Align then Fusion: Generalized Large-scale Multi-view Clustering with Anchor Matching Correspondences
Authors: Siwei Wang
Xinwang Liu
Suyuan Liu
Jiaqi Jin
Wenxuan Tu
Xinzhong Zhu
En Zhu
Venue: NeurIPS 2022 Accept (2022)
Keywords: multi-view graph clustering
anchor graph clustering

Abstract: Multi-view anchor graph clustering selects representative anchors to avoid full pair-wise similarities and therefore reduce the complexity of graph methods. Although widely applied in large-scale applications, existing approaches do not pay sufficient attention to establishing correct correspondences between the anchor sets across views. To be specific, anchor graphs obtained from different views are not aligned column-wisely. Such an Anchor-Unaligned Problem (AUP) would cause inaccurate graph fusion and degrade the clustering performance. Under multi-view scenarios, generating correct correspondences could be extremely difficult since anchors are not consistent in feature dimensions. To solve this challenging issue, we propose the first study of the generalized and flexible anchor graph fusion framework termed Fast Multi-View Anchor-Correspondence Clustering (FMVACC). Specifically, we show how to find anchor correspondence with both feature and structure information, after which anchor graph fusion is performed column-wisely. Moreover, we theoretically show the connection between FMVACC and existing multi-view late fusion and partial view-aligned clustering, which further demonstrates our generality. Extensive experiments on seven benchmark datasets demonstrate the effectiveness and efficiency of our proposed method. Moreover, the proposed alignment module also shows significant performance improvement applying to existing multi-view anchor graph competitors indicating the importance of anchor alignment. Our code is available at \url{https://github.com/wangsiwei2010/NeurIPS22-FMVACC}.

Align then Fusion: Generalized Large-scale Multi-view Clustering with Anchor Matching Correspondences
Siwei Wang', Xinwang Liu!.*, Suyuan Liu', Jiaqi Jin', Wenxuan Tu', Xinzhong Zhu?, En Zhu' School of Computer, National University of Defense Technology, Changsha, China
7Zhejiang Normal University (wangsiwei13, xinwangliu, suyuanliu, jinjiaqi, juyualliu,jiniaatt
twx, enzhu)@nudt... edu.
Abstract
Multi-view anchor graph clustering selects representative anchors to avoid full pair-wise similarities and therefore reduce the complexity of graph methods. Al- though widely applied large-scale applications, existing approaches do not pay sufficient attention to establishing correct correspondences between the an- chor sets across views. be specific, anchor graphs obtained from different views are not aligned column-wise Such Anchor- Unaligned Problem (AUP) would cause inaccurate graph fusion and degrade the clustering performance. Un- der multi-view scenarios, generating correct correspondences could be extremely difficult since anchors are not consistent in feature dimensions. To solve this challenging issue, propose the first study of the generalized and flexible an- chor graph fusion framework termed Fast Multi- View Anchor- Correspondence Clustering (FMVACC). Specifically, show how find anchor correspondence with both feature and structure information, after which anchor graph fusion is performed column-wisely: Moreover, theoretically show the connection be- tween FMVACC and existing multi-view late fusion [18) and partial view-aligned clustering Z which further demonstrates our generality. Extensive experiments on seven benchmark datasets demonstrate the effectiveness and efficiency of our proposed method. Moreover, the proposed alignment module also shows signif- icant performance improvement applying to existing multi-view anchor graph competitors indicating the importance of anchor alignment. Our code available at / gig.h....... com/ http://wwiiii110///eerrrrr 1PS2222F FMVACC
Anchor-Unaligned
1
Introduction
As an effective unsupervised multi-view learning technology, multi-view graph clustering (MVGC) comprehensively utilizes multiple pair-wise instance similarities into optimal flexible graph structures
23 5 41 In general, MVGC firstly constructs graph structures for every single view and then 45). For example, Zhang al. explore latent representation from multiple views with linear models and nonlinear networks in [4]. 440 optimizes the consensus graph by minimizing disagreement of individual graphs and then reach an agreement with rank constraint. (5 further proposes to utilize partition level information rather than graph structure agreement and achieve promising performance.
26
refines individual graphs with ideal fused graph
Although massive MVGC approaches have been proposed, one major issue is scalability in real
large-scale applications 26 23 5) For existing MVGC approaches, the complexity is quadratic or
cubic respecting to instance number which is unbearable for big data 36]. As an effective way for large-scale problem, anchor graph strategy firstly selects m individual anchors to represent full
*Corresponding author
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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View
View
0
Columm Permutation Matrix P
Anchor Graph
Anchor Graph
Anchor Graphs Fusion
Clustering
Anchor- Unaligee Problem
Matrix Form Aligned Graphs Fusion
Figure 1: An example of the Anchor-Unaligned Problem (AUP) for two views. (Left) AUP of view 1 and view 2: wrong correspondences between anchor sets; (Right) Columns of anchor graphs Z₁ and Zz are not orderly arranged, which often leads to inaccurate graph fusion clustering tasks. Therefore, is necessary find correct correspondences before fusion and then align them.
Generally, the quality of anchors is of central importance for performance and the sampling and k-means two strategies are commonly-used For example, Kang et operate k-means on each view separately and fuse into an optimal graph [ 34 proposes to jointly optimize anchors and the respective graph among multiple views. The space and time complexity have been reduced into O (rnm) and Onm²) linear instance numbers, which
instances among views 14
32
proves to be efficient for large-scale tasks [17
Despite being widely applied in large-scale applications, one vital factor of successful multi-view anchor graph clustering is to build correct correspondences for anchor sets. The selected anchor sets generated by different views may mismatch without guarantees of correspondences, since k-means or sampling is performed on each view separately. Taking two anchor graphs Z1 E Raxm and Ze E R= showcase Fig. [ the anchor graphs are not aligned tiiullyysssly,, which we refer Problem (AUP). Such an unaligned issue would cause inappropriate graph
as Anchor-Unaligne fusion and degrade the clustering performance in return. One pioneer work, SFMC [3] provides an intuitive way to select samples with same indexes to implicitly avoid wrong correspondences. However, this way could destroy the flexibility of anchors. To the best of our knowledge, no generalized framework for flexible multi-view anchor correspondences has been proposed so far since it is difficult to directly compute anchor distances with inconsistent dimensions for different views.
To this end, we propose a generalized and flexible multi-view anchor graph fusion framework termed Fast Multi View Anchor-Correspondee Correspondence Clustering (FMVACC) this paper. Unlike the existing rigid style of fixed indexes, FMVACC flexibly selects anchors with more discriminate inner-view structures. Then, we elegantly solve the unmeasurable representations. Based on this, FMVACC captures both feature and structure information to help establish accurate anchor correspondences. After that, the anchor graph fusion is polumn-wisedd performed with cross-view consistency. Further, we show that the late fusion [18] and the PVC [ are our special variants. Extensive experiments clearly demonstrate the effectiveness of our method. Moreover, our proposed alignment module also shows significant clustering performance improve- ment applying to existing multi-view anchor graph competitors, which indicating the importance of anchor alignment. We summarize the contributions of this work as follows,
anchor issue by relational
We study a new paradigm for large-scale multi-view anchor graph clustering, termed as Anchor- Enabor-nnlligned Problem (AUP). The selected anchor sets in multi-view data are not aligned, which may lead to inaccurate graph fusion and degrade the clustering performance.
We propose a flexible anchor graph fusion framework termed FMVACC to tackle the AUP problem. After generating flexible anchor candidates, an anchor alignment module is proposed to solve AUP with both feature and structure information. To the best of our knowledge, it is the first study of the flexible anchor correspondence fusion framework.
Extensive experiments demonstrate the effectiveness and efficiency of our proposed frame- work. Moreover, the proposed alignment module also shows significant performance improvement existing approaches which indicates the importance of anchor alignment.

