24±0.65 |\n|       $k$=9      | 52.44±1.56 | 53.10±0.52 | 72.43±0.50 | 59.74±1.47 | 92.33±0.07 | 22.64±0.44 |\n|      $k$=15      | 52.17±2.43 | 56.59±1.34 | 71.03±0.57 | 60.37±0.94 | 93.24±0.00 | 22.94±0.29 |\n|      $k$=19      | 52.47±2.61 | 54.82±0.27 | 71.14±0.68 | 59.34±1.12 | 92.63±2.39 | 22.78±0.56 |\n|      $k$=25      | 51.72±1.98 | 57.44±0.15 | 71.04±0.67 | 59.49±0.99 | 92.00±2.72 | 23.41±0.50 |\n  \n**Q4: Without explicit correspondence, what principle underlies the computation of cross-view similarity between non-aligned samples?**  \n**A4**: Thanks for your insightful comments. Our proposed method addresses this challenge through a carefully designed two-step strategy. First, within each view, every unaligned sample is represented by its affinity relative to the aligned samples in the same view, effectively embedding the unaligned data into a consistent reference criterion defined by these aligned samples. Since the aligned samples are assumed to correspond across views, cross-view similarity between unaligned samples can then be evaluated indirectly by comparing their affinities over the common aligned samples. This approach builds on the fact that non-aligned samples displaying similar affinity patterns with a common set of aligned samples across their respective views are likely to share semantic consistency, even without explicit correspondence. Furthermore, the experimental results demonstrate the effectiveness of the proposed strategies.  \n  \n**Q5: What is the rationale for enforcing orthogonality on the alignment matrix M? A clear justification is currently lacking and should be provided.**  \n**A5**: We appreciate your insightful question. In our framework, we aim to establish soft correspondences between unaligned samples by reconstructing them based on structurally relevant information within their respective subspaces, rather than relying on hard 0-1 alignment. The orthogonality constraint is introduced to preserve the intrinsic geometric structure of the data during this reconstruction process. Specifically, orthogonality helps ensure that the transformation encoded by $\\mathbf{M}$ behaves in a structure-preserving manner, avoiding distortion of sample relationships that could arise from degenerate or overly flexible alignments. This is particularly important in the presence of noise and semantic inconsistencies across views, where maintaining relative distances and semantic separability becomes challenging. Additionally, orthogonality serves as a form of regularization that stabilizes the optimization process and prevents trivial or collapsed solutions."}]

Author rebuttal:

Meta-review: This paper proposes a scalable sample-alignment-based multi-view clustering (MVC) method to address the challenge of partially aligned views. The key innovation lies in a cluster-label matching (CLM) strategy to select a benchmark view, followed by constructing cross-view similarity graphs to represent non-aligned samples. The method integrates alignment into a late-fusion framework, enabling clustering without strict sample correspondence. Experiments on eight benchmarks demonstrate its effectiveness, particularly in partial alignment scenarios. Reviewers highlight the novelty of the alignment strategy, its scalability, and competitive performance against traditional MVC methods. Combining their consensus acceptance recommendation after rebuttal, the paper is recommended to be accepted.
Decision: Accept (spotlight)