Robust Consensus Anchor Learning for Efficient Multi-view Subspace Clustering
Authors: Yalan Qin
Nan Pu
Guorui Feng
Nicu Sebe
Venue: ICML 2025 poster (2025)
Keywords: Efficient multi-view clustering
consensus anchors
a unified framework.
Primary area: general_machine_learning->clustering
TL;DR: We first show that if the data are sufficiently sampled from independent subspaces, and the objective function meets some conditions, the achieved anchor graph has the block-diagonal structure.

Abstract: As a leading unsupervised classification algorithm in artificial intelligence, multi-view subspace clustering segments unlabeled data from different subspaces. Recent works based on the anchor have been proposed to decrease the computation complexity for the datasets with large scales in multi-view clustering. The major differences among these methods lie on the objective functions they define. Despite considerable success, these works pay few attention to guaranting the robustness of learned consensus anchors via effective manner for efficient multi-view clustering and investigating the specific local distribution of cluster in the affine subspace. Besides, the robust consensus anchors as well as the common cluster structure shared by different views are not able to be simultaneously learned. In this paper, we propose Robust Consensus anchors learning for efficient multi-view Subspace Clustering (RCSC). We first show that if the data are sufficiently sampled from independent subspaces, and the objective function meets some conditions, the achieved anchor graph has the block-diagonal structure. As a special case, we provide a model based on Frobenius norm, non-negative and affine constraints in consensus anchors learning, which guarantees the robustness of learned consensus anchors for efficient multi-view clustering and investigates the specific local distribution of cluster in the affine subspace. While it is simple, we theoretically give the geometric analysis regarding the formulated RCSC. The union of these three constraints is able to restrict how each data point is described in the affine subspace with specific local distribution of cluster for guaranting the robustness of learned consensus anchors. RCSC takes full advantages of correlation among consensus anchors, which encourages the grouping effect and groups highly correlated consensus anchors together with the guidance of view-specific projection. The anchor graph construction, partition and robust anchor learning are jointly integrated into a unified framework. It ensures the mutual enhancement for these procedures and helps lead to more discriminative consensus anchors as well as the cluster indicator. We then adopt an alternative optimization strategy for solving the formulated problem. Experiments performed on eight multi-view datasets confirm the superiority of RCSC based on the effectiveness and efficiency.

Robust Consensus Anchor Learning for Efficient Multi-view Subspace Clustering
Yalan Qin
Nan Pu
Guorui Feng
|
Nicu Sebe
Abstract
1. Introduction
As a leading unsupervised classification algo- rithm in artificial intelligence, multi-view sub- space clustering segments unlabeled data from different subspaces. Recent works based on the anchor have been proposed to decrease the com- putation complexity for the datasets with large scales in multi-view clustering. The major dif- ferences among these methods lie on the objec- tive functions they define. Despite considerable success, these works pay few attention to guar- anting the robustness of learned consensus an- chors via effective manner for efficient multi- view clustering and investigating the specific lo- cal distribution of cluster in the affine subspace. Besides, the robust consensus anchors as well as the common cluster structure shared by dif- ferent views are not able to be simultaneously learned. In this paper, we propose Robust Con- sensus anchors learning for efficient multi-view Subspace Clustering (RCSC). We first show that if the data are sufficiently sampled from indepen- dent subspaces, and the objective function meets some conditions, the achieved anchor graph has the block-diagonal structure. As a special case, we provide a model based on Frobenius norm, non-negative and affine constraints in consensus anchors learning, which guarantees the robust- ness of learned consensus anchors for efficient multi-view clustering and investigates the spe- cific local distribution of cluster in the affine sub- space. Experiments performed on eight multi- view datasets confirm the superiority of RCSC based on the effectiveness and efficiency.
Clustering is an important field in artificial intelligence and machine learning (Jain, 2008). As the information technol- ogy develops, large amounts of data from multiple views or channels (Liu et al., 2025; Qin et al., 2023d) can be col- lected in real-world scenarios. This type of data is termed as multi-view data and it widely exists the world. For instance, we can depict an image by multiple represen- tations, such as local binary pattern (LBP), histogram of oriented gradient (HOG) and Gabor feature representation. For dealing with the multi-view data in clustering tasks, various multi-view clustering approaches have been pre- sented in the literature. As opposed to clustering for the data with single view (Qin et al., 2023e;a; 2021; 2022b; 2025b; et al., 2024), multi-view clustering can achieve more reasonable performance in practice (Qin et al., 2022a; Liu al., 2024; Qin al., 2023b; Liu al., 2023b; Qin et al., 2023c; Sun al., 2024: Qin al., 2024a; Liu al., 2022a; Qin et al., 2024d; Liu et al., 2023a; Qin et al., 2025c; Liu et al., 2022b; Qin et al., 2025e). It is able to integrate diverse feature representations of an object obtained by dif- ferent views and provide more comprehensive object infor- mation. Among the existing mueti-view clustering works, methods based the graph have gained significant atten- tion in recent years.
The graph-based methods for multi-view clustering (Nie et al., 201 7b; Zhan et al., 2019; Wang et al., 2020) can re- flect the relationships among samples in multi-view data by constructing graph structures. Then the final results can be achieved based on the partiton of the obtained graph. These methods usually adopt formulate the relationships among data points. Cao et al. (Cao et al., 2015) presented multi-view clustering method based on diversity and smoothness, which investigates the complementarity among different representations. Wang et al. (Wang et al., 2020) learned the unified global graph and view-specific graphs by adopting the mutual reinforce- ment technique. Chen al. (Chen al., 2020) simultane- ously learned the global self-representation, latent embed- ding space and cluster structure for subspace learning on multi-view data. Liang et al. (Liang et al., 2020) utilized the multi-view graph learning to learn a unified graph and
adjacent graph to
School of Communication and Information Engineering. Shanghai University, Shanghai 200444, China *Department of Information Engineering and Computer Science, University of Trento, Trento 40128, Italy. Correspondence to: Guorui Feng <fgr2082 (a0u82allye all00022llyyu..om..
Proceedings of the 1nnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Robust Consensus Anchor Learning for Efficient Multi-view Subspace Clustering
leveraged the incosistency and consisteney among multi- ple view-specific graphs. Nie et al. (Nie al., 2017a) presented to simultaneously perform local structure learn- ing and semi-supervised classffcationncluuternng. Wang et al. (Wang et al., 2020) coupled unified graph, graph in- duced by similarity across views and indicator into a uni- fied framework. Despite significant progress, most of these multi-view clustering approaches tend suffer from high computation complexity. It takes O(n²) to construct n adjacent graph and needs O(n") in partitioning this graph, which limits their scalability for multi-view datasets with large scales.