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multi-view graph 2 Related Work
2.1 Multi-view Graph Clustering
As a representative unsupervised multi-view learning method 6
37
16
29
47
clustering (MVGC) represents single-view structures with graphs and then operates graph fusion individual graphs [5] 28
3 42 Many graph approaches have been proposed with the differences of these two parts. For constructing stage, self-expressive representation and local similarity graph are proved to be effective on finding pairwise relationships [4 45 44
24
43 40). Moreover, by imposing various regularization terms on the optimal graph, graph fusion stage reach consensus agreement
on similarity and partition information [22 10 g 20]. Zhan et al. optimize the consensus graph by minimizing disagreement of individual graphs and then optimizes with low-rank constraint [40]. Notice that the AUP problem does not happen under full pair-wise graph settings since the anchors are instances themselves and therefore naturally aligned.
2.2 Multi-view Anchor Graph Clustering
Although existing multi-view graph clustering methods have been efficient in improving the perfor- mance, the high computational complexity prevents their application large-scale datasets. The space and time complexity of majority graph approaches are O(un²) and (nn)) quadratic and cubic to instance numbers 25 [4]. To tackle the above issue, multi-view anchor graph clustering has been widely studied to effectively reduce the time and space consumption by constructing anchor graphs with selected anchors instead of the entire instances [30].
Specifically, the first step of multi-view anchor graph clustering is to construct individual anchor graphs each view by solving the following problem,
min X; Z. |F + 3 1 s.t. Z, 2 0, Z,1m 1, 2.
where A; E Rmxd: is the anchor matrix for the i-th view. Z; R is the anchor graph for the i-th view and 7 = 1, ensures that the similarity sum of each sample to all the anchors equals 1. Noticed that A, is fixed before the construction process, so the quality of anchors plays an essential
part in the success of multi-view anchor graph clustering. Li et al. [14] and Kang et al. [I]] propose
to get the anchor set by performing k-means on the original data of individual views. Random
sampling is also a common strategy for selecting anchor points (21]. Moreover, Li et al. 13 provide
a directly alternate sampling method determine the anchors based on scores.
After constructing the view-specific anchor graphs {Z,) the second step is to get the fused anchor graph G Raxm as follows,
1 Σ a.
min a.G
G F
s.t. G1m =1, a T1. = 1, 12220 20
(2)
where E R2 is the weight coefficient which measures the impact of each view and a T 1, 1 guarantees that G1. = 1,, The weight coefficient a and fused graph G are alternatively optimized in unified framework. However, solving Eq. (2) will lead to trivial solution. To avoid the above problem, Kang et al. [11 directly assign the same weight to all views and Li et al. [13] further solve by imposing rank constraint on the fused anchor graph. After obtaining the fusion graph G, we perform k- m-masss on its left singular vector to get the final clustering result. The space and time complexity of multi-view anchor graph clustering have been reduced into (vnm) and Onnm²) which can be applied to large-scale tasks effectively.
3 Anchor-Unalignel Clustering for Large-scale Multi-view Data
In this section, we propose a generalized and flexible anchor graph fusion framework, termed fast multi-view rorchorccrreepondencc clustering (FMVACC), which could elegantly solve the anchor- unmeasurable issue for different views and therefore perform anchor graph fusion correctly. We firstly show how FMVACC captures the feature and structure information for multi-view anchor set matching. Then we present our FMVACC framework with theoretical analysis and show the connection between late fusion strategy and partial view-aligned clustering.
3

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Algorithm Obtain single-view anchors and anchor graphs
Input: Multi-view dataset (X, and anchor number
1: Initialize m. anchor points in the i-th view.
2: repeat
3: Update {Zr] by solving Eq. (4
4: Update {A,]==1 by solving Eq. 5
5: until converged.
Output: (Aih== and { as the obtained anchors and respective anchor graphs.
3.1 Flexible Anchor Generation
Given multi-view data matrices {Xi]== with n samples, v views and d, dimension for the i-th view. By selecting anchors in each view and following anchor graph framework described in Eq. (I) in Section 2.2 we define the obtained anchor set and the respective individual anchor graph for the 2-th view A, E ==xxd and Z E ==xm
Firstly, is widely accepted that the quality of anchors plays an essential part in the success of multi-view anchor graph clustering, and adaptive optimizing anchors is proven to be effective in clustering contrary to existing fixed strategy. In our method, different from no constraint of (A;) ) Eq. ( the anchors are further imposed to be orthogonal that A,A! Im each view to make the learned anchors more discriminative and diverse.
min X; Z,A, + 3 I s.t. A,A, = Im Z: 0, 1,1m =1,
(3)
The optimization Eq. 33 can solved by two-step alternative approach as follows,
Updating Z;: Denoting
(t) as the I-th element in j-th row, Eq. can be solved by row as the 3j
following form,
10)1 Yis F S.t. 23, 0, 1,
min
(4)
where y which can be solved efficiently at global minimum with O(nmd;).
B). Eq. 44 is a projection capped simplex problem defined in 35
Updating A,; By denoting B = Z X,, updating anchor matrices {A,)._. equivalent as follows,
max TexXTr(A,D B), s.t. A:A, Im
A.
The optimum of A; can be analytically obtained with rank m truncated SVD by 33 The complexity to update each A, is O(rm² + md,,). > dj, we simply remove the orthogonal constraints into unconstrained optimization problems.
After optimization, we can obtain flexibly more representative single view anchors {A.),- and anchor graphs. However, the anchor sets {A:). for individual views may mismatch with other views' counterparts, and therefore the formed anchor graphs are not aligned column-wisely. To achieve correct anchor graph clustering, we first need establish the matching correspondences of the pair of anchor sets by applying graph matching approaches, which mathematically means to seek the matching correspondence or the permutation matrix P {0, 1 m shown in Fig. [ After that, anchor graph fusion is column- performed the same as the conventional pipelines.
3.2 Generalized Multi-view Anchor Matching Framework
The main difficulty of solving the AUP problem is unmeasurable anchor sets under multi- dimensional metric space. The obtained anchors in multiple views are naturally presented with different dimensions, making it difficult directly compute their distances and unmeasurable in various metric spaces. Therefore, a significant problem arises: how to flexibly find correspondences for multiple anchor sets under different data metric spaces?
One intuitive solution implicitly avoid correspondence issues, proposed SFMC K3 representative instances with the same indexes among multiple views. However, this sampling style
sample

