AF-UMC: An Alignment-Free Fusion Framework for Unaligned Multi-View Clustering
Authors: Bohang Sun
Yuena Lin
Tao Yang
Zhen Zhu
Zhen Yang
Gengyu Lyu
Venue: NeurIPS 2025 poster ()
Keywords: Multi-view clustering
alignment-free fusion
cross-view consistent representation
Primary area: general_machine_learning
TL;DR: We propose an alignment-free consistency fusion framework for unaligned multi-view clustering problem.

Abstract: The Unaligned Multi-view Clustering (UMC) aims to learn a discriminative cluster structure from unaligned multi-view data, where the features of samples are not completely aligned across multiple views. Most existing methods usually prioritize employing various alignment strategies to align sample representations across views and then conduct cross-view fusion on aligned representations for subsequent clustering. However, ***due to the heterogeneity of representations across different views, these alignment strategies often fail to achieve ideal view-alignment results, inevitably leading to unreliable alignment-based fusion.*** To address this issue, we propose an alignment-free consistency fusion framework named AF-UMC, which bypasses the traditional view-alignment operation and directly extracts consistent representations from each view to perform global cross-view consistency fusion. Specifically, we first construct a cross-view consistent basis space by a cross-view reconstruction loss and a designed Structural Clarity Regularization (SCR), where autoencoders extract consistent representations from each view through projecting view-specific data to the constructed basis space. Afterwards, these extracted representations are globally pulled together for further cross-view fusion according to a designed Instance Global Contrastive Fusion (IGCF). Compared with previous methods, AF-UMC directly extracts consistent representations from each view for global fusion instead of alignment for fusion, which significantly mitigates the degraded fusion performance caused by undesired view-alignment results while greatly reducing algorithm complexity and enhancing its efficiency. Extensive experiments on various datasets demonstrate that our AF-UMC exhibits superior performance against other state-of-the-art methods.

AF-UMC: An Alignment-Free Fusion Framework for Unaligned Multi-View Clustering
Bohang Sun 1.2, Yuena Lin', Tao Yang', Zhen Zhu Zhen Yang', Gengyu Lyu
'College of Computer Science, Beijing University of Technology
EEninneering Research Center of Intelligent Perception and Autonomous Control, Ministry of Education
IIdealism Beijing Technology Co., Ltd.
*School of Computer Science and Technology, Zhejiang Sci-tech University, China KEYI College, Zhejiang Sci-tech University, China
sunbohang@emails bjut edu. cn, yuenalin@126 com, yangtao@ilxzy. cn hzzhuzhen@yeah. net, yangzhenābjut edu. en, lyugengyudgmail. com
Abstract
The Unaligned Multi-view Clustering (UMC) aims to learn discriminative cluster structure from unaligned multi-view data, where the features of samples are not completely aligned across multiple views. Most existing methods usually prioritize employing various alignment strategies to align sample representations across views and then conduct tross-view fusion aligned representations for subsequent clustering. However, due to the heterogeneity of representations across different views, these alignment strategies often fail to achieve ideal view-alignment results, inevitably leading to unreliable alignment-based fusion. To address this issue, we propose alignment-free consistency fusion framework named AF-UMC, which bypasses the traditional view-alignment operation and directly extracts consistent representations from each view to perform global cross-view consistency fusion. Specifically, we first construct a cross-view consistent basis space by cross-view reconstruction loss and a designed Structural Clarity Regularization (SCR), where autoencoders extract consistent representations from each view through projecting view-specific data the constructed basis space. Afterwards, these extracted representations are globally pulled together for further cross-view fusion according a designed Instance Global Contrastive Fusion (IGCF). Compared with previous methods, AF-UMC directly extracts consistent representations from each view for global fusion instead of alignment for fusion, which significantly mitigates the degraded fusion performance caused by undesired view-alignment results while greatly reducing algorithm complexity and enhancing its efficiency. Extensive experiments various datasets demonstrate that our AF-UMC exhibits superior performance against other state-of-the-arr methods.
1 Introduction
Multi-view data is usually collected from multiple sources, which is represented by several het- erogeneous features. For instance, in personalized online recommendations, diversified individual preferences are collected from various e-commerce platforms. To recommend reliable products, is necessary comprehensively integrate all these preferences. However, in practical scenarios, the collected multi-vier preference data is usually unaligned since different platforms do not store data in unified order in general. Under such conditions, the traditional multi-view learning methods lose their capability to fuse the unaligned multi-view data 38 37
'Gengyu Lyu the corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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H
(X's5)
(K,S5))
Projecting
23
Resulting in
Heterageneous
Mismatched Basis Spaces
Structurally Equivalent
Feature
Representations
Non-Unique Matching Results
(x² 5")
Projecting
(X
(X
dh
Heterogeneous
Ideal
Structural Mismatch
Structure
1
Matching
(X--s*) )
x
Representations
(a) Feature-Based Methods
(b) Structure-Base Methods
Figure 1: The drawbacks of existing feature-based and structure-based methods. (a) The unintended mismatched basis spaces of feature-based methods. Triangles in different colors denote samples of different categories, and solid arrows in different colors denote different basis vectors {z]) / Since Z and Z are independently constructed in each view without any cross-view constraints, the correspondences between basis vectors {(z) 2 S is c} are often incorrect across views (i.e., Z' and Z are often mismatched across views), which induces the heterogeneity of projected feature representations (H') H across views. (b) The undesired structural equivalence across basis vectors of structure- structure-based methods. Circles in different colors denote different basis vectors {x" ) (i.e., the nodes structure S"), and thicker black lines indicate closer structural relationship between the basis vectors. x and x3 are structurally equivalent when they have the coincident structural correspondence $ (i.e., they share the coincident neighbor node xf and have the coincident relationship with x)) [39]. When x tries to match the corresponding basis vector x on S² through structural correspondence, it will find correspondence on S which induces non-unique matching results. This easily leads to structural mismatched basis spaces (X2, X²) and heterogencous structure representations (S² S²).
candidates (x². x3) that have the coincident structural
The key learn from unaligned multi-view data lies in how fuse cross-view information with the unaligned sample features across different views. Existing unaligned multi-view clustering (UMC) methods, mainly divided into feature-based and structure-basedd provide effective solution, which prioritizes employing various alignment strategies on sample representations and then fuses these aligned representations into cross-view consistent representation for clustering. For instance, feature- based UMC methods
(35 3 8J construct an orthogonal matrix Z" E RexD, to represent a basis space with basis vectors {z 1. le in each view v, and obtain c-dimensional feature representation H E IR Nxe xc by projecting samples X Nx onto Z", formally expressed | Z || After that, H" is used for cross-view representation alignment through various alignment strategies, such introducing a learnable alignment matrix with O(N2) memory or Hungarian algorithm, and then the aligned feature representations are used to fuse the cross-view consistent representation H
min
X
H
H (H
s.t. Vo, ( II
In Eq. [I. (f-) indicates the associated alignment strategies, V and N are the number of views and samples, respectively. However, the constructed basis spaces ( l==1 are often mismatched across views due to the independent construction process |X" H"Z"" | without any cross-view constraints, as shown in Figure [ (a), which induces the heterogeneity of projected representations [H"YY and the heterogeneity makes trouble for subsequent alignment strategies to achieve ideal view-alignment results, leading to unreliable cross-view fusion. For structure- based methods [18 40 32). they project structure representation S" by a self-representation term ||X S| methods. Consequently, each row vector X can regarded as both sample feature and a basis vector, and the obtained structure representation S for X indicates the structure of both samples and basis vectors. In this case, their alignment strategies (b.) on {S") \V simultaneously perform both multi-view sample alignment and basis space matching, and then the aligned sample structures ($((S*) ) V are used to fuse the
"| which is an analogue
of the projection operation ||X" H 1| in feature-