To cope with the above issues, we propose a novel Robust Consensus anchor learning for efficient multi-view Sub- space Clustering (RCSC). We first theoretically show that block-diagonal anchor graph can be obtained if the ob- jective function meets certain conditions based on the in- dependent subspace assumption. As a special case, we provide a model based on Frobenius norm, non-negative and affine constraints in consensus anchors learning, which guarantees the robustness of learned consensus anchors for efficient multi-view clustering and investigates the specific local distribution of cluster in the affine subspace. While it is simple, we theoretically give the geometric analysis regarding the formulated RCSC. The union of these three constraints is able to restrict how each data point is de- scribed in the affine subspace with specific local distribu- tion of cluster for guaranting the robustness of learned con- sensus anchors. We can fully explore the correlation among the learned consensus anchors with the view-specific pro- jection, which encourages the grouping effect and groups highly correlated anchors together. The robust anchor learning, partition and anchor graph construction are jointly modeled in unified framework. Then the robust consensus anchors and the common cluster structure shared by differ- ent views able to be simultancously learned. We can guarantee the mutual enhancement for these procedures in this manner and achieve more discriminative consensus an- chors as well as the cluster indicator. By imposing the or- thogonal constraints on the actual bases, we constrain a factor matrix to be the cluster indicator matrix based on the rigorous clustering interpretation. We then develop an alternate minimizing algorithm for solving the formulated problem. The major contributions in this paper are:
For dealing with the computation complexity issue, multi- view clustering methods based on the anchor have been given (Qin et al., 2025a;d; Qin & Qian, 2024; Qin et al., 2024c;b), which show promising capability in real appli- cations. Different from constructing an x n graph, the methods based on the anchor typically produce small num- ber of anchors from the dataset and represent the structure of data by building an n x I anchor graph, where I de- notes the total number of anchors. In general, I is lower than n, which enables the scale of data to be greatly de- creased for reaching the goal of increasing the clustering efficiency. Specifically. Yang et al. (Yang et al., 2021) in- creased the efficiency and all views are required to yield the same anchors. Kang et al. (Kang et al., 2020) em- ployed the subspace learning based on the anchor to learn a anchor graph for each view and then heuristically com- bined different anchor graphs into a unified one. Li et al. (Li al., 2022) presented scalable multi-view clustering method by fusing anchor graphs. It is able to employ the discrete cluster structure to adaptively learn unified graph by anchor graph fusion. Wang et al. (Wang et al., 2022) employed multiple projection matrices and a set of latent consensus anchors to learn a unified anchor graph. Yang et al. (Yang et al., 2022) seeked for I anchors on the original data with the guidance of K-means. The produced cen- troids are regarded as anchors and sparse anchor graph is constructed between the obtained anchors and the original dataset. Despite great success, these methods pay few at- tention to ensuring the robustness of learned consensus an- chors for efficient multi-view clustering and investigating the specific local distribution of cluster in the affine sub- space. The correlation among the learned consensus an- chors, which encourages the grouping effect and tends to group highly correlated anchors together, is not able to be fully explored. Besides, the robust consensus anchors and the common cluster structure shared by different views are not able to be simultancously learned in a unified frame- work. Then the mutual enhancement for these procedures not guaranteed in this manner and more discriminative consensus anchors as well as cluster indicator are not ob- tained.
1. We propose novel Robust Consensus anchor learn- ing for efficient multi-view Subspace Clustering (RCSC). We first theoretically demonstrate that an anchor graph with block-diagonal structure can be achieved if the objective function satisfies certain con- ditions. As a special case, we give a model based on Frobenius norm, non-negative and affine constraints in consensus anchors learning, which guarantees the robustness of learned consensus anchors for efficient multi-view clustering and investigates the specific lo- cal distribution of cluster in the affine subspace.
2. We are able to fully explore the correlation among the learned consensus anchors with the guidance of view- specific projection, which encourages the grouping ef- feet and tends to group highly correlated anchors to- gether. We jointly perform the robust anchor learn- ing, partition and anchor graph construction in uni- fied framework. Then, the robust consensus anchors and the common cluster structure shared by different views are simultaneously learned, which ensures the
2

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Robust Consensus Anchor Learning for Efficient Multi-view Subspace Clustering
mutual enhancement for these procedures and helps lead to more discriminative consensus anchors well as the cluster indicator.
across views, respectively. UPA R4,x1 represents the basis matrix. We then theoretically demonstrate that block-diagonal anchor graph can be achieved if the cor- responding objective function satisifies certain conditions based on the independent subspace assumption, which is shown as Theorem in the following.
3.
We impose the orthogonal constraints on the actual bases and constrain factor matrix to be the cluster indicator matrix built on the rigorous clustering inter- pretation. Extentive experiments on different multi- view datasets validate the effectivenss and efficiency of RCSC, especially on the datasets with large scales.
Theorem Assume the subspaces {8: are indepen- dent, X a matrix with columns consisting of some vec- tors from Q2 and U. A; is a matrix with columns consisting of a basis of 9s where (UP)TUP = I and A = The solution S to the following
2. Methodology
In this section, we present the motivation and formulation of RCSC, followed by the optimization process and the re- lated analysis of computation complexity for RCSC.
-U*
(2)
is block-diagonal and unique.
Proof. For data point x² A 0 and xf E 94 we just need to prove that there exists a unique s. x (U*A)s. A)s, where = [sf .... , + with 8; * 0 and 8j
2.1. Motivation
for all A i. Since (UP)TUP I and A A = I, we can obtain A (UP)"UPA = 7. Thus, UPA is or-
The anchor strategy is usually employed to find the under- lying structure by choosing a small number of data points as anchor bases. Some existing mutli-view clustering meth- ods based on the anchor conduct K K-means to achieve clus- tering centroids with the anchor bases being fixed. Despite great success, these methods pay few attention to guarant- ing the robustness of learned consensus anchors for effi cient multi-view clustering and investigating the specific local distribution of cluster in the affine subspace. The cor- relation among the learned consensus anchors, which en- courages the grouping effect and groups highly correlated anchors together, is ignored to be fully explored. More- over, the robust consensus anchors and the common cluster structure shared by multiple views not simultancously learned. Therefore, the mutual enhancement for these pro- cedures is not effectively ensured and more discriminative consensus anchors as well as cluster indicator are not ac- quired.
(UP A) (U* thogonal. Due to the orthogonality among subspaces. there exists a unique decomposition for x" as follows:
x² + ....... x + +..
......... +U?A,)ss + + (URAR)
where (Uf..)); E Q and i = 1...., k. Therefore, (U A,)s; = x" and (UZ A.)s, 0 for all j # Since U A, basis of Sr have 8; o 0. 8 is unique, and 8, 0 for all j +..
Accoring to Theorem 1, a basis of X can be learned and we use it as the dictionary. It is easy to obtain true so- lution by solving the problem in Eq. (1) if subspaces are independent. To deal with more general multi-view clus- tering issue in the affine subspace, we introduce the affine constraint S 1 = 1 the original objective function and obtain the optimization problem as:
2.2. Formulation
We generate view-specific data points via certain gener- ation model based on a latent space. Given multi-view dataset (X" E Rdxxa]y > p=1 with dp and n being the di- mension and size of dataset, we first assume that the data noise free and formulate the corresponding objective function as:
|X- - AS. s.t. ST =
min U.S.S
(4)
p=1
(U A-A=1,
where 1 denotes the vector with all entries being one. To locally depict the distribution of contaminated data points from different subspaces, we add a non-negative constraint for and achieve the problem as:
Σ - UPASS p=1 s.L. (U ATA=1,
min A,S
(1)
where S R(x" the shared affinity matrix, {UP} e Rd,xd indicates projection matrix as the consensus an- chor guidance, A E pdxt represents the unified anchors, I and d are the number of anchors and shared dimension
min UP.A.S
| - U AS s.l. s. 0, S = 1,
(UP)TUP=1, A=A=I.