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Feature Cerrespondenee: mis R2, 2,P15
View 1
View
Anchor Graph Z
Serciuaal Gruph
S,.5p
Manching ma Manin
Unmeasurable Anchors with Different Dimensions
Anchor Graph Z
Structure Cerrespendenne:
min PTS,P|,
Figure 2: Overview of the proposed FMVACC consisting of two parts: feature and structure corre- spondence. (Left)Anchor sets are represented by different dimensions, which makes them unmea- surable in various metric spaces; (Right)In FMVACC, each column of the graph is taken n--dimensional feature of each anchor, and the optimal matching matrix P is obtained by minimizing the both the feature and structure correspondences.
could destroy the flexibility and effectiveness of anchors and ignore the local individual view structure for clustering. It has been shown that not sampling but learning anchors in individual views has achieved more satisfactory performance, demonstrating the necessity of introducing flexibility into anchor frameworks. Another solution is to project nalti-dimennional data into commonly-shand space and then explicitly optimize consensus anchors in latent space [34]. However, the projection operator may lead information loss and extra dimension hyperparameter selection. Flexible anchor sets naturally induce the unaligned matching problem, which is often ignored in the existing literature. To the best of our knowledge, no generalized framework of flexible anchor correspondences for multi-view clustering has not been formally proposed so far. Clearly, the clustering performance will be improved by establishing the correct correspondences of anchors during the fusion stage.
The first step is how to make the obtained multiple anchor sets (Ai);=1 measurable. Going back to Eq. (3. the respective single view anchor graphs {Z,)==> represent the similarities between instances and the anchors. Elegantly, each column of the anchor graphs {Z:) S R is taken as a new n-dimensional feature representation of each anchor, which makes the anchors between different views measurable, As shown in Fig. 2 we transform the anchor alignment problem into graph matching problem of nodes with consensus embedding features. Under such transformation, the multiple anchor sets are measurable in I= space and therefore can calculate their similarities to find set correspondences. Followed by the traditional graph matching problem (38), our proposed FMVACC retains both first-order (feature) and second-orde (structure) information for better matching in the following subsections.
Feature Correspondence: For the first-order feature correspondence, we consider the principle that the correspondence probability of anchor points a and ( should be high if their features are more similar than other pairs. a.")) denoted as the j-th anchor in i-th view. Therefore, the feature correspondence can be fulfilled by minimizing the optimization goal as follows.
min 7
Z1Pl > max Tr Z.P)
max
K K.P
maxp_k.
=1
s.t. P1 P1 1, 10.1)axm
where K p Z2 vec 10,1)²² k= vec(K) Rm² to denote the vectorization of matrices respectively. K.y means the similarity between anchor pair (1) and a which can be as the opposite of the pair distances. is easy prove that by minimizing feature correspondence in Eq. (6), higher feature similarity value of K, will lead to bigger value the assignment value Pj--
Remark 1. Minimizing feature correspondence in Eq. (6) can be regarded as seeking the optimal transport plan for anchor sets A₁ and Az. Define the pair distance D, C is a large enough number to ensure Dij 2 0. Then max mm E,
Kij, where max P
EME m D.P,i >
min E D Ex Dip where indicates the optimal transport plan.
5

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Algorithm 2 Fast Multi-View Anchor-Correspooddenc Clustering (FMVACC)
Input: Multi- view dataset (X, ) cluster number k and anchor number m.
1: Obtain anchor sets {A, i==1 and respective anchor graphs {Z, 1=m
2: for = 2 - do
3: Calculate K z Z; and S S, Z Z. for Eq. (
4: Obtaining permutation matrix P, by solving Eq.
5: end for
Output: Fused aligned graph Z Aligned and permutation matrices (P:)----
Structure Correspondence: With the mentioned first-order feature correspondence part, we further consider the second-order graph structure regularization. To find better matching, the inner structures of anchor sets should also be comparable after matching, which indicates the matching between pair-wise edges. The second-order structure correspondence can be achieved by minimizing the following function,
min || - P S
max TrxXTr(S PS PAx max max PS PTPPPPPl -
maxp (Si @ S2) p.
s.t. 1,P 1 1, 1,P (0,1)mxx
(7)
where $₁ @ S2 is the Kronecker product of S and S2, and S = - Z1. S2 = Z Zz. Therefore, S and S2 represent the inner graph structures two anchor sets A and Ag. By minimizing Eq. 77 the inner graph structures can reach a maximum agreement.
Unified Correspondence Objective: Taking both first-order feature and second-order structure into consideration, the unified optimization goal for two anchor sets can be formulated as follows,
min Z2 Z,Pli +
P STP J
max Tr Z.P P PT
s.t. P1 -
{0, 1)
where À is the balanced hypermeter. Problem a quadratic assignment problem (QAP) and proved to be NP-hard under most circumstances 12]. The feasible region constraint often relaxes
into its convex hull, the Birkhoff polytope with double stochastic region where P1 1, P, E [0, 1j Then the optimization problem transforms into,
max Tr (Z A S2P), s.t. P1 1
1, [0, 1|mx
(9)
(9) as the multi-view anchor correspondence framework. To efficiently solve Eq. we adopt the Projected Fixed-Point Algorithm [9 to update P as follows,
We refer Eq.
(
p((+)) (1 (1- (1-a)p of (Vf (p()) = (1- + ar(K 2 e [0, 1].
(10)
where is the step size parameter, l. denotes the number of iterations and r denotes the double stochastic projection operator. The convergence of the algorithm has been proved in [19], and we set 0.5 in this paper. Details of solving Eq. (9) and Remark 2 can be found in supplementary.
Remark 2. The algorithm to solve Eq. converges at rate 2 + A |[[ @ S2)
After obtaining (P:)i we achieve the fused aligned anchor graph
Aligmed = (Z) + Z.P.)/u.
(II)
The final clustering result can be obtained by simply calculating rank-k truncated SVD Z Aligned and output for k-means. The whole process is summarized in Algorithm 2
3.3 Complexity Analysis
In this subsection, analyze our proposed algorithm in terms of space/time complexity.

