Fast Incomplete Multi-view Clustering by Flexible Anchor Learning
Authors: Yalan Qin
Guorui Feng
Xinpeng Zhang
Venue: ICML 2025 poster (2025)
Keywords: Multi-view clustering
anchor learning
Primary area: general_machine_learning->clustering
TL;DR: This paper proposes a novel fast incomplete multi-view clustering method for the data with large scales.

Abstract: Multi-view clustering aims to improve the final performance by taking advantages of complementary and consistent information of all views. In real world, data samples with partially available information are common and the issue regarding the clustering for incomplete multi-view data is inevitably raised. To deal with the partial data with large scales, some fast clustering approaches for incomplete multi-view data have been presented. Despite the significant success, few of these methods pay attention to learning anchors with high quality in a unified framework for incomplete multi-view clustering, while ensuring the scalability for large-scale incomplete datasets. In addition, most existing approaches based on incomplete multi-view clustering ignore to build the relation between anchor graph and similarity matrix in symmetric nonnegative matrix factorization and then directly conduct graph partition based on the anchor graph to reduce the space and time consumption. In this paper, we propose a novel fast incomplete multi-view clustering method for the data with large scales, termed Fast Incomplete Multi-view clustering by flexible anchor Learning (FIML), where graph construction, anchor learning and graph partition are simultaneously integrated into a unified framework for fast incomplete multi-view clustering. To be specific, we learn a shared anchor graph to guarantee the consistency among multiple views and employ a adaptive weight coefficient to balance the impact for each view. The relation between anchor graph and similarity matrix in symmetric nonnegative matrix factorization can also be built, i.e., each entry in the anchor graph can characterize the similarity between the anchor and original data sample. We then adopt an alternative algorithm for solving the formulated problem. Experiments conducted on different datasets confirm the superiority of FIML compared with other clustering methods for incomplete multi-view data.

Fast Incomplete Multi-view Clustering by Flexible Anchor Learning
Yalan Qin Guorui Feng'
Xinpeng Zhang'
Abstract
1. Introduction
Multi-view clustering aims to improve the final performance by taking advantages of comple- mentary and consistent information of all views. In real world, data samples with partially avail- able information are common and the issue re- garding the clustering for incomplete multi-view data is inevitably raised. To deal with the par- tial data with large scales, some fast clustering approaches for incomplete multi-view data have been presented. Despite the significant success, few of these methods pay attention to learning anchors with high quality in unified frame- work for incomplete multí-view clustering, while ensuring the scalability for large-scale incom- plete datasets. In addition, most existing ap- proaches based on incomplete multi view clus- tering ignore build the relation between an- chor graph and similarity matrix in symmet- ric nonnegative matrix factorization and then di- rectly conduct graph partition based on the an- chor graph to reduce the space and time con- sumption. In this paper, we propose a novel fast incomplete multi-view clustering method for the data with large scales, termed Fast Incomplete Multi-view clustering by flexible anchor Learn- ing (FIML), where graph construction, anchor learning and graph partition are simultaneously integrated into a unified framework for fast in- complete multi-view clustering. To be specific, we learn a shared anchor graph to guarantee the consistency among multiple views. The relation between anchor graph and similarity matrix in symmetric nonnegative matrix factorization can also be built. Experiments conducted on different datasets confirm the superiority of FIML com- pared with other clustering methods for incom- plete miew data.
In real application, data are usually represented with dif- ferent features from multiple views (Liu al., 2025). This kind of data is usually named multi-view data and integrat- ing the various information for clustering has shown to be critical task in the unsupervised learning field. By investi- gating the complementary and diverse information among different views, a large number of clustering methods for multi-view data have been given (Zhang et al., 202 Ku- mar et al., 2011; 1; Qin et al., 2022a; Chen et al., 2022: Zhao et al., 2023; Wang al., 2023; Yu et al., 2023; Jia al., 2023; Liu et al., 2024; Qin al., 2023b; Liu et al., 2023b; Sun et al., 2024; Qin et al., 2024a; Liu et al., 2022a; Qin et al., 2024d; Liu et al., 2023a; Qin et al., 2025c; Liu et al., 2022b; Qin et al., 2025e) in recent years, which is differ- ent from the clustering for single view (Qin et al., 2023d;a; 2021; 2022b; 2025b; Pu al., 2024). For instance, Ku- mar et al. (Kumar et al., 2011) guaranteed that different representations able to agree with each other by regularizing the clustering hypotheses. Ye al. (Ye al., 2016) maximized the sum of weighted similarities among multiple clusterings to study the underlying clustering. Nie et al. (Nic et al., 2017) simultancously learned the local structure as well as performed cemi-supervised classifica- tion or clustering. Luo al. (Luo et al., 2018) studied specificity and consistency in the representations from dif- ferent views. Chen et al. (Chen et al., 2020) jointly ex- plored the affinity matrix as well as the low-rank represen- tation tensor. Zhou et al. (Zhou et al., 2020) utilized the predefined kernels to learn a consistent representation or a shared kernel and then achieved the unified clustering re- sults. The vital part of clustering for multi-view data to study the consistency of different views by learning unified representation. Most existing multi-view cluster- ing works make the assumption that data samples from dif- ferent views are available (Zhao et al., 2017; Zhang et al., 2021b).
However, data samples in most applications often lack the information for some views (Xu et al., 2023; 2019; Qin et al., 2023c; Lv al., 2022). Then the approaches based integrity have difficulty in dealing with incomplete data from multiple views. In order to handle such problem, several methods of incomplete multi-view clustering have been presented (Zhang et al., 2020). We can conclude these
School of Communication and Information Engineering, Shanghai University, Shanghai 200444, China. Correspondence to: Guorui Feng <fgr2082 aalisaa@aaammm.m......
Proceedings of the 1nnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Fast Incomplete Multi-view Clustering by Flexible Anchor Learning
Ap
BrZ
Bp
Z
F
x
0000
X²
Input Incomplete Features
Anchor Learning
Graph
Graph Partition
Construction
Figure 1. Framework of FIML. It jointly models graph construction, anchor learning and graph partition in a unified framework for fast incomplete multi-view clustering. To be specifie, X and X" are incomplete multi-view datasets as input, Bp denotes the projection as the anchor guidance, Ap is the indicator representation for the missing data, Z refers to the shared anchor graph, G and F denote the centroid matrix and cluster assignment, respectively.
methods into three strategies including graph construction, matrix factorization and deep learning. The methods based on graph construction aim to produce a similarity matrix shared by different views. For instance, Liu et al. (Liu et al., 2017) simultancously learned a representation and filled in the blank value. The methods based on matrix fac- torization make full use of L constraint and nonnegative matrix factorization to learn a consensus representation (Li et al., 2014). Shao et al. (Shao et al., 2015) integrated weighted matrix factorization and L2,1 regularization to obtain better clustering performance. The methods based on deep learning use deep neural network to recover the missing data and the feature representation. As represen- tative, Lin et al. (Lin et al., 2021) employed contrastive learning for integrating data recovery and feature learning. However, most existing methods easily suffer from the high computation and space cost, which inevitably restricts their availability on the datasets with large scales.
high-quality anchors in a unified framework for incomplete multi-view clustering, while ensuring the scalability for in- complete datasets with large scales. In addition, few of the existing approaches based incomplete multi-view clus- tering pay attention to building the relation between anchor graph and similarity matrix in symmetric nonnegative ma- trix factorization and then directly performing graph parti- tion based on the anchor graph for reducing the space and computation consumption.