(5)
3

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Robust Consensus Anchor Learning for Efficient Multi-view Subspace Clustering
Thus, the non-negative constraint endows the learned S with the probabilistic meaning. In real applications, the data are usually contaminated with the possible noise. We then adopt the Frobenius norm for penalizing the noise based on affine and non-negative constraint in Eq. (5), for- mulated as:
Then we have ||uça, - uLaall2 v2. Note that 8* is the optimal solution to Eq. (7), we obtain
||=0" Un 112 + >|| L(s*) = L(0) = ||>"
(15)
Therefore, 1| UP As* 1|2 S ||>0|e
Thus, Eq. (15) im-
plies
min Σ - AS B s.t. S20,
A.S p=1 1, 1I,
(6)
|0 1
83 l2
V2 A
(16)
The grouping effect in the above theorem demonstrates that the obtained solution is correlation dependent. Theorem 2 shows that the difference between sf and sf is nearly zero if up a; and uL,aj are highly correlated. We then fully ex- plore the correlation among the learned consensus anchors A with the guidance of view-specific projection UP, which encourages the grouping effect and groups highly corre- lated anchors together.
where À > 0 is a parameter to balance different parts. We then guarantee the robustness for efficient multi-view clus- tering and investigate the specific local distribution of clus- ter in the affine subspace. The grouping effect is stated in the theorem as follows.
Theorem 2. Given data point x? E Rdr, matrix UPA E Rd x and parameter X. Assume each column of UP A is normalized. Let s* be the optimal solution to the following problem:
As abovementioned, we introduce the Frobenius norm, non-negative and affine constraints into the shared affinity matrix learning. While it simple, the union of these three constraints can restrict how each data point is described in the affine subspace with specific local distribution of cluster for guaranting the robustness of learned consensus anchors. We then denote the basis matrix U A as
min UP
|000 - U As + A||5|12 s.t. 22.22 0,
(7)
S 1, Ur = I, A=
We have
U [(UPA)), (URA)), (UPA)*), (17)
|| مازز2
(8) where (UP is the basis matrix lying in the I-th affine subspace. We adopt (URA)"; to denote basis matrices in the t-th affine subspace except the basis (U²A),
- 2(1 r),
|P 2|
()=A)- is employed to indicate basis matrices in all affine sub- spaces except the 1-th affine subspace. The sets of ba- sis in (UP A)', (UPA)- and (UPA)L, can be denoted by ((UPA)) (((UPA)))) and I((UPA)),) respectively. A basis tends to locate in three possible positions re- garding (UP A)), i.e., edge, inside and vertex. We adopt (UPide((UAA)) and vertex((UP to in-
where: r (u (((fo)) denotes the basis correlation.
Proof. Let Ls)) ||>0 [s + A||=|[2- Since s* is the optimal solution to Eq. (7), it meets
0s(s)
= 0.
(9)
Ösk
edge((U* ((UP))'), dicate these three positions. For non-vertex (edge and in- side) basis, we first give the lemma and then have the theo- rem as follows:
Since UP I and A = I, we can obtain
(UPA)T(UP = (U (UU))TPAA UPA = I. Thus, UPA is or-
thogonal. Then we have
22( (00 UP As*) + +270 = 0,
(10)
Lemma 1. Assume that A and D are square matrices, we can obtain
- (2" (2---10 -UPAs") + +2A8} O,
(11)
(A matrices B and C with compatible dimension.
B | D
= =1|A[l- + D|]| for (A9
A
0 D
0
Thus
1 5 = ((ufaa)) - (1) As*). (12)
Theorem 3. For any non-vertex basis (U*A), edge (UP (U)))) and (UP E inside A)*) with 1 = .,k, kk, the optimal solution to Eq. (6) is block diago- nal if I((UPA))) does not intersect with I((UPA) A)-
1|ufa:11= = |lufas11 =1,
(13) 1,2,... ...
and
(14) Proof We prove the above theorem by the contrapositive
(uFa))²uffa = O.
and assume that there an optimal solution S to Eq. (6)
4

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Robust Consensus Anchor Learning for Efficient Multi-view Subspace Clustering
and S does have the block diagonal property. We set the block diagonal matrix W as
the centroid matrix and F R represents the cluster assignment with Fi if j-th data point is not belonged to the i-th cluster and 1 otherwise.
1 {o,
Sij, (UP (UPA), A), and (U> in same subspace, 0,
Wi =
otherwise.
2.3. Optimization
(18)
For solving the problem in Eq. (21), we design an alter- nate optimization algorithm to seek for the solution to each variable while fixing the other variables.
Then Q = S - W is adopted to denote the difference between S and W. We employ = {j (UPA), E I((UPA))) and k = {j= (U*A), & }((U)A))) to indi- cate indices of basis for the 1-th affine subspace and other affine subspaces, respectively. Each edge and inside ba- sis (U A), lying in r((UPA)) ()*) which causes S against the block diagonal structure can be written as
UP subppooobob:: With the other variables being fixed, the objective function regarding UP is
min U
2
U - s.t. (U") =1
(0, = (UPA); - Σ (UP (19)
1=1
(22)
JEL
JEC
We then transform the above optimization problem by trace as follows:
Since Ejec Wij + Ejer Qij 1, we divide two sides of Eq. (19) by Σjek Qi and obtain:
max Tr( s.t. (U") =I, (23)
ΣjEk (UPAU)))
(UPA): - EEEc WG, (UP(UA))
UP
EjerQ Qij
ESer Qij
where C -XPSTA" Assuming the singular value de- composition (SVD) of C" is U. we can easily ob- tain the optimal U by calculating V'VV".
(UP A); - EjEc W (UP(
1- EjEE Wis
(20)
S-subproblem: With the other variables being fixed, the objective function regarding S is
We observe that the left and right sides of Eq. (20) are basis in T((URA)- ")= and T(UUAA) respectively. It can be concluded that I(UUAA) does intersect with T(UUP A)²). Therefore, according to Lemmal, we have ||S||| 2 |W]]- Then, W is the optimal and owns the block diagonal structure. Though it is hard to give sufficient con- ditons of the similar structure for vertices as Thcorem 3, we find that the proportions of vertices are far more less than edge and inside basis in practice.
min
a 1 UP USAS + A|SS|A + 311 GF GF
p=
s.t. S20, 1-1=1.
(24)
We then rewrite it by the quadratic programming (QP) problem as follows:
To integrate the partition into the unified framework, we adopt the orthogonal and nonnegative factorization to di- rectly assign clusters to the data. Then extra post- processing steps are not needed in recovering cluster struc- tures based on the factor matrix. Specifically, we impose the orthogonal constraint on the actual bases. The above process is formulated
min (TS.. + 1
1 ST ST.. s.t. St. 0, ST. 1111,
(25)
where 1²= = = =2 -X
CT and W
2 2(2n aj + Thus, we tackle the QP problem to achieve the optimization for each column in S.
A A-subproblem: With the other variables being fixed, the objective function regarding A is
min A.S.G.F
2 X" - USS 2
+ - G s.. s1.520, 0, 1=1=1,
2II - UPASI s.t. A
(26)
a 2 1, (UP)TUP I, A A- G²G (21)
Likewise, Eq. (26) is equal to the problem as follows:
Fg E (0.1), Σ 1 Vj = 1,2, ,,
maxTr(ATB) B). s.t. ATA
(27)
i=1
where of denotes the learned coefficients, 8 > 0 is pa- rameter for balancing different terms, G E Rixk stands for
where B = a XPST Then the optimal A is
UTVT where B= Vg.DVT.
5

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Robust Consensus Anchor Learning for Efficient Multi-view Subspace Clustering
F-subproblem: With the other variables being fixed, the objective function regarding F is
Algorithm 1 Algorithm of RCSC
Input: Multi-view dataset {X"))==>" 1p=1" parameter X, B, number of clusters k.
min B G s.t. Fig E 10,
Output: Cluster assignment F.
k F.y = =j Σ
(28)
1: Initialize: Initialize A, UP, {appp=1. S, Fndd G.