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Space Complexity: In our paper, the major memory costs of our method are matrices (A. Rmxd, {P:) IR and {= E Rmxx Thus the space complexity of our FMVACC is O(md + mnn + m²v). In our algorithm, m < n and d < n. Therefore, the space complexity of FMVACC is (n))
Time complexity: The computational complexity of FMVACC is composed of three steps as mentioned before. When updating {Z.);=1 it costs (nnmd) to get the optimal value. Updating {A.)-- needs O(nm² + m2dd.. Then the time cost of alignment module is (mm)). Therefore, the total time cost of the optimization process is Onndd + nmn + dddd + m²). Consequently, the computational complexity of our proposed optimization algorithm linear complexity O(n).
After the optimization, we perform SVD on Z to obtain the spectral embedding and output the discrete clustering labels by k-means [34]. The post-procees needs (nm²). which is also linear complexity respecting to samples. In total, our algorithm achieves MVC with both linear space and time complexity, which demonstrates the efficiency of FMVACC.
3.4 Theoretical Analysis
In this subsection, we further show theoretical analysis of our proposed FMVACC with some existing widely-studied multi-view clustering settings.
Connection with Multi-view Late Fusion Strategy [18] Multi-view late fusion strategy seeks to fuse partition level information where multiple clustering results can reach a maximum agreement on the final partition matrix. It is easy to show that this strategy is special case of FMVACC where the m anchors are clustering centers (m k) and each single Z; represents the discrete partition. However, the differences are: (i) Late fusion only considers the feature correspondence in Eq. (6) with spectral relaxations PPT In while the structure counterpart is ignored; (ii) FMVACC shows more generality of late fusion with no limitation of anchor type (clustering centers) and numbers (k).
Connection with PVC ( PVC assumes little different multi-view setting where the data matrices are not aligned with some rows. Taking existing correspondences as supervision signals, PVC recovers other unknown correspondences with a sinkhorn layer in the neural network. We can easily show that PVC is also special case of FMVACC where only the structure loss in Eq. 77) is measured with some correspondence known and the anchors are instances themselves.
These two connections with existing late fusion strategy and PVC settings further theoretically demonstrate the generality of FMVACC. We summarize contributions as: (i) We novelly solve multi-view anchor sets unmeasurable issues by introducing both feature and structure correspondence. (ii) FMVACC is pioneering work that firstly studies the multi-view anchor unaligned issue for anchor graph fusion tasks. (iii) By following our proposed flexible alignment module, it is quite interesting rethink current multi-view anchor graph fusion with correct correspondences and further benefit the community.
Experiments
In this section, we conduct experiments to verify the effectiveness of the proposed FMVACC and aligmment module. A simulated dataset is used verify the AUP issue and the results on seven real-world multi-view datasets are reported. By the way, all the experimental environment are implemented on a desktop computer with an Intel Core i7-7820X CPU and 64GB RAM, MATLAB 2020b (64-bit). More experimental settings can be found in the appendix.
4.1 Simulated Multi-view Dataset
Data Generation We construct a simulated dual-view dataset, as shown in the left half of Fig. The data have 4 independent clusters where each cluster consists of 50 woo-dinensional instances following the respective Gaussian distribution.
Compared Algorithms We compare with the two SOTA multi-view anchor graph representatives LMVSC [T] and SFMC [[3]. Further, we introduce the proposed flexible anchor selection and alignment modules into the two methods to illustrate the effectiveness of two counterparts.
7

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View2
LMVSC
View
Zz2
(c) Zunnligned
Aligned
(e)
View2
Ours
View
103
(1) Zi
Z2
(h) nallingeed
AAkigned
i View2
SEMC
View1
(k) Z
(1) Zz
(m) ZOriginal
(n) Z
(o) P
Figure 3: Visualization of the anchor graphs and permutation matrices obtained from LMVSC/SFMC and Ours on the simulated dataset. The first and second rows of LMVSC and Ours clearly verify the effectiveness of the alignment module on better clustering results. The third row illustrates that the flexible strategy with alignment enjoys more preferable performance than the fixed indexes (SFMC).
Table Clustering comparison of LMVSC, SFMC and our FMVACC on simulated dataset (mean std). The better results are highlighted in bold.
Method
ACC(%)
NMI(%)
Fscore (%)
Unaligned
84.06+0.16
82. 1820.04
79.69+0.07
LMVSC
Aligned
100.00=0.00
100.00=0.00
100.00=0.00
Unaligned
70.35
72.89=0.68
70.28+0.81
Ours
Aligned
100.00:0.00
100.00=0.00
100.00:0.00
Original
64.50+0.00
10=0.00
73.41+0.00
SFMC
Flexible
97.50=0.00
93.15+0.00
95.12+0.00
Visualization and Results Analysis According to Table and Fig. we have the following observations: (i) For LMVSC and our FMVACC, aligning anchors between views can greatly improve the clustering effect, which verifies the effectiveness of our alignment module. Specifically, LMVSC with our alignment module achieves about 15.94% (ACC), 17.82% (NMI) and 20.31 % (Fscore) improvement compared with the column- column-unaligne, unaligned results. (ii) From the results of SFMC, can be seen that ficxible anchor selection strategy achieves better clustering performance, Numerically, adopting flexible strategy provides 33, 33.00% (ACC) and 21.71% % (Fscore) on the simulated dataset. (ii)) For LMVSC and our method in Fig. 3 Z Aligned obtained after anchors alignment contains less noise and more accurate structural information, and the column permutation matrix P in the third row is the expected identity matrix I, which verifies the correctness our method.
4.2 Real-world Multi-view Datasets
Benchmark Datasets We further perform experiments on eight real-world datasets ranging from 169 to 63896 instances, diverse dimensions and views. 3-Sources contains 169 instances clusters for 3 views. UCI-Digit refers to 2000 handwritten images in 10 clusters for views. BDGP haves 2500 instances for views while SUNRGBD collects 10335 samples. MNIST a commonly-used dataset and YTF-10/2 are collected from [5].