In this paper, we propose a novel fast incomplete multi- view clustering method for the data with large scales, termed Fast Incomplete Multi-view clustering by flexible anchor Learning (FIML), where graph construction, anchor learning and graph partition are simultaneously considered in a unified framework for fast incomplete multi-view clus- tering as Fig. These three parts can boost each other, which promotes the quality of the clustering and improves the efficiency for large scale datasets. To be specific, we learn shared anchor graph to guarantee the consistency among multiple views and adopt a adaptive weight coef- ficient to balance the impact for each view. The relation between anchor graph and similarity matrix in symmetric nonnegative matrix factorization can also be built, i.e., each entry in the anchor graph can describe the similarity be- tween the anchor and original data sample. In particular, constrain the factor matrix to be a cluster indicator rep- resentation by introducing the orthogonal constraint on the actual bases. Furthermore, we adopt the alternative algo- rithm for solving the optimization problem.
To cope with the above issue, many methods for the data with large scales have been proposed (Qin et al., 2025a;d; Qin & Qian, 2024; Qin et al., 2024c;b). Wang et al. (Wang et al., 2011) built a constrained factor matrix for explor- ing the cluster structure. Kang et al. (Kang et al., 2020) employed K-means to obtain the anchors and then collo- cated them for a unified representation. Li al. (Li et al., 2022) adopted the consistent learned anchors for handling the clustering problems of incomplete multi-view data. Nie et al. (Nie et al., 2020) simultaneously performed cluster- ing on column and row of the original dataset. Wang et al. (Wang et al., 2022b) used the guidance of consensus an- chors to study the anchor graph shared by different views. Sun al. (Sun al., 2021) exploited the underlying distri- bution of the data to construct the anchor graph. Among these existing methods, the approaches based on anchor have achieved attention due to the scalability and efficiency. This kind of methods usually employs the original data and the generated anchors to build the corresponding anchor graph, resulting in more satisfied clustering performance. Despite great success, the above methods neglect learning
As a summary, the proposed FIML has the main contribu- tions in the following:
We give new insight to the community of incom- plete multi-view clustering for large scale datasets, i.c., graph construction, anchor learning and graph partition in fast incomplete multi-view clustering can boost each other, which are able to be integrated into a problem. The combination of these three issues
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Fast Incomplete Multi-view Clustering by Flexible Anchor Learning
the focus in our work. While most existing work treat graph construction, anchor learning and graph parti- tion as separated problems in incomplete multi-view clustering for the datasets with large scales.
gorithm (spectral clustering) is usually needed achieve the final clustering results. This multiple-stage process sig- nificantly affects the quality of the anchors. Besides, few of the existing methods pay attention to building the re- lation between anchor and similarity matrix in symmetric nonnegative matrix factorization. As is known, each entry of a similarity matrix can describe the similarity between data samples in the dataset. Performing symmetric non- negative matrix factorization for the similarity matrix can directly lead to the final partition. Then building the rela- tion between anchor and similarity matrix can take advan- tages of directly obtaining the final clustering results in in- complete multi-view clustering. How to learn anchors with high quality in a unified framework and build the relation between anchor graph and similarity matrix in symmetric nonnegative matrix factorization to ensure the scalability large-scale dataset for incomplete multi-view clustering remains considerably challenging issue.
We propose a novel fast incomplete multi-view clus- tering method for large scale data termed as FIML, where graph construction, anchor learning and graph partition are simultaneously considered in a unified framework for fast incomplete multi-view clustering. The relation between anchor graph and similarity ma- trix in symmetric nonnegative matrix factorization is also built, i.e., each entry in the anchor graph is able to characterize the similarity between the anchor and original data sample.
Based on the relation between anchor graph and simi- larity matrix, we constrain the factor matrix with rig- orous interpretation to be cluster indicator represen- tation by introducing the orthogonal constraint on the actual bases and use the alternative algorithm for solv- ing the formulated problem. Extensive experiments are performed on different datasets to demonstrate the superiority of FIML in terms of effectiveness and effi- ciency.
Formulation: Different from most existing works for in- complete multi-view clustering. we learn anchors instead of selecting them based on the available data samples in the dataset. The proposed FIML integrates graph construction, anchor learning and graph partition into a unified frame- work for fast incomplete multi-view clustering. Then the discriminative anchors are automatically learned and the final partition can be achieved in this manner. Based on the assumption that multiple views are sampled from a la- tent space, the anchors from multiple views are expected to be consistent in this space. Given multi-view dataset { (Xp xx))==>> we construct the projection Bd E Rd as the consensus anchor guidance to integrate complementary information from different views into the shared anchor graph Z E Rm where dp and m are the dimension of the data and the total number of anchors for the p-th view, re- spectively. The indicator representation Ap E {0, 1) 1 is adopted to mark the unavailable data samples. The above process can be formulated as follows:
2. Fast Incomplete Multi-view Clustering by Flexible Anchor Learning
This section begins with introducing the motivation and formulation of the proposed FIML, then moves on to the detailed optimization process for FIML. We lastly conduct the analysis about the computation complexity to demon- strate the efficiency of FIML.
For large-scale incomplete data clustering, re- ducing the redundancy of the data is the key to increase effi ciency. Some existing works denote each data sample with a linear combination of the others and the global relation can be well exploited in this manner. However, the rela- tively high storage and computation time produced in this way inevitably limit the scalability of incomplete multi- view clustering for large-scale dataset. Actually, relatively less data samples are enough to reconstruct the latent space. Therefore, selecting some data samples from the original dataset as anchors or landmarks for reconstructing the rela- tion structure is commonly used in the existing works.
Motivation:
min a, 2, pml
X ByZA,
(I)
s.t. 11,, 2 0, 1-1
where az is the coefficient of each view. It can be learned based on the contribution to the shared anchor graph. X,, denotes the available data samples for the p-th view. Since the space complexity of anchor graph Z is O( x)), we can relate Z with similarity matrix in symmetric non- negative matrix factorization for directly obtaining the final partition. As is known, symmetric nonnegative similarity matrix with the space complexity O(n x n)) can be adopted to achieve the final clustering results based on factorization. Each entry in the anchor graph Z describes the similarity between data sample and anchor. Since the symmetric con- straint on Z E RmxR not guaranteed in factorization
Nevertheless, some existing incomplete multi-view cluster- ing approaches usually conduct strategies based on heuris- tic sampling, where the anchor selection and graph con- struction are separated. Then the graph is constructed after selecting the anchors for different views. In this manner, the complementary information among different views is not able to be well explored and further algorithm is needed to obtain a shared graph. Afterwards, the clustering al-
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where Ap = (X, Hp) @ denotes the Hadamard prod- uct, Hp = ap, ap = [ap.1. ... 4p.n] and ap. = Ei" Ap..j- After conducting the singular value decom- Fast Incomplete Multi-view Clustering by Flexible Anchor Learning
with < ni, we remove such constraint anchor graph 7 and this is the main difference between anchor graph and similarity matrix in symmetric nonnegative matrix factor- ization. We then introduce the centroid matrix G E R x and the cluster assignment F E R with & being the total number of clusters in the dataset as follows:
the first m singular vectors of Ap in the left and right, re- spectively.
(2) Optimize Z With others being fixed, the objective for Z turns to solve the problem as:
min
- B,ZA, + -
min Z -GF s.t. G² G.F
(6)
(2)
s.t. 2 2 0, 2²1
Fg E {0,1}, =, 2, n, Fij 1,
We then remove the irrelevant items and rewrite Eq. (6) as follows:
where P.j if the j-th data sample not belonged to the i-th cluster and I otherwise. Note that imposing the orthogonal constraint on the actual bases can guide learning the factor matrix with rigorous clustering interpretation. To combine the partition information into the unified model, we formulate the total objective function as:
min Σ a
TT(2"
oz 1
A
p=1
2x, B,ZQ, - -2 - t" GF), s.t. Z2. z=1=1 = 1,
ap
Σ 1 B,ZAp1i + Z - GFI
min B.
i. where Op = Ap Since 2; can be denoted as a vec- tor with 2j,4 being the j-th entry, can solve Eq. (7) by column as follows:
p=1
s.t. G==1, Σ Pg Fg E (0,1), Vj = 2,
minl Val s.t. 21 2 0, 2, 11
1=1
a 1 = 1, By 1, 0, 111=1
(3)
where v/ = EE of HpH P.t.
Bp H.H.s
We then rewrite the Lagrangian function of Eq. (8) as:
where A > 0 denotes the parameter for balancing differ- ent terms. Then graph construction, anchor learning and graph partition are jointly integrated into a unified frame- work for incomplete multi-view clustering in this manner, where these three parts can boost each other to achieve ef- fective and efficient clustering results for incomplete large- scale multi-view dataset.
L(z,, 0s, (1) = || - 1 of (2 1), (9)
where o, and 7/2 correspond to Lagrangian multipliers. With KKT conditions, we have the equation:
2, V14 0,1 - 7 =
(10)
We then design an alternating algorithm for optimizing each variable in Eq. (3) by fixing the others.
Optimization:
& 2 0.
Combining the constraint tion as follows:
TT = 1, we can obtain the equa-
(1) Optimize {Bp}p= With other variables being fixed, the objective function for {Bp]--1 can be rewritten as
1 + w,
= m
+ o,,,0),
(11)
min (Bp) pm
X BBZZA,li s.t. Bt.BB.D BY Bp I.
With other variables being fixed, the ob- jective function for G is transformed into the problem as follows:
(3) Optimize G:
(4)
We can remove the irrelevant items and transform Eq. (4)
into the form as follows:
min A||Z - GF s.t. GTG
(12)
max mrxxTr(BAAp) s.t. B Bp -I B
(5)
The optimization for G can be written as
max Tr J) s.t. G
(13)
where J = ZFT Then the optimal solution for G is U,V with J = =UEEVV based on singular value de- composition (SVD).
position on Ap. the optimal solution of B, can be derived as E, T
my where E. and V,, represent the matrices with
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Fast Incomplete Multi-view Clustering by Flexible Anchor Learning
(4) Optimize F: With other variables being fixed, the ob- jective function for F can be formulated as the minimiza- tion problem as:
Algorithm 1 Algorithm of FIML
Input: Incomplete multi-view dataset {Xp] p the to- tal number of clusters k and the missing indicator
p A1IZ GF
pp=
Output: The final cluster assignment F.
(14)
1: Initialize: Initialize Z, ppp and (pp]p=1
s.t. Fij E (0.1), 1,
Fij = 1,
2: repeat
3: Update Z by solving the problem in Eq. (6);
We then have the minimization problem as
4: Update { by solving the problem in Eq. (4);
Update G by solving the problem in Eq. (12);
min Fig
AIl - GF.,ll? s.t. F.,, E {0, 135, 1Pyll:=1
6: Update F by solving the problem in Eq. (14);
7: Update by solving the problem in Eq. (17);
(15)
8: until convergence
Then the optimal row can be achieved by
8" = arg : min||S., G=
(16)
According to Eq. (16), we can find that the optimal clus- ter assignment is achieved by the cluster centroid and the object.
3.1. Compared Methods
The experiments are conducted on several widely adopted datasets including news groups (NGs), WebKB, ORL, STLIO, MNIST and Cifar100. We compare FIML with some representive incomplete multi-view clustering ap- proaches as follows: BSV (Ng et al., 2001), MIC (Shao et al., 2015), MKKM-IK (Ma et al., 2021), DAIMIC (Hu & Chen, 2018), APMC (Guo & Ye, 2019), PIC (Wang al., 2019), EEIMVC (Liu al., 2021), V (Fang et al., 2020), IMVC-CBG (Wang et al., 2022a), and FIMVC- VIA (Liu et al., 2022c).
(5) Optimize p. With other variables being fixed, the ob- jective function for 0% is:
min Nop s.t. a 1 = =1=, 1, a 2 0, p=1
(17)
where Kip
= 1Xp4, BB We can obtain the opti-
mal a based on Cauchy-Schwarz inequality as:
5
In the experiment, use four metrics to evaluate the perimental results, which include accuracy (ACC), NMI, F1-score and Purity. We repeat each algorithm for total 20 times and then report the mean and standard deviation of the results. The parameters for the compared methods of incomplete multi-view clustering are set as their recom- mended ones. We run all experiments on AMD Ryzen 5 Six-Core Processor 3.60 GHz.
(18)
12 Ep=1 8p'
where 5 = [61, ... 8z] with op = 1 1/kip
2.1. Complexity Analysis
The computation burden of FIML consists of the optimiza- tion cost of each variable. To be specific, the time complex- ity for optimizing B, is O(m²d + mnd) at each iteration. Optimizing the weight a of each view costs O(mdd). The time cost to learn the shared anchor graph Z is ((mn)). For optimizing F, the time cost is (mnn). The time cost to update G is Omm)²). Then the total time complexity of the proposed FIML is O(m²d d + mnd + mnk + mk?) where I denotes the number of iterations for these parts. Due to n >> m and n >> k, the computation cost of FIML. is nearly linear to the size of the dataset Onn).
3.2. Parameter Selection
There are total two parameters appeared in FIML, includ- ing the trade-off parameter À and the number of anchors m. We then perform experiments on different datasets to study how these two parameters influence the final clustering performance. We set X and m in the range of [0.001, 0.1, 1, 10, 100, 1000] and [k, 2k, 3k, 5k, 7k), re- spectively. Here, A: corresponds to the total number of clus- ters in dataset. According to Figs. 2-3, we find that better performance is achieved when X = 1 under the same m on different datasets. Besides, the clustering result of FIML is relatively stable over different parameter values on these datasets, which shows that FIML is generally robust to the trade-off parameter X. It can also be observed that different number of anchors m has relatively little influence on the clustering performance under the same À for these datasets.
3. Experiments
In this section, we perform experiments to validate the ef- fectiveness and efficiency of FIML on several widely used multi-view datasets. Among these datasets, there are some large datasets for better verifying the clustering per- formance and running time of FIMI..
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Fast Incomplete Multi-view Clustering by Flexible Anchor Learning
(a) ACC
(b) NMI
(c) Fl Fcore
(d) Purity
Figure 2. Parameter Study of m À on NGs terms four metries.
ACC
(b) NMI
(c) Fl-score
(d) Purity
Figure 3. Parameter Study of m and À on ORL in terms of four metries.
100
Visting
(a) ORL
(b) NGs
(c) WebKB
(d) STL.00
(c) Cifar100
Figure 4. Clustering Performance in terms of ACC on datasets with different missing ratios.
Makingg
Vastin
Masinn
(a) ORL
NGs
(c) WebKB
(d) STL10
(c) Cifar100
Figure 5. Clustering Performance in terms of NMI on datasets with different missing ratios.
BACZKT
Masingg
Massing
Visiing
Maxding
(a) ORL
NGs
WebKB
(d) STLIO
(c) Cifar100
Figure 6. Clustering Performance in terms of F1-score on datasets with different missing ratios.
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Fast Incomplete Multi-view Clustering by Flexible Anchor Learning
Table Clustering Performance based ACC (%) datasets. "N/A
denotes out of memory.
Dataset
BSV
MIC
MKKM-IK
DAIMC
APMC
PIC
EEIMVC
V
IMVC-CBG FIMVC-VIA
Ours
ORL
24.30.10.50
37.60:1: 1.50
19.90-122.00
68.00-12..30
65.50 1.60 69.00 1.50 73.20:1..400
67.00.1:1.00 1.30
69.50:1.2.00
76.30:12.70
78.84.0.50
NGs
2.000:22.00
21. 21.50:50.50
80.20:10.00
59.50-10.05 89. 0.05 82. 02.00-0.20 77.90:10.15
79.90:10.400 4.40
88.90:10.15
89.70.10.05
91.40:0.40
WebKB
$7.00:12.20
63.50:00.50
68.000110.00
N/A
85.30=0.05 71.60=0.00 61.80:13.40
0..20:1:050
04.50:10.50
91.50:10.50
93,00:0.26
STLI0
11.20:10.10
N/A
75.80-L0.30
23.00:E 50 27.00-10:50 2..20 2..70.1...00 18.50:10.50
55.60:10.880
76.00:10.30
78, 78.30110.60
MNIST
N/A
N/A
N/A
77.60+0.550 50
N/A
N/A
98,20:+:0.05
98.75-:0.30
98 98.9000.000
Cifar100
N/A
N/A
N/A
89.68+00500
N/A
N/A
N/A
N/A
93,00=+:1.20
98.90=0.60
09.50==0.26
Table 2. Clustering Performance based on NMI (%) on datascts. "N/A
denotes out of memory.
Dataset
BSV
MIC
MKKM-IK
DAIMC
APMC
PIC
FEIMVC
V-3
IMVC-CBG FIMVC-VIA
Ours
ORL
48.52:40.80
06.50
76,20+11000 ,0 83,00+11100 10 30-30=00000 80 03,20=0.50 85.40:t 1.30 81.00 $1.00+:0.50
81.20:+:1.50
88.00:1 1.30 90.15=0.00
NGs
1.320=1..00
2.30:10.50
13.10::0.10
73.40:10.05
73.41
60.60=0.110 57.20
20 59.00:::0440
73.00:10.05
75.50:10.05
17.00=0.11
WebKB
0.85=0800
3.30=t0.50
4.00=t0.10
N/A
47,90=0.20 1.70:t0.00
3,50=+0.50
23.60 23.60=+1.000
37. 37.20=#0.15
48.90=b0.20
51.20=0.500
SIL10
0.16:t0.20
N/A
60.30:10:40
5.00=1120 11.00=0.90 14.20=0.115 29.80=33.00 5.90=00.00
27.20:+:.20
57.35-+0.20
59,80=0.50
MNIST
NIA
N/A
N/A
93.90:10.50
N/A
NIA
NIA
N/A
94,90100.10
96.20:10.300
97.30110.10
CifarI00
N/A
N/A
N/A
98.20:10:20
N/A
98.60:10.30
99.70:1:0.10
99.80-10.20
3.3. Experimental Results
3.4. Running Time
We list the detailed clustering results of FIML and the com- pared approaches on different datasets in terms of four met- rics in Tables 1 1-4. Note that N/A is adopted to indicate that the method suffers from the error due out of memory. We also compare FIML with IMVC-CBG and FIMVC-VIA under different missing ratios several datasets under dif- ferent metrics. According to Tables 1-4 and Figs. 4-7, we draw the following conclusions:
In this part, we show the running time of FIML and the compared approaches on different benchmark datasets. Based on Table 5, we have the observations as follows:
Our FIML needs less running time than other meth- ods for incomplete multi-view clustering on differ- ent datasets in terms of ACC, which indicates its effi- ciency for computation cost. Some other methods for comparison suffer from the memory issue on MNIST and Cifar100 based on the running time, which further shows the efficiency of our FIML.
The proposed FIML can provide better performance than other compared methods for incomplete multi- view clustering in terms of different metries. For in- stance, FIML gains better clustering performance of 9.84% than PIC in terms of ACC on ORL, which shows that combining graph construction, anchor learning and graph partition in a unified framework of incomplete multi-view clustering is able to boost each other and result in effective clustering results.
Some compared methods are able to consume less running time on some small dataset, i.c., EEIMVC uses less computation cost than our FIML on some datasets. However, these methods do not perform as well as ours when running large-scale datasets. It can be explained by the fact that using a unified frame- work integrated by graph construction, anchor learn- ing and graph partition can improve the efficiency for incomplete multi-viww clustering, especially on the datasets with large scales.
Compared with other methods for incomplete multi- view clustering, FIML shows better clustering perfor- mance with different missing ratios on several datasets under four metrics, which shows that the learned an- chors for representing all data samples are relatively informative for these datasets and methods based on kemne or graph do not show the same satisfied perfor- mance.
The dataset with larger dimensions tends to need more running time when these datasets are close in the size, i.e, IMVC-CBG needs more running time on Cifar100 than MNIST and the latter dataset has smaller dimen- sion than the former dataset. As the size of dataset increases, our FIML and the compared methods usu- ally consume more running time on different datasets.
FIML produces more satisfied clustering performance than FIMVC-VIA on different datasets, showing that using the unified framework integrated by graph con- struction, anchor learning and graph partition can help achieving better cluster assignment matrix and this matrix can directly result in the final results.
3.5. Ablation Study
In this section, we perform ablation study to validate the superiority of adopting a unified framework integrated by graph construction, anchor learning and graph partition. In
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Fast Incomplete Multi-view Clustering by Flexible Anchor Learning
Table 3. Clustering Performance based Fi-score (%) on datasets. "N/A
denotes out of memory.
Dataset
BSV
MIC
MKKM-IK
DAIMC
APMC
PIC
EEIMVC
V
IMVC-CBG FIMVC-VIA
Ours
ORL
9.00:10.50
17.50:1
1.00
2..3001...30
56.80 8.61 50.50-12.40 2.40 57,70 1,30 2.950:1
55.00.1:..00
46. 36.00.1.3.00
68,20113.00
71.20.1..50
NGs
32 30 1.00
32.80 02.80:10.20
68, 18.70:10.00
80. 30.00.05
80. 0.06 02.80=0020 64.00:10.20
65.6 65.60:10.40
4.0
79.50:1:005
80,80:10.000
83.20:10.00
WebKB
1.55
62.00::0.50
64.60:1:0300
N/A
85.00=0.055
03.60 62.80:10.20
7..90:10.40
83.00:10.07
88.70:10.05
90.50:00055
STLI0
11.70:10.05
N/A
57.80:10.30
13.20 1 80
18.60 1.20 21,40:10.10 29.90.1..00
17.05.10.550
34.60:10..07
59.90:10.00
62.50.1.500
MNIST
N/A
N/A
N/A
35.50++0.30
N/A
96.20:4:0110
97.50:10.50
09.20=4.11
Cifar100
N/A
N/A
N/A
90.50++0:50
N/A
N/A
N/A
N/A
91,90=+0.500
99.00=0.50
99.60=00200
Table 4. Clustering Performance based on Purity (%) on datasets. "N/A denotes out of memory.
Dataset
BSV
MIC
MKKM-IK
DAIMC
APMC
PIC
FRIMVC
V
IMVC-CBG FIMVC-VIA
Ours
ORL
26.90=00900 40.50+ 1,50 63. $3.00++2.00 0.0 90:1 60 30:3001122
72.30=1.00
76.00-t2.10 10 70.20-
69.30--11.88
79.10:42.000 2.00
82.50=0.29
NGs
43.10 1.50 21 0.:000055
79.60 09.60:00:00
50.50--0.05
0.05 82.40=0020
77.80:10.10 79. 0.40
88,70:10.05
90.00:1:0.66
91.12:0.05
WebKB
78. 0.200 78. 08.2440.60
78. 08.00
N/A
90. 15=0.08
78, 18,20=0.40
78. 18:00.000.. 30 91. 91.70=+3.00 84.60=+0.05
91.60::0.20
04.10=0.20
SILI0
11.30:0.05
N/A
05.80:00.30
27.60=1.20 29.30=00115 46.90: 18.60=+0.50
55.60::0.08
76.00:1:020
78.90=0.55
MNIST
N/A
N/A
N/A
97.50:10.30
N/A
N/A
NIA
NIA
98.00:10.100
98,50.1...50
09.20110110
Cifar100
N/A
92.50:10.50
94.90:10.50
99.00:10.550
99.55.10.20
Q4
Wessing
Mosingg
Mannorate
(a) ORL
(b) NGS
(c)
WebKB
(d)
STLIO
(e) Cifar100
Figure Clustcring Performance in terms of Purity on datasets with different missing ratios.
Table 5. Running time all methads on different datasets. "N/A
ddenotes out of memory.
Damaset
BSV
MIC MKKM-IK
APMC
PIC
EEIMYC
V-3
IMVCFIMYC
ORL
0.15
425.80 :00
0.50
0.50
0.30
0.55
3.00
1.20
NGs 0.05
145.00
0.50
0.20
0.28
0.25
0.15
14.50
1.50
0.30
0.25
WehKB
0.15
340,50
3.20
N/A
0.28
1,30
0.22
32.00
0.6
0.28
0.34
(a) ORL
(b) NGs
(c) WebKB
SILIO
66.90
NIA
1666.00
72.00
335000
68.50
4529000
18.50
6:20
MNIST
N/A
N/A
M/A
5000.20
N/A
NIA
NIA
N/A
$52.00
30:20
Gfar100 N/A
N/A
N/A
25200.00
N/A
N/A
N/A
N/A
81500
47.00
35.00
Figure 8. Convergence on different datasets. (a) ORL. (b)
NGs. (c) WebKB.
Table 6. Ablation study based on separated or unified manner
Metries
Manner
ORL
NGs
WebKB
STLIO
MNIST
Cifar100
3.6. Convergence Analysis
ACC
Separated
Unified
70.60: 20 82.45 22.45-0.00 30 82,00A077
0..54 91 10000000 40 93.00-0..66
71 0:55 84.022000 40400045
0.66
98 00000000
99.50:0.226
We conduct convergence analysis of FIML on different datasets by showing the evolution process of the objec- tive function with iterations in terms of ACC. According to Fig. 8, we observe that FIML monotonically decreases with iterations and tends to converge in about some iter- ations on these datasets, which demonstrates the conver- gence of FIML.
NMI
Separated 75 15 70 35 44.0110.78
Unified 90. 60: 77 17:00:::10 18 $1.2010.50 50 59 80 30 97. 10:30:1:10 99.90 90.50:1..00
48 20:20.27
92.00:1::70 91 00:000
F1-40000
Separated 62,
Unified 71.20:1
76 20-20----000
80.20--0.00
00.50110.05
04.90
0.50 10020:1:0110 99.60:1:0.0
90.504-000 90.49:4-..55
Separated 70.85 0.39 84, 04.60:10.64
82.70:1::20
09.00:20.90
08.50.1:0.05 82.40:10.11
Parity
Unified 02.500=:0.29 93. 12:50000
94.108:4.20
0:55 99:00-b0.10 99.55
comparative experiments, we first learn anchors and con- struct the graph obtain informative representation. Then the graph partition is isolated from the above two processes in the designed experiment. According Table 6, we can find that the clustering performance of the proposed FIML. significantly outperforms than the case in separated man- ner, demonstrating the necessarity of using unified frame- work integrated by graph construction, anchor learning and graph partition to directly achieve the final assignment.
4. Conclusion
we propose FIML in this work for fast incomplete multi- view clustering. It simultaneously considers graph con- struction, anchor learning and graph partition in a unified framework, in which these parts boost each other for im- proving the effectiveness and efficiency for datasets with large scales. To be specific, a shared anchor graph for guar-
8

---

Fast Incomplete Multi-view Clustering by Flexible Anchor Learning
anteeing the consistency among multiple views is learned and the adaptive weight coefficient is adopted to balance the impact for each view. We then adopt the alternative algorithm to solve the optimization problem. Extensive ex- periments on several benchmark datasets show the effec- tiveness and efficiency of FIML under different metrics.
Kumar, A., Rai, P., and III, H. D. Co-regularized multi- view spectral clustering. In Advances in Neural Infor- mation Processing Systems, pp. 1413-1421, 2011.
Li, M., Wang, S., Liu, X., and Liu, S. Parameter-free and scalable incomplete multiview clustering with pro- totype graph. IEEE Transactions on Neural Networks and Learning Systems, pp. 1-11, 2022.
Acknowledgments
Li, S., Jiang, Y., and Zhou, Z. Partial multi-view cluster- ing. In Proceedings of the AAAI Conference Artificial Intelligence, pp. 1968-198 1974,, 2014.
The work was partially supported by Eastern Talent Plan Leading Project under Grant BJKJ2024011 I and National Natural Science Foundation of China (62402303).
Lin, Y. Gou, Y., Liu, Z., Li, B., Lv, J., and Peng, X. COMPLETER: incomplete multi-view clustering via contrastive prediction. In IEEE Conference on Computer Vision and Pattern Recognition, pp. 11174-11183, 2021.
Impact Statement
This paper presents work whose goal advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Liu, C., Wu, Z., Wen, J., Xu, Y., and Huang, C. Localized sparse incomplete multi-view clustering. IEEE Transac- tions on Multimedia, 25:5539-555 55519,555111111 2023a.
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Reviews:
[{"review_id": "Ku5SZ5GZVl", "reviewer": "Reviewer_agLA", "summary": "This paper proposes a novel fast incomplete multi-view clustering method for large scale data termed as FIML, where graph construction, anchor learning and graph partition are simultaneously considered in a unified framework for fast incomplete multi-view clustering. The relation between anchor graph and similarity matrix in symmetric nonnegative matrix factorization is also built, i.e., each entry in the anchor graph is able to characterize the similarity between the anchor and original data sample.", "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 authors should add more recent works for comparison in the experiment.\n\nA1: Thanks for the comment! We have added a recent method for comparison in the experimental section, i.e., OMVCDR [a].\n\n[a] One-Step Multi-View Clustering With Diverse Representation, 2024\n\nThe clustering results of OMVCDR based on ACC for all datasets are 77.00±0.00, 89.85±0.10, 91.75±0.00, 76.50±0.15, 98.80±0.00, 98.95±0.10\n\nThe clustering results of OMVCDR based on NMI for all datasets are 89.50±0.00, 75.85±0.15, 49.00±0.05, 58.00±0.00, 96.50±0.10, 99.72±0.00\n\nThe clustering results of OMVCDR based on F1-score for all datasets are 69.00±0.00, 81.15±0.10, 88.95±0.00, 60.00±0.05, 97.80±0.00, 99.25±0.00\n\nThe clustering results of OMVCDR based on Purity for all datasets are 79.56±0.00, 91.25±0.20, 92.00±0.00, 76.50±0.50, 98.79±0.00, 99.20±0.05\n\nQ2: The authors should give more detailed analysis for the experimental results based on ACC, NMI, F1-score and Purity.\n\nA2: Good question! We have added more related detailed analysis for the experimental results as shown in the following. On the STL10 dataset, our method outperforms the last five multi-view clustering methods in tables by achieving improvements of 49.5%, 31.6%, 59.8%, 22.7% and 2.3%. We can also find that the anchor based algorithms are capable of handling the bigger data compared with the traditional multi-view clustering methods.\n\nQ3: The best clustering results in the experiment should be bold.\n\nA3: Thanks for the comment! We will bold the best clustering performance based on ACC, NMI, F1-score and Purity in Table 1-4 in the experiment for the camera-ready version to make the performance gains more obvious.\n\nQ4: The brief optimization process before the detailed steps should be given.\n\nA4: Good question! The algorithm of FIML consists of the steps shown in the following. It contains the input, output and initialization of the algorithm. The detailed optimization steps regarding each variable are also given in this algorithm. Then the optimization stops when the convergence condition achieves. We will add such brief optimization process before the detailed steps in Optimization part to make the whole optimization routine more clear.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes. There are no proofs for theoretical claims and there is no need to check the correctness.", "experimental_designs_or_analyses": "Yes. The dataset used in the experiment covers a wide range of categories and quantities, enabling a comprehensive presentation for the final results. Additionally, the comparison of experimental methods consists of advanced models from the past three years, enhancing the credibility of the experimental results.", "relation_to_broader_scientific_literature": "Compared to the previous studies, the highlights of this research constrain the factor matrix with rigorous interpretation to be cluster indicator representation by introducing the orthogonal constraint on the actual bases and use the alternative algorithm for solving the formulated problem. Extensive experiments are performed on different datasets to demonstrate the superiority of FIML in terms of effectiveness and efficiency.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strength: This paper builds the relation between anchor graph and similarity matrix in symmetric nonnegative matrix factorization, i.e., each entry in the anchor graph is able to characterize the similarity between the anchor and original data sample.  \n\nWeakness: \n\n1. The authors should add more recent fast multi-view clustering methods for comparison in the experiment to validate the novelty of this work.\n\n2. The authors can give more detailed experimental analysis for the experimental results based on ACC, NMI, F1-score and Purity in Table 1-Table 4.\n\n3. The best clustering performance based on ACC, NMI, F1-score and Purity in Table 1-Table 4 in the experiment should be bold.\n\n4. The authors should give the brief optimization process before the following optimization steps in Optimization part. Then the whole optimization routine becomes more clear.", "other_comments_or_suggestions": "No.", "questions_for_authors": "1. The authors should give more detailed experimental analysis for the experimental results based on ACC, NMI, F1-score and Purity in Table 1-Table 4.\n2. The authors are expected to give the brief optimization process before the following optimization steps in Optimization part. Then the whole optimization routine becomes more clear.", "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"}}, {"review_id": "C1vtarLvxn", "reviewer": "Reviewer_Tzg8", "summary": "This paper proposes a novel fast incomplete multi-view clustering method for the data with large scales, termed Fast Incomplete Multi-view clustering by flexible anchor Learning (FIML), where graph construction, anchor learning and graph partition are simultaneously considered in a unified framework for fast incomplete multi-view clustering. These three parts can boost each other, which promotes the quality of the clustering and improves the efficiency for large scale datasets. To be specific, the authors learn a shared anchor graph to guarantee the consistency among multiple views and adopt a adaptive weight coefficient to balance the impact for each view.", "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: Compare the proposed FIML with the most related works in Introduction.\n\nA1: Thanks for the comment! The existing works most related to FIML are FPMVS-CAG and SMVSC. 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, are that we learn a shared anchor graph to guarantee the consistency among multiple views and employ a adaptive weight coefficient to balance the impact for each view. The relation between anchor graph and similarity matrix in symmetric nonnegative matrix factorization can also be built, i.e., each entry in the anchor graph can characterize the similarity between the anchor and original data sample.\n\nQ2: Add more recent compared methods in the experiment. \n\nA2: Good question! We have added a recent method for comparison in the experimental section, i.e., OMVCDR [a].\n\n[a] One-Step Multi-View Clustering With Diverse Representation, 2024\n\nThe clustering results of OMVCDR based on ACC for all datasets are 77.00±0.00, 89.85±0.10, 91.75±0.00, 76.50±0.15, 98.80±0.00, 98.95±0.10\n\nThe clustering results of OMVCDR based on NMI for all datasets are 89.50±0.00, 75.85±0.15, 49.00±0.05, 58.00±0.00, 96.50±0.10, 99.72±0.00\n\nThe clustering results of OMVCDR based on F1-score for all datasets are 69.00±0.00, 81.15±0.10, 88.95±0.00, 60.00±0.05, 97.80±0.00, 99.25±0.00\n\nThe clustering results of OMVCDR based on Purity for all datasets are 79.56±0.00, 91.25±0.20, 92.00±0.00, 76.50±0.50, 98.79±0.00, 99.20±0.05\n\nQ3: The memory of the adopted device should be given.\n\nA3: Thanks for the comment! Since the running time is listed in the experiment, we should give the memory of the used device in the experiment. The memory of the adopted device in the experiment is 8G in the experiment and we will add this description for the camera-ready version.\n\nQ4: The best and second best results should be bold in the experiment.\n\nA4: Good question! It is important to bold the best and highlight the second best clustering performance for tables in the experiment in making the performance gains more obvious for the readers. We will bold the best and highlight the second best clustering performance for tables of the camera-ready version in the experiment to make the performance gains more obvious.\n\nQ5: What is the unit of running time in the experiment, second or log second?\n\nA5: Thanks for the comment! It is needed to give more detailed running time description in the related parts. The unit of running time in the experiment is log second and we will add this description for the camera-ready version.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes. The effectiveness of the proposed FIML is better demonstrated in the experiment.", "theoretical_claims": "Yes. The authors do not give proofs for theoretical claims in this work and there is no need to check them.", "experimental_designs_or_analyses": "Yes. the experimental designs are reasonable.", "relation_to_broader_scientific_literature": "The authors propose a novel fast incomplete multi-view clustering method for the data with large scales, termed Fast Incomplete Multi-view clustering by flexible anchor Learning (FIML), where graph construction, anchor learning and graph partition are simultaneously considered in a unified framework for fast incomplete multi-view clustering.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strength: \n\nThis paper learns a shared anchor graph to guarantee the consistency among multiple views and adopt a adaptive weight coefficient to balance the impact for each view. The relation between anchor graph and similarity matrix in symmetric nonnegative matrix factorization can also be built.\n\nWeakness: \n\n1. The authors propose a novel fast incomplete multi-view clustering method for the data with large scales, termed Fast Incomplete Multi-view clustering by flexible anchor Learning (FIML). The authors are expected to compare the proposed FIML with the most related works in Introduction part. Then the novelty of the proposed FIML is more clear for the readers.\n2. The authors should add more recent related works in multi-view clustering for comparison in the experiment. Then the effectiveness of the proposed FIML is better demonstrated in the experiment.\n3. Considering the running time is given in the experiment, the authors should give the memory the used device in the experiment.\n4. The authors should bold the best and highlight the second best clustering performance for tables in the experiment to make the performance gains more obvious.\n5. The authors are expected to give more detailed running time description in the related parts, i.e., what is the unit of running time in the experiment , second (s) or log second?", "other_comments_or_suggestions": "No.", "questions_for_authors": "1. The authors should give more detailed running time description in the related parts, i.e., what is the unit of running time in the experiment , second (s) or log second?\n2. The authors are expected to add more recent related works in multi-view clustering for comparison in the experiment. Then the effectiveness of the proposed FIML is better demonstrated in the experiment.", "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": "KNiNwbpDqO", "reviewer": "Reviewer_dTzz", "summary": "This paper proposes a novel fast incomplete multi-view clustering method for the data with large scales, termed Fast Incomplete Multi-view clustering via flexible anchor Learning (FIML), where graph construction, anchor learning and graph partition are simultaneously integrated into a unified framework for efficient incomplete multi-view clustering. To be specific, the authors learn a shared anchor graph to guarantee the consistency among multiple views and employ a adaptive weight coefficient to balance the impact for each view. The relation between anchor graph and similarity matrix in symmetric nonnegative matrix factorization can also be built, i.e., each entry in the anchor graph can characterize the similarity between the anchor and original data sample.", "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 each entry in the anchor graph describes the similarity between data sample and anchor.\n\nA1: Thanks for the comment! The reason why each entry in the anchor graph Z describes the similarity between data sample and anchor can be explained by the fact that dimension of the anchor graph Z is m \\times n. It corresponds to m anchors and n data samples and larger entry in the anchor graph Z indicates larger similarity between data sample and anchor. We will add the above explanations for the camera-ready version.\n\nQ2: The authors do not bold the best results in the experiment.\n\nA2: Good question! It is needed to bold the best clustering results on tables for the experiment to make the performance achievements more obvious. We will bold the best clustering results on tables for the camera-ready version in the experiment to make the performance achievements more obvious.\n\nQ3: The reason why the [0.001, 0.1, 1, 10, 100, 1000] is chosen for parameter selection should be given.\n\nA3: Thanks for the comment! The reason why the [0.001, 0.1, 1, 10, 100, 1000] is chosen for parameter selection is that values in such range represent different representative magnitudes in parameter selection for \\lambda in the experiment. We will add this explanation for the camera-ready version.\n\nQ4: Giving the detailed specific values analysis for the experimental results.\n\nA4: Good question! It is needed to give the detailed specific values analysis regarding the superiority of adopting a unified framework integrated by graph construction, anchor learning and graph partition. On the STL10 dataset, our method outperforms the last five multi-view clustering methods in tables by achieving improvements of 49.5%, 31.6%, 59.8%, 22.7% and 2.3%. We will add the above detailed specific values analysis for the experimental results in the camera-ready version.\n\nQ5: How many iterations are needed for the proposed FIML in reaching convergence?\n\nA5: Thanks for the comment! It needs about 20 iterations for the proposed FIML in reaching convergence. We will add this description regarding the iteration convergence for the camera-ready version.\n\nQ6: What is the memory value of the adopted device in experiment?\n\nA6: Good question! It is needed to give the memory description of the adopted device considering the running time of FIML and the compared approaches on different benchmark datasets are shown in Table 5. The memory value of the used device is 8G and we will add this description for the camera-ready version.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "Yes. There exist no proofs for theoretical claims in the paper and it is not needed to check them.", "experimental_designs_or_analyses": "Yes. The used compared methods include the works from the recent years, increasing the credibility of the final results.", "relation_to_broader_scientific_literature": "This paper gives a new insight to the community of incomplete multi-view clustering for large scale datasets compared to the existing methods, i.e., graph construction, anchor learning and graph partition in efficient incomplete multi-view clustering can boost each other, which are able to be integrated into a problem. The combination of these three issues is the focus in our work. While most existing work treat graph construction, anchor learning and graph partition as separated problems in incomplete multi-view clustering for the datasets with large scales.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "**Strength**\n\nThe authors constrain the factor matrix with rigorous interpretation to be cluster indicator representation by introducing the orthogonal constraint on the actual bases and use the alternative algorithm for solving the formulated problem. Extensive experiments are performed on different datasets to demonstrate the superiority of FIML in terms of effectiveness and efficiency.\n\n**Weakness**\n\n1. The authors state that each entry in the anchor graph $ Z $ describes the similarity between data sample and anchor. Since the symmetric constraint on $ Z\\in R^{m\\times n} $ are not guaranteed in factorization with $ m\\ll n $, the authors remove such constraint on anchor graph $ Z $ and this is the main difference between anchor graph and similarity matrix in symmetric nonnegative matrix factorization. In this part, the authors are expected to give the reason why each entry in the anchor graph $ Z $ describes the similarity between data sample and anchor.\n2. The authors list the detailed clustering results of FIML and the compared approaches on different datasets in terms of four metrics in Tables 1-4. The authors also compare FIML with IMVC-CBG and FIMVC-VIA under different missing ratios on several datasets under different metrics. According to Tables 1-4 and Figs. 4-7, the authors then draw some following conclusions. However, the authors do not bold the best clustering performance in Tables 1-4 in terms of four metrics for this paper. \n3. The authors perform the parameter selection for the trade-off parameter $ \\lambda $ in the range of $ [0.001,0.1,1,10,100,1000] $ to study how these this parameter influences the final clustering performance and find that better performance is achieved when $ \\lambda=1 $ under the same $ m $ on different datasets. Besides, the clustering result of FIML is relatively stable over different parameter values on these datasets, which shows that FIML is generally robust to the trade-off parameter $ \\lambda $. However, the authors do not give the detailed reason why the $ [0.001,0.1,1,10,100,1000] $ is chosen for parameter selection in this paper.\n4. The authors perform ablation study to validate the superiority of adopting a unified framework integrated by graph construction, anchor learning and graph partition. In comparative experiments, the authors first learn anchors and construct the graph to obtain informative representation. Then the graph partition is isolated from the above two processes in the designed experiment. However, the authors do not give the detailed specific values analysis regarding the superiority of adopting a unified framework integrated by graph construction, anchor learning and graph partition.", "other_comments_or_suggestions": "No", "questions_for_authors": "1. The authors conduct convergence analysis of FIML on different datasets by showing the evolution process of the objective function with iterations in terms of ACC and observe that FIML monotonically decreases with iterations and tends to converge in about some iterations on these datasets. Here, how many iterations are needed for the proposed FIML in reaching convergence?\n2. The authors show the running time of FIML and the compared approaches on different benchmark datasets in Table 5. However, the authors do not give the memory description of the used device. What is the memory value of the used device?", "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": "xmxk4yJWD2", "reviewer": "Reviewer_y35w", "summary": "This paper proposes FIML in this work for fast incomplete multi-view clustering. It simultaneously considers graph construction, anchor learning and graph partition in a unified framework, in which these parts boost each other for improving the effectiveness and efficiency for datasets with large scales. To be specific, a shared anchor graph for guaranteeing the consistency among multiple views is learned and the adaptive weight coefficient is adopted to balance the impact for each view.", "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 summarization of the objective function of the proposed FIML after Eq. (3) should be given.\n\nA1: Thanks for the comment! The related summarization of the objective function for the proposed FIML after Eq. (3) is shown in the following. Then graph construction, anchor learning and graph partition are jointly integrated into a unified framework for incomplete multi-view clustering in this manner, where these three parts can boost each other to achieve effective and efficient clustering results for incomplete large-scale multi-view dataset. The discriminative anchors can be automatically learned and the final partition is achieved in this manner. We will add the above summarization for the camera-ready version.\n\n\nQ2: What is KKT condition before Eq. (10)?\n\nA2: Good question! It is needed to explain KKT and the meaning is not clear without the related description. KKT is the abbreviation for Karush-Kuhn-Tucker conditions and we will add this explanation for the camera-ready version.\n\nQ3: More recent methods should be added for comparison in the experiment. \n\nA3: Thanks for the comment! We have added a recent method for comparison in the experimental section, i.e., OMVCDR [a].\n\n[a] One-Step Multi-View Clustering With Diverse Representation, 2024\n\nThe clustering results of OMVCDR based on ACC for all datasets are 77.00±0.00, 89.85±0.10, 91.75±0.00, 76.50±0.15, 98.80±0.00, 98.95±0.10\n\nThe clustering results of OMVCDR based on NMI for all datasets are 89.50±0.00, 75.85±0.15, 49.00±0.05, 58.00±0.00, 96.50±0.10, 99.72±0.00\n\nThe clustering results of OMVCDR based on F1-score for all datasets are 69.00±0.00, 81.15±0.10, 88.95±0.00, 60.00±0.05, 97.80±0.00, 99.25±0.00\n\nThe clustering results of OMVCDR based on Purity for all datasets are 79.56±0.00, 91.25±0.20, 92.00±0.00, 76.50±0.50, 98.79±0.00, 99.20±0.05\n\nQ4: The most important part in motivation should be highlighted.\n\nA4: Good question! It is needed to highlight the most important part in motivation, which is presented as “Actually, relatively less data samples are enough to reconstruct the latent space. Therefore, selecting some data samples from the original dataset as anchors or landmarks for reconstructing the relation structure is commonly used in the existing works.” We will highlight the above details for the camera-ready version.\n\nQ5: The formats of works in references should be consistent.\n\nA5: Thanks for the comment! We will check the formats of works in references part for consistency in increasing the presentation of the paper.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes. There are no proofs for theoretical claims in this work and I do not need to check them.", "experimental_designs_or_analyses": "Yes. The adopted experimental methods for comparison contain methods from the recent years, which enhances the credibility of the experimental performance.", "relation_to_broader_scientific_literature": "Compared to the existing works, this paper proposes FIML in this work for fast incomplete multi-view clustering. It simultaneously considers graph construction, anchor learning and graph partition in a unified framework, in which these parts boost each other for improving the effectiveness and efficiency for datasets with large scales.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strength: The authors learn a shared anchor graph for guaranteeing the consistency among multiple views and adopt the adaptive weight coefficient to balance the impact for each view.\n\nWeakness: \n1. After giving the total objective function for the proposed FIML, the authors should show the summarization of this objective function after Eq. (3) to make the formulation of FIML better explained.\n2. The authors should further what is KKT condition before Eq. (10) for the Optimization part of the proposed FIML.\n\n3.The compared methods in the experiment are not enough and the authors are expected to add more recent works for comparison in Experiment part.\n\n4. The length of motivation part in the work is oversized in Section 2 for the paper. Then the authors should highlight the most important part in this part to make the motivation more obvious.\n\n5. The formats of works in references part should be consistent to increase the presentation of this paper.", "other_comments_or_suggestions": "No.", "questions_for_authors": "1. The authors are expected to add more recent works for comparison in Experiment part.\n\n2. The authors should highlight the most important part in motivation part to make the motivation more obvious.", "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": {}}]

Author rebuttal:

Meta-review: This paper presents a fast incomplete multi-view clustering method by flexible anchor learning. Graph construction, anchor learning and graph partition are simultaneously integrated into a unified framework for fast incomplete multi-view clustering. Experiments on several datasets demonstrate the superiority. After the rebuttal, it receives three accept and one weak accept. Its merits, including the interesting idea, extensive experiments, and good results, are well recognized by the reviewers. The response well addresses the reviewers' concerns about the additional comparison, detailed analysis, and so on. I think the current manuscript meets the requirement of this top conference and recommend for acceptance. Please incorporate the revision in the updated manuscript.
Decision: Accept (poster)