2: repeat
3: Update S by solving Eq. (24);
4: Update (Up)p=1 by solving Eq. (22);
We then independently solve each object for the optimiza- tion problem and obtain
5: Update A by solving Eq. (26);
Update G by solving Eq. (31);
min
B| GF,,l., st. F., (0, Fiy
11
Update F by solving Eq. (28);
8: Update by solving Eq. (33);
(29)
until convergence
We can find the optimal row by
= arg min S G.all?
(30)
complexity of O(Ink) is needed to optimize F. It takes O(I22 + 1k2) to update G, which consists of the time cost in SVD and matrix multiplication. It needs ((1) to update Op. The total time cost of our method is ((pd? + pdkk + dlk= +Ik2 + + di2 Ink)o) with being the total number of iterations, where p Σp=1 dp. Since n > k and > I, the computation cost of our method is nearly linear to O()).
With the other variables being fixed, the objective function regarding G is
G-subproblem:
min B|SS- - GFI s.t. G
(31)
G
Then the optimization problem regarding G is rewritten as
max TrxGT) J), s.t. G= G
(32)
3. Experiments
where J = SFT Then, the optimal G is equal to U'VT.
In this part, we evaluate the proposed method against the representive methods eight multi-view datasets under different metrics in terms of effectiveness and efficiency.
where J = UTE,Yf
subsubprobl:m: With the other variables being fixed, the objective function regarding O₂ is
3.1. Datasets and Experimental Settings
min - -U s.t. a1-== = 1.
(33)
For the experimental evaluation, we use eight real-world multi-view datasets, namely, ORL, Mfeat, Caltech101- 20, 1altech101-all, SUNRGBD (Song et al., 2015), NUSWIDEOBJ (Chua et al., 2009), AWA and Youtube- Face. Eight representive multi-view clustering methods are employed for comparison, including AMGL (Nie al., 2016), SFMC (Li et al., 2022), BMVC (Zhang et al., 2019), LMVSC (Kang al., 2020), MSGL (Kang et al., 2022), FRMVS (Wang et al., 2022), EOMSC-CA (Liu et al., 2022c) and OMSC (Chen et al., 2022).
Based on optimal Op can be obtained by
inequality, the
X U
0p
(34)
| ASII
The objective function monotonically decreases in each iteration until convergence since the convex property for each sub- problem. We list the procedure of RCSC in Algo- rithm 1.
We need to determine the anchor number in evaluating the clustering performance of all methods. For ensuring the fairness, the best parameters are used for compared meth- ods. The anchor number of our method is tuned in the range of [2k, 3k, ... 7k]. where i denotes the total number of clusters in dataset. To reduce the randomness, we repeat each experiment for 20 times and report their mean values and variances in the experiment. We evaluate the cluster- ing results by three widely adopted metrics, which consists of accuracy (ACC), normalized mutual information (NMI)
2.4. Complexity Analysis
The computation cost of method includes the costs brought by optimizing all variables. Specificlly, it costs O(In)) to update S. In optimizing UP, conducting ma- trix multiplication needs (dd,kk?) and SVD consumes (d,aP) for each view. It needs (ddkk) in matrix mul- tiplication and (dd?) in SVD for optimizing A. The
6

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Robust Consensus Anchor Learning for Efficient Multi-view Subspace Clustering
(a)
(b)
(c)
(d)
(h)
1
(k)
(m)
(n)
(0)
(
(u)
(w)
(x)
Figure 1. The first and second line are parameter selection of X, 3 eight datasets, respectively. The third line is sensity investigation of anchor number on eight datasets.
Table 1. Clustering results based on ACC (%) on all datasets. "N/A denotes out of memory.
Dataset
AMGL
SFMC
BMVC
LMVSC
MSGL
FPMVS
EOMSC-CA
OMSC
Ours
ORL,
04.50=0.01
61 61.40=0.05
48. 48.70=:0.05
58.60=0.022
21 21.00-10.05
52.00-t:0.50
62.20-10.05
63.80=0.00
65.20=:0.00
Micat
02.60=0.02
75.50=0.20
69 69.30=0.005
81.75=#0.05
75:40:1:002
82.20++0.05
82.25+0003
84.00=0005
85.60=0.00
Caltech101-20
28.70=0.20
59.40=0.05
16.80:10.05
29.00:0.30
48.00:10.02
66.15:1-0.10
64.10:10.50
65.00=0.10
6..40::0000
Caltech101 all
1..0010.01
17.70
21.20.10.03
15.50.10.01
14.10.10.02
27.50.10005
22.30.10.03
24.00-0.00
28.00 28.0010..50
SUNRGBD
9.80-10.01
11.30=0.05
06.70:::001
18:00:1:0.05
13.00:10:01
23.40:1:0.05
23.70:1:.05
25.20=0.00
27.00=:0.00
NUSWIDEOBJ
N/A
12.20=0.05
12 02.90=00.05
14:70=:0.05
12.00:10.05
19.20=1:0.0
19.60=20.05
21.00=0.05
23.50=0.00
N/A
3.92-20.03
8.60=0005
7.20=0.03
8.00=00022
8.90=0.01
8.65+:0,55
0000
10.50::0.10
Youtbbeaace
N/A
N/A
8 8.90=0.05
14.00=00.02
16.70=0.01
23.00+t0.03
26.45=00055
26.50=0.00
27.80=00.00
to negotiate the importances of partition term and Frobe- nious norm term. We illustrate the clustering performance of the proposed method with varying parameters À and B in Fig. 1. It is observed that appropriate values for these two parameters are generally beneficial to the clustering re- sults on different datasets. According to Fig. 1, we observe that relatively desired clustering results are achieved when 3 0.1 and À = 0.1 on various datasets. Moreover, the results of the proposed method are generally stable over varying values within the range of parameters 3 and À on different datasets, which shows that RCSC is generally ro- bust to these two parameters.
(a) ORI.
(b) Mfeat
(c) Cal-20
(d) Cal-all
Figure 2. Robustness study of our method datasets under ACC.
ESCROCEN
3.2. Experimental results and analyis
In this section, the proposed method is compared with the eight representive methods on several multi-view datasets. To be specific, we report the clustering results with respect to ACC, NMI and F1-score of all multi-view clustering methods in Tables 1-3, respectively. We adopt N/A in- dicate that the method is not able to be computationally feasible the dataset caused by out of memory. Based on the obtained clustering results in Tables 1-3, we can draw some conclusions as follows:
Figure 3. Logarithm of running time on different datasets.
and F1-score. A high value for each of these metrics indi- cates better clustering performance on the dataset.
We first study how parameters /3 and X influence the final clustering performance. These two parameters are adopted
7

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Robust Consensus Anchor Learning for Efficient Multi-view Subspace Clustering
Table 2. Clustering results based on NMI (%) on all datasets.
"N/A
"denotes out of memory.