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Compared Algorithms Nine state-of-ther the-trt multi-view clustering methods are introduced: MSC- IAS 36
PMSC [T],, RMKM [,, BMVC [46]. LMVSC (TT) FMCNOF(39) MSGL_ 2 SFMC 13 The first three algorithms are graph methods and the others are repre- sentative large-scale competitors. Notice that SFMC selects anchor number from 0.1n to n so that can not be applied in large-scale scenarios.
AMGL (2T].
Table 2: Performance on the seven benchmarks. 'OM' indicates the out-of-memory failure.
Sumplec
MSC-IAS
AMXII
PMSC
RMKM
LMVSC
FMCNOF
MSGI
Prrpooal
NCC
47.040.000
60.37a7
UCK-Dige
2000
62.50-5.00
58.55+3.50
04.850000
59.4765.41
BDGP
2500
52.0964.99
35.17a1.
26.41a0119
41,44g0.00
29,48,0.00
50.1920.03
31.05.0.00
40.7789007
20,080000
58.63e2.74
SUNRGBD
10335
OM
0M
18.3550000
16.94820000
18.6600255
19.7:0000
13,74690222
12,346000
22.17=0.64 66
MINIST
60000
OM
0M
0M
86.21:0.00
74.9800.00
05.47:5.39
76.20e0.00
07.25x2.88 84
OM
98.67=1.83
YTF-10
39654
OM
QM
OM
75 75.68:0.00
60:4360.000
69. 69.29G4.46
43.42:0.00
68. 68.3065.57 57
OM
76.64:4.21
Y7F-20
65896
OM
QM
OM
57.62s0.00
60,0950.00
63.02=3.56
38.6120.00
65.51=2.36
OM
72.54:2.73
NTL L153
Sources
169
15
WA220222
47. 47.5367.79 75
21. 31-728000
19.7921.00
319620000
26
10
56,8525.44
UCT Digs
2000
57,5722 85
86.142l.32 .39
43 20c1 155
76 76.0440000
56 36.0M80.00 00
85.1221.04
55.1700.00
64.8911 57
89.40 28:0000
54.3322.37 37
SUNRGBD
BDGP
10335
2500
33.0722 S1
OM
18.19en.36 38
176121 20
5 6o 19
CON
26.1110000
25. 12:0000
20.3920.00 00
4.0080U00
15.41:6007
25.75:0:20
10.29:0.000 0.000
15666a1000
1875
18 171779077
5:350000
25440000
M6.1=2.00
MNIST
60000
OM
OM
CON
68.58a0.00
95.61:2.00
04.0311 40
ON
96.74:9775
YIF 10
39654
OM
OM
89:2240000
58.91200000
75.4221.95
29 15
74
ON
77.1321557
63896
OM
OM
CIM
71.44+000
71.6240.00
74.00-1.00 A88
45 45:0.00000
74.614082
78.4761.01
Sources
169
25
S4.0E5
00180000
42.7730000
181551110 10
120000
UCI-Diga
$1.6766558
70 84L3
75
72.100000
80.92-2..2
50 00:00.00 D0
68.37=2.44
83,88efu00
13.33a4.29
2500
40.43
67700.77
29.55am.10
36.2840.00
26.51a0.00
37.82:004
2889000000 D1
43.88
44.25a3.43
SUNRGBD
6.40ta0.20
0M
10.5440.00
11.6800:14
1400500 00
20
12.14
12.9ka0.29
88.2840000
97.20+2.00
YTF-10
38654
QM
79.27:0.00
53.1500.00
64.41:4.05
88
74.98:1.55
11.29-4.2
YTF-20
GM
53.80s0.00
48.0660.00
47.94:3.59
25.84:0.00
70.70:1.37
66.40:2.16
Experimental Results We conduct comparative experiments of eight multi-view clustering algo- rithms on seven commonly-used multi-view datasets, and the experimental results are displayed in Table 2
From these results, we observe that:
Our method also outperforms the other seven algorithms on most clustering metrics when applied to the first three small-scale datasets. Specifically, in terms of ACC, FMVACC achieves 39.53 % (3-Sources), 7.21 % (UCI-Digit) and 23.46 % (BDGP) progress compared to the classical multi-view graph algorithm AMGL.
When applied on the last four large-scale datasets, compared to well-known large-scale methods RMKM, BMVC, LMVSC, FMCNOF and MSGL, our FMVACC outperforms them on most metrics. Note that the OM failure of SFMC on the last three datasets is due to the large number of anchors it needs select. The above experimental results show the effectiveness and superiority of our FMVACC.
Visualization We visualize the effectiveness of alignment module on UCI-Digit in Fig. For LMVSCUnaligned and LMVSC Aligned on UCI-digits, the biggest value of similarity values have increased form 0.7 to 0.9, and the Aligned version contains much less noises than unaligned based on correct anchor graph fusion (ACC 66.78% to 82.97%). We can also find similar phenomenon with our proposed approach and conclude that aligned fusion can reduce the noise and enhance clustering performance. For LMVSC and ours, the fused anchor graphs obtained after alignment have clearer structures and therefore achieve better performance, which demonstrates the effectiveness and superiority of our anchor alignment module.
ACC = 66.78%
ACC 82.97%
ACC = 81.43%
ACC 89.47%
01
100
1980
2000
1000
1990
2000
800
1530
1500
2000
853
1006
1803
2000
LMVSC L.MVSCUnndggeee
LMVSC
Ours Unaiigned
Ours Allgned
Figure 4: Visualization of the aligned and unaligned anchor graphs (LMVSC and Ours) UCI-Digit.
Execution Time: We evaluate the running time of the proposed FMVACC and seven other MVC comparison algorithms on the benchmark multi-view datasets, which is displayed in Table 3 We can
9