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cross-view consistent structure S min Σ EI - S'X
S ?
Nevertheless, since the basis space X" is directly constructed using view niw-speccific sample features, the view-specific inherent heterogeneity is completely reserved and induces the heterogeneity for structures {S²) y across different views, where the alignment strategies on { also lose their expected capability. In addition, it is difficult to obtain ideally matched basis spaces through structural match, since structurally equívalent basis vectors disrupt the ideal matching results, as shown in Figure (b), which also induces the heterogeneity of representations {S") Je=I. deceiving the alignment and the subsequent multi-view fusion towards a biased direction. To sum up. the current methods, whether feature-based or structure- suffer from the common limitation: Their cross-view fusion operation depends on aligned representations, but the ideal view-alignment results often fail to be obtained by alignment strategies due to the heterogeneity of representations across views, inevitably inducing unreliable alignment-basee fusion.
To address the above issues, in this paper, we propose alignment-free consistency fusion framework AF-UMC for unaligned multi-view clustering, which directly extracts consistent representations from each view for globally fusing cross-view consistent representation and does not require additional alignment strategies. Specifically, we first construct a cross-view consistent basis space. On one hand, the basis space is designed to capture cross-view shared information from multiple views, where exclusive diversity is filtered out and the shared consistency is reserved. On the other hand, a Structural Clarity Regularization (SCR) is designed to prevent the basis space from learning structurally equivalent basis vectors and to encourage the basis space to capture matched information from different views. Afterwards, autoencoders are employed to extract consistent representations from each view by projecting view-specific data to the constructed basis space. Finally, these extracted representations are globally pulled together for further fusing cross-view consistent representation by a designed Instance Global Contrastive Fusion (IGCF), and then the clustering results are obtained by K-means clustering. During the whole process, different from previous methods that utilize alignment strategies with ((N2) complexity to align representations for cross-view fusion, AF-UMC directly extracts consistent representations from each view for global fusion, avoiding the additional cost of alignment strategies while mitigating the risk of fusing non-corresponding representations. In summary, the main contributions of this paper lie in:
We analyze a common problem in existing unaligned multi-view clustering methods: align- ment strategies often fail to achieve ideal view-alignment results due to the inherent hetero- geneity of representations across different views, inevitably leading their alignment-base cross-view fusion toward a biased direction.
We propose an alignment-free consistency fusion framework AF-UMC, which does not require additional alignment strategies and directly extracts consistent representations from each view by projecting view-specific data to a constructed cross-view consistent basis space, and then globally fuses them into cross-view consistent representation.
Extensive experimental results on various datasets demonstrate that our proposed model exhibits superior performance against other state-of-the-arr algorithms. Moreover, we conduct comprehensive ablation studies on both loss functions and model components, clearly demonstrating their effectiveness within our AF-UMC.
2 Related works
Multi-view clustering. Multi-view clustering aims unsupervisedly fuse multi-view data to
differentiate crucial clusters, and is a fundamental task in the fields of data mining (20 [4, 30 31 25
22] pattern recognition [19 29 4|l 13
9 6 28), etc. The key to dealing with such a problem lies in how to fuse cross-view information and obtain consistent representation for clustering. Current multi-view clustering methods are mainly divided into two categories, i.e., shallow methods and deep dearning-based methods. For instance, Wu et al. [34] propose a shallow method, which integrates multi-view samples into a unified tensor through matrix factorization and then utilizes a low-rank kernel tensor constraint to fuse cross-view consistent representation. Wang et al. 27 propose deep learning-basee method, which employs graph autoencoders to pull together structurally similar samples and then introduces contrastive learning for fusing cross-vier consistent representation.
3

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Alignment- Freem Free Consistency Extraction
Unaligned Samples
Consistent reprosentation H!
Global Contrastive Fusion
IGCF
H core
Consistent Space
2: SCR
Consistent Rasis Space
Structurally Equivalent
Facilitate
H
K-means
Xprv
Mitigate
Z
H
Cross- View Reconstruction
Basis Vectors
Projection of Samples
Global Center
Push
Pull Together
The overview of AF-UMC, which consists of two main stages: alignment-free consistency extraction and global contrastive fusion. In the first stage, construct cross-view consistent basis space through a cross-view reconstruction and a designed Structural Clarity Regularization (SCR), where autoencoders extract consistent representations from each view through projecting view-specific data to the constructed basis space. In the second stage, these extracted representations are globally pulled together for fusing cross-view consistent representation oore according to designed Instance Global Contrastive Fusion (IGCF), and then the final clustering results are obtained by K-means clustering.
Figure 2:
Unaligned multi-view elustering. Unaligned multi-view clustering aims cluster the multi-view data where the sample features are not completely aligned across views 35
3
18,
40,
32 21).
The key to dealing with such problem lies in how to fuse cross-view information for clustering under the unaligned sample features. Existing methods can be divided into two main categories, ie., feature-based methods and structure-based methods. For instance, Ji et al. S propose a feature-based method, which first introduces learnable alignment matrix with O(N)) memory to align multi-view feature representations, and then utilizes a low-rank kernel tensor constraint to capture cross- view consistency while fusing cross-viee view consistent feature representation. Xin et al. [35 also propose feature-based method, which employs the Hungarian algorithm to align multi-view feature representations and then introduces cross-view contrastive loss to pull together cross-view positive representations for fusing a cross-view consistent feature representation. Wen al. 32 propose structure-based method, which first extracts structure representation by self-representationn function in each view and introduces a learnable alignment matrix with (N memory structurally align cross-view samples, and then introduces low-rank kernel constraint to fuse aligned sample structures into cross-view consistent structure. Although these methods have achieved competitive performance, they still suffer from the same drawback: Their alignment strategies often fail to achieve ideal view-alignment results due to the heterogeneity of representations across different views, inevitably leading unreliable cross-view view fusion.
3 Method
In this paper, we propose an alignment- consistency fusion framework AF-UMC for unaligned multi-view clustering, which eliminates the requirement for alignment strategies and directly extracts consistent representations from each view to perform global cross-view consistency fusion. The AF-UMC is decomposed into two stages: Alignment-Free Consistency Extraction and Global Contrastive Fusion. the first stage, we extract consistent representations from each view by projecting view-specific samples to the constructed cross-view consistent basis space. In the second stage, these extracted representations are globally pulled together fuse the cross-view consistent representation for clustering. Figure illusstatees the overview of our proposed method.

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3.1 Problem definition
Given an unaligned multi-view dataset X = [X with V views, where X {x]}== E RNDD, denotes the sample features of u-th view, D, is the dimension of X" and is the number of samples. The goal of AF-UMC is to fuse multi-view information for separating multi-view data to pre-define clusters. Notably, the multi-view sample features cannot be directly fused since [xx] x²), p # q) are often derived from different samples in the unaligned multi-view dataset.
3.2
Alignment-free consistency extraction
In this stage, we construct a cross-view consistent basis space = {zi)i=1 E Rexd to capture cross- view consistency from multiple views and extract consistent representations from each view through projecting view-specific samples to the basis space. Specifically, we first construct the cross-view consistent basis space through multi-view samples reconstruction. As shown in Figure 2 Z is involved in decoding samples from multiple views and captures cross-view shared consistency, while filtering out view-specific diversity that does not overlap across the views. The decoding process of view v is denoted by (H", Z) H + X pre E IR xD., where H {br) E R
indicates the latent representation that is extracted by encoder E" (x") X + H". After that, we aim learn cross-view consistent structure Scon {S 1 E Rexe to constrain the structural consistency of captured information, which further facilitates Z to capture consistent information. To achieve this purpose, we relax the widely used orthogonal constraint Z be linearly independent, since it is difficult for orthogonal Z to learn structural relationships. This relaxation is formulated as:
zz = = rank(Z) =
where the similarity structure = 1= E Rex of is obtained by a scaled exponential form of cosine similarity cco((t))/T) and introduced to learn the consistent structure Scon where cos(2; 2;) indicates the cosine similarity between z; and zy, the exponential function is used to magnify the difference across similarity scores for obtaining a clearer similarity structure and Ty is a temperature coefficient. However, directly introducing S structural constraint may cause Z to learn structurally equivalent basis vectors, inducing structural mismatch of captured information (39]. To address this issue, we design Structure Clarity Regularization (SCR) to mitigate structural equivalence of { on S. Considering that structurally equivalent basis vectors (Z,, 2j) share the coincident
neighbor nodes and the same structural relationships with each neighbor node Zς [39] i.e., 8i,k 8j,kr we measure the structural equivalence between 2; and Zj as follows:
( sT)
Σ
(Si.6- 8,,,)
(4)
where the higher value of s5), indicates the lower structural equivalence between 2; and Zj- After measuring the structural equivalence across {2,)s=1 our Structure Clarity Regularization is designed to penalize the structural equivalence follows:
C,
Σ 1EKCGE
e-pl...))/71,
(5)
where the negative exponential function =P() encourages sp() toward higher values penalize structural equvalence across {zi]e=1- Through the above operations, Z can effectively capture cross-view consistency from multiple views, promoting autoencoders to directly extract consistent representations from each view through projecting view-specific samples to the basis space. To strengthen the capability of autoencoders in extracting consistent representations, a reconstruction loss between {D" (H", Z) and [X"]V
I is introduced as follows:
V
C,
X - D" H ZJ
Σ
x - DE (X"
(6)
The extracted consistent representations ( Joz1 are used in the next stage for cross-view fusion.
5

