Original: Align then Fusion: Generalized Large-scale Multi-view Clustering with Anchor Matching Correspondences
Revised: Align then Fusion: Generalized Large-scale Multi-view Clustering with Anchor Matching Correspondences

Original: Abstract
Revised: Abstract

Original: Multi-view anchor graph clustering selects representative anchors to avoid full pairwise similarities and therefore reduce the complexity of graph methods.
Revised: Multi-view anchor graph clustering selects representative anchors to avoid full pair-wise similarities and therefore reduce the complexity of graph methods.

Original: Althoughwidely applied in large-scale applications, existing approaches do not pay sufficientattention to establishing correct correspondences between the anchor sets acrossviews.
Revised: Although widely applied in large-scale applications, existing approaches do not pay sufficient attention to establishing correct correspondences between the anchor sets across views.

Original: To be specific, anchor graphs obtained from different views are not alignedcolumn-wisely.
Revised: To be specific, anchor graphs obtained from different views are not aligned column-wisely.

Original: Such an A nchor- U naligned P roblem (AUP) would cause inaccurategraph fusion and degrade the clustering performance.
Revised: Such an A nchor- U naligned P roblem (AUP) would cause inaccurate graph fusion and degrade the clustering performance.

Original: Under multi-view scenarios,generating correct correspondences could be extremely difficult since anchors arenot consistent in feature dimensions.
Revised: Under multi-view scenarios, generating correct correspondences could be extremely difficult since anchors are not consistent in feature dimensions.

Original: To solve this challenging issue, we propose thefirst study of a generalized and flexible anchor graph fusion framework termed F ast M ulti-
Revised: To solve this challenging issue, we propose the first study of the generalized and flexible anchor graph fusion framework termed F ast M ulti-

Original: V iew A nchor- C orrespondence C lustering (FMVACC).
Revised: V iew A nchor- C orrespondence C lustering (FMVACC).

Original: Specifically, we showhow to find anchor correspondence with both feature and structure information,after which anchor graph fusion is performed column-wisely.
Revised: Specifically, we show how to find anchor correspondence with both feature and structure information, after which anchor graph fusion is performed column-wisely.

Original: Moreover, wetheoretically show the connection between FMVACC and existing multi-view latefusion [19] and partial view-aligned clustering [7], which further demonstratesour generality.
Revised: Moreover, we theoretically show the connection between FMVACC and existing multi-view late fusion [18] and partial view-aligned clustering [7], which further demonstrates our generality.

Original: Extensive experiments on seven benchmark datasets demonstratethe effectiveness and efficiency of our proposed method.
Revised: Extensive experiments on seven benchmark datasets demonstrate the effectiveness and efficiency of our proposed method.

Original: Moreover, the proposedalignment module also shows significant performance improvement applying toexisting multi-view anchor graph competitors indicating the importance of anchoralignment.
Revised: Moreover, the proposed alignment module also shows significant performance improvement applying to existing multi-view anchor graph competitors indicating the importance of anchor alignment.

Revised: Our code is available at https://github.com/wangsiwei2010/NeurIPS22-FMVACC .

Original:  Introduction
Revised:  Introduction

Original: 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[11, 33, 27, 31, 45].
Revised: 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 [26, 23, 5, 41].

Original: In general, MVGC firstly constructs graph structures for every single view andthen refines individual graphs with ideal fused graph [48, 44, 47, 49, 42].
Revised: In general, MVGC firstly constructs graph structures for every single view and then refines individual graphs with ideal fused graph [44, 40, 43, 45].

Original: For example, Zhang et al.explore latent representation from multiple views with linear models and nonlinear networks in [48].[44] optimizes the consensus graph by minimizing disagreement of individual graphs and then reachan agreement with rank constraint.
Revised: For example, Zhang et al. explore latent representation from multiple views with linear models and nonlinear networks in [44]. [40] optimizes the consensus graph by minimizing disagreement of individual graphs and then reach an agreement with rank constraint.

Original: [49] further proposes to utilize partition level information ratherthan graph structure agreement and achieve promising performance.
Revised: [45] further proposes to utilize partition level information rather than graph structure agreement and achieve promising performance.

Original: Although massive MVGC approaches have been proposed, one major issue is scalability in reallarge-scale applications [28, 24, 5].
Revised: Although massive MVGC approaches have been proposed, one major issue is scalability in real large-scale applications [26, 23, 5].

Original: For existing MVGC approaches, the complexity is quadratic orcubic respecting to instance number n , which is unbearable for big data [10, 40].
Revised: For existing MVGC approaches, the complexity is quadratic or cubic respecting to instance number n , which is unbearable for big data [9, 36].

Original: As an effective wayfor large-scale problem, anchor graph strategy firstly selects m individual anchors to represent fullinstances among views [16, 35, 13, 15, 30, 38, 36]. Generally, the quality of anchors is of centralimportance for performance and the sampling and k -means two strategies are commonly-used. Forexample, Kang et al. operate k -means on each view separately and fuse into an optimal graph [13].[38] proposes to jointly optimize anchors and the respective graph among multiple views.
Revised: As an effective way for large-scale problem, anchor graph strategy firstly selects m individual anchors to represent full

Original: The spaceand time complexity have been reduced into O ( vnm ) and O ( nm 2 ) linear to instance numbers, whichproves to be efficient for large-scale tasks [18].

Original: View
Revised: View

Original: View
Revised: View

Original: Anchor-Unaligned Problem
Revised: Anchor-Unaligned Problem

Original: Matrix Form of Aligned Graphs Fusion (cid:20) (cid:21) (cid:22) (cid:23)
Revised: Matrix Form of Aligned Graphs Fusion (cid:20) (cid:21) (cid:22) (cid:23)

Original: Column Permutation Matrix (cid:1790)               0 (cid:3400)
Revised: Column Permutation Matrix (cid:1790)               0 (cid:3400)

Original: Anchor Graph (cid:1800) (cid:2778) (cid:23) (cid:21) (cid:20) (cid:22)
Revised: Anchor Graph (cid:1800) (cid:2778) (cid:23) (cid:21) (cid:20) (cid:22)

Original: Anchor Graph (cid:1800) (cid:2779) Anchor Graphs Fusion
Revised: Anchor Graph (cid:1800) (cid:2779) Anchor Graphs Fusion

Original: Clustering (cid:20) (cid:21) (cid:22) (cid:23) (cid:20) (cid:21) (cid:22) (cid:23)
Revised: Clustering (cid:20) (cid:21) (cid:22) (cid:23) (cid:20) (cid:21) (cid:22) (cid:23)

Original: Therefore, it is necessary to find correct correspondences before fusion and then align them.
Revised: Therefore, it is necessary to find correct correspondences before fusion and then align them.

Revised: instances among views [14, 31, 11, 13, 27, 34, 32].

Revised: Generally, the quality of anchors is of central importance for performance and the sampling and k -means two strategies are commonly-used.

Revised: For example, Kang et al.

Original:  Despite being widely applied in large-scale applications, one vital factor of successful multi-viewanchor graph clustering is to build correct correspondences for anchor sets.
Revised: operate k -means on each view separately and fuse into an optimal graph [11]. [34] proposes to jointly optimize anchors and the respective graph among multiple views. The space and time complexity have been reduced into O ( vnm ) and O ( nm 2 ) linear to instance numbers, which 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.

Original: The selected anchor setsgenerated by different views may mismatch without guarantees of correspondences, since k -meansor sampling is performed on each view separately.
Revised: 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.

Original: Taking two anchor graphs Z 1 ∈ R n × m and
Revised: Taking two anchor graphs Z 1 ∈ R n × m and

Original: Z 2 ∈ R n × m as showcase in Fig.
Revised: Z 2 ∈ R n × m as showcase in Fig.

Original: 1, the anchor graphs are not aligned column-wisely, which we referas A nchor- U naligned
Revised: 1, the anchor graphs are not aligned column-wisely, which we refer as A nchor- U naligned

Original: P roblem (AUP).
Revised: P roblem (AUP).

Original: Such an unaligned issue would cause inappropriate graphfusion and degrade the clustering performance in return.
Revised: Such an unaligned issue would cause inappropriate graph fusion and degrade the clustering performance in return.

Original: One pioneer work, SFMC [15] providesan intuitive way to select samples with same indexes to implicitly avoid wrong correspondences.
Revised: One pioneer work, SFMC [13] provides an intuitive way to select samples with same indexes to implicitly avoid wrong correspondences.

Original: However, this way could destroy the flexibility of anchors.
Revised: However, this way could destroy the flexibility of anchors.

Original: To the best of our knowledge, nogeneralized framework for flexible multi-view anchor correspondences has been proposed so far sinceit is difficult to directly compute anchor distances with inconsistent dimensions for different views.
Revised: 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.

Original: To this end, we propose a generalized and flexible multi-view anchor graph fusion framework termed
Revised: To this end, we propose a generalized and flexible multi-view anchor graph fusion framework termed

Original: F ast M ulti-
Revised: F ast M ulti-

Original: V iew A nchor- C orrespondence C lustering (FMVACC) in this paper.
Revised: V iew A nchor- C orrespondence C lustering (FMVACC) in this paper.

Original: Unlike the existingrigid style of fixed indexes, FMVACC flexibly selects anchors with more discriminate inner-viewstructures.
Revised: Unlike the existing rigid style of fixed indexes, FMVACC flexibly selects anchors with more discriminate inner-view structures.

Original: Then, we elegantly solve the unmeasurable multi-dimensional anchor issue by relationalrepresentations.
Revised: Then, we elegantly solve the unmeasurable multi-dimensional anchor issue by relational representations.

Original: Based on this, FMVACC captures both feature and structure information to helpestablish accurate anchor correspondences.
Revised: Based on this, FMVACC captures both feature and structure information to help establish accurate anchor correspondences.

Original: After that, the anchor graph fusion is column-wiselyperformed with cross-view consistency.
Revised: After that, the anchor graph fusion is column-wisely performed with cross-view consistency.

Original: Further, we show that the late fusion [19] and the PVC [7]are our special variants.
Revised: Further, we show that the late fusion [18] and the PVC [7] are our special variants.

Original: Extensive experiments clearly demonstrate the effectiveness of our method.
Revised: Extensive experiments clearly demonstrate the effectiveness of our method.

Original: Moreover, our proposed alignment module also shows significant clustering performance improvement applying to existing multi-view anchor graph competitors, which indicating the importance ofanchor alignment.
Revised: Moreover, our proposed alignment module also shows significant clustering performance improvement applying to existing multi-view anchor graph competitors, which indicating the importance of anchor alignment.

Original: We summarize the contributions of this work as follows,
Revised: We summarize the contributions of this work as follows,

Original: • We study a new paradigm for large-scale multi-view anchor graph clustering, termed as A nchor- U naligned P roblem (AUP).
Revised: • We study a new paradigm for large-scale multi-view anchor graph clustering, termed as A nchor- U naligned P roblem (AUP).

Original: The selected anchor sets in multi-view data are notaligned, which may lead to inaccurate graph fusion and degrade the clustering performance.
Revised: The selected anchor sets in multi-view data are not aligned, which may lead to inaccurate graph fusion and degrade the clustering performance.

Original: • We propose a flexible anchor graph fusion framework termed FMVACC to tackle the AUPproblem.
Revised: • We propose a flexible anchor graph fusion framework termed FMVACC to tackle the AUP problem.

Original: After generating flexible anchor candidates, an anchor alignment module is proposed to solve AUP with both feature and structure information.
Revised: After generating flexible anchor candidates, an anchor alignment module is proposed to solve AUP with both feature and structure information.

Original: To the best of our knowledge, it is the first study of the flexible anchor correspondence fusion framework.
Revised: To the best of our knowledge, it is the first study of the flexible anchor correspondence fusion framework.

Original: • Extensive experiments demonstrate the effectiveness and efficiency of our proposed frame- work.
Revised: • Extensive experiments demonstrate the effectiveness and efficiency of our proposed framework.

Original: Moreover, the proposed alignment module also shows significant performanceimprovement on existing approaches which indicates the importance of anchor alignment
Revised: Moreover, the proposed alignment module also shows significant performance improvement on existing approaches which indicates the importance of anchor alignment.

Original:  Related Work 
Revised:  Related Work 2.1 Multi-view Graph Clustering

Original: 2.1 Multi-view Graph Clustering

Original: As a representative unsupervised multi-view learning method [29, 6, 8, 41], multi-view graphclustering (MVGC) represents single-view structures with graphs and then operates graph fusion on individual graphs [17, 32, 25, 3, 46].
Revised: As a representative unsupervised multi-view learning method [6, 37, 16, 29, 47], multi-view graph clustering (MVGC) represents single-view structures with graphs and then operates graph fusion on individual graphs [15, 28, 24, 3, 42].

Original: Many graph approaches have been proposed with the differencesof these two parts.
Revised: Many graph approaches have been proposed with the differences of these two parts.

Original: For constructing stage, self-expressive representation and local similarity graph areproved to be effective on finding pairwise relationships [4, 49, 2, 48, 47, 44].
Revised: For constructing stage, self-expressive representation and local similarity graph are proved to be effective on finding pairwise relationships [4, 45, 2, 44, 43, 40].

Original: Moreover, by imposingvarious regularization terms on the optimal graph, graph fusion stage reach consensus agreementon similarity and partition information [23, 12, 9, 21].
Revised: Moreover, by imposing various regularization terms on the optimal graph, graph fusion stage reach consensus agreement on similarity and partition information [22, 10, 8, 20].

Original: Zhan et al.
Revised: Zhan et al.

Original: optimize the consensus graphby minimizing disagreement of individual graphs and then optimizes with low-rank constraint [44].
Revised: optimize the consensus graph by minimizing disagreement of individual graphs and then optimizes with low-rank constraint [40].

Original: Notice that the AUP problem does not happen under full pair-wise graph settings since the anchorsare instances themselves and therefore naturally aligned.
Revised: Notice that the AUP problem does not happen under full pair-wise graph settings since the anchors are instances themselves and therefore naturally aligned.

Original: 2.2 Multi-view Anchor Graph Clustering
Revised: 2.2 Multi-view Anchor Graph Clustering

Original: Although existing multi-view graph clustering methods have been efficient in improving the performance, the high computational complexity prevents their application to large-scale datasets.
Revised: Although existing multi-view graph clustering methods have been efficient in improving the performance, the high computational complexity prevents their application to large-scale datasets.

Original: Thespace and time complexity of majority graph approaches are O ( vn 2 ) and O ( n 3 ) quadratic and cubicto instance numbers [26, 16].
Revised: The space and time complexity of majority graph approaches are O ( vn 2 ) and O ( n 3 ) quadratic and cubic to instance numbers [25, 14].

Original: To tackle the above issue, multi-view anchor graph clustering has beenwidely studied to effectively reduce the time and space consumption by constructing anchor graphswith selected m anchors instead of the entire instances [34].
Revised: 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 m anchors instead of the entire instances [30].

Original: Specifically, the first step of multi-view anchor graph clustering is to construct individual anchorgraphs on each view by solving the following problem,
Revised: Specifically, the first step of multi-view anchor graph clustering is to construct individual anchor graphs on each view by solving the following problem,

Original: [Equation]
Revised: [Equation]

Original: (1) where A i ∈ R m × d i is the anchor matrix for the i -th view.
Revised: where A i ∈ R m × d i is the anchor matrix for the i -th view.

Original: Z i ∈ R n × m is the anchor graph for thei -th view and Z i 1 m = 1 n ensures that the similarity sum of each sample to all the anchors equals 1.
Revised: Z i ∈ R n × m is the anchor graph for the i -th view and Z i 1 m = 1 n ensures that the similarity sum of each sample to all the anchors equals 1.

Original: Noticed that A i is fixed before the construction process, so the quality of anchors plays an essentialpart in the success of multi-view anchor graph clustering.
Revised: Noticed that A i is fixed before the construction process, so the quality of anchors plays an essential part in the success of multi-view anchor graph clustering.

Original: Li et al.
Revised: Li et al.

Original: [16] and Kang et al.
Revised: [14] and Kang et al.

Original: [13] proposeto get the anchor set by performing k -means on the original data of individual views.
Revised: [11] propose to get the anchor set by performing k -means on the original data of individual views.

Original: Randomsampling is also a common strategy for selecting anchor points [22].
Revised: Random sampling is also a common strategy for selecting anchor points [21].

Original: Moreover, Li et al.
Revised: Moreover, Li et al.

Original: [15] providea directly alternate sampling method to determine the anchors based on scores.
Revised: [13] provide a directly alternate sampling method to determine the anchors based on scores.

Original: After constructing the view-specific anchor graphs { Z i } vi =1 , the second step is to get the fused anchorgraph G ∈ R n × m as follows,
Revised: After constructing the view-specific anchor graphs { Z i } vi =1 , the second step is to get the fused anchor graph G ∈ R n × m as follows,

Original: [Equation]
Revised: [Equation]

Original: where α ∈ R v is the weight coefficient which measures the impact of each view and α ⊤ 1 v = 1guarantees that G1 m = 1 n .
Revised: where α ∈ R v is the weight coefficient which measures the impact of each view and α ⊤ 1 v =guarantees that G1 m = 1 n .

Original: The weight coefficient α and fused graph G are alternatively optimizedin a unified framework.
Revised: The weight coefficient α and fused graph G are alternatively optimized in a unified framework.

Original: However, solving Eq.
Revised: However, solving Eq.

Original: (2) will lead to a trivial solution.
Revised: (2) will lead to a trivial solution.

Original: To avoid the aboveproblem, Kang et al.
Revised: To avoid the above problem, Kang et al.

Original: [13] directly assign the same weight to all views and Li et al.
Revised: [11] directly assign the same weight to all views and Li et al.

Original: [15] further solveit by imposing rank constraint on the fused anchor graph.
Revised: [13] further solve it by imposing rank constraint on the fused anchor graph.

Original: After obtaining the fusion graph G , weperform k -means on its left singular vector to get the final clustering result.
Revised: After obtaining the fusion graph G , we perform k -means on its left singular vector to get the final clustering result.

Original: The space and timecomplexity of multi-view anchor graph clustering have been reduced into O ( vnm ) and O ( nm 2 ) ,which can be applied to large-scale tasks effectively.
Revised: The space and time complexity of multi-view anchor graph clustering have been reduced into O ( vnm ) and O ( nm 2 ) , which can be applied to large-scale tasks effectively.

Original:  Anchor-Unaligned Clustering for Large-scale Multi-view Data
Revised:  Anchor-Unaligned Clustering for Large-scale Multi-view Data

Original: In this section, we propose a generalized and flexible anchor graph fusion framework, termed fastmulti-view anchor-correspondence clustering (FMVACC), which could elegantly solve the anchorunmeasurable issue for different views and therefore perform anchor graph fusion correctly.
Revised: In this section, we propose a generalized and flexible anchor graph fusion framework, termed fast multi-view anchor-correspondence clustering (FMVACC), which could elegantly solve the anchorunmeasurable issue for different views and therefore perform anchor graph fusion correctly.

Original: Wefirstly show how FMVACC captures the feature and structure information for multi-view anchorset matching.
Revised: We firstly show how FMVACC captures the feature and structure information for multi-view anchor set matching.

Original: Then we present our FMVACC framework with theoretical analysis and show theconnection between 
Revised: Then we present our FMVACC framework with theoretical analysis and show the connection between late fusion strategy and partial view-aligned clustering.

Original: 3.1 Flexible Anchor Generation
Revised: 3.1 Flexible Anchor Generation

Original: Given multi-view data matrices { X i } vi =1 with n samples, v views and d i dimension for the i -th view.
Revised: Given multi-view data matrices { X i } vi =1 with n samples, v views and d i dimension for the i -th view.

Original: By selecting m anchors in each view and following anchor graph framework described in Eq.
Revised: By selecting m anchors in each view and following anchor graph framework described in Eq.

Original: (1) in
Revised: (1) in

Original: Section 2.2, we define the obtained anchor set and the respective individual anchor graph for the i -thview are A i ∈ R m × d i and Z i ∈ R n × m .
Revised: Section 2.2, we define the obtained anchor set and the respective individual anchor graph for the i -th view are A i ∈ R m × d i and Z i ∈ R n × m .

Original: Firstly, it is widely accepted that the quality of anchors plays an essential part in the success ofmulti-view anchor graph clustering, and adaptive optimizing anchors is proven to be effective inclustering contrary to existing fixed strategy.
Revised: Firstly, it 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.

Original: In our method, different from no constraint of { A i } vi =1in Eq.
Revised: In our method, different from no constraint of { A i } vi = in Eq.

Original: (1), the anchors are further imposed to be orthogonal
Revised: (1), the anchors are further imposed to be orthogonal

Original: that A i
Revised: that A i

Original: A ⊤ i
Revised: A ⊤ i

Original: = I m in each view to makethe learned anchors more discriminative and diverse.
Revised: = I m in each view to make the learned anchors more discriminative and diverse.

Original: [Equation]
Revised: [Equation]

Original: The optimization of Eq.
Revised: The optimization of Eq.

Original: (3) can be solved by a two-step alternative approach as follows,
Revised: (3) can be solved by a two-step alternative approach as follows,

Original: Updating Z i :
Revised: Updating Z i :

Original: Denoting z ( i ) j,l as the l -th element in j -th row, Eq.
Revised: Denoting z ( i ) j,l as the l -th element in j -th row, Eq.

Original: (3) can be solved by row as thefollowing form,
Revised: (3) can be solved by row as the following form,

Original: [Equation]
Revised: [Equation]

Original: (4) where y ( i ) j
Revised: where y ( i ) j

Original: = ( X i
Revised: = ( X i

Original: A ⊤ i ) j, : / (1 + β ) .
Revised: A ⊤ i ) j, : / (1 + β ) .

Original: Eq.
Revised: Eq.

Original: (4) is a projection capped simplex problem defined in [39],which can be solved efficiently at global minimum with O ( nmd i ) .
Revised: (4) is a projection capped simplex problem defined in [35], which can be solved efficiently at global minimum with O ( nmd i ) .

Original: Updating A i : By denoting B
Revised: Updating A i : By denoting B

Original: =
Revised: =

Original: Z ⊤ i
Revised: Z ⊤ i

Original: X i , updating anchor matrices { A i } vi =1 is equivalent as follows,
Revised: X i , updating anchor matrices { A i } vi =1 is equivalent as follows,

Original: [Equation]
Revised: [Equation]

Original: The optimum of A i can be analytically obtained with rank m truncated SVD by [37].
Revised: The optimum of A i can be analytically obtained with rank m truncated SVD by [33].

Original: The complexityto update each A i is O ( nm 2 + m 2 d i ) .
Revised: The complexity to update each A i is O ( nm 2 + m 2 d i ) .

Original: If m > d i , we simply remove the orthogonal constraints intounconstrained optimization problems.
Revised: If m > d i , we simply remove the orthogonal constraints into unconstrained optimization problems.

Original: After optimization, we can obtain flexibly more representative single view anchors { A i } vi =1 andanchor graphs.
Revised: After optimization, we can obtain flexibly more representative single view anchors { A i } v i =1 and anchor graphs.

Original: However, the anchor sets { A i } vi =1 for individual views may mismatch with otherviews’ counterparts, and therefore the formed anchor graphs are not aligned column-wisely.
Revised: However, the anchor sets { A i } vi =1 for individual views may mismatch with other views’ counterparts, and therefore the formed anchor graphs are not aligned column-wisely.

Original: Toachieve correct anchor graph clustering, we first need to establish the matching correspondences ofthe pair of anchor sets by applying graph matching approaches, which mathematically means to seekthe matching correspondence or the permutation matrix P ∈ { 0 , 1 } m × m shown in Fig.
Revised: To achieve correct anchor graph clustering, we first need to 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 × m shown in Fig.

Original: After that,anchor graph fusion is column-wisely performed the same as the conventional pipelines.
Revised: After that, anchor graph fusion is column-wisely performed the same as the conventional pipelines.

Original: 3.2 Generalized Multi-view Anchor Matching Framework
Revised: 3.2 Generalized Multi-view Anchor Matching Framework

Original: The main difficulty of solving the AUP problem is unmeasurable anchor sets under multidimensional metric space .
Revised: The main difficulty of solving the AUP problem is unmeasurable anchor sets under multidimensional metric space .

Original: The obtained anchors in multiple views are naturally presented withdifferent dimensions, making it difficult to directly compute their distances and unmeasurable invarious metric spaces.
Revised: The obtained anchors in multiple views are naturally presented with different dimensions, making it difficult to directly compute their distances and unmeasurable in various metric spaces.

Original: Therefore, a significant problem arises: how to flexibly find correspondencesfor multiple anchor sets under different data metric spaces?
Revised: Therefore, a significant problem arises: how to flexibly find correspondences for multiple anchor sets under different data metric spaces?

Original: One intuitive solution to implicitly avoid correspondence issues, proposed in SFMC [15], is to samplerepresentative instances with the same indexes among multiple views.
Revised: One intuitive solution to implicitly avoid correspondence issues, proposed in SFMC [13], is to sample representative instances with the same indexes among multiple views.

Original: However, this sampling stylecould destroy the flexibility and effectiveness of anchors and ignore the local individual view structurefor clustering. It has been shown that not sampling but learning anchors in individual views hasachieved more satisfactory performance, demonstrating the necessity of introducing flexibility into View
Revised: However, this sampling style

Original: [Figure]
Revised: [Table]

Original: (cid:28611) (cid:28611) (cid:28612) (cid:28611) (cid:28611) (cid:28612) (cid:28611) (cid:28611) (cid:28611) (cid:28611) (cid:28611) (cid:28612) (cid:28612) (cid:28611) (cid:28611)(cid:28595) (cid:28611)

Original: [Figure]
Revised: [Table]

Original: (cid:1800) (cid:2779) (cid:1866) (cid:1866) (cid:1865) (cid:1865) (cid:1853) (cid:2869)(cid:4666)(cid:2869)(cid:4667) (cid:1853) (cid:2870)(cid:4666)(cid:2869)(cid:4667) (cid:1853) (cid:2869)(cid:4666)(cid:2870)(cid:4667) (cid:1853) (cid:2869)(cid:4666)(cid:2871)(cid:4667) (cid:1853) (cid:2869)(cid:4666)(cid:2872)(cid:4667) (cid:1853) (cid:2870)(cid:4666)(cid:2870)(cid:4667) (cid:1853) (cid:2870)(cid:4666)(cid:2871)(cid:4667) (cid:1853) (cid:2870)(cid:4666)(cid:2872)(cid:4667)
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: (cid:1800) (cid:2778) [Figure] [Figure]
Revised: could destroy the flexibility and effectiveness of anchors and ignore the local individual view structure for clustering.

Original: [Figure] (cid:1793) [Figure] anchor frameworks.
Revised: 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.

Original: Another solution is to project multi-dimensional data into a commonly-sharedspace and then explicitly optimize consensus anchors in latent space [38].
Revised: Another solution is to project multi-dimensional data into a commonly-shared space and then explicitly optimize consensus anchors in latent space [34].

Original: However, the projectionoperator may lead to information loss and extra dimension hyperparameter selection.
Revised: However, the projection operator may lead to information loss and extra dimension hyperparameter selection.

Original: Flexible anchorsets naturally induce the unaligned matching problem, which is often ignored in the existing literature.
Revised: Flexible anchor sets naturally induce the unaligned matching problem, which is often ignored in the existing literature.

Original: To the best of our knowledge, no generalized framework of flexible anchor correspondences formulti-view clustering has not been formally proposed so far.
Revised: To the best of our knowledge, no generalized framework of flexible anchor correspondences for multi-view clustering has not been formally proposed so far.

Original: Clearly, the clustering performance willbe improved by establishing the correct correspondences of anchors during the fusion stage.
Revised: Clearly, the clustering performance will be improved by establishing the correct correspondences of anchors during the fusion stage.

Original: The first step is how to make the obtained multiple anchor sets { A i } vi =1 measurable .
Revised: The first step is how to make the obtained multiple anchor sets { A i } vi =1 measurable .

Original: Going backto Eq.
Revised: Going back to Eq.

Original: (3), the respective single view anchor graphs { Z i } vi =1 represent the similarities betweenthe instances and the anchors.
Revised: (3), the respective single view anchor graphs { Z i } vi =1 represent the similarities between the instances and the anchors.

Original: Elegantly, each column of the anchor graphs { Z i } vi =1 ⊆ R n × mis taken as a new n -dimensional feature representation of each anchor, which makes the anchorsbetween different views measurable.
Revised: Elegantly, each column of the anchor graphs { Z i } vi =1 ⊆ R n × m is taken as a new n -dimensional feature representation of each anchor, which makes the anchors between different views measurable.

Original: As shown in Fig.
Revised: As shown in Fig.

Original: 2, we transform the anchor alignment probleminto a graph matching problem of m nodes with consensus n embedding features.
Revised: 2, we transform the anchor alignment problem into a graph matching problem of m nodes with consensus n embedding features.

Original: Under suchtransformation, the multiple anchor sets are measurable in R n space and therefore can calculate theirsimilarities to find set correspondences.
Revised: Under such transformation, the multiple anchor sets are measurable in R n space and therefore can calculate their similarities to find set correspondences.

Original: Followed by the traditional graph matching problem [43],our proposed FMVACC retains both first-order (feature) and second-order (structure) information forbetter matching in the following subsections.
Revised: Followed by the traditional graph matching problem [38], our proposed FMVACC retains both first-order (feature) and second-order (structure) information for better matching in the following subsections.

Original: Feature Correspondence:
Revised: Feature Correspondence:

Original: For the first-order feature correspondence, we consider the principle thatthe correspondence probability of anchor points a ( i ) 1 and a ( j ) 2 should be high if their features aremore similar than other pairs.
Revised: For the first-order feature correspondence, we consider the principle that the correspondence probability of anchor points a ( i ) 1 and a ( j ) 2 should be high if their features are more similar than other pairs.

Original: a ( j ) i is denoted as the j -th anchor in i -th view.
Revised: a ( j ) i is denoted as the j -th anchor in i -th view.

Original: Therefore, the featurecorrespondence can be fulfilled by minimizing the optimization goal as follows,
Revised: Therefore, the feature correspondence can be fulfilled by minimizing the optimization goal as follows,

Original: [Equation]
Revised: [Equation]

Original: where K = Z ⊤ 1 Z 2 , p = vec( P ⊤ ) ∈ { 0 , 1 } m 2 , k = vec( K ) ∈ R m 2 to denote the vectorization ofmatrices respectively.
Revised: where K = Z ⊤ 1 Z 2 , p = vec( P ⊤ ) ∈ { 0 , 1 } m 2 , k = vec( K ) ∈ R m 2 to denote the vectorization of matrices respectively.

Original: K ij means the similarity between anchor pair a ( i ) 1 and a ( j ) 2 which can be asthe opposite of the pair distances.
Revised: K ij means the similarity between anchor pair a ( i ) 1 and a ( j ) 2 which can be as the opposite of the pair distances.

Original: It is easy to prove that by minimizing feature correspondence in
Revised: It is easy to prove that by minimizing feature correspondence in

Original: Eq.
Revised: Eq.

Original: (6), higher feature similarity value of K ij will lead to bigger value of the assignment value P ji .
Revised: (6), higher feature similarity value of K ij will lead to bigger value of the assignment value P ji .

Original: Remark 1.
Revised: Remark 1.

Original: Minimizing feature correspondence in Eq.
Revised: Minimizing feature correspondence in Eq.

Original: (6) can be regarded as seeking the optimaltransport plan for anchor sets A 1 and A 2 .
Revised: (6) can be regarded as seeking the optimal transport plan for anchor sets A 1 and A 2 .

Original: Define the pair distance D ij =
Revised: Define the pair distance D ij =

Original: C − K ij , where
Revised: C − K ij , where

Original: C is a large enough number to ensure D ij ≥ 0 .
Revised: C is a large enough number to ensure D ij ≥ 0 .

Original: Then max P (cid:80) mi =1 (cid:80) mj
Revised: Then max P (cid:80) [Equation] [Table]

Original: =1 K ij P ji = max P m C −(cid:80) mi =1 (cid:80) mj =1 D ij P ji ⇐⇒ min P (cid:80) mi =1 (cid:80) mj =1 D ij P ji , where P indicates the optimal transport plan.
Revised: [Table]

Original: Structure Correspondence:
Revised: Structure Correspondence:

Original: With the mentioned first-order feature correspondence part, we furtherconsider the second-order graph structure regularization.
Revised: With the mentioned first-order feature correspondence part, we further consider the second-order graph structure regularization.

Original: To find better matching, the inner structures of anchor sets should also be comparable after matching, which indicates the matching betweenpair-wise edges.
Revised: To find better matching, the inner structures of anchor sets should also be comparable after matching, which indicates the matching between pair-wise edges.

Original: The second-order structure correspondence can be achieved by minimizing thefollowing function,
Revised: The second-order structure correspondence can be achieved by minimizing the following function,

Original: [Equation]
Revised: [Equation]

Original: where S 1 ⊗ S 2 is the Kronecker product of S 1 and S 2 , and S 1 =
Revised: where S 1 ⊗ S 2 is the Kronecker product of S 1 and S 2 , and S 1 =

Original: Z ⊤ 1 Z 1 , S 2 =
Revised: Z ⊤ 1 Z 1 , S 2 =

Original: Z ⊤ 2 Z 2 .
Revised: Z ⊤ 2 Z 2 .

Original: Therefore, S 1and S 2 represent the inner graph structures of two anchor sets A 1 and A 2 .
Revised: Therefore, Sand S 2 represent the inner graph structures of two anchor sets A 1 and A 2 .

Original: By minimizing Eq.
Revised: By minimizing Eq.

Original: (7),the inner graph structures can reach a maximum agreement.
Revised: (7), the inner graph structures can reach a maximum agreement.

Original: Unified Correspondence Objective: Taking both first-order feature and second-order structure intoconsideration, the unified optimization goal for two anchor sets can be formulated as follows,
Revised: 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,

Original: [Equation]
Revised: [Equation]

Original: where λ is the balanced hypermeter.
Revised: where λ is the balanced hypermeter.

Original: Problem (8) is a quadratic assignment problem (QAP) and isproved to be NP-hard under most circumstances [14].
Revised: Problem (8) is a quadratic assignment problem (QAP) and is proved to be NP-hard under most circumstances [12].

Original: The feasible region constraint often relaxesinto its convex hull, the Birkhoff polytope with double stochastic region  P1 = 1 , P ⊤ 1 = 1 , P ∈[0 , 1] m × m .
Revised: The feasible region constraint often relaxes into its convex hull, the Birkhoff polytope with double stochastic region where P1 = 1 , P ⊤ 1 = 1 , P ∈ [0 , 1] m × m .

Original: Then the optimization problem transforms into,
Revised: Then the optimization problem transforms into,

Original: [Equation]
Revised: [Equation]

Original: We refer Eq.
Revised: We refer Eq.

Original: (9) as the multi-view anchor correspondence framework.
Revised: (9) as the multi-view anchor correspondence framework.

Original: To efficiently solve Eq.
Revised: To efficiently solve Eq.

Original: (9),we adopt the Projected Fixed-Point Algorithm [20] to update P as follows,[Equation]
Revised: (9), we adopt the Projected Fixed-Point Algorithm [19] to update P as follows,

Original: [Equation]
Revised: [Equation]

Original: where α is the step size parameter, t denotes the number of iterations and Γ denotes the doublestochastic projection operator.
Revised: where α is the step size parameter, t denotes the number of iterations and Γ denotes the double stochastic projection operator.

Original: The convergence of the algorithm has been proved in [20], and we setα = 0 .
Revised: The convergence of the algorithm has been proved in [19], and we set α = 0 .

Original: 5 in this paper.
Revised: 5 in this paper.

Original: Details of solving Eq.
Revised: Details of solving Eq.

Original: (9) and Remark 2 can be found in supplementary.
Revised: (9) and Remark 2 can be found in supplementary.

Original: Remark 2.
Revised: Remark 2.

Original: The algorithm to solve Eq.
Revised: The algorithm to solve Eq.

Original: (9) converges at rate 12 + λ ∥ ( S 1 ⊗ S 2 )
Revised: (9) converges at rate 12 + λ ∥ ( S 1 ⊗ S 2 )

Original: ∥ F .
Revised: ∥ F .

Original: After obtaining { P i } vi =2 , we achieve the fused aligned anchor graph as
Revised: After obtaining { P i } vi =2 , we achieve the fused aligned anchor graph as

Original: [Equation]
Revised: [Equation]

Original: The final clustering result can be obtained by simply calculating rank- k truncated SVD on Z Alignedand output for k -means.
Revised: The final clustering result can be obtained by simply calculating rank- k truncated SVD on Z Aligned and output for k -means.

Original: The whole process is summarized in Algorithm 2.
Revised: The whole process is summarized in Algorithm 2.

Original: 3.3 Theoretical Analysis
Revised: 3.3 Complexity Analysis

Revised: In this subsection, we analyze our proposed algorithm in terms of space/time complexity.

Revised: Space Complexity:

Revised: In our paper, the major memory costs of our method are matrices { A i } vi =1 ∈ R m ×

Revised: d i , { P i } vi =1 ∈ R m × m and { Z i } vi =1 ∈ R m × n .

Revised: Thus the space complexity of our FMVACC is

Revised: O ( md + mnv + m 2 v ) .

Revised: In our algorithm, m ≪ n and d ≪ n .

Revised: Therefore, the space complexity of

Revised: FMVACC is O ( n ) .

Revised: Time complexity: The computational complexity of FMVACC is composed of three steps as mentioned before.

Revised: When updating { Z i } vi =1 , it costs O ( nmd ) to get the optimal value.

Revised: Updating { A i } vi =1 needs O ( nm 2 + m 2 d ) .

Revised: Then the time cost of alignment module is O ( m 3 ) .

Revised: Therefore, the total time cost of the optimization process is O ( nmd + nm 2 + m 2 d + m 3 ) .

Revised: Consequently, the computational complexity of our proposed optimization algorithm is linear complexity O ( n ) .

Revised: After the optimization, we perform SVD on Z to obtain the spectral embedding and output the discrete clustering labels by k -means [34].

Revised: The post-process needs O ( nm 2 ) , which is also linear complexity respecting to samples.

Revised: In total, our algorithm achieves MVC with both linear space and time complexity, which demonstrates the efficiency of FMVACC.

Revised: 3.4 Theoretical Analysis

Original: In this subsection, we further show theoretical analysis of our proposed FMVACC with some existingwidely-studied multi-view clustering settings.
Revised: In this subsection, we further show theoretical analysis of our proposed FMVACC with some existing widely-studied multi-view clustering settings.

Original: Connection with Multi-view Late Fusion Strategy [19] Multi-view late fusion strategy seeks tofuse partition level information where multiple clustering results can reach a maximum agreement onthe final partition matrix.
Revised: 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.

Original: It is easy to show that this strategy is a special case of FMVACC wherethe m anchors are clustering centers ( m = k ) and each single Z i represents the discrete partition.
Revised: It is easy to show that this strategy is a special case of FMVACC where the m anchors are clustering centers ( m = k ) and each single Z i represents the discrete partition.

Original: However, the differences are: (i) Late fusion only considers the feature correspondence in Eq.
Revised: However, the differences are: (i) Late fusion only considers the feature correspondence in Eq.

Original: (6)with spectral relaxations PP ⊤ = I m while the structure counterpart is ignored; (ii) FMVACC showsmore generality of late fusion with no limitation of anchor type (clustering centers) and numbers ( k ).
Revised: (6) with spectral relaxations PP ⊤ = I m 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 ).

Original: Connection with PVC [7] PVC assumes a little different multi-view setting where the data matricesare not aligned with some rows.
Revised: Connection with PVC [7] PVC assumes a little different multi-view setting where the data matrices are not aligned with some rows.

Original: Taking existing correspondences as supervision signals, PVCrecovers other unknown correspondences with a sinkhorn layer in the neural network.
Revised: Taking existing correspondences as supervision signals, PVC recovers other unknown correspondences with a sinkhorn layer in the neural network.

Original: We can easilyshow that PVC is also a special case of FMVACC where only the structure loss in Eq.
Revised: We can easily show that PVC is also a special case of FMVACC where only the structure loss in Eq.

Original: (7) is measuredwith some correspondence is known and the anchors are instances themselves.
Revised: (7) is measured with some correspondence is known and the anchors are instances themselves.

Original: These two connections with existing late fusion strategy and PVC settings further theoreticallydemonstrate the generality of FMVACC.
Revised: These two connections with existing late fusion strategy and PVC settings further theoretically demonstrate the generality of FMVACC.

Original: We summarize our contributions as: (i) We novelly solvemulti-view anchor sets unmeasurable issues by introducing both feature and structure correspondence.(ii) FMVACC is a pioneering work that firstly studies the AUP for anchor graph fusion tasks.
Revised: We summarize our contributions as: (i) We novelly solve multi-view anchor sets unmeasurable issues by introducing both feature and structure correspondence. (ii) FMVACC is a pioneering work that firstly studies the multi-view anchor unaligned issue for anchor graph fusion tasks.

Original: (iii) Byfollowing our proposed flexible alignment module, it is quite interesting to rethink current multi-viewanchor graph fusion with correct correspondences and further benefit the community.
Revised: (iii) By following our proposed flexible alignment module, it is quite interesting to rethink current multi-view anchor graph fusion with correct correspondences and further benefit the community.

Original:  Experiments
Revised:  Experiments

Original: In this section, we conduct experiments to verify the effectiveness of the proposed FMVACC andalignment module.
Revised: In this section, we conduct experiments to verify the effectiveness of the proposed FMVACC and alignment module.

Original: A simulated dataset is used to verify the AUP issue and the results on sevenreal-world multi-view datasets are reported.
Revised: A simulated dataset is used to verify the AUP issue and the results on seven real-world multi-view datasets are reported.

Revised: 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).

Original: More experimental settings can be found in the appendix.
Revised: More experimental settings can be found in the appendix.

Original: 4.1 Simulated Multi-view Dataset
Revised: 4.1 Simulated Multi-view Dataset

Original: Data Generation We construct a simulated dual-view dataset, as shown in the left half of Fig. 1.
Revised: Data Generation We construct a simulated dual-view dataset, as shown in the left half of Fig. 1.

Original: The data have 4 independent clusters where each cluster consists of 50 two-dimensional instancesfollowing the respective Gaussian distribution.
Revised: The data have 4 independent clusters where each cluster consists of 50 two-dimensional instances following the respective Gaussian distribution.

Original: Compared Algorithms We compare with the two SOTA multi-view anchor graph representatives
Revised: Compared Algorithms We compare with the two SOTA multi-view anchor graph representatives

Original: LMVSC [13] and SFMC [15].
Revised: LMVSC [11] and SFMC [13].

Original: Further, we introduce the proposed flexible anchor selection andalignment modules into the two methods to illustrate the effectiveness of two counterparts.
Revised: Further, we introduce the proposed flexible anchor selection and alignment modules into the two methods to illustrate the effectiveness of two counterparts.

Original: [Table]
Revised: [Table]

Original: Visualization and Results Analysis
Revised: Visualization and Results Analysis

Original: According to Table 1 and Fig.
Revised: According to Table 1 and Fig.

Original: 3, we have the followingobservations: (i) For LMVSC and our FMVACC, aligning anchors between views can greatlyimprove the clustering effect, which verifies the effectiveness of our alignment module.
Revised: [Table] 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.

Original: Specifically,
Revised: Specifically,

Original: LMVSC with our alignment module achieves about 15.94% (ACC), 17.82% (NMI) and 20.31%(Fscore) progress compared with the column-unaligned results.
Revised: LMVSC with our alignment module achieves about 15.94% (ACC), 17.82% (NMI) and 20.31% (Fscore) improvement compared with the column-unaligned results.

Original: (ii)
Revised: (ii)

Original: From the results of SFMC, it canbe seen that flexible anchor selection strategy achieves better clustering performance.
Revised: From the results of SFMC, it can be seen that flexible anchor selection strategy achieves better clustering performance.

Original: Numerically,adopting flexible strategy provides 33.00% (ACC) and 21.71%
Revised: Numerically, adopting flexible strategy provides 33.00% (ACC) and 21.71%

Original: (Fscore) on the simulated dataset.
Revised: (Fscore) on the simulated dataset.

Original: (iii)
Revised: (iii)

Original: For LMVSC and our method in Fig.
Revised: For LMVSC and our method in Fig.

Original: 3, Z Aligned obtained after anchors alignment contains less noiseand
Revised: 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 of our method.

Original: 4.2 Real-world Multi-view Datasets
Revised: 4.2 Real-world Multi-view Datasets

Original: Benchmark Datasets We further perform experiments on eight real-world datasets ranging from 169to 63896 instances, multiple dimensions and views.
Revised: Benchmark Datasets We further perform experiments on eight real-world datasets ranging from 169 to 63896 instances, diverse dimensions and views.

Original: 3-Sources contains 169 instances in 6 clustersfor 3 views.
Revised: 3-Sources contains 169 instances in 6 clusters for 3 views.

Original: UCI-Digit refers to 2000 handwritten images in 10 clusters for 3 views.
Revised: UCI-Digit refers to 2000 handwritten images in 10 clusters for 3 views.

Original: BDGP haves2500 instances for 3 views while SUNRGBD collects 10335 samples.
Revised: BDGP haves 2500 instances for 3 views while SUNRGBD collects 10335 samples.

Original: MNIST is the commonly-useddataset and YTF-10/20 are collected from [5].
Revised: MNIST is a commonly-used dataset and YTF-10/20 are collected from [5].

Original: Compared Algorithms Seven state-of-the-art multi-view clustering methods are introduced: MSCIAS [40], AMGL [22], PMSC [12], RMKM [1], BMVC [50], LMVSC [13], SFMC [15].
Revised: Compared Algorithms Nine state-of-the-art multi-view clustering methods are introduced: MSCIAS [36], AMGL [21], PMSC [10], RMKM [1], BMVC [46], LMVSC [11], FMCNOF[39], MSGL[9], SFMC [13].

Original: The firstthree algorithms are graph methods and the others are representative large-scale competitors.
Revised: The first three algorithms are graph methods and the others are representative large-scale competitors.

Original: Noticethat SFMC selects anchor number from 0 .
Revised: Notice that SFMC selects anchor number from 0 .

Original: 1 n to n so that it can not be applied in large-scale scenarios.
Revised: 1 n to n so that it can not be applied in large-scale scenarios.

Original: [Table]
Revised: [Table]

Original: Experimental Results We conduct comparative experiments of eight multi-view clustering algorithms on seven commonly-used multi-view datasets, and the experimental results are displayed in
Revised: Experimental Results We conduct comparative experiments of eight multi-view clustering algorithms on seven commonly-used multi-view datasets, and the experimental results are displayed in

Original: Table 2.
Revised: Table 2.

Original: From these results, we observe that:
Revised: From these results, we observe that:

Original: • Our method also outperforms the other seven algorithms on most clustering metrics when applied to the first three small-scale datasets.
Revised: • Our method also outperforms the other seven algorithms on most clustering metrics when applied to the first three small-scale datasets.

Original: Specifically, in terms of ACC, FMVACCachieves 39.02 % (3-Sources), 6.43 % (UCI-Digit) and 27.19 % (BDGP) progress comparedto the classical multi-view graph algorithm AMGL.• When applied on the last four large-scale datasets, compared to well-known large-scalemethods RMKM, BMVC and LMVSC,  our FMVACC outperforms them on most metrics.
Revised: Specifically, in terms of ACC, FMVACC achieves 39.02 % (3-Sources), 6.43 % (UCI-Digit) and 27.19 % (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.

Original: Note that the OM failure of SFMC on the last three datasets is due to the large number of anchors it needs to select.
Revised: Note that the OM failure of SFMC on the last three datasets is due to the large number of anchors it needs to select.

Original: The above experimental results show the effectiveness and superiority of our 
Revised: The above experimental results show the effectiveness and superiority of our FMVACC.

Original: FMVACC.

Original: Note that due to the space limitation, the complexity analysis and related experimental results areplaced in the appendix.

Original: Visualization We visualize the effectiveness of alignment module on UCI-Digit in Fig.
Revised: Visualization We visualize the effectiveness of alignment module on UCI-Digit in Fig.

Original: For
Revised: For

Revised: LMVSC Unaligned 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%).

Revised: We can also find similar phenomenon with our proposed approach and conclude that aligned fusion can reduce the noise and enhance clustering performance.

Original:  LMVSC and ours, the fused anchor graphs obtained after alignment have clearer structures andtherefore achieve better performance, which demonstrates the effectiveness and superiority of ouranchor alignment module.
Revised: 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.

Original: The alignment clearly finds correspondences between views and thereforecontributes to improving clustering.
Revised: and unaligned anchor graphs (LMVSC and Ours) on 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 discover that half of the algorithms cannot be applied on the latter three datasets due to 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 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.

Original:  Effect of Alignment Module In Table 3, we report the accuracy obtained from ablation study on
Revised: [Table] Effect of Alignment Module In Table 4, we report the accuracy obtained from ablation study on

Original: LMVSC and our method.
Revised: LMVSC and our method.

Original: It can be found that for the above two algorithms on all seven benchmarks,the clustering performance achieved by using the alignment module is better than its opponent.
Revised: It can be found that for the above two algorithms on all seven benchmarks, the clustering performance achieved by using the alignment module is better than its opponent

Original: [Table]
Revised: [Table]

Revised: [Table]

Original:  Conclusion
Revised:  Conclusion

Original: In this paper, we present the first study of multi-view anchor unaligned problem.
Revised: In this paper [Table] we present the first study of multi-view anchor unaligned problem.

Original: A flexible multiview anchor graph fusion framework termed F ast M ulti-
Revised: A flexible multiview anchor graph fusion framework termed F ast M ulti-

Original: V iew A nchor- C orrespondence C lustering(FMVACC) is proposed.
Revised: V iew A nchor- C orrespondence C lustering (FMVACC) is proposed.

Original: FMVACC firstly constructs flexible individual anchors and then capturescorrespondences with both feature and structure information.
Revised: FMVACC firstly constructs flexible individual anchors and then captures correspondences with both feature and structure information.

Original: Theoretical analysis between FMVACCand some existing multi-view strategies is also provided.
Revised: Theoretical analysis between FMVACC and some existing multi-view strategies is also provided.

Original: Extensive experiments are conducted todemonstrate the effectiveness of anchor alignment and our proposed framework FMVACC.
Revised: Extensive experiments are conducted to demonstrate the effectiveness of anchor alignment and our proposed framework FMVACC.

Original: It is quiteinteresting to rethink current multi-view anchor graph fusion with correct correspondences and furtherbenefit the research community.
Revised: It is quite interesting to rethink current multi-view anchor graph fusion with correct correspondences and further benefit the research community.

Original: Further, the success of anchor graph clustering heavily dependson the quality of anchor representation.
Revised: Further, the success of anchor graph clustering heavily depends on the quality of anchor representation.

Original: In the future, we will explore how to obtain more-qualifiedanchors by deep learning. 
Revised: In the future, we will explore how to obtain more-qualified anchors by deep learning. Acknowledgment

Revised: This work was supported by the National Key R&D Program of China 2020AAA0107100, the

Revised: Natural Science Foundation of China (project no. 61922088, 61773392, 62006237 and 61976196).

Original: Checklist
Revised: Checklist

Original: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’scontributions and scope?
Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?

Original: [Yes] Please see the abstract and introduction sections.
Revised: [Yes] Please see the abstract and introduction sections.

Revised: (b) Did you describe the limitations of your work?

Revised: [Yes] Please see the conclusion section.

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [No] There isno negative societal impacts.
Revised: [No] There is no negative societal impacts.

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?

Original: [Yes] 
Revised: [Yes] 2.

Original: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?
Revised: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes] Please seesection 4.
Revised: [Yes] Please see section 4.

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes] The proofs areavailable in the appendix.
Revised: [Yes] The proofs are available in the appendix.

Original: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?
Revised: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Original: [Yes] The code, data are available in the supplemental materials.
Revised: [Yes] The code, data are available in the supplemental materials.

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [Yes] All the training details are specified.
Revised: [Yes] All the training details are specified.

Original: (c) Did you report error bars (e g., with respect to the random seed after running experi- ments multiple times)?
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [Yes] Please see Table 1 and .(d)
Revised: [Yes] Please see Table 1 and .

Original:  Did you include the total amount of compute and the type of resources used (e g., typeof GPUs, internal cluster, or cloud provider)?
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?

Original: [Yes] The sources are mentioned in theappendix.
Revised: [Yes] The sources are mentioned in the appendix.

Original: If you are using existing assets (e g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?

Original: [Yes] The creators are all cited.
Revised: [Yes] The creators are all cited.

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [Yes] The assets are all mentioned.
Revised: [Yes] The assets are all mentioned.

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [No] (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?
Revised: [No] (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?

Original: [Yes] The data and codes are all permitted.
Revised: [Yes] The data and codes are all permitted.

Original: (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
Revised: (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?

Original: [No] The used data do not contain such content.
Revised: [No] The used data do not contain such content.

Original: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?
Revised: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
Revised: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Original: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?
Revised: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?

Original: [N/A]
Revised: [N/A]