---

discover that half of the algorithms cannot be applied on the latter three datasets due to out-of out-of-memory failure. Remarkably, our FMVACC method performs more efficiently than most MVC algorithms. especially on the MNIST dataset with 60000 samples, FMVACC consumes only 571.10 10 seconds of running time, second only to the BMVC method, while RMKM and LMVSC respectively consume 1468.30 seconds and 604.68 seconds of running times.
Table 3: Running time the seven benchmarks. 'OM' indicates the out-of-memory failure.
Samples
MSC-IAS
AMGI.
RMKM
RMVC
LMMSCC
FMCNOP
MSGI.
SFMC
Proposed
S-Sources
169
6.51
0.15
34,00
1.01
3.49
0.35
0.33
0.70
0.43
1.64
Digit
2000
10.62
27.64
722.99
3.06
0.34
2.26
0.72
3.50
$1.43
2.31
BDGP
2500
13.26
73.71
15215.00
7.53
2.85
1.77
3.80
38.99
5.47
SUNRGED
10335
0M
4039.
OM
233.92
1.79
39.94
18.23
61.40
5313.32
256.71
MNNSTT
60000
OM
OM
OM
1468.30
17.63
604.68
63.84
684.51
OM
571.10
YTF-10
38654
OM
OM
OM
675,42
108.22
196.70
39.88
488.63
OM
248.47
YTH-20
63896
OM
OM
OM
1780.50
80.53
513.52
822.92
OM
628.20
Effect of Alignment Module Table we report the accuracy obtained from ablation study on LMVSC and our method. It can be found that for the above two algorithms on all seven benchmarks, the clustering performance achieved by using the alignment module better than opponent.
Table 4: Clustering performance comparison LMVSC and our FMVACC on benchmarks (ACC(%))
Method
3-Sources
UCI-Digit
BDGP
SUNRGBD
MNIST
YTF-10
YTF-20
Unaligned
46.25+1.18
66.7813.74
40.77:0.07
13.74=0.22
97.25=2.88
68.30=3.57
65.51
LMVSC
Aligned
47.11+0.97
82.97+6.23
55.49:0.34
17.41=0.39
97.85+3.42
72.09+2.8
67.27+2.84
Unaligned
52.8144.13
81.4345.46
49.9724.17
18.0020.80
98.35=1.85
71.6924.80
71.4543.8
Aligned
60.3747.13
89.4745.41
58.6342.74
22.17=0.66
98.67=1.93
76.64s4.21
72.54+2.73
5 Conclusion
In this paper, we present the first study of multi-view anchor unaligned problem. A flexible multi- view anchor graph fusion framework termed Fast Multi View Anchor- Correpoonnee Clustering (FMVACC) is proposed. FMVACC firstly constructs flexible individual anchors and then captures correspondences with both feature and structure information. Theoretical analysis between FMVACC and some existing multi-view strategies is also provided. Extensive experiments are conducted to demonstrate the effectiveness of anchor alignment and our proposed framework FMVACC. quite interesting to rethink current multi-view anchor graph fusion with correct correspondences and further benefit the research community. Further, the success of anchor graph clustering heavily depends on the quality of anchor representation. In the future, we will explore how to obtain more-qualified anchors by deep learning.
Acknowledgment
This work was supported by the National Key R&D Program of China 2020AAA0107100,, the Natural Science Foundation of China (project no. 61922088, 61773392, 62006237 and 61976196).
References
[1] Xiao Cai, Feiping Nie, and Heng Huang. Multi-view k-means clustering on big data. In Twenty-Third International Joint conference artificial intelligence. Citeseer, 2013.
[2] Xiaochun Cao, Changqing Zhang, Huazhu Fu, Si Liu, and Hua Zhang. Diversity-induce multi-view subspace clustering. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 586-594, 2015.
[3] Man-Sheng Chen, Ling Huang, Chang-Dong Wang, and Dong Huang. Multi-view clustering latent embedding space. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 3513-3520, 2020.
10

---

[4] Hongchang Gao, Feiping Nie, Xuelong Li, and Heng Huang. Multi-view subspace clustering. In Proceedings of the IEEE international conference on computer vision, pages 4238 4246,
2015.
[5] Dong Huang, Chang-Dong Wang, and Jian-Huang Lai. Fast multi-view clustering via ensembles: Towards scalability, superiority, and simplicity. arXiv preprint arXiv:2203. /572, 2022.
[6] Shudong Huang, Zhao Kang, Ivor W Tsang, and Zenglin Xu. Auto- multi-view clustering via kernelized graph learning. Pattern Recognition, 88:174-184, 2019.
[7] Zhenyu Huang, Peng Hu, Joey Tianyi Zhou, Jiancheng Lv, and Xi Peng. Partially view-aligned clustering. Advances Neural Information Processing Systems, 33:2892-29022 2020.
[8] Zhao Kang, Zipeng Guo, Shudong Huang, Siying Wang, Wenyu Chen, Yuanzhang Su, and Zenglin Xu. Multiple partitions aligned clustering. In Proceedings of the Twenty-Eighti Inter- national Joint Conference on Artificial Intelligence, IJCAI-19, pages 2701-2707. International Joint Conferences on Artificial Intelligence Organization, 7 2019.
[9] Zhao Kang, Zhiping Lin, Xiaofeng Zhu, and Wenbo Xu. Structured graph learning for scalable subspace clustering: From single view to multiview. IEEE Transactions on Cybernetics, 2021.
[10] Zhao Kang, Xinjia Zhao, Chong Peng, Hongyuan Zhu, Joey Tianyi Zhou, Xi Peng, Wenyu Chen, and Zenglin Xu. Partition level multiview subspace clustering. Neural Networks, 122:279-288, 2020.
[11] Zhao Kang, Wangtao Zhou, Zhitong Zhao, Junming Shao, Meng Han, and Zenglin Xu. Large- scale mieti-view subspace clustering in linear time. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 4412-4419 2020.
[12] Eugene L Lawler. The quadratic assignment problem. Management science, 9(4):866-599 1963.
[13] Xuelong Li, Han Zhang, Rong Wang, and Feiping Nie. Multi-view clustering: A scalable and parameter-free bipartite graph fusion method. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2020.
[14] Yeqing Li, Feiping Nie, Heng Huang, and Junzhou Huang. Large-scale multi-viev spectral clustering via bipartite graph. In Twenty-Ninth AAAI Conference on Artificial Intelligence, 2015.
[15] Zhenglai Li, Chang Tang, Xinwang Liu, Xiao Zheng, Guanghui Yue, Wei Zhang, and En Zhu. Consensus graph learning for multi-view clustering. IEEE Transactions on Multimedia, 2021.
[16] Hongfu Liu, Junjie Wu, Tongliang Liu, Dacheng Tao, and Yun Fu. Spectral ensemble clustering via weighted k-means: Theoretical and practical evidence. IEEE transactions on knowledge and data engineering, 29(5): 11111229-110 1143, 2017.
[17] Wei Liu, Junfeng He, and Shih-Fu Chang. Large graph construction for scalable semi-supervised learning. In ICML, 2010.
[18] Xinwang Liu, Xinzhong Zhu, Miaomiao Li. Lei Wang. Chang Tang, Jianping Yin, Dinggang Shen, Huaimin Wang, and Wen Gao. Late fusion incomplete multi-view clustering. IEEE transactions on pattern analysis and machine intelligence, 41(10):2410-244 2018.
[19] Yao Lu, Kaizhu Huang, and Cheng-Lin Liu, A fast projected fixed-point algorithm for large graph matching. Pattern Recognition, 60:971-982, 2016.
[20] Shirui Luo, Changging Zhang, Wei Zhang, and Xiaochun Cao. Consistent and specific multi- view subspace clustering. In Proceedings of the AAAI Conference Artificial Intelligence, volume 32, 2018.
[21] Feiping Nie, Jing Li, Xuelong Li, et al. Parameter-fres aetg-weighted multiple graph learning: a framework for multiview clustering and supi-vuee classification. IJCAI, pages 18887, 2016.
11