Dataset
AMGL
SFMC
BMVC
LMVSC
MSGL
FPMVS
BOMSC-CA
OMSC
Ours
ORL
87.10.10.07
02.70.00.01
67.70.10.03
78.50.10.03
43.70.10.02
7..00:10.05
68.10:10.02
86.50_00.10
90.00:10.00
Micat
84.70=00.05
84.80=0.10
66.05=0.115
76.00=00.20
76.54:1:005
79.40:t0.01
83,20:20.15
84,20=0.10
85,32:=0.15
Callech101-20
0..50.1.200
42.80-0.00
16.20:10.03
41.20110.10
31.50:10.05
63.30.1-0.05
51.10110.05
51.80=0.30
52.40:10.00
Caltech1Ol-al
35.30
26. 1000.03
42.50:10.04
33.30.1.002
26.10:10:02
34.10-10.05
24.65
30.00=0.00
31 31.30:1..00
SUNRGBD
18
2.30==0.05
19.50=00.05
24.50==0.05
9.30=0.05
24.10+t0.05
22.50=t0.01
24.30=0.00
25.00==0.10
NUSWIDEOBJ
N/A
0.96:2:001
12.90=:0.02
12.80=20.05
5.70=20.03
13.20-20.005
13.20:20.115
14.00=0.00
15.20==0.00
AWA
N/A
0.30:10.05
9.70::0.02
8.50==0.55
7.90:10.03
8.50:10.33
9.70:10003
10.00=0.02
10.22::0.00
YoutabeFace
N/A
N/A
5.90
11.80.10.01
0.07:10.01
2.40:10.01
0.32:10.01
0.37:10.00
0.50:1..000
Table 3. Clustering results based on Fi-score (%) on all datasets.
"N/A denotes out of memory.
Dataset
AMGL.
SFMC
BMVC
LMVSC
MSGL
FPMVS
FOMSC-CA
OMSC
Ours
ORL
51.20=0.03
30.60=0.05
30.50=:0.04
45.90=00009
51.50==:.20
38.40=+0.15
62.10::0.00
63.20=0.10
65.00=:0.00
Mieut
79.80:20.05
71.10=0.15
58.80:10.01
72.50:00.02
70.10:10.02
76.00:1:0:40
77.00:10.01
78.20=0..00
79.90:00.00
Callech101-20
21.80.10.05
3..60-0.02
11.40.10.20
25.60.10.50
41.80.10.05
66.00:10.05
64.70.10.20
65.00-0.10
66.79.10.20
Caltech10l-all
4.05
4.65
18.00=0.005
10.50=0.05
8.60=00044
17.90-+0.03
10.80-00.03
15.00=0.00
18.20:00115
SUNRGBD
6.40
12.10.10.00
10.20.10.01
11.60.10.20
9.50.10.15
16.00.10.05
15.30.10.05
17.00-0.00
19.20 0.00
NUSWIDEORJ
N/A
11.50=0.01
8.80=0.02
9.30=0005
8.50=20.05
13:50:1:007
13.60:10.05
14.50=0:00
15.60::0.10
AWA
N/A
4.60::0.03
5.5920.02
3.60=0.05
4.20=#0.01
6.20=00005
5,90=:0.05
6,20=#0.00
7.00=20.20
YoutubeFace
N/A
N/A
5.80:00U2
8.30=0.01
15.00:10.10
14.00=50.05
16.40::0.01
17.10=0.00
18.50:00.00
For most datasets, the proposed method achieves more desired performance under different metrics and still behaves well on multi-view datasets with relatively large scale. For example, the clustering performance gain of the proposed method is about 2.32% higher than MSGL in terms of NMI on AWA.
different datasets. To be specific, we randomly select half of the multi-view dataset to be corrupted with white Gaus- sian noise. This type of noise is added to the selected data point xf notes the corrupted ratio and j is the noise satisfying the standard Gaussian distribution. According to Fig. 2, we can observe that the proposed method is robust on different datasets compared with other methods and performs better on these datasets, which can be explained by the fact that the ensuring the robustness of learned consensus anchors in the affine subspace for efficient multi-view subspace clus- tering is helpful in achieving satisfied performance.
via x² = x! + pj. where x! E [0, 255], de-
Methods based on anchor tend to generate better per- formance under three metries in most cases on multi- view datasets with large scales compared with general graph-based methods, which demonstrates that buid- ing the graph based the anchor is helpful to handle the multi-view datasets with large scales.
Our method produces consitently better results than other methods based on the anchor for most of the multi-view datasets, which validates the necessarity of ensuring the robustness of learned consensus an- chors for efficient multi-view subspace clustering and exploring the correlation among the learned consensus anchors with the guidance of view view-specific projection in the manner of encouraging the grouping effect and grouping highly correlated anchors together.
3.4. Running Time
We report the execution times of the compared methods and ours on different datasets. Note that Caltech101-20 and Caltech101-all are two versions of Caltech101 dataset and we just list the running time of Caltech 101-all for sim- plicity. As shown in Fig. 3, it is observed that the proposed method has shown comparable logarithm of running time cost to the existing efficient methods on most of the multi- view datasets, i.e., MSGL.. Thus, our method can obtain advantageous clustering results on various datasets while maintaining relatively competitive efficiency. It can be ex- plained by the fact that jointly modeling the robust consen- sus anchors and the common cluster structure in unified framework is crucial to guide the efficiency for multi-view clustering. The extra clustering algorithm is not needed to obtain the final results, i.c., spectral clustering.
3.3. Sensitivity Investigation and Robustness Study
We investigate how the total number of anchors impacts the clustering results in this part. For simplicity, we fix the shared dimension and conduct the sensity analysis for the number of anchors on several datasets in terms of different metrics. According to Fig. 1, we find that the proposed method is not significantly influenced by the number of an- chors and the clustering results with different number of anchors are relatively stable.
4. Conclusion
We propose a novel RCSC in this work. We theoretically demonstrate that block-diagonal anchor graph is obtained
We also study the robustness of the proposed method on
8

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Robust Consensus Anchor Learning for Efficient Multi-view Subspace Clustering
if the objective function satisfies certain conditions. As special case, we give model based Frohenius norm, non-negative and affine constraints in consensus anchors learning which guarantees the robustness of learned con- sensus anchors for efficient multi-view clustering and in- vestigates the specific local distribution of cluster the affine subspace. Extensive experiments verify the effec- tiveness and efficiency of the proposed method on different multi-view datasets under three metrics.
view multiview. JEEE Trans. Cybern., 52(9):8976- 8986, 2022.
Li, X., Zhang, H., Wang, R., and Nie, F. Multiview cluster- ing: A scalable and parameter-free bipartite graph fusion method. JEEE Trans. Pattern Anal. Mach. Intell., 44(1): 330-344, 2022.
Liang, Y., Huang, D., Wang, C., and Yu, P. S. Multi-view graph learning by joint modeling of consistency and in- consistency. CoRR, abs/2008. 10208, 2020.
Acknowledgments
Liu, C.. Wu, Z., Wen, J., Xu, Y., and Huang, C. Localized sparse incomplete multi-view clustering. IEEE Transac- tions on Multimedia, 25:5539-5555
The work was partially supported by Eastern Talent Plan Leading Project under Grant BJKJ2024011 I and National Natural Science Foundation of China (62402303).
2023a.
Liu, C., Wen, J., Wu, Z., Luo, X., Huang, C., and Xu, Y. Information recovery-driven deep incomplete mul- tiview clustering network. IEEE Transactions on Neural Networks and Learning Systems, 35(11):1544211555, 2024.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must specifically highlighted here.
Liu, C., Wen, Xu, Y., Zhang, B., Nie, L., and Zhang, M. Reliable representation learning for incomplete multi- view missing multi-label classification. IEEE Transac- tions on Putlern Analysis and Machine Intelligence, pp. 2025.
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Reviews:
[{"review_id": "j3cYQ0nj3u", "reviewer": "Reviewer_Y9MR", "summary": "This study proposes a novel method named RCSC, which aims to improve the clustering effectiveness on multi-view datasets by jointly addressing anchor graph construction, partitioning, and robust anchor learning. A key finding is that when data are adequately sampled from independent subspaces and the objective function satisfies certain conditions, the resulting anchor graph exhibits a block diagonal structure. This structure facilitates the revelation of specific local distributions of clusters within affine subspaces. Additionally, the research underscores the importance of robustness in consensus anchors. This emphasis on robustness ensures that the consensus anchors can reliably represent the underlying data structures across different views, thereby enhancing the overall clustering quality.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1: Streamline and differentiate some details in the abstract.\n\nA1: Thanks for the comment! It is needed to streamline and differentiate some details in the abstract, which is able to improve the overall readability and appeal of the manuscript. We will remove the related details regarding the significant overlapped parts with the contribution summary section for the camera-ready version.\n\n\nQ2: Incorporate more schematic diagrams or graphical illustrations at appropriate places within the text.\n\nA2: Good question! It is needed to incorporate more schematic diagrams or graphical illustrations at appropriate places within the text for further enhancing the readers’ understanding. It will also help in providing a more intuitive presentation of complex concepts and relationships. For example, we will add the specific names of the datasets adopted for parameter selection and sensity investigation of anchor number in the title of Figure 1.\n\n\nQ3: Optimize the presentation of figures in the paper.\n\nA3: Thanks for the comment! We will consider employing more effective layout strategies or alternative solutions as reviewer mentioned to optimize the presentation of the figures to ensure that all critical data are clear visible, which is able to avoid that many figures are too small to examine the results thoroughly. For example, we will use more lines and increase the size of all figures in Figure 1 for parameter selection and sensity investigation of anchor number.", "claims_and_evidence": "When proving Theorem 3, it is mentioned at the end that \"we find that the proportions of vertices are far more less than edge and inside basis in practice.\" Although this observation is pointed out, there is no in-depth exploration of the reasons or mechanisms behind it. Providing some tentative theoretical explanations, even if preliminary, would also enhance the persuasiveness of the argument.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are indeed well-suited for addressing the problem.", "theoretical_claims": "Yes, I have checked the correctness of any proofs for theoretical claims, such as Theorem 1-3, and verified the correctness of all formulas. Specifically, a more detailed description of how Eq.(16) was derived can be provided", "experimental_designs_or_analyses": "Yes, I have reviewed the soundness and validity of the experimental designs and analyses conducted in this study.", "relation_to_broader_scientific_literature": "The paper emphasizes the importance of the robustness of consensus anchors and reducing computational complexity. These improvements are crucial for dealing with large-scale datasets, aligning with the research objectives in the fields of machine learning and data mining - developing algorithms that can operate efficiently while maintaining high accuracy. Moreover, the RCSC method promotes a grouping effect, which means combining highly correlated consensus anchors to generate more distinctive cluster indicators. This approach is significant for exploring the deeper structures within data.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths:\n1. The paper demonstrates a clear writing approach, providing detailed and well-organized formula derivations that guide the reader from the problem definition to the gradual introduction of various constraints. \n\n2.The paper develops a novel approach for multi-view subspace clustering.\n\nWeaknesses:\n1.The abstract is somewhat verbose and lacks conciseness, with significant overlap with the contributions summary section. It is recommended that the authors streamline and differentiate these sections to improve the overall readability and appeal of the manuscript.\n\n2.To further enhance the readers' understanding, it is suggested to incorporate more schematic diagrams or graphical illustrations at appropriate places within the text. This will help in providing a more intuitive presentation of complex concepts and relationships.\n\n3. While the extensive experimentation conducted is noteworthy, the sheer volume of experimental data has resulted in many figures being too small to examine the results thoroughly. The authors should consider employing more effective layout strategies or alternative solutions to optimize the presentation of these figures, ensuring that all critical data are clearly visible.", "other_comments_or_suggestions": "See weaknesses above.", "questions_for_authors": "See weaknesses above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "g3GPYk5GZC", "reviewer": "Reviewer_y2ao", "summary": "This paper proposes Robust Consensus anchors learning for efficient multi-view Subspace Clustering (RCSC). The authors first show that if the data are sufficiently sampled from independent subspaces, and the objective function meets some conditions, the achieved anchor graph has the block-diagonal structure. As a special case, the authors provide a model based on Frobenius norm, non-negative and affine constraints in consensus anchors learning, which guarantees the robustness of learned consensus anchors for efficient multi-view clustering and investigates the specific local distribution of cluster in the affine subspace. While it is simple, we theoretically give the geometric analysis regarding the formulated RCSC. The union of these three constraints is able to restrict how each data point is described in the affine subspace with specific local distribution of cluster for guaranting the robustness of learned consensus anchors.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1: The reason why the partition can be integrated into the unified framework.\n\nA1: Thanks for the comment! To integrate the partition into the unified framework, we adopt the orthogonal and nonnegative factorization to directly assign clusters to the data. The reason why we integrate the partition into the unified framework is that extra post-processing steps are not needed in recovering cluster structures based on the factor matrix. Specifically, we impose the orthogonal constraint on the actual bases. We will add such explanation for the camera-ready version.\n\n\nQ2: The second best results in Tables 1-3 should be highlighted.\n\nA2: Good question! It is needed to highlight the second best clustering performance in Tables 1-3 to make the performance gains of the proposed method more obvious. We will highlight the second best clustering performance in Tables 1-3 of the experiment for the camera-ready version.\n\n\nQ3: The detailed analysis regarding how the anchor number impacts the results.\n\nA3: Thanks for the comment! According to Fig. 1, we find that the proposed method is not significantly influenced by the number of anchors and the clustering results with different number of anchors are relatively stable. Besides, larger number of anchors tends to achieve more desired clustering performance and not too large anchor number will help produce relatively satisfied clustering performance. We will add the above descriptions for the camera-ready version.\n\n\nQ4: The memory value of the used device should be given in running time analysis.\n\nA4: Good question! It is needed to give the memory value of the used device in running time analysis. The memory of the adopted device in running time analysis for the experiment is 8G and we will add it for the camera-ready version.\n\n\nQ5: The consistency for the formants of reference should be ensured.\n\nA5: Thanks for the comment! We will correct the issue mentioned by the reviewer for the inconsistency issue for the formants of reference and correct the whole reference part to avoid the similar issues for the camera-ready version.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes. There are proofs for theoretical claims in the paper and I have checked them.", "experimental_designs_or_analyses": "Yes. The adopted methods for comparison include the works from the recent years, increasing the credibility of the final results.", "relation_to_broader_scientific_literature": "This paper first shows that if the data are sufficiently sampled from independent subspaces, and the objective function meets some conditions, the achieved anchor graph has the block-diagonal structure. As a special case, the authors provide a model based on Frobenius norm, non-negative and affine constraints in consensus anchors learning, which guarantees the robustness of learned consensus anchors for efficient multi-view clustering and investigates the specific local distribution of cluster in the affine subspace.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strength: It is able to ensure the mutual enhancement for these procedures and helps lead to more discriminative consensus anchors as well as the cluster indicator. We then adopt an alternative optimization strategy for solving the formulated problem.\n\n\nWeakness: \n1. The authors adopt the orthogonal and nonnegative factorization to directly assign clusters to the data for integrating the partition into the unified framework. The authors should explain why the partition can be integrated into the unified framework based on the orthogonal and nonnegative factorization.\n\n2. The authors report the clustering results with respect to ACC, NMI and F1-score of all multi-view clustering methods in Tables 1-3, respectively. Then they adopt N/A to indicate that the method is not able to be computationally feasible on the dataset caused by out of memory. The authors are expected to highlight the second best clustering performance in Tables 1-3 to make the performance gains of the proposed method more obvious.\n\n3. The authors fix the shared dimension and conduct the sensity analysis for the number of anchors on several datasets in terms of different metrics. According to Fig. 3, The authors find that the proposed method is not significantly influenced by the number of anchors and the clustering results with different number of anchors are relatively stable. In this part, the authors should give more detailed analysis regarding how the total number of anchors impacts the clustering results in this part.", "other_comments_or_suggestions": "No.", "questions_for_authors": "1. The authors report the execution times of the compared methods and theirs on different datasets. As shown in Fig. 5, it is observed that the proposed method has shown comparable logarithm of running time cost to the existing efficient methods on most of the multi-view datasets, i.e., MSGL. However, the authors do not give the memory value of the used device, which is important in running time analysis part.\n\n2. In this reference part, some names of publication are termed for short, i.e., Artif. Intell., however, some names of publications are fully termed, i.e., IEEE Conference on Computer Vision and Pattern Recognition. I think the authors should carefully check this issue to improve the whole presentation of this paper.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "M2LY6qfESR", "reviewer": "Reviewer_1Fds", "summary": "To improve the scalability of the multi-view subspace clustering to large-scale data, this paper proposes Robust Consensus anchors learning for efficient multi-view Subspace Clustering (RCSC), which joints the robust anchor learning, anchor graph construction, and partition into a unified framework. This paper theoretically demonstrates that a optimal anchor graph with block-diagonal structure is obtained if the objective function satisfies certain conditions. The RCSC is simple and effective in terms of performance.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1: It is important to emphasize the connection and differences between the following works: i.e., FPMVS-CAG and SMVSC, especially SMVSC.\n\nA1: Good question! FPMVS-CAG jointly performs anchor selection and subspace graph construction into a framework. Then the two processes can be negotiated with each other to improve the clustering performance. SMVSC integrates anchor learning and graph construction into a unified optimization process. A more discriminative clustering structure can be achieved in this manner. The connection between the above two works and ours is that anchor learning and subspace graph construction are simultaneously conducted in a unified framework. The differences between these two works and ours, especially SMVSC, are that we mainly focus on the robust consensus anchor learning as the highlighted in the title. We first theoretically demonstrate that an anchor graph with block-diagonal structure can be achieved if the objective function satisfies certain conditions. As a special case, we give a model based on Frobenius norm, non-negative and affine constraints in consensus anchors learning, which guarantees the robustness of learned consensus anchors for efficient multi-view clustering and investigates the specific local distribution of cluster in the affine subspace. We will add these analysis in the camera-ready version.\n\nQ2: In Lemma 1, the symbol ||A||_* is the nuclear norm? What is the function of Lemma 1? It seems that there is no nuclear norm in the objective function.\n\nA2: Good question! The symbol ||A|| _* is the nuclear norm in Lemma 1. Lemma 1 mainly gives an inequation to support the conclusion with ||S|| _ *>=||W|| _ * in Theorem 3. Then W is optimal and owns the block diagonal structure. Both left and right of inequation in Lemma1 are nuclear norms and we will further highlight it for the camera-ready version.\n\nQ3: Some symbol definitions are unclear, such as X_i^p, A_i.\n\nA3: Thanks for the comment! As reviewer mentioned, some symbol definitions are unclear, i.e., X_i^p denotes the data points belonging to the i-th cluster for p-th view. A_i is the anchors belonging to the i-th cluster. We will add these explanations and check the whole paper to avoid the similar issues.\n\nQ4: The data used in the experiment is unclear, i.e., the scale of YoutubeFace.\n\nA4: Good question! It is needed to clearly show the data in the experiment as reviewer pointed. YoutubeFace has total 101499 instances generated from YoutTube and we will add it for the camera-ready version.\n\nQ5: The experimental section lacks the latest and SOTA methods for comparison.\n\nA5: Thanks for the comment! We have added two latest and SOTA methods for comparison in the experimental section, i.e., OMVCDR [a] and RCAGL [b].\n\n[a] One-Step Multi-View Clustering With Diverse Representation, 2024\n\n[b] Robust and Consistent Anchor Graph Learning for Multi-View Clustering, 2024\n\nThe clustering results of OMVCDR based on ACC for all datasets are 64.00±0.00, 84.20±0.00, 65.75±0.00, 24.32±0.00, 25.50±0.10, 21.26±0.00, 9.18±0.00, and 26.79±0.00.\n\nThe clustering results of RCAGL based on ACC for all datasets are 64.13±0.05, 84.17±0.00, 65.28±0.10, 25.00±0.00, 25.35±0.00, 22.00±0.05, 9.25±0.00, and 26.92±0.05.\n\nThe clustering results of OMVCDR based on NMI for all datasets are 88.72±0.00, 84.35±0.00, 51.95±0.00, 30.27±0.00, 24.27±0.00, 14.18±0.00, 9.98±0.00, and 0.34±0.00.\n\nThe clustering results of RCAGL based on NMI for all datasets are 88.90±0.00, 84.69±0.05, 51.86±0.00, 30.50±0.05, 24.68±0.00, 14.40±0.00, 10.10±0.00, and 0.35±0.02.\n\nThe clustering results of OMVCDR based on F1-score for all datasets are 63.15±0.00, 78.30±0.00, 65.20±0.00, 15.80±0.00, 17.59±0.00, 14.82±0.00, 6.42±0.00, and 17.35±0.00.\n\nThe clustering results of RCAGL based on F1-score for all datasets are 63.80±0.00, 78.85±0.00, 65.14±0.10, 16.37±0.00, 18.10±0.00, 15.00±0.00, 6.57±0.00, and 17.70±0.00.\n\nQ6: In Figure 4, the horizontal axis is Corrupted ratio, but why is 2k, 3k, 4k, 5k, 6k, 7k in (c) and (d)?\n\nA6: Thanks very much for carefully reading our paper! The horizontal axis should be Corrupted ratio in Figure 4 and we wrongly wrote is as 2k, 3k, 4k, 5k, 6k, 7k in (c) and (d). We will correct this typo and check the whole paper to avoid the similar issues.\n\nQ7: Reduce the length of the Abstract.\n\nA7: Thanks for the comment! It is needed to streamline and differentiate some details in the abstract, which is able to improve the overall readability and appeal of the manuscript. We will remove the related details regarding the significant overlapped parts with the contribution summary section for the camera-ready version.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes. The authors give proofs for theoretical claims in this work and I have carefully checked them.", "experimental_designs_or_analyses": "Yes. The authors perform extensive experiments on different benchmark datasets to demonstrate the effectiveness and efficiency of the proposed method.", "relation_to_broader_scientific_literature": "This paper theoretically demonstrates that an optimal anchor graph with block-diagonal structure is obtained if the objective function satisfies certain conditions. The RCSC is simple and effective in terms of performance.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strength: \n\nTo improve the scalability of the multi-view subspace clustering to large-scale data, this paper proposes Robust Consensus anchors learning for efficient multi-view Subspace Clustering (RCSC), which joints the robust anchor learning, anchor graph construction, and partition into a unified framework.\n\nWeakness: \n\n1. Authors claim that \"Besides, the robust consensus anchors and the common cluster structure shared by different views are not able to be simultaneously learned in a unified framework\". However, some methods can already achieve this, for example, FPMVS-CAG(Fast Parameter-free Multi-view Subspace Clustering with Consensus Anchor Guidance) and SMVSC(Scalable Multi-view Subspace Clustering with Unified Anchors). It is crucial to analyze and explain the connection and differences with the above methods, especially the SMVSC.\n\n2. In Lemma 1, the symbol ||A||_* is the nuclear norm? What is the function of Lemma 1? It seems that there is no nuclear norm in the objective function.\n\n3. Some symbol definitions are unclear, such as X_i^P, A_i.\n\n4. The data used in the experiment is unclear, such as the scale of the YoutubeFace.\n\n5. The experimental section lacks the latest and SOTA methods for comparison.\n\n6. In Figure 4, the horizontal axis is Corrupted ratio, but why is 2k, 3k, 4k, 5k,6k, 7k in (c) and (d)?", "other_comments_or_suggestions": "The Abstract is too long to read.", "questions_for_authors": "1. The experimental section lacks the latest and SOTA methods for comparison.\n2. In Figure 4, the horizontal axis is Corrupted ratio, but why is 2k, 3k, 4k, 5k,6k, 7k in (c) and (d)?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "AHnkVJWIc9", "reviewer": "Reviewer_MKbn", "summary": "This paper proposes a novel Robust Consensus anchor learning for efficient multi-view Subspace Clustering (RCSC). The authors first theoretically demonstrate that an anchor graph with block-diagonal structure can be achieved if the objective function satisfies certain conditions. The authors impose the orthogonal constraints on the actual bases and constrain a factor matrix to be the cluster indicator matrix built on the rigorous clustering interpretation. Extensive experiments on different multi-view datasets validate the effectiveness and efficiency of RCSC, especially on the datasets with large scales.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1: The explanation why UpA∈Rdp×1 can represent the basis matrix and the related analysis can be given here.\n\nA1: Good question! As reviewer mentioned, we adopt A∈Rd×l to represent the unified anchors, l and d are the number of anchors and shared dimension across views, respectively. UpA∈Rdp×l represents the basis matrix. We then theoretically demonstrate that a block-diagonal anchor graph can be achieved if the corresponding objective function satisfies certain conditions based on the independent subspace assumption, which is shown as Theorem 1 in the following. However, we do not give the explanation why UpA∈Rdp×l can represent the basis matrix and the related analysis can be given here. The reason why we use UpA∈Rdp×l to represent the basis matrix is that the dimension of UpA corresponds to the dimension of the basis matrix for the p-th view. We will add this explanation in the camera-ready version.\n\nQ2: What kind of noise is dealt with by the Frobenius norm in this work?\n\nA2: Thanks for the comment! The data are usually contaminated with the possible noise in real applications and we then adopt the Frobenius norm for penalizing the noise based on affine and non-negative constraint in Eq. (5). The Frobenius norm is adopted to deal with the Gaussian noise in this work.\n\nQ3: The authors should give some explanation for the procedure of RCSC in Algorithm 1.\n\nA3: Good question! As reviewer pointed, we state that the objective function monotonically decreases in each iteration until convergence since the convex property for each sub-problem and the procedure of RCSC is listed in Algorithm 1. We are expected to give some explanation for the procedure of RCSC in Algorithm 1. Given multi-view dataset, parameter \\lambda , \\beta ,number of clusters k ,we first initialize A , U^{p} , {\\alpha{p}}_{p=1}^{v} , S , F , G and achieve the cluster assignment F after repeating the six updating steps in Algorithm 1. We will add these details in the camera-ready version.\n\nQ4: The reason why the range of [2k,3k,…,7k] is adopted in the experiment should be given.\n\nA4: Thanks for the comment! We tune the number of the proposed method in the range of [2k,3k,…,7k], where k denotes the total number of clusters in the dataset. The reason why we choose [2k,3k,…,7k] as the range instead of the others is that such range represents the representative magnitudes in the experiment.\n\nQ5: The dimension of (UpA)t can be given to better show the reason why it can be adopted as the basis matrix lying in the t-th affine subspace.\n\nA5: Good question! As reviewer pointed, we should give the dimension of (UpA)t to better show the reason why it can be adopted as the basis matrix lying in the t-th affine subspace. The dimension of (UpA)t is dp×l, which corresponds to the dimension of the basis matrix lying in the t-th affine subspace. We will add this explanation in the camera-ready version.\n\nQ6: The authors need to the give the notation table in the paper.\n\nA6: Thanks for the comment! It is needed to give the notation table throughout the paper to make the physical meaning of each variable very clear. We will give the notation table for the camera-ready version and omit the details here for simplicity.", "claims_and_evidence": "Yes,", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes.  There have proofs for theoretical claims in the paper and they are checked the correctness.", "experimental_designs_or_analyses": "Yes. The comparison of experimental methods consists of methods from the recent years, which increases the credibility of the experimental results.", "relation_to_broader_scientific_literature": "Compared to the previous studies, the highlights of this paper fully explore the correlation among the learned consensus anchors with the guidance of view-specific projection, which encourages the grouping effect and tends to group highly correlated anchors together.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths: \nThis paper imposes the orthogonal constraints on the actual bases and constrain a factor matrix to be the cluster indicator matrix built on the rigorous clustering interpretation. Extensive experiments on different multi-view datasets validate the effectiveness and efficiency of RCSC, especially on the datasets with large scales.\n\n\nWeaknesses: \n1. The authors adopt $ A\\in R^{d\\times l} $ to represent the unified anchors, $ l $ and $ d $ are the number of anchors and shared dimension across views, respectively. $ U^{p}A\\in R^{d_{p}\\times l} $ represents the basis matrix. The authors then theoretically demonstrate that a block-diagonal anchor graph can be achieved if the corresponding objective function satisfies certain conditions based on the independent subspace assumption, which is shown as Theorem 1 in the following. However, the authors do not give the explanation why $ U^{p}A\\in R^{d_{p}\\times l} $ can represent the basis matrix and the related analysis can be given here.\n2. The authors state that the data are usually contaminated with the possible noise in real applications and then adopt the Frobenius norm for penalizing the noise based on affine and non-negative constraint in Eq. (5). Here, the authors need to explain what kind of noise is dealt with by the Frobenius norm.\n3. The authors state that the objective function monotonically decreases in each iteration until convergence since the convex property for each sub-problem and then list the procedure of RCSC in Algorithm 1. The authors are expected to give some explanation for the procedure of RCSC in Algorithm 1.\n4. The authors tune the number of the proposed method in the range of $ [2k,3k,\\cdots,7k] $, where $ k $ denotes the total number of clusters in dataset. The authors are expected to give some analysis why choose $ [2k,3k,\\cdots,7k] $ as the range instead of the others in this part.", "other_comments_or_suggestions": "No.", "questions_for_authors": "1. The authors use $ (U^{p}A)^{t} $ as the basis matrix lying in the $ t $-th affine subspace. Here, the dimension of $ (U^{p}A)^{t} $ is expected to be given to better show the reason why it can be adopted as the basis matrix lying in the $ t $-th affine subspace.\n2. The authors do not give the notations table throughout the paper and the physical meaning of each variable is not very clear. Then the authors need to give the notation table in this paper.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "N/A."}}]

Author rebuttal:

Meta-review: To address anchor graph construction, partitioning, and robust anchor learning, this paper proposes the RCSC method to improve the clustering effectiveness on multi-view datasets. All reviewers give positive scores. This paper has clear writing. The experiment is reasonable.
Decision: Accept (poster)