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3.3 Global contrastive fusion
To further fuse the extracted representations /- while avoiding fusing non-cor con-corresponding representations that derive from different sample instances, we bypass instance-to to-ettttn-ntneee fusion and perform cross-view fusion at global level. Specifically, we first calculate the global center h" of H" each view v, where h" = Σ /// After that, we select a view as central view core and bring hore closer to global centers { (("))e Te=1, v o core} for promoting Hoore to be globally consistent with {(H") J==1 e * core}:
LE =
core halle.
(7)
vyfcore
where core is set to the view with the largest original feature dimension since it usually provides more comprehensive description of samples and more representative global center for facilitating cross-view global fusion, and Heore is treated the fused cross-view consistent representation for subsequent clustering. However, such global-to-global operation only directly influences the global center h oore of Hoore failing ensure that each instance he effectively fuses multi-view global information to achieve global consistency. To solve this issue, we design the Instance Global Contrastive Fusion (IGCF) to introduce instance-to to-global contrast, where { {(h .h),, core # v} serves positive pairs for bringing each instance of Heare closer to global centers {h")V= core} while {hhçoe,, h hoore), # j} serves as negative pairs for reinforcing the discriminability across [hrore]] -i-1- Such strategy encourages each instance of Hore to effectively fuse multi-view global information while prompting the extracted consistent cluster structure Hoore be clearer. In addition, we mask the normally used cross-view negative pairs { {hhgrr,, "h"",, h,),, core & v}, since they often include the representation pairs from the same sample in unaligned multi-view data and hinder {(Horre H"), core * e) from achieving global consistency. Accordingly, our Instance Global Contrastive Fusion formulated as follows:
C. Σ
Σ log Σ JEn e
lggor)) / + (D h////'
(8)
where 7i is temperature coefficient, ( (hore .h")/// in denominator is used to prevent all instances (hoore) AN F==1 from collapsing onto centers to avoid poor separability.
=1 Theorem Assuming that H" [h hy = h 1h 1,2, ..., N,, and there exists a constant such that p(h (grre) > 8, = 1,2, .........
N holds, then
Σ 7( H 2 (V -1)log N 8Les
Theorem indicates that minimizing contrastive loss Le equal to maximizing mutual information between and global centers [[h"]V Tom1, * core], where the detailed proof is provided in Appendix D Finally, the fused cross-view consistent representation Hoore used for clustering with K-means. The whole loss function in our method AF-UMC is represented as:
L= AL, + 7Ce:
(10)
where À and y are trade-off coefficients.
3.4 Optimization
Our designed AF-UMC, consisting of multiple autoencoders and a basis matrix that indicates cross-view consistent basis space, is optimized by gradient descent algorithm. Specifically, the autoencoders and basis matrix are trained for reconstructing original samples, where the basis matrix optimized by Eqs. (5) and (6) for capturing structurally matched consistency, and autoencoders are optimized by Eq. (6) for extracting consistent representations from each view. Afterwards, global contrastive fusion operation conducted to fuse cross-view consistent representation by Eq. (8).
Finally, the cross-view consistent representation is used for clustering with the K-means algorithm.

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Table 1: Statistical characteristics of the ten datasets.
Data
Samples
Clusters
View dimensions
Caltech7-5
1400
40/254/1984/512/999
Handwritten
2000
10
240/76/216/47/64/6
Scene
4485
15
20/59/40
Caltech1024 102-5
9144
102
48/40/254/5 2/928
Hdigit
10000
10
784/256
Aloi
10800
100
77/13/64/125
NUSWIDEOBJ
30000
31
65/22//145774/129
NNISYMNIST
50000
10
784/784
Cifar10
50000
10
512/2048/1024
YoutubeFace
101499
31
64/5121664/647/838
4 Experiments
4.1 Experimental settings
Datasets. We employ ten widely-used multi-view datasets for comparative studies, which includes
six small-scale datasets of Caltech7-5 (4), Handwritten [26). Scene [3]. Caltech102-5 [4). Hdigir
[1] Aloi [15] and four large-scale datasets NUSWIDEOBJ [16], NoisyMNIST (24], CifarI0 (2].
YautubeFace 77. The specific characteristics of these datasets are listed in Table
The compared methods. In order to verify the effectivenesss of AF-UMC, employ six state-
of-the-art unaligned multi-view clustering methods for comparative experiments on small-scale
datasets, including MVC-UM (KDD, 2021) 40). T-UMC (TCYB, 2022) [1]. UPMGC (TNNLS,
2023) 32), FUMC (UCAL, 2024) 14] VuC (TMM, 2024) M. TUMCR (KDD, 2024) [8] Besides, considering that most unaligned multi-view clustering methods cannot be employed on large-scale datasets due to its excessive complexity, except for FUMC and OpVuC, we additionally employ four state-of-the-art aligned multi-view clustering methods for unaligned large-scale datasets, including LMVSC (AAAI, 2020) [T]. MFLVC (CVPR, 2022) 377. GCFAgg (CVPR, 2023) (38 and SCMVC (TMM, 2024) B3 Moreover, for the reliability our comparative experiments, all compared methods are implemented according to the source codes released by the authors, and the optimal parameters are set according to the suggestions in the corresponding literature.
Evaluation metrics. There are four widely-used metrics applied to quantitatively evaluate the performance of unaligned multi-view clustering methods, including Accuracy (ACC), Normalized Mutual Information (NMI), Purity (Pur) and Adjusted Rand Index (ARI), whose detailed definitions For each of the above metrics, the higher value indicates the better performance.
are illustrated
[7]
Implementation details. The encoder E" and decoder D are respectively formulated by ML.s with dimensions (D.,, 500, 500, 2000, 512, c) and {d, 2000, 500, 500, D,), where the activation function is ReLU, The consistent basis space Z is set to a matrix of e x d, where e is set to the number categories k and d is mini-batches of size 256 by using Adam optimizer [I] with learning rate of 0,0003 in Py Torch and respectively. All experiments are
512. During the whole process, AF-UMC trains 50 epochs on
23 framework. The hyperparameters y and \ are conducted on the same machine with the Intel(R) Xeon(R) Gold 6148 2.40GHz CPU, 8 GeForce RTX 3090 GPUs, and 512GB RAM.
4.2 Experimental results
Table 2 and Table 3 respectively r record the experimental comparisons on small-scale datasets and large-scale datasets, where the best and the second-best performance underlined, respectively. addition, FigureBJillsstn illustrates the visualization of clustering results each method the Handwritten dataset. According to Tables 2 and Figure 3 we can observe that:
highlighted in bold and
(1) Tables 23 except for ARI on Scene dataset, our AF-UMC is superior methods on all evaluation metrics, even has significant leading gap compared with second-best
comparing
7

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Table 2: Comparative results between AF-UMC and 6 state-of-the-ane art methods six small-scale datasets, where the best results are presented in bold and the second-best are in underline.
Method
Dataset
Metric
MVC-UM
UPMGC
FUMC
OpVuC
TUMCR
T-UMC
AF-UMC
ACC
0.2785
0.8079
0.2044
0.3279
0.2557
0.4079
0.8721
NMI
0.1038
0.7137
0.0269
0.1229
0.0721
0.3271
0.7798
Caltech7-5
ARI
0.0875
0.7034
0.0137
0.0820
0.0612
0.3180
0.7485
PUR
0.3014
0.8079
0.2098
0,3851
0.2771
0.2771
0.8721
ACC
0.7465
0,6270
0.1946
0.1465
0,4830
0.7720
0.9035
NMI
0.7230
0,5860
0.0652
0.0181
0.3853
0.6703
0.8205
Handwritten
ARI
0.6305
0.5014
0.0431
0.0052
0.3627
0.6564
0.8002
PUR
0.7465
0.6346
0.2004
0.1705
0.4980
0.7720
0.9035
ACC
0.2608
0,1386
0.1647
0.3275
0.2990
0.3882
0,4190
NMI
0.2787
0,0576
0,0904
0.3409
0.2488
0.3816
0,4217
Scene
ARI
0.1947
0.0434
0.0731
0.1836
0.2171
0.2856
0.2548
PUR
0.2791
0.1503
0.1727
0.3741
0.3398
0.4239
0.4593
ACC
0.0638
0.0930
0.0576
0.1355
0.0977
0.1017
0.2275
NMI
0.2150
0,1883
0,1472
0.2974
0.2066
0.2512
0,4528
Caltech102-5
ARI
0.0571
0.0742
0.0431
0.0958
0.0639
0.0741
0.1755
PUR
0.1846
0.1704
0.1226
0.2794
0.1943
0,2348
0.4269
ACC
0.4627
0.4087
0.3603
0,3994
0.1551
0.4993
0.6950
NMI
0,4418
0,3700
0.3355
0,3151
0.0255
0,4387
0,6000
Hdigit
ARI
0.3987
0.2943
0.0216
0.2054
0.0203
0.3708
0.5194
PUR
0.5031
0.4545
0.4289
0.4173
0.1679
0.5398
0.6980
ACC
0.3543
0.0383
0.0874
0,1432
0.2330
0,5057
0.5399
NMI
0.6039
0.1118
0.2220
0,4105
0.3363
0.6536
0.7590
Aloi
ARI
0.2381
0,0213
0,0351
0.0897
0.1853
0.3859
0,4151
PUR
0.3692
0.0402
0.0897
0.1747
0.2429
0.5238
0.5816
MVC-UM
UPMGC
FUMC
OpVuC
TUMCR
T-UMC
AF-UMC
Figure 3: The visualizations of the clustering results of different methods on Handwritten dataset.
methods. Especially the Hdigit dataset, the improvements the second-best method 19.57%, 15.57%, 10.99%, and 15.82% on ACC, NMI, ARI and PUR, respectively. These experimental results demonstrate the effectiveness of AF-UMC and attribute such success designed alignment- free consistency fusion framework, which bypasses undesired alignment strategies and obtains cross-view consistent representation with a clearer cluster structure through global fusion.
(2) In Figure 3 we select unaligned multi-view clustering methods to conduct the visualization comparisons of clustering results with our proposed AF-UMC. We can observe that our AF-UMC exhibits a clearer cluster structure than all other methods, which demonstrates the superiority of AF-UMC in fusing consistent representation from unaligned multi-vien data.
4.3 Model analysis
Convergence analysis. Figure shows shows the convergence curves of AF-UMC on Caltech7-5, NoisyM NIST datasets, where the values of loss and evaluation metrics are illustrated in each subfigure. According Figure 4 we can observe that loss drops significantly at the beginning of the iteration process and then gradually reaches stable value as the number of iterations increases. And the evaluation metrics gradually increase and fluctuate in a narrow range. These results verify the convergence of our proposed AF-UMC.

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Table 3: Comparative results between AF-UMC and 6 state-of-the-ie methods on four large-scale
datasets. "_" means that the code can't 't be run due to its excessive time or space complexity.
Method
Dataset
Metric
LMVSC
MFLVC
GCFAgg
SCMVC
FUMC
OpVuC
AF-UMC
ACC
0.0674
0.0973
0.0455
0.0474
0.0945
0.1016
0.1216
NMI
0.0263
0.0047
0.0057
0.0089
0.0775
0.0864
0.1041
NUSWIDEOBJ
ARI
0.0158
0.0004
0.0002
0,0008
0.0193
0.0213
0.0306
PUR
0.0842
0.1268
0.1223
0.1269
0.1914
0.2041
0.2208
ACC 0.2416
0.1131
0.1078
0.1311
0.2843
0.5111
0.5899
NMI
0.1512
0.0015
0.0006
0.0099
0.2329
0.4241
0.4982
NoiyMNIST
ARI
0.1835
0.0007
0.0001
0.0051
0.2231
0.3308
0.4154
PUR
0.2908
0.1137
0.1088
0,1357
0.3457
0.5377
0.6247
ACC 0,3961
0.3550
0.1284
0,3831
0,2174
0.8008
0.8453
NMI
0,3323
0.0067
0.1975
0,0892
0.6872
0.7025
Cifar10
ARI
0.3178
0.1074
0.0033
0.1464
0.0747
0.6284
0.6932
PUR
0.4966
0.3552
0.1324
0,3969
0.2191
0.8008
0.8453
ACC
0.0405
0.0737
0.0414
0,0510
0.0717
0.1625
NMI
0.0169
0.0049
0.0029
0.0187
0.0366
0.1444
YoutubeFace
0.0105
0.0008
0.0001
0.0018
0.0068
0.0270
PUR
0.1132
0.2662
0.2662
0.2662
0,2662
0.2851
ACC
ACC
p.5
35
25.8 0.8
MMI
0.7
ARI
25.6
ARI
7.5
p6
b5
0.2
0.3
33
0.
0.4
loss
25.4 0.4
loss
7.0
9A
40
01
0.2
25.2
0.2
6.5
10
0.0
10
20
30
50
25.0
0.0
10
20
30
40
58.0
10
10
10
10
10
(a) Caltech7-5
(b) NoiSyMNIST
(c) Caltech7-5
(d) NoiSy
Figure 4: The convergence analysis and parameter analysis Caltech7-5 and NoisyMNIST datasets.
Parameter sensitivity analysis. We experimentally evaluate the effect of hyperparameters the clustering performance of AF-UMC, which includes and A. Figure shows the NMI metric value of AF-UMC on Caltech7-5, NoisyMNIST datasets, where y and varied from 10-3 to 10 According to Figure the clustering results of AF-UMC are insensitive to both 7 and À ranging from 0.1 10. In our experiments, set y and 1.
Ablation study. We conduct two series of ablation studics from the perspective of loss functions and model components Caltech7-5 and NoisyMNIST datasets. Table records the ablation studies of different loss functions, where L, is the loss to reconstruct original samples, L, is the loss to capture structurally matched consistency and C. is the loss to globally fuse the extracted representations. Table records the ablation studies of different model components, where BAE represents the autoencoders with consistent basis space and Ins-Glo represents the instance-to to-tncee---glbball contrast operation. According to Tables we can find that:
(1) In Table 4 (C) is superior to (B), which indicates that capturing structurally matched consistency into basis space helpful autoencoders extracting consistent representations from each view and further improves the performance of cross-view fusion. Meanwhile, (C) also shows better clustering performance than (A), which indicates that our globally fused cross-view consistent representation contains clearer cluster structure for achieving better clustering performance.
(2) In Table 5 (a) replaces the designed instance-to to-g-lbal contrast operation with global-to- to-global operation as Eq. (7). and (b) replaces the BAE with traditional autoencoders. According to Table (c) shows better performance than (a), which indicates that our designed instance-to-globe contrast operation effectively fuses multi-view samples into a cross-view consistent representation for more effective clustering. Meanwhile, (c) outperforms (b), which demonstrates that the BAE successfully extracts consistent representations from each view for promoting subsequent global cross-view fusion.

---

Table 4: Ablation studies loss functions of AF-UMC Caltech7-5 and NoisyMNIST datasets.
Loss
Caltech7-5
NoisyMNIST
C.
La Ce
ACC
NMI
PUR
ARI
ACC
NMI
PUR
ARI
(A)
0.8079
0.7422
0.8164
0.6880
0,4860
0.4063
0.5257
0.2835
B
0.8014
0.6983
0.8014
0.6406
0.5046
0.4573
0.5501
0.3297
0.8721
0.7798
0.8721
0.7485
0.5899
0.4982
0.6247
0.4154
Table 5: Ablation studies on model components of AF-UMC on Caltech7-5 and NoisyMNIST datasets.
Components
Caltech7-5
NoisyMMIST
BAE
Ins-Glo
ACC
NMI
PUR
ARI
ACC
NMI
PUR
ARI
(8)
0.8107
0.7498
0.8107
0.6820
0,4826
0.4256
0,5328
0.2919
(b)
0.7107
0,5989
0.7107
0,5312
0.4793
0.4136
0,5202
0,2862
(c)
/
0.8721
0.7798
0.8721
0.7485
0.5899
0,4982
0.6247
0.4154
5
Conclusion
In this paper. we propose an alignment-free consistency fusion framework named AF-UMC for unaligned multi-view clustering. Different from previous methods that conduct view-alignment then fuse aligned feature representations, our proposed method directly extracts consistent representations from each view for global multi-view fusion. Our proposed method significantly mitigates the degraded performance caused by undesired view-alignment results in previous methods while greatly reducing algorithm complexity and enhancing its efficiency. Extensive experimental results on various datasets have verified the effectiveness of our proposed method.
6
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 62306020), the Young Elite Scientist Sponsorship Program by BAST (No. BYESS2024199,, the Beijing Natural Science Foundation (No. L244009). the National Key Research and Development Program of China (No. 2023YFB3107100, and the Central Guidance for Local Scientific and Technological Development Fund (No.2024ZY0124).
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Align while fusion: generalized nonaligned IEEE Transactions on Neural Networks and
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Appendix
We provide more details and results about our work in the appendices. Here are the contents:
Appendix
Commonly used notations.
Appendix
Training process of AF-UMC.
Appendix
Complexity analysis of AF-UMC.
Appendix
Proof of Theorem |
Appendix
Additional experiment results.
A
Commonly used notations
T shows the commonly used notations and the associated definitions.
Table Notations and Definitions
Notation
Definition
X
The samples of the w-th view The basis space of the v-th view
Z
Z
The cross-view consistent basis space
e E" D H
The e basis vectors in Z The encoder of the v-th view The decoder of the v-th view The extracted representation of the v-th view The similarity structure across basis vectors {z) The global center of H"
h
Training process of AF-UMC
Algorithm outlines the execution flow for AF-UMC. At each training epoch t, autoencoders first extract consistent representations {H") v=1 by projecting multi-view samples {X"} Je=1 onto the cross-view consistent basis space Z. Then, global contrastive fusion globally pulled together these extracted representations (H* to fuse a cross-view consistent representation Horr After T training epochs, the final clustering results are obtained by performing K K-means clustering on Hore
Ju=1
Algorithm 1 The Training Process of AF-UMC.
Input: Unaligned multi-view data X
(X") number of clusters training epochs T.
Output:
Clustering results
1: Initialize autoencoders (-"()), ())) and cross-view consistent basis space Z.
2: for epoch t 1 to T:
3: for view 1 to V:
4: Extract consistent representation H by projecting X" onto Z.
5: end for
6: Globally bring {H") J closer to fuse a cross-view consistent representation Hoore
7: Optimize model by Lr, C, and L...
8: end for
9: Perform K-means clustering Hoore
C Complexity analysis of AF-UMC
We analyze our proposed AF-UMC in terms of space/time complexity.
Space Complexity: our method, the memory costs contain a basis space matrix E Rexd V autoencoders and V representation matrices [H'
Ju=1 E Ng where the space complexity of an
21

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autoencoder is d( and / is the number of MLP layers. As result, the total space complexity of our AF-UMC is Oedd VINd+ VNcc..
Time Complexity: The time cost of AF-UMC arises from three parts: (1) O(VNd + e² + dV) the cost of computing three loss functions. (2) O(VINd), the cost of optimizing V autoencoders. (3) O(ed), the cost of optimizing cross-view consistent basis space Z. Therefore, the total time cost of AF-UMC is OV Nd + c + +NV2d VINd + cd).
D Proof of theorem 1
In this part, we want to prove that minimizing contrastive loss L. is equal to maximizing mutual
information. For expressing more clearly, we first construct H" = [hh]] 1.N
1j=1 where H" E NR and
hy he indicates the j-th row of H The proof is motivated by 22 43
Proof. Le is our designed contrastive loss, which is formulated as:
( //1
L.
log
N
N (h .ggor / + Ne (hgoree, h/) Σ
VS ustrore
1,1= 14
We assume that p(h
h.)
p(hfore) ) p(h)), i + j and let N₁ EE N
1i=1 p(h (php)
plafore .i))
we have:
N
N
p (h core hī)) log (hfore, hz)
HD
(hgore) p (hz)
i=1
1=1
(hrare hy)
p (hoore, h2) p (h. core .n.) log
p (h. core h2) log
+ ΣΣ
(hçore) p (h:)
p (hoore) ) (hz)
jti
N
p (hgore, h!))
Σ (hoore, h2) log
P (hgore) (h:) N₁ N₁
i=1
p(hgore R
N Σ (hoore, h:) log
pl p(hgore )p (b))
(hoore, .h:) log.
N Since positive pairs are correlated, we have the estimate: p(h core hT) 2 p(h (((p (h))). According to (36], we have popeo ~ # ....... N, and d(beare /T x p(h. u by))
then:
p(h (.) p(hgore (hb)) (hg)
[(H core HD =
(hoore, h log
w=l
V
N
p (hoore, h: (hoore) (h:)
(h core hit)) log
sylcore
p(h phoure b2) ((h
h h core) log
N₂
cml
1=1
uplcore
V
p (hrore, h
log
N - 1 +
p (hoore) p (h))
sgloare
log
simmmsvn (hoore h/)/T + (V log N
N
Σ Ejji siu(h (ngore hg3/ sim
s=1
1=1
ugfeore
(V - log ôC..
22

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E
Additional experiment results
Visual comparison on large-scale dataset. Figure 5 shows the visual comparison between our AF-UMC and the existing SOTA methods (LMVSC TO]. MFLVC (37.. GCEAgg [8), SCMVC [3]. FUMC [4] OpVuC (3) on the large-scale dataset Cifar 10. We can observe that our AF-UMC
exhibits clearer cluster structure than all other methods, which demonstrates the superiority of AF-UMC in fusing large-scale unaligned multi-view data.
LMVSC
MFLVC
GCFAgg
SCMVC
FUMC
Vp
AF-UMC
Figure 5: The visualizations of the clustering results of different methods Cifart0 dataset.
Additional Ablation study. (1) Ablation study reconstruction loss L,: Table shows the ablation study on L, From Table 7 (B) shows a sigmificant performance improvement over (A), indicating that L, plays critical role in improving representation quality. (2) Ablation study cross-view consistent basis space Z: Considering that ablating also removes the loss L, defined on Z, is difficult to directly evaluate the impact of the individual basis space Z. To address this issue, we design the following ablation study in Table where (a) ablates both L, and Z. and (b) only ablates L.. From Table (b) shows better performance than (a), demonstrating the effectiveness of cross-view consistent basis space in prompting autoencoders to extract consistent representations from each view.
Table Ablation studies on loss functions of AF-UMC on Caltech7-5 and NoisyMNIST datasets.
Loss E.
Caltech7-5
Noisy MNIST
ACC
NMI
PUR
ARI
ACC
NMI
PUR
ARI
(A)
0.5014
0.4366
0.5429
0.3056 0.7485
0.4637
0.3924
0.5057
0.2826
0.8721
0.7798
0.8721
0.5899
0.4982
0.6247
0.4154
Table 8: Ablation studies on model components of AF-UMC on Caltech7-5 and NoisyMNIST datasets.
Components
Caltech7-5
NoiSyMNIST
ACC
NMI
PUR
ARI
ACC
NMI
PUR
ARI
(a)
w/o (C, & Z)
0.7107
0.5989
0.7107
0.5312
0.4793
0.4136
0.5202
0.2862
(b)
w/o C.
0.8014
0.6983
0.8014
0.6406
0.5046
0.4573
0.5501
0.3297
23


Reviews:
[{"review_id": "0FkGFGpjvk", "reviewer": "Reviewer_pBuo", "summary": "The work explores unaligned multi-view clustering problem and propose that due to heterogeneity of various views, current alignment strategies are not fully effective. To address this issue, the authors proposed an alignment-free method, which builds a cross-view consistent basis space and extracts consistent representations from each view by projecting view-specific data onto the consistent basis space. Several experiments are conducted to support the efficacy of the method.", "questions": "1. Please give a more detailed explanation to demonstrate how the low-rank constraint helps to capture cross-view consistency into the basis space.\n2. What are the justifications of setting the central view $core$ to the view with the largest original feature dimension?\n3. What are the differences between the structurally equivalent basis vectors and the normal basis vectors?", "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": "##  Strengths:\n\n1. The authors proposed a new problem of the ineffectiveness of existing alignment strategies due to view heterogeneity.\n2. The authors provided a novel alignment-free solution by directly learning a consistent representation for global fusion;\n3. Sufficient analyses and experiments are provided;\n\n## Weaknesses:\n\n1. Some details are not fully explained. Especially, the reason for the mismatched constructed basis spaces is not well analyzed.\n2. It seems that the ablation study does not consider all the cases. Please conduct the missing cases or explain the potential reasons for not covering this case.\n3. The clarity might need further improvement by specifying some sentences to make them clearer.", "was_revised": true, "final_justification": "The authors have addressed my primary questions, and by comprehensively considering the novelty and technical soundness, I will maintain my previous rating.", "rebuttal": "************ **For Weakness #1:** ************\n---------------------------------------------------------\n\nExisting unaligned multi-view clustering methods fall into two main categories: feature-based methods and structure-based methods, whose reasons for the mismatched constructed basis spaces are as follows: (1) **Feature-based methods:** The basis space $\\mathbf{Z}^v$ is constructed by intra-view sample reconstruction and only captures information from a single view $v$, as reflected by $\\\\|\\mathbf{X}^v - \\mathbf{H}^v\\mathbf{Z}^v\\\\|\\_{F}^{2}$. Due to the discrepancy in information across different views, it is unlikely for both basis vectors $(\\mathbf{z}\\_i^v, \\mathbf{z}\\_i^u)$ to coincidentally capture the matching information from different views $(v, u)$, respectively, thereby leading to mismatches across basis spaces $\\\\{\\mathbf{Z}^v\\\\}\\_{v=1}^V$. (2) **Structure-based methods:** As self-representation function $\\\\|\\mathbf{X}^v - \\mathbf{S}^v\\mathbf{X}^v\\\\|\\_{F}^{2}$, the basis space $\\mathbf{X}^v$ (right) is directly constructed using the samples $\\mathbf{X}^v$ (left) in each view $v$. Due to the misalignment across samples $\\\\{\\mathbf{X}^v\\\\}_{v=1}^V$, the basis spaces constructed by different $\\mathbf{X}^v$ are naturally mismatched across views. \n\n---------------------------------------------------------\n\n\n************ **For Weakness #2:** ************\n---------------------------------------------------------\n\nThank you for the suggestions. We supplement the missing cases as follows. **(1) Ablation study on reconstruction loss $\\mathcal{L}_{r}$:** Table 1 shows the ablation study on $\\mathcal{L}\\_{r}$. From Table 1, (b) shows a significant performance improvement over (a), indicating that $\\mathcal{L}\\_{r}$ plays a critical role in improving representation quality. **(2) Ablation study on cross-view consistent basis space $\\mathbf{B}$:** Considering that ablating $\\mathbf{B}$ also removes the loss $\\mathcal{L}\\_{s}$ defined on $\\mathbf{B}$, it is difficult to directly evaluate the impact of the individual basis space $\\mathbf{B}$. To address this issue, we design the following ablation study in Table 2, where (a) ablates both $\\mathcal{L}\\_{s}$ and $\\mathbf{B}$, and (b) only ablates $\\mathcal{L}\\_{s}$. From Table 2, (b) shows better performance than (a), demonstrating the effectiveness of cross-view consistent basis space in prompting autoencoders to extract consistent representations from each view.\n\n**Table 1**\n|         | Loss functions      | ***Caltech7-5***  |    |  |                                | ***NoisyMNIST***|    |    |    |\n|:---------:|:---------------:|:--------:|:------:|:------:|:------:|:--------:|:------:|:------:|:------:|\n|         | $\\mathcal{L}\\_{r}$   | **ACC**  | **NMI** | **PUR** | **ARI** | **ACC**  | **NMI** | **PUR** | **ARI** |\n| (a)     |    | 0.5014   | 0.4366 | 0.5429 | 0.3056 | 0.4637   | 0.3924 | 0.5057 | 0.2826 |\n| (b)    | ✓ | **0.8721** | **0.7798** | **0.8721** | **0.7485** | **0.5899** | **0.4982** | **0.6247** | **0.4154** |\n\n**Table 2**\n|         |   | ***Caltech7-5***   |        |  |                     | ***NoisyMNIST***  |       |           |           |\n|:---------:|:---------:|:----:|:--------:|:-------:|:--------:|:------:|:------:|:------:|:------:|\n|         | Without (w/o) | **ACC**  | **NMI** | **PUR** | **ARI** | **ACC**  | **NMI** | **PUR** | **ARI** |\n| (a)     |   w/o $\\mathcal{L}_{s}$ & $\\mathbf{B}$| 0.7107   | 0.5989 | 0.7107 | 0.5312 | 0.4793   | 0.4136 | 0.5202 | 0.2862 |\n| (b)     | w/o $\\mathcal{L}_{s}$      | **0.8014** | **0.6983** | **0.8014** | **0.6406** | **0.5046** | **0.4573** | **0.5501** | **0.3297** |\n\n---------------------------------------------------------\n\n\n************ **For Weakness #3:** ************\n---------------------------------------------------------\n\nWe will revise the relevant sentences to make them clearer in the revised manuscript.\n\n---------------------------------------------------------\n\n************ **For Question #1:** ************\n---------------------------------------------------------\n\nLow-rank constraint can reduce the diversity components from different views and promote the basis space to focus more on capturing cross-view consistency. Specifically, consider $V$ representations $\\\\{\\mathbf{H}^v\\\\}\\_{v=1}^V$ from $V$ views, where $\\mathbf{H}^v \\in \\mathbb{R}^{N \\times d}$. We vertically concatenate these representations into a matrix $\\mathbf{H}^{all}=[\\mathbf{H}^1; \\cdots; \\mathbf{H}^V] \\in \\mathbb{R}^{VN \\times d}$, and then partition it into $K$ blocks according to the ground-truth class labels:\n$$\\mathbf{M}=\\begin{bmatrix}\\mathbf{M}^1;\n\\\\ \\cdots ;\n\\\\ \\mathbf{M}^k;\n\\\\ \\cdots ;\n\\\\ \\mathbf{M}^K\n\\end{bmatrix}\n$$\nwhere $\\mathbf{M}^k \\in \\mathbb{R}^{N_k \\times d}$ denotes the representations of the $N_k$ samples in class $k$. In each block $\\mathbf{M}^k$, diversity components encourage the representations to be dissimilar across views and drive $\\mathbf{M}^k$ to be high-rank, which indicates that diversity components tend to push $\\mathbf{M}^k$ into a high-rank space since the diversity components cannot be effectively represented in a low-rank space. In contrast, consistency components encourage the representations to be similar across views within each block $\\mathbf{M}^k$ and can be effectively represented in a low-rank space. Therefore, by appropriately lowering the rank of the basis space $\\mathbf{B}$, the diversity components can be suppressed, enabling $\\mathbf{B}$ to capture cross-view consistency more effectively.\n\n---------------------------------------------------------\n\n************ **For Question #2:** ************\n---------------------------------------------------------\n\nIn multi-view datasets, lower-dimensional views often provide a more limited and fragmented description of the samples, and the representations extracted from these views usually exhibit a biased or unstable global center. Consequently, these views are less suitable as the central view for cross-view global fusion. Instead, the view with the largest original feature dimension provides a more comprehensive description of the samples, promoting the extracted representations with a more stable global center. Therefore, the view with the largest original feature dimension is more suitable as the central view for cross-view global fusion.\n\n---------------------------------------------------------\n\n************ **For Question #3:** ************\n---------------------------------------------------------\n\nThere are two differences between the structurally equivalent basis vectors and the normal basis vectors. (1) **Intrinsic property:** Structurally equivalent basis vectors can be interchanged without changing the graph structure [1], while normal basis vectors cannot. Specifically, consider a graph $G^1=(\\mathbf{Z}^1, \\mathbf{S}^1)$, where $\\mathbf{Z}^1$ denotes the vertices (i.e., basis vectors) and $\\mathbf{S}^1$ denotes the graph structure. Structurally equivalent basis vectors $(\\mathbf{z}_i^1, \\mathbf{z}_j^1)$ can be interchanged in $\\mathbf{Z}^1$ without changing $\\mathbf{S}^1$, while normal basis vectors cannot [1]. (2) **Influence on structure matching:** When matching $G^1$ with another graph $G^2=(\\mathbf{Z}^2, \\mathbf{S}^2)$ using a structure-based matching function $\\varPhi(\\mathbf{S}^1, \\mathbf{S}^2)$, multiple matching results $\\\\{\\mathbf{Z}^1 \\xrightarrow{m} \\mathbf{Z}^2\\\\}$ arise by interchanging structurally equivalent $(\\mathbf{z}_i^1, \\mathbf{z}_j^1)$ without changing the value of $\\varPhi(\\mathbf{S}^1, \\mathbf{S}^2)$. In this case, structurally equivalent basis vectors hinder the ideal and unique matching between $\\mathbf{Z}^1$ and $\\mathbf{Z}^2$, whereas normal basis vectors do not have this influence.\n\n[1] Yang, Dominic, Yurun Ge, Thien Nguyen, Denali Molitor, Jacob D. Moorman, and Andrea L. Bertozzi. \"Structural equivalence in subgraph matching.\" IEEE Transactions on Network Science and Engineering 10, no. 4 (2023): 1846-1862."}, {"review_id": "14bFa4mIfH", "reviewer": "Reviewer_UEME", "summary": "This paper finds that current Unaligned Multi-view Clustering (UMC) methods often fail to achieve ideal view-alignment results, inevitably inducing their alignment-based cross-view fusion towards a biased direction. Different from current methods, this paper provides an alignment-free consistency fusion framework named AF-UMC, which first builds a cross-view consistent basis space and extracts consistent representations by projecting original samples of each view into this basis space, and then fuses cross-view consistent representations by pulling together these extracted representations at a global level, which realizes the alignment-free consistency fusion.", "questions": "1.\tWhy do the authors constrain the basis vectors to be linearly independent? Are there any negative effects if there exist linearly dependent basis vectors in the basis space?\n2.\tIn my opinion, the cosine similarity function could quantify the similarity across basis vectors as their structural similarity. Why do the authors apply an additional exponential function to the obtained cosine similarity?\n3.\tIn global contrastive fusion, each instance of fused cross-view consistent representation is encouraged to be closer to the global centers. Won’t this operation destroy the cluster distributions and make the distributions indistinguishable?", "limitations": "The authors only provide the visualizations of clustering results on small-scale datasets, but the visualizations on large-scale datasets are ignored. It is crucial to provide the visualizations on large-scale datasets in the manuscript to further verify the effectiveness of AF-UMC.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "***Strengths***\nThe work is well-motivated by analyzing the problem of the view-alignment operation existing in current UMC methods. Besides, the designed alignment-free consistency fusion framework is novel and effective in mitigating the degraded fusion performance caused by the undesired view-alignment operation. The extensive experimental results on various multi-view datasets demonstrate the superiority of the proposed AF-UMC.\n\n***Weaknesses***\n1.\tThe authors merely point out that current methods often fail to ideally align heterogeneous representations across views, but the specific process is not provided in this paper. The authors should introduce the alignment process of existing methods in detail and further note how the heterogeneous representations hinder the alignment process from achieving ideal view-alignment results.\n2.\tThe cross-view consistent basis space is designed to participate in sample reconstructions of multiple views, where it seems inevitable to receive view-specific diversity from each view. I want to know whether the view-specific diversity will weaken the consistency within the basis space.\n3.\tFor the ablation studies on loss functions of AF-UMC, an ablation study on $ \\mathcal{L}_{r} $ is missing here.", "was_revised": true, "final_justification": "The rebuttal solves most of my concerns, and therefore I decide to maintain my positive score.", "rebuttal": "************ **For Weakness #1:** ************\n------------------------------------------------------------\n\n\nThe alignment processes of existing methods are mainly divided into two categories: **Learning an alignment matrix $\\mathbf{P}$** and **Employing the Hungarian algorithm**. Assuming there are two unaligned and heterogeneous representations $(\\mathbf{H}^1, \\mathbf{H}^2)$ from two views, and for a specific sample instance $k$, $((\\mathbf{h}^1\\_i)\\_k, (\\mathbf{h}^2\\_j)\\_k)$ denote its corresponding unaligned representations in $(\\mathbf{H}^1, \\mathbf{H}^2)$. (1) **Learning an alignment matrix $\\mathbf{P}$:** These approaches assume that the consistent representations from each instance $k$ have a lower featural discrepancy, and attempt to learn an alignment matrix $\\mathbf{P}$ by minimizing the discrepancy loss between $\\mathbf{P}\\mathbf{H}^1$ and $\\mathbf{H}^2$. However, due to the inherent heterogeneity between $\\mathbf{H}^1$ and $\\mathbf{H}^2$, the representations from the same instance often exhibit a large featural discrepancy, making it difficult to align them by discrepancy minimization, thereby hindering the alignment process. (2) **Employing the Hungarian algorithm:** Unlike the above methods aligning representations by extra loss terms, these methods directly employ the existing Hungarian algorithm, which calculates the alignment costs of each possible pair $(\\mathbf{h}^1_i, \\mathbf{h}^2_j)$ from $\\mathbf{H}^1=\\\\{\\mathbf{h}^1_i\\\\}\\_{i=1}^N$ to $\\mathbf{H}^2=\\\\{\\mathbf{h}^2_j\\\\}_{j=1}^N$ by a relevant cost function, e.g., Euclidean distance, cosine distance, etc, and then select the alignment result with the lowest total costs. However, due to the heterogeneity between $\\mathbf{H}^1$ and $\\mathbf{H}^2$, the alignment cost for some true corresponding pairs (e.g., $((\\mathbf{h}^1_i)_k, (\\mathbf{h}^2_j)_k)$) is often higher than that for some false corresponding pairs (e.g., $((\\mathbf{h}^1\\_i)\\_k, (\\mathbf{h}^2\\_p)\\_o), k \\neq o$), where it is hard for Hungarian algorithm to select the true corresponding pairs for each instance, thereby hindering the alignment process.\n\n------------------------------------------------------------\n\n************ **For Weakness #2:** ************\n------------------------------------------------------------\n\n\nThe basis space only receives diversity at the beginning of model training. As training progresses, the diversity is gradually filtered out and will not weaken the consistency within the basis space. Specifically, the diversity only represents the information from a specific view, and has limited capacity in reconstructing samples from multiple views. Therefore, when the basis space is designed to reconstruct samples from multiple views, it tends to prioritize capturing cross-view consistency, facilitating the diverse information to fade over the model training. Besides, we constrain the basis space to be low-rank for capturing more compact consistency, which further reduces view-specific diversity.\n\n------------------------------------------------------------\n\n\n************ **For Weakness #3:** ************\n------------------------------------------------------------\n\n\nTable 1 shows the ablation study for reconstruction loss $\\mathcal{L}\\_{r}$ on ***Caltech7-5*** and ***NoisyMNIST*** datasets, where (b) shows a significant performance improvement over (a), indicating that $\\mathcal{L}_{r}$ plays a critical role in improving representation quality.\n\n**Table 1**\n|         | Loss functions      | ***Caltech7-5***  |    |  |                                | ***NoisyMNIST***|    |    |    |\n|:---------:|:---------------:|:--------:|:-------------:|:-----------:|:------:|:--------:|:------:|:------:|:------:|\n|         | $\\mathcal{L}_{r}$   |          **ACC**  | **NMI** | **PUR** | **ARI** | **ACC**  | **NMI** | **PUR** | **ARI** |\n| (a)     |    | 0.5014   | 0.4366 | 0.5429 | 0.3056 | 0.4637   | 0.3924 | 0.5057 | 0.2826 |\n| (b)    | ✓ | **0.8721** | **0.7798** | **0.8721** | **0.7485** | **0.5899** | **0.4982** | **0.6247** | **0.4154** |\n\n------------------------------------------------------------\n\n************ **For Question #1:** ************\n------------------------------------------------------------\n\n\nThe linearly independent basis vectors have a characteristic that each of them cannot be linearly represented by others, which helps eliminate repeated consistent information (i.e., linearly dependent information) in the consistent basis space. If linearly dependent basis vectors exist, some consistent information will be repeatedly encoded across different basis vectors and samples with consistent information can be represented by different basis vectors, which intensifies the difference in sample representations and degrades the consistency of representations.\n\n\n------------------------------------------------------------\n\n************ **For Question #2:** ************\n------------------------------------------------------------\n\n\nThe exponential function $e^{(\\cdot)/\\tau_g}$ maps the value range $(-1, 1)$ of cosine similarity into a broader range $(0, e^{1/\\tau_g})$, which amplifies structural differences among basis vectors, encouraging a clearer and more distinguishable similarity structure.\n\n------------------------------------------------------------\n\n************ **For Question #3:** ************\n------------------------------------------------------------\n\nThis operation will not destroy the cluster distribution and will not make the distribution indistinguishable. In addition to pulling instances toward global centers, the global contrastive fusion also introduces negative pairs to push dissimilar instances apart, preventing instances from being overly concentrated around the centers and preserving the distributional separability. Besides, Figure 3 provides the visualizations of the clustering results (see paper), where our AF-UMC shows a clear cluster distribution, confirming that this operation will not destroy the cluster distribution and will not make the distribution indistinguishable.\n\n------------------------------------------------------------\n\n************ **For Limitation #1:** ************\n------------------------------------------------------------\n\n\nWe perform the visualization comparison with other SOTA methods (FUMC, GCFAgg, LMVSC, MFLVC, OpVuC, SCMVC) on the large-scale dataset ***Cifar10***. Since we cannot directly provide a link to the visual results, we instead describe their distribution as follows: (1) FUMC, LMVSC, GCFAgg, MFLVC, OpVuC and SCMVC show a poor inter-class separation. (2) Our AF-UMC achieves well-separated clusters. **We will provide the visualization comparison in the revised manuscript.**"}, {"review_id": "8MeKMAtJ5Q", "reviewer": "Reviewer_gxAQ", "summary": "This paper focuses on the unaligned multi-view clustering (UMC) problem and analyzes the shortcomings of existing UMC methods in fusing multi-view information. This paper points out that the alignment-based cross-view fusion strategies of those UMC methods may be unreliable since the alignment strategies fail to achieve ideal view-alignment results. Therefore, this paper provides an alignment-free consistency fusion framework, which eliminates the requirement for alignment strategies and directly extracts a consistent representation from each view to perform global cross-view fusion. Sufficient experiments demonstrate the excellent performance of the proposed model.", "questions": "Most of my questions are listed in the section above.  Additionally, for the ablation studies in Table 5, (b) conducts an ablation on the cross-view consistent basis space, where the loss constraint $ \\mathcal{L}_{s} $ on the basis space seems to be ablated simultaneously, which is insufficient in evaluating the impact of the individual cross-view consistency basis space.", "limitations": "The model should be compared with the baseline models in 2025.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The analysis that alignment strategies fail to achieve ideal view-alignment results has been adequately explained in the Introduction and is understandable. \n2. The idea of alignment-free cross-view fusion is interesting and the proposed alignment-free consistency fusion framework reasonably realizes this idea.\n3. The experiments are sufficient. This paper evaluates the method on both small-scale and large-scale datasets, all of which show excellent clustering performance when compared with SOTA models.", "was_revised": true, "final_justification": "The authors propose a novel fusion framework for MVC, the rebuttal have addressed my score, so I decide to raise my score.", "rebuttal": "************ **For Question #1:** ************\n--------------------------------------------------------------\n\nWhen the cross-view consistent basis space is ablated, the loss function $\\mathcal{L}\\_{s}$ on the basis space is indeed ablated as well. To evaluate the individual impact of the consistent basis space $\\mathbf{B}$, we provide additional ablation studies in Table 1, where (1) ablates both $\\mathcal{L}\\_{s}$ and $\\mathbf{B}$, and (2) only ablates $\\mathcal{L}\\_{s}$. From Table 1, (2) shows better performance than (1), demonstrating the effectiveness of cross-view consistent basis space in prompting autoencoders to extract consistent representations from each view.\n\n**Table 1**\n|         |   | ***Caltech7-5***   |        |  |                     | ***NoisyMNIST***  |       |           |           |\n|:---------:|:---------:|:------:|:------:|:------:|:--------:|:------:|:------:|:------:|:-----:|\n|         | Without (w/o) | **ACC**  | **NMI** | **PUR** | **ARI** | **ACC**  | **NMI** | **PUR** | **ARI** |\n| (1)     |   w/o $\\mathcal{L}\\_{s}$ & $\\mathbf{B}$      | 0.7107   | 0.5989 | 0.7107 | 0.5312 | 0.4793   | 0.4136 | 0.5202 | 0.2862 |\n| (2)     | w/o $\\mathcal{L}\\_{s}$      | **0.8014** | **0.6983** | **0.8014** | **0.6406** | **0.5046** | **0.4573** | **0.5501** | **0.3297** |\n\n--------------------------------------------------------------\n\n************ **For Limitation #1:** ************\n--------------------------------------------------------------\n\nThe model should be compared with the baseline models in 2025.\n\nWe provide an experimental comparison between our AF-UMC and the baseline model MGCCFF [1] (2025) in Table 2, where the incomplete rate is set to 0 and other parameters are set to the optimal values provided by the authors. Due to the excessive complexity of MGCCFF, Table 2 only shows the experimental comparison on small-scale datasets, with ACC as the metric. As shown in Table 2, AF-UMC outperforms MGCCFF on all small-scale datasets, demonstrating its excellent performance.\n\n**Table 2**\n|       |  ***Caltech7-5*** | ***Handwritten***   |   ***Scene***     | ***Caltech102-5*** |   ***Hdigit***    | ***Aloi***  |\n|:---------:|:---------:|:------:|:------:|:------:|:--------:|:--------:|\n|    MGCCFF  | 0.7857 | 0.8248 |  0.3713  | 0.1708 |  0.6167 | 0.5176 |\n| AF-UMC |   **0.8721**  |**0.9035**   |**0.4190** | **0.2275** | **0.6950** |**0.5399** | \n\n[1] Zhao, Liang, Ziyue Wang, Xiao Wang, Zhikui Chen, and Bo Xu. \"Incomplete and Unpaired Multi-View Graph Clustering with Cross-View Feature Fusion.\" In Proceedings of the AAAI Conference on Artificial Intelligence, pp. 22786-22794. 2025."}, {"review_id": "2cPlXOROyW", "reviewer": "Reviewer_SXo8", "summary": "For the task of unaligned multi-view clustering (UMC), the authors present a new alignment-free consistency fusion framework (AF-UMC). Unlike traditional methods that employ various alignment strategies to obtain aligned sample representations for cross-view fusion, AF-UMC is designed to directly extract a consistent representation from each view for global cross-view contrastive fusion. Through the operation of alignment-free consistency extraction and global contrastive fusion, AF-UMC obtains a consistent representation with a clearer clustering structure. Experimental results also demonstrate the superior performance of AF-UMC.", "questions": "Q1. The authors constrain the cross-view consistent basis space to reconstruct samples of multiple views in Section 3.2, and they state to use this operation for capturing the cross-view consistency and filtering out the view-specific diversity. Why is such an operation able to encourage the basis space to capture consistency while filtering out diversity?\nQ2. Why do the authors implement a cross-view consistent basis space? What are the advantages of the cross-view consistent basis space in comparison to current methods that respectively construct a basis space in each view?\nQ3. Why do the authors set the limitation that the dimension $d$ of the basis vectors is greater than the number $c$ of basis vectors in Eq. (3)? Are there any negative effects if $c > d$?", "limitations": "None.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\nS1. Interesting model design: AF-UMC bypasses the traditional view-alignment operation and directly extracts a consistent representation from each view for global fusion rather than traditional alignment-based fusion.\n\nS2. Comprehensive experimental settings: various experimental results on benchmark datasets are provided to demonstrate the effectiveness of AF-UMC, particularly the ablation study highlighting the contribution of different loss functions and model components for clustering performance in Tables 4-5.\n\n\nWeaknesses:\nW1. Some mathematical symbols are explained repeatedly in both Introduction and Method, which seem redundant. The authors can introduce these symbols in a single subsection or directly provide a symbol list to help readers understand.\nW2. The authors divide their proposed AF-UMC into two stages: alignment-free consistency extraction and global contrastive fusion, but do not clarify how the two stages are trained, such as firstly extracting consistent representations until the reconstruction loss converges and then realizing global contrastive fusion, or fulfilling consistent representation extraction and global fusion at each training epoch. The authors should provide the complete process of model training.\nW3. The authors note that AF-UMC saves the additional cost for the alignment strategy, but the paper doesn’t provide the model complexity analysis.", "was_revised": true, "final_justification": "All my concerns has been addressed and I decide to improve my ratings to 5.", "rebuttal": "************ **For Weakness #1:** ************\n--------------------------------------------------------------------\n\nWe provide a symbol list in the following table and we will add it in the final version.\n\n|    **Notation**     |  **Definition**    |\n|----------------|----------------|\n|   $\\mathbf{X}^v$    | The samples of $v$-th view  |\n|  $\\mathbf{Z}^v$  |    The basis space of $v$-th view  | \n|   $\\mathbf{B}$  |  The cross-view consistent basis space  |\n|   $\\\\{\\mathbf{b}\\_i\\\\}\\_{i=1}^c$ |  The $c$ basis vectors in $\\mathbf{B}$|\n| $E^v(\\cdot)$ | The encoder of $v$-th view |\n| $D^v(\\cdot)$ | The decoder of $v$-th view |\n|   $\\mathbf{H}^v$  |  The extracted representation of $v$-th view |\n|   $\\mathbf{S}$  |  The similarity structure across basis vectors $\\\\{\\mathbf{b}_i\\\\}\\_{i=1}^c$ |\n| $\\bar{\\mathbf{h}}^v$ | The global center of $\\mathbf{H}^v$ |\n\n--------------------------------------------------------------------\n\n************ **For Weakness #2:** ************\n--------------------------------------------------------------------\n\nWe provide the detailed training process of our proposed AF-UMC in Algorithm 1.\n\n\n--------------------------------------------------------------------\n\n**Algorithm 1: Training process of AF-UMC** \n\n**Input:** Unaligned multi-view data $\\mathbf{X} = \\left\\\\{\\mathbf{X}^v\\right\\\\}\\_{v=1}^V$, number of clusters $c$, number of training epochs $T$.\n\n**Output:** Clustering results.\n\na. Initialize autoencoders $\\\\{E^{v}(\\cdot), D^v(\\cdot)\\\\}\\_{v=1}^V$ and cross-view consistent basis space $\\mathbf{B}$.\n\nb. **for** epoch $t = 1$ to $T$:\n\n&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**for** view $v = 1$ to $V$:\n\n&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Extract consistent representation $\\mathbf{H}^v$ by projecting $\\mathbf{X}^v$ into $\\mathbf{B}$\n\n&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**end for**\n\n&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Globally bring $\\\\{\\mathbf{H}^v\\\\}\\_{v=1}^V$ closer to fuse a cross-view consistent representation $\\mathbf{H}^{core}$.\n\n&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Optimize model by $\\mathcal{L}\\_{r}$,  $\\mathcal{L}\\_{s}$ and $\\mathcal{L}\\_{c}$.\n\n&nbsp;&nbsp;&nbsp;**end for**\n\nc. Perform k-means clustering on $\\mathbf{H}^{core}$.\n\n--------------------------------------------------------------------\n\n************ **For Weakness #3:** ************\n--------------------------------------------------------------------\n\nWe analyze our proposed AF-UMC in terms of space/time complexity. \n\n**Space Complexity:** In our method, the memory costs contain a basis space matrix $\\mathbf{B} \\in \\mathbb{R}^{c\\times d}$, $V$ autoencoders and $V$ representation matrices $\\\\{\\mathbf{H}^v\\\\}\\_{v=1}^V \\in \\mathbb{R}^{n\\times c}$, where the space complexity of an autoencoder is $\\mathcal{O}(lnd)$ and $l$ is the number of MLP layers. As a result, the total space complexity of our AF-UMC is $\\mathcal{O}(cd+Vlnd+Vnc)$.\n\n**Time Complexity:** The time cost of AF-UMC arises from three parts: (1) $\\mathcal{O}(Vnd+c^3+nV^2d)$, the cost of computing three loss functions. (2) $\\mathcal{O}(Vlnd)$, the cost of optimizing $V$ autoencoders. (3) $\\mathcal{O}(cd)$, the cost of optimizing a cross-view consistent basis space $\\mathbf{B}$. Therefore, the total time cost of AF-UMC is $\\mathcal{O}(Vnd+c^3+nV^2d+Vlnd+cd)$.\n\n--------------------------------------------------------------------\n\n************ **For Question #1:** ************\n--------------------------------------------------------------------\n\nSince the basis space is shared across multiple views and optimized to minimize the overall reconstruction loss, components that are consistently present in all views contribute more effectively to reducing the overall loss. Therefore, when the basis space is designed to reconstruct samples from multiple views, it naturally prioritizes capturing cross-view consistency while gradually reducing the diversity with the model training, thereby capturing consistency while filtering out diversity.\n\n--------------------------------------------------------------------\n\n************ **For Question #2:** ************\n--------------------------------------------------------------------\n\n\n\nCompared with existing methods that construct a basis space to capture intra-view information in each view, our cross-view consistent basis space can directly capture cross-view consistent information, facilitating autoencoders to directly extract consistent representations from each view for promoting subsequent global contrastive fusion across views.\n\n--------------------------------------------------------------------\n\n************ **For Question #3:** ************\n--------------------------------------------------------------------\n\n\nIf the number $c$ of basis vectors is greater than their dimension $d$, at least one basis vector $\\mathbf{b}_i$ can be linearly represented by the others. As a result, samples represented by $\\mathbf{b}_i$ can also be represented by other basis vectors, which intensifies the differences in sample representations and reduces the consistency of representations."}]

Author rebuttal:

Meta-review: This paper proposes AF-UMC, a novel alignment-free fusion framework for unaligned multi-view clustering. Unlike existing methods that rely on complex and often unreliable alignment strategies to match samples across views before fusion, AF-UMC bypasses explicit alignment entirely. Instead, it directly extracts consistent representations from each view by projecting view-specific data onto a shared, low-rank, cross-view consistent basis space, which is constructed using a cross-view reconstruction loss and a structural clarity regularization to avoid learning redundant basis vectors. These representations are then globally fused using an Instance global contrastive enhancement mechanism, which pulls the global centers of views together and enhances each instance's consistency via contrastive learning. The method significantly reduces computational complexity and improves clustering performance by avoiding alignment errors. Experiments on ten datasets show state-of-the-art results.

I have checked the author-reviewer discussions, and all the reviewers acknowledged the rebuttal and replied to the authors. This paper finally receives the scores of 4, 4, 5, 5, which means all the reviewers would like to be on the positive side. 

This paper is well-motivated, which addresses a fundamental limitation in unaligned multi-view clustering: the reliance on error-prone alignment steps. The proposed AF-UMC framework is novel, efficient, and empirically validated across a wide range of datasets.

Based on the above situations, I vote for acceptance.
Decision: Accept (poster)