---

[22] Feiping Nie, Jing Li, Xuelong Li, et al. Self-weighted multiview clustering with multiple graphs.
In IJCAI, pages 2564-2570, 2017.
[23] Feiping Nie, Xiaoqian Wang. Cheng Deng, and Heng Huang. Learning a structured optimal bipartite graph for co-clustering. Advances Neural Information Processing Systems, 30, 2017.
[24] Feiping Nie, Xiaoqian Wang, Michael Jordan, and Heng Huang. The constrained laplacian rank algorithm for graph-based clustering. In Proceedings of the AAAI conference on artificial intelligence, volume 30, 2016.
[25] Feiping Nie, Wei Zhu, and Xuelong Li. Unsupervised large graph embedding, In Thirty-first AAAI conference on artificial intelligence, 2017.
[26] Xi Peng, Huajin Tang, Lei Zhang, Zhang Yi, and Shijie Xiao. A unified framework for eepresentation-baseee subspace clustering of -ut-ll-sample and large-scale data. IEEE transac- tions on neural networks and learning systems, (7()):2499 2512, 2015.
[27] Mengjing Sun, Pei Zhang, Siwei Wang, Sihang Zhou, Wenxuan Tu, Xinwang Liu, En Zhu, and Changjian Wang. Scalable multi-vier subspace clustering with unified anchors. In Proceedings 2021.
of the 29th ACM International Conference Multimedia, pages
[28] Chang Tang, Xinwang Liu, Xinzhong Zhu, En Zhu, Zhigang Luo, Lizhe Wang, and Wen Gao. Cgd: Multi-view clustering via cross-view graph diffusion. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 5924-5931, 2020.
[29] Zhiqiang Tao, Hongfu Liu, Sheng Li, Zhengming Ding, and Yun Fu, From ensemble clustering to multi-view clustering. In IJCAI, 2017.
[30] Meng Wang, Weijie Fu, Shijie Hao, Hengchang Liu, and Xindong Wu. Learning on big graph: Label inference and regularization with anchor hierarchy. IEEE transactions on knowledge and data engineering, 29(5): 101-1114, 2017.
[31] Meng Wang, Weijie Fu, Shijie Hao, Dacheng Tao, and Xindong Wu. Scalable semi-supervised learning by efficient anchor graph regularization. IEEE Transactions Knowledge and Data Engineering,
2016.
[32] Siwei Wang, Xinwang Liu, Li Liu, Wenxuan Tu, Xinzhong Zhu, Jiyuan Liu, Sihang Zhou, and En Zhu. Highly-efficient incomplete large-scale multi-view clustering with consensus bipartite graph. In Proceedings of the IEEE conference computer vision and pattern recognition, 2022.
[33] Siwei Wang, Xinwang Liu, En Zhu, Chang Tang, Jiyuan Liu, Jingtao Hu, Jingyuan Xia, and Jianping Yin. Multi-view clustering via late fusion alignment maximization. In Proceedings of the Tiehty-Eighth International Joint Conference on Artificial Intelligence, pages 3778-3784 2019.
[34] Siwei Wang, Xinwang Liu, Xinzhong Zhu, Pei Zhang, Yi Zhang, Feng Gao, and En Zhu. Fast parameter-frre multi-view subspace clustering with consensus anchor guidance. IEEE Trans. Image Process., 31:556-568, 2022.
[35] Weiran Wang and Canyi Lu. Projection onto the capped simplex. arXiv preprint 1rXiv:1503.01002, 2015.
[36] Xiaobo Wang, Zhen Lei, Xiaojie Guo, Changqing Zhang, Hailin Shi, and Stan Z Multi-view subspace clustering with intactnessawaar similarity. Pattern Recognition, 88:50-63, 2019.
[37] Yang Wang, Xuemin Lin, Lin Wu, Wenjie Zhang, Qing Zhang, and Xiaodi Huang. Robust subspace clustering for multi-view data by exploiting correlation consensus. IEEE Transactions on Image Processing, 24(1 1) 1):3939-3944 39499 2015.
[38] Junchi Yan, Xu-Cheng Yin, Weiyao Lin, Cheng Deng, Hongyuan Zha, and Xiaokang Yang. short survey of recent advances in graph matching. In Proceedings of the 2016 ACM on International Conference on Multimedia Retrieval, pages 167-174, 2016.
12

---

[39] Ben Yang, Xuetao Zhang, Feiping Nie, Fei Wang, Weizhong Yu, and Rong Wang. Fast multi- view clustering via nonnegative and orthogonal factorization. IEEE Transactions on Image Processing, 30:2575-22555 2586, 2020.
[40] Kun Zhan, Feiping Nie, Jing Wang, and Yi Yang. Multiview consensus graph clustering. IEEE Trunsuctions on Image Processing, 28(3):1261-122 2018.
[41] Kun Zhan, Chaoxi Niu, Changlu Chen, Feiping Nie, Changqing Zhang, and Yi Yang. Graph structure fusion for multiview clustering. IEEE Transactions Knowledge and Data Engineer- ing, 31(10):194419999 2018.
[42] Kun Zhan, Changqing Zhang, Junpeng Guan, and Junsheng Wang. Graph learning for multiview clustering. IEEE transactions on cybernetics, 48(10))2887-288) 2017.
[43] Changqing Zhang, Huazhu Fu, Qinghua Hu, Xiaochun Cao, Yuan Xie, Dacheng Tao, and Dong Xu. Generalized latent multi-view subspace clustering. IEFE transactions on pattern analysis and machine intelligence, 42(1):866999) 2018.
[44] Changqing Zhang. Qinghua Hu, Huazhu Fu, Pengfei Zhu, and Xiaochun Cao. Latent multi-view subspace clustering. In Proceedings of the IEEE Conference on Computer Vision and Patlern Recognition, pages 4279-4287, 2017.
[45] Zhang, X. Liu, Xiong, Zhou, W. Zhao, E. Zhu, and Z. Cai. Consensus one-step multi- view subspace clustering. IEEE Transactions on Knowledge and Data Engineering, pages 1-1, 2020.
[46] Zheng Zhang. Li Liu, Fumin Shen, Heng Tao Shen, and Ling Shao. Binary multi view clustering. IEEE transactions pattern analysis and machine intelligence, (1(7))1774-782, 2018.
[47] Handong Zhao, Zhengming Ding, and Yun Fu. Multi-view clustering via deep matrix factoriza- tion. In Thirty-first AAAI conference on artificial intelligence, 2017. 2017.
Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
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14


Reviews:
[{"review_id": "uVQV8IILv4Z", "reviewer": "Reviewer_Rr2L", "summary": "This paper focuses on multi-view graph clustering. The author studied a practical problem in multi-view graph clustering, i.e., anchor-unaligned problem. To solve this, the author propose a new anchor graph fusion framework including an anchor alignment module to solve AUP. Experiments on several datasets show reasonable performance improvement and effectiveness of the proposed method.", "questions": "1. What is the difference between Eq.8 and Eq.9？Detailed proof is expected.\n2. In training, the permutation matrix is computed by solving Eq.9, which is actually relaxed. How to get the permutation matrix (binary value) in the inference period?", "limitations": "N/A", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The motivation is strong and practical.\n2. The solution to obtain the permutation matrix is interesting, which is proved effective theoretically and experimentally.\n3. This paper is the first work to study the AUP in multi-view graph clustering.\n\nWeaknesses:\n1. The reviewer suggests improving Fig.2 for better clarity. Current figures cannot deliver the main idea of the proposed method.\n2. The authors do not clearly define the experimental settings in the real-world multi-view datasets and how they tune the parameters. Do these datasets still suffer from the AUP problem? \n3. The visualizations of the anchor graphs on UCI digits show little difference between the aligned and unaligned cases, which is less convincing.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "bABADCSpOLG", "reviewer": "Reviewer_U2ZL", "summary": "This paper focuses on the large-scale multi-view anchor graph fusion strategies where the anchor graphs in individual views are not naturally aligned. The authors discover this phenomenon and provide a matching framework to address this issue. The paper is well-written and easy to read. The idea is well-motivated with clear contributions of illustrating necessity of multi-view anchor alignment. The comprehensive experimental results not only show performance improvements with the proposed FMVACC but also enhances existing large-scale baselines with more flexibility, indicating the effectiveness of multi-view anchor alignment.", "questions": "1.Some details should be clarified in Table 2 with representative large-scale multi-view methods. RMKM is not a graph method and should be introduced in the related work sections.\n2.The authors should illustrate how to initialize single-view anchors since initializations are important.\n3.There are some formatting inconsistencies in the text that the authors should check carefully, i.e. AUP and \\mathcal for complexity analysis.\n4.Some other strategies of multi-view clustering should be discussed in the introduction part, for example, multi-view NMF and multi-view ensemble clustering.\n5.I suggest the authors can compare the performance of some latest multi-view scalable graph clustering methods [1][2]. [1] is an extension version of LMVSC mentioned in the manuscript and [2] utilizes NMF into multi-view anchor graph study.\n[1]Structured Graph Learning for Scalable Subspace Clustering: From Single View to Multiview. IEEE TCYB, 2021.\n[2]Fast Multi-View Clustering via Nonnegative and Orthogonal Factorization. IEEE TIP, 2021.\n", "limitations": "Yes. The authors have adequately addressed the limitations and potential negative societal impact of their work.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n1.\tThe paper is well-written and easy to read. The idea of recognizing the multi-view anchor alignment problem has been overlooked in the existing literature which it can benefit further research and community.\n2.\tThe authors propose the first study of generalized flexible multi-view anchor graph framework. Different from sampling fixed anchors for AUP problem in SOTA, a matching framework has been made with more flexibility and performance improvement. \n3.\tThe effectiveness of the alignment module has been proved both on simulated and real-world datasets. Moreover, existing baselines enjoy considerable performance improvement with the module.\n4.\tThe method can be applied to large-scale scenarios where the scalability is proved by theory analysis and experiments.\n5.\tExperiments compared with recent works and ablation studies are well presented. \n\nWeakness:\n1.\tAlthough the proposed method focuses on multi-view graph method, the literature review can be enlarged with other strategies of multi-view clustering approaches, for example, multi-view NMF and multi-view ensemble clustering.\n2.\tThe authors should illustrate how to initialize single-view anchors since initializations are important.\n3.\tThere are some formatting inconsistencies in the text that the authors should check them carefully, i.e. AUP.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "40AV9J-7qRR", "reviewer": "Reviewer_nLLC", "summary": "This paper studies an important problem, i.e., anchor-based large-scale multi-view clustering-Anchor Unaligned problem (AUC). The authors review existing multi-view anchor fusion strategies and propose to establish alignment between multi-view anchor sets. The results are promising, and the effectiveness of anchor alignment has been proved.", "questions": "1. Although the settings are a little different, the authors could discuss the differences between PVC and the proposed method in details. How can they apply to PVC settings?\n2. The paper needs careful proofreading and some typos should be corrected, e.g. Eq.(3). Definitions for the constraints should be presented.\n3.The complexity analysis is missing and I suggest the authors should also list with existing baselines (i.e, SFMC and LMVSC) both space and time complexity.\n4. Figure (e)(j)(o) in Figure 3 are the obtained permutation matrices and more discussions can be introduced.\n5. The authors should survey not only multi-graph clustering models but also with other multi-view approaches in the part of related work.\n", "limitations": "YES", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n1.\tThe idea is interesting and novel. Figure 1 and Table 1 illustrate the issue of anchor-unaligned problem for existing multi-view anchor graph clustering.\n2.\tThis paper introduces a flexible anchor graph fusion framework termed FMVACC to tackle the AUP problem, which is a generalized large-scale multi-view anchor graph study.\n3.\tThe authors solve the unmeasurable multi-dimensional anchor matching problem by introducing two parts: feature and structure correspondences in Algorithm 2.\n4.\tThe experimental results seem promising. The effectiveness of the proposed anchor-aligned module has been proved in Table 1/3 and Figure 3/4.\n5.\tThe code is available and easy to reproduce.\n\nWeakness:\n1.\tThe complexity analysis is missing and I suggest the authors should also list with existing baselines (i.e, SFMC and LMVSC) both space and time complexity.\n2.\tThe paper needs careful proofreading and some typos should be corrected.\n3.\tFor better representation, Figure (e)(j)(o) in Figure 3 are the obtained permutation matrices and more discussions can be introduced.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept