Alleviate Anchor-Shift: Explore Blind Spots with Cross-View Reconstruction for Incomplete Multi-View Clustering
Authors: Suyuan Liu
Siwei Wang
KE LIANG
Junpu Zhang
Zhibin Dong
Tianrui Liu
En Zhu
Xinwang Liu
Kunlun He
Venue: NeurIPS 2024 poster (2024)
Keywords: Incomplete Multi-View Clustering
Anchor-Based Multi-View Clustering
Primary area: learning_theory

Abstract: Incomplete multi-view clustering aims to learn complete correlations among samples by leveraging complementary information across multiple views for clustering. Anchor-based methods further establish sample-level similarities for representative anchor generation, effectively addressing scalability issues in large-scale scenarios. Despite efficiency improvements, existing methods overlook the misguidance in anchors learning induced by partial missing samples, i.e., the absence of samples results in shift of learned anchors, further leading to sub-optimal clustering performance. To conquer the challenges, our solution involves a cross-view reconstruction strategy that not only alleviate the anchor shift problem through a carefully designed cross-view learning process, but also reconstructs missing samples in a way that transcends the limitations imposed by convex combinations. By employing affine combinations, our method explores areas beyond the convex hull defined by anchors, thereby illuminating blind spots in the reconstruction of missing samples. Experimental results on four benchmark datasets and three large-scale datasets validate the effectiveness of our proposed method.

Alleviate Anchor-Shift: Explore Blind Spots with Cross-Vice Reconstruction for Incomplete Multi-View Clustering
Suyuan Liu?
Siwei Wang*
Ke Liang' Junpu Zhang' En Zhu' Kunlun He Xinwang Liu I+
Zhibin Dong'
Tianrui Liu'
National University of Defense Technology, Changsha, China
PAcademy of Military Sciences, Beijing, China
3 Chinese PLA General hospital, Beijing, China
{suyuanliu, enzhu, xinwangliul@nudd edu,
Abstract
Incomplete multi-view clustering aims learn complete correlations among sam- ples by leveraging complementary information across multiple views for clustering. Anchor-based methods further establish sample-level similarities for representative anchor generation, effectively addressing scalability issues in large-scale scenarios. Despite efficiency improvements, existing methods overlook the misguidance in anchors learning induced by partial missing samples, i.e., the absence of sam- ples results in shift of learned anchors, further leading to sub-optimal clustering performance. To conquer the challenges, our solution involves a cross-view re- construction strategy that not only alleviate the anchor shift problem through a carefully designed cross-view learning process, but also reconstructs missing sam- ples way that transcends the limitations imposed by convex combinations. By employing affine combinations, our method explores areas beyond the convex hull defined by anchors, thereby illuminating blind spots the reconstruction of missing samples. Experimental results on four benchmark datasets and three large-scale datasets validate the effectiveness of our proposed method.
1
Introduction
In multi-view learning, data collected from different sensors or media often suffer from missing values For example, remote sensing images collected from various sensors may experience partial missing due to channel noise. Traditional multi-view learning methods cannot directly handle these missing values [5] 6 To address this issue, incomplete multi-view lcarning methods have been developed to leverage the available data from all views to perform downstream tasks
2 3
g 10. Among these, incomplete multi-view clustering (IMC) relies view consistency and complementarity, effectively enabling the partitioning of data with missing values [1]
12 B3
Existing IMC methods be categorized into three types based on their approach: similarity-baat based, imputation- based, and matrix decomposition based methods. Similarity-based methods recover relationship matrix among all samples using available data in each view [14
15 T6 17 18)
Imputation-ban based methods first fill in the missing parts, transforming the problem into a complete multi-view clustering problem [19 20 21). Matrix decomposition--asee methods map samples from all views into common latent space to construct a unified representation for clustering 24 25). However, these methods all require constructing an n x n similarity matrix,
22 23
24
*Corresponding Author
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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0.5
(a)
(b)
(c)
(d)
Figure 1: (a)Anchors learned in complete data. (b)Anchors initialized in incomplete data. (c)Data
reconstructed with convex combination. (d)Data reconstructed with affine combination.
resulting in On1²) space complexity, which limits their application to large-scale scenarios
26 27 28.
In contrast, anchor- based IMC methods reduce space complexity to (n) by learning a small number of representative anchors from the data and replacing the full similarity matrix with relationships between anchors and samples [29 30 29 31. Despite addressing scalability, these methods overlook the anchor-shift problem caused by missing data. As shown in Fig. [a)(b), anchor learning can be misled by missing data, resulting in discrepancy between learned anchors and those from complete data. This problem diminishes the representational capacity of anchors and causes the anchor graphs to be misaligned across views, thereby compromising the clustering performance.
In this work, we propose novel Anchor-based Incomplete Multi-view Clustering with Cross- View Reconstruction strategy, termed AIMC-CVR. AIMC-CVR encompasses two key modules designed to resolve the anchor-shift problem anchor-based IMC. The first module, the cross-view anchor learning module, is dedicated to mitigating the anchor-shiff problem by learning a complete anchor graph. Specifically, we designed symmetrized cross-view projection mechanism to ensure dimensional consistency across view pairs. By leveraging the relationships between anchors and samples across different view pairs, we constructed a complete anchor graph.
The second module, the affine combination-basel reconstruction module, iteratively updates the anchors with available and reconstructed data. Existing anchor-based IMC methods often convex constraints to build relationships between anchors and samples 32
leading to blind spots in sample reconstruction. As shown in Fig. [Tc, samples reconstructed based on convex combinations are restricted to the convex hull of the anchors. Instead of relying on convex combinations, our innovative approach utilizes an affine combination-base reconstruction strategy. This strategy broadens the scope of sample reconstruction, as depicted in Fig. [d), allowing for more comprehensive and accurate representation of missing data.
33
34
35
Our main contributions are as follows:
By employing cross-view anchor learning and affine combination- based reconstruction, we effectively alleviate the mnchor-shfft problem in missing data scenarios.
Unlike traditional sample-level imputation methods, we reconstruct missing samples with anchors and anchor graphs, significantly reducing reconstruction complexity, Affine combi- nations further explore blind spots in sample reconstruction, as demonstrated by theoretical analysis and experimental results.
Comparative experiments with sate-of-the-ard IMC and anchor-based IMC methods validate the effectiveness and superiority of AIMC-CVR.
2
Related Work
Cross-View Learning. Cross-view learning specialized form of multi-view learning that leverages interactions between paired views to learn cross-view representations, enabling the exploration of finer-grate finer-grained inter-view relationships [36] 37 38). For instance, Tang et al. ensure local structural

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consistency across views by simultaneously constructing similarity graphs between pairs of views and within each view [39] Similarly, Liu al. leverage the fact that samples are always present in at least one view, constructing a complete cross-view similarity matrix based on relationships within and between views 40] These methods demonstrate the potential of cross-view learning to maintain structural consistency and fully utilize the available data. In the next section, we will explore the application of cross-view view learning strategies to multi-view anchor learning, focusing on view complementarity to construct a complete anchor graph and mitigate the anchor-shift problem in scenarios with missing data.
Sample-Level Imputation. Imputation- based multi-view clustering methods first impute the missing data before clustering [41 [2 (1). Typically, existing methods perform sample-level imputation by leveraging mueti-view information to learn complete similarity relationships between samples. For example, Yin et reconstruct missing data based on the decomposition matrix and sample similarity relationships [20]. Liu et al. integrate data imputation and clustering within unified optimization framework [9] However, these sample-level imputation methods typically rely on similarity matrices of size O( which limits their scalability for large datasets.
3
Method
In this section, we introduce the Anchor- based Incomplete Multi-view Clustering with Cross- View Reconstruction (AIMC-CVR) strategy. We begin by introducing the two core modules: the cross-view anchor learning module and the affine combination- retical analysis that demonstrate the merits of our proposed affine combination- strategy. Finally, we present the overall objective function of the algorithm and propose four-step alternating iterative algorithm to solve the corresponding optimization problem.
reconstruction module. We provide a theo- reconstruction
3.1 Cross- View Anchor Learning Module
p=1' where d, and n denotes the dimension of the data and the total number of samples, respectively.
Given multi-view data (x(r) (P) [[xx")" J
In the context of incomplete data scenarios, the data matrix X (() for the p-th view can be partitioned
[x!"), x(#'), where x(P) xad representing the existing portion of into two distinct subsets, i.e.,
the data, and x E [d-x((=npp) is the missing portion. The observed part is obtained by applying
(p)
an index matrix (P) {0, 13 1 to the complete data matrix, such that x(") X (r)), The index matrix (p) encodes the presence of samples, where indicates that the i-th sample in the complete dataset corresponds to the j-th ranked existing sample in the observed subset !P)) Based on the samples present in each view, we can learn the anchors and their corresponding anchor graphs as follows:
min AP).Z)) p=1
Σ (p)
A (vl (P))
F
s.t. (p)
1, Z (P)
The anchor graph (P)) is incomplete, representing only the relationship between existing samples and anchors in the p-th view. While complete anchor graph be synthesized from all views through late-fusion, this process is compromised by the inherent misalignment of anchor graphs caused by view discrepancy. This discrepancy, known as the anchor-shift problem, arises because anchors learned from np samples differ from those learned from complete data. Consequently, varying missing samples across views lead to inconsistent anchor-shift, resulting in misaligned anchor graphs at the representation level.
To address this issue, we propose cross- -iew anchor learning module. This module constructs complete anchor graph under the assumption that each sample appears in at least one view in the incomplete multi-view scenario. Specifically, for the anchor graph (P)) in the p-th view, we update with the following objective:
3

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min W Z(0)
W(4) x!
W(()
W(r))
W(P))
I, ((2 1, Z (P] 2
where W(r)) is the projection matrix, which projects the dimension-reduced data into a higher- dimensional space. Unlike previous approaches that reduce all data to the same lower dimension, we aim to preserve high-dimensional features across views as much as possible to ensure the effectiveness of cross-view learning- A symmetric cross-view projection mechanism is designed to ensure dimensional consistency between different view pairs: if dp dg. W(P)) is the projection matrix that needs to be optimized; otherwise, W(r)) equals the identity matrix, shown below:
W(Pg) E R4,
dp > dg.
W(
IE Rd,dd,
dp S dg-
By sequentially learning the similarity between the ny points present in the q-th view (where q ,...., u) and the anchors in the p-th view, can obtain anchor graph with size of m x n. Furthermore, the compiete anchor graph can guide the learning of anchors, thereby implicitly alleviating the anchor-shift problem.
3.2 Affine Combination-basee Reconstruction Module
The cross-view anchor learning module fills the missing columns the anchor graph by measuring the distance between the existing samples in other views and the anchors of the current view in the same space with a projection matrix, which avoids recovering the missing samples. However, the measurement of similarity between views-view representations overly dependent the reliability of the projection matrix and the consistency between views. Additionally, simply relying on cross-view information to implicitly correct anchor shifts is insufficient. Therefore, propose recover the incomplete data in each view to directly correct the anchors affected by the missing parts, Traditional imputation methods, which reconstruct missing data based the similarity among all samples, require an additional quadratic space complexity, making them impractical for large-scale problems. Thus, we propose fast reconstruction module, as follows:
min x A.) Z(P)
Σ
X
X (p)
A (z) (0)
(4)
s.t. Z (p)
1 = 1, Z 2 0,
where x(() E xd, represents the existing samples in the view, and x(() E Rd represents the missing samples be reconstructed. x is constructed from the anchors and their corresponding anchor graphs. The reconstructed x(P) then participates in the next iteration of anchor learning, with the reconstruction of missing samples and anchor learning iterating and mutually reinforcing each other. Accurately reconstructed x(P) enables the learned anchors in the next iteration to be closer to the true global anchors, whereas inaccurate reconstruction exacerbates the anchor-shif problem. However, the reconstructed missing samples are constrained within the convex hull of the anchor set in Eq. (4. According Theorem I there always exist samples that cannot be reconstructed by the convex combination of anchors.
Theorem Suppose A {aa....,,am) is an anchor set composed of cluster centers from the dataset X = {II,.... Rn). J. Then, there always exist a sample belonging to X that lies outside the convex hull of the anchor set A. Mathematically
min |E fra - -cl/2 0, Ec X,
(5)
s.t. fT fT 0.
When most missing samples are outside the convex hull of the anchors, the next iteration of an- chor learning erroncously shifts inward, worsening anchor-shift. Therefore, propose an affine-

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combination based reconstruction strategy, which relaxes convex constraints on anchor graph rows to affine ones.
min X (O)
Σ
(p) x())
A (O))
F
(6)
s.t. PP) 1-
Theorem 2.
For sample e lying outside the convex hull of the anchor set A, the representation constructed by the affine combination of set A is always closer to than that constructed by the convex combination of sei A. Mathematically,
min | 9:0; - cl2 min
D> fla - -ll22
g 1= 1,f2
According to Theorem 2 utilizing the affine combination of anchors facilitates the recovery of more accurate missing samples. Ultimately, by combining the cross-view anchor learning module with the alfine-combination based reconstruction module, the objective of AIMC-CVR is as follows:
min Σ
(p) - A A + Σ
+1
W (pg) x(() W(ep) A (p) Z G
(8)
#=1
s.t. W(m))
W(m) I, Im) 1-
where (p) Xm W(P9), (P) (p)
The hyperparameter 33 helps adjust the sparsity of the anchor graph, and À is hyperparameter balancing the influence of the two modules. Finally, we concatenate the anchor graph from each view to obtain the common one, Z = [Z())... [)), which avoids the anchor alignment problem present in other fusion methods 33 Note that the
columns of Z still consist of ones, ensuring that its recovered transition probability matrix S is doubly stochastic matrix, as proven in the appendix. Therefore, directly performing k-means
clustering on the left singular vectors of Z yields the final clustering results [26].
3.3
Optimization
To solve the optimization problem in Eq. (), we propose four step alternating iterative algo- rithm. When optimizing variable, the other variables are fixed to their previous iteration values. Additionally, since each variable is independent across views, we update them sequentially by view.
Step 1: Update x Fixing ((+)), A (p), (PP) and removing terms unrelated to (2) we have the following optimization problem:
min
2x(P)) (p)
A (p) (p) E(p)
(9)
where E(P) E (0, (0 is the index matrix, by
1 indicates that the i-th sample is ranked j-th
among the missing samples. By taking the derivative of Eq. (9) with respect to (p) and setting it to zero, obtain the closed-form solution for X(2) follows:
(P) A (p) (p)
(10)
The solution for (!P) shows that incorporates the anchors and anchor graphs from the previous iteration, contributing the update of anchors in the next iteration.
5

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Step 2: Update W(r), Fixing X (p) A (p), Z(2 and removing terms unrelated to W(P4), we have the following optimization problem when dp < dgi
x (p)
W(rs) A (z) (() G W(r x! -
2
min
F'
(11)
s.t. W(r) W(rr) = I.
By further simplification, Eq. can transform into the following form:
Tr (Wirg) B(xr)) (
max
(12)
s.t. W(rg) W(Pg) = I.
where ((P)) x (A
+ Z G G x!") According to reference 43 the
optimal Wles is derived from the product of the left and right singular vectors of B(P4),
Step 3: Update A (p), Fixing (()) W(Pg), (()) and removing terms unrelated to A (P), we have the
following optimization problem:
min Tr ((P) -2A (DO) (
(13)
where C = Z (m) + > Z GOO Z D
X (p) (p)
A DE== W(T)) W((4) x (0) (
Z(w) By taking the derivative of Eq. (3) with respect to
(p) and setting it to zero, we obtain the closed-form solution for A (p) as follows:
A (p) D(P)
(14)
Step4: Update ())) Since each column of (0) is independent of others, we optimize the i-th column z, (p) of (P)) while fixing x!P) ( A (p) follows:
min (p)
Hz (p)
2t zi
(p)
(15)
(p) T
s.t. 2
==
where H (A + 1)A (P) (p)
31, t; =
= > x()) ((9)) T ((r (p)
x(#) (v) When i-th sample exists in the p-th view (4) 1, else o.1) 0. We employ the
Lagrange multiplier method to tackle the above problem. Firstly, the Lagrangian function for Eq.
15) as follows:
L
Hz, (P)
2t, z; (p) + az(zz 1 1),
(p) (p)
(16)
where is the Lagrangian multiplier. The corresponding KTT conditions is
Hz +; 9 0,
(17)
T 1-1.
By substituting the first term into the second, we can get = HHH
1-11 .Ten we have 2,
(x)
H (t, $1). The entire optimization procedure for AIMC-CVR is summarized in Algorithm

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Algorithm The proposed AIMC-CVR
Input: Multi-view dataset (X(p) 1"
p=1" anchors number m, clusters number k, parameters 8, and {Z(m) pm1
1: Initialize {A (x)) [A))) p-11 (W(rg)
p.gm1
2: while not converged do
3: for p = -> do
4: Update X! (P)
with Eq. 10).
5: q = 1 do
6: Update W(()) by solving Eq. 12
7: end for
8: Update A (P) with Eq. 14
9: Update (p) by solving Eq. 15
end for
11: end while
Concatenate Z(P) obtain Z.
Output: Performing k-means on Z to get the final cluster results.
3.4 Complexity Analysis
The computational complexity of AIMC-CVR primarily consists of solving four variables. When updating x the complexity of matrix multiplication is O(nmd,). For updating ((r)), the complexity of matrix multiplication is O(nmdp + ndp² + dp ), and the SVD decomposi- tion complexity is Odd,²). ). When optimizing A ((r), the complexity of matrix multiplication is Onndd + m2d + ndzdg), and the complexity of inversion is (mm)). When optimizing Z(r), the complexity matrix multiplication is (n sonmmd+nddpp tdd Therefore, each iteration, AIMC- CVR consumes a time complexity of On Ep=r (mdd + dp + dp EØ= dg) + Ep= (m2 dp do)), which is linear with respect to nI.
The space complexity of AIMC-CVR is O(n(m + E/= dp) + m Σn= dp + Ep=EE E dgdg). primarily stemming from storing relevant matrix variables, also scales linearly with the number of samples.
3.5 Convergence Analysis
In this section, we provide a theoretical analysis of the convergence our proposed AIMC-CVR. The objective function in Eq. (8) is non-convex when considering all variables simultaneously. To address this, we employ a four-step iterative optimization algorithm, detailed in Algorithm [ where each variable is optimized sequentially while keeping the others fixed. During each iteration, the variables being optimized have analytical solutions, ensuring that the objective function of AIMC- CVR decreases monotonically with successive iterations. Furthermore, since the objective function in Eq. (8) is bounded below by zero, our proposed AIMC-CVR is guaranteed to converge to local minimum.
Experiments
4.1 Experimental Setup
Datasets description. Seven widely used multi-view datasets are employed evaluate the per- formance of AIMC-CVR, including: MSRCV [44] is composed of images from seven categories. Benkaiccontain text and citations collected from the website. Wiki 45) is dual-view dataset with text-image pairs. Hdigi is dataset composed of handwritten digit images. YTF10 and YTF20

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are two subsets extracted from the YouTubeFace dataset. MNIS is subset extracted from
larger dataset supplied by NIST. Details of these datasets are list the Table [J Following [26]. randomly remove 10% to 90% of the samples in 10% intervals to create missing versions of the above datasets, which ensures each sample exists in least one view.
Table Employed datasets in experiments.
Datasets
#Samples (n)
#Vieww (V)
#Clusters (k)
#Dimensionality (d_p)
MSRCV
210
3
7
256
512
1302
WebKB
1051
2
2
334
2949
Wiki
2866
2
10
10
128
Hdigit
10000
2
10
256
784
YTFIO
38654
10
512
576
640
YTF20
63896
4
20
512
576
640
944
MNIST 70000
4
9
512
576
640
944
Compared methods. We compared proposed method with the following eight state-of-the-as -art incomplete multi-view clustering algorithms: DAIMC[22), UEAF[2], EEIMVC FLSD[24] V9H(15), IMVC-CBG(26) SCBGL[29), DVSAIBI]. The first five comparison methods are tradi- tional similarity- -baed approaches, while the latter three are large-scale anchor-base methods.
Implementation details. For all comparison algorithms, we set the parameters according to their descriptions in the corresponding literature. In our method, the anchors number is searched in [k; 2k: 3k), and the parameter À and 3 are both searched in [0.001, 0.01, 1, 10, 100). To evaluate the clustering performance, we employ accuracy (ACC), normalized mutual information (NMI), Purity, and Fscore for comparison. Each compared algorithm missing rates, and the results for each missing rate were averaged to obtain the clustering results. Additionally, conduct 20 repetitions of the k-means step for all algorithms and calculate the mean and variance for the final experimental result. All experiments were conducted on desktop computer equipped with an Intel Core 9--1000XX CPU, 64GB of RAM.
tested across datasets with different
4.2 Clustering Performance Comparison
We compare AIMC-CVR with eight state-of-the-an algorithms across seven datasets, as shown in Table Our algorithm demonstrates superior competitive clustering performance across all datasets, highlighting its effectiveness. On smaller datasets such as MSRCV, WebKB, Wiki, and Hdigit, method achieves the highest ACC in all cases, surpassing the second-best algorithms by 3.65% 7.89%, 0.19% and 3.96%. This showcases our algorithm's 's robustness and efficacy in handling incomplete multi-view datasets. Furthermore, Fig. 2 shows the clustering accuracy (ACC) curves for all algorithms across different datasets as the missing rate varies, Our algorithm consistently outperforms all others at every missing rate, demonstrating its superiority.
Additionally, our method demonstrates excellent scalability, successfully processing three large-scale datasets that the traditional methods cannot handle due to memory constraints. By addressing the anchor-shift problem, which was overlooked by other three anchor-basee methods (IMVC-CBG, SCBGL, DVSAD). achieved notable enhancement in clustering performance. Specifically, on YTF10, YTF20, and MNIST datasets, our algorithm achieves ACC improvements over the second-best algorithms by 3.18%, 0.42% and 7.47%.
4.3 Ablation Study
To showcase the effectiveness of different modules and strategies, we constructed five variants of
AIMC-CVR as follows: (1) AIMC-CVR-vl removes the cross-view anchor learning module by
setting À = O. (2) AIMC-CVR-V removes the affine combination-based reconstruction module. (3)
AIMC-CVR-V3 removes the sparsity regularization term by setting / 0. (4) AIMC-CVR-VE keeps the initialized A fixed and does not update it during subsequent optimization. (5) AIMC-CVR-v5 replaces affine combination with convex combination by adding non-negative constraints to (())
'htps://www.micc unifi wtt.a//ww....oeeee///sttttteeeee http://yaann.ecum.com/eec

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Table 2: Clustering performance on seven datasets, '-' means unable rum due to memory limitation.
DAIMC
UEAF
KHIMVC
V"H
IMVC-CHG
SCBGI.
Proposed
ACCVS)
MSRCV
66,0447
55.4744.77
69.622487
58 $416.33
71.11a5.2
63.5121.73
69.9430.70
79,8750.10
83.52=1.00
78.3625.77
78.21:0.00
54.69.10.53
8344g0.144
75.2144.02
68.6110.00
91.3710.00
Wiki
46.07=1.11
47.75=0.05
48.4500.14
48 40.0000330
32.57=0.46
47.2780.02
46.00=0.077
35.8120.13
48.64220.15
Hdigit
56.13 13,4344
65.01=3.34
58.35=3.00
63.91=4.32
78.33+6.35
62 02.36+0.08
68 68.0540.0 10
76. 11+0.00
82.29+0001
YTFI0
80.55
78.3810.00
YTT20
70.74g0.99
76.56+2.38
76.00s0.33
76.98:2.35
MNIST
53,24+0.30
60.58+0.00
11,9500.00
69.42=0001
NMI(S)
MSRCV
58.88-500
21.11
57 57.38-3117
52 89:3.24
64 24
56,25e2122
59 90H1 19
69 31+0.16
72.17=1.20
WebKB
18.29:10.60
23 52:0.77
3.5720.56
1.561-4.4
21.8211.13
21.9260001
16. 16.6000000
16.44.0.00
51.6510.00
32.31=0.83
39.71
40.01=0.15
19.03=0.21
35.91=0.02
33.5540.07
19.3560.11
Hdigi
47.26+282
55.04=1.48
47.58-1.32
57.94-1.00
81.0522188
51.33+0.11
54.1740.05
62. 94+0.00
67.0540.22
YTFIO
73 73.2260.81
81,5840.88 8.80
83.79x0.01
81,81
YTF20
76.25e0.76
80.2440.91
83.20+0.97
MNIST
50.59+0.01
57.62+0.01
55.86+0.02
61.6820.00
Punkyc
$7.37+3.99
61.19-5.17
73.664405
71,24+1.9K
79,87+011
83.52=1.00
WebKB
81.7523.68
82.9120.09
78.12=0.00
78 18.72:1.15
31.3823
83.48:0.14 14
78.12:0.00
79 09.0540.00
91.37220000
Wiki
49.6240.85
31.71:0.04
50 10.
51.7100.11
38.4320.37
5144260.02
50.3910.05
16.77.0.11
$1.8810008
Hdigit
58.89=3.79
66.97=2.50
60.60-1955
66.90=2.77
81.37=4.22
79H0
82.29+0001
YIHIO
75. 15.
83.72+0.99
82 8540.00
84.43+1.21
YTF20
75.40=1.01
80.1241.38
81 $1.2340.21
81.27-181
58.0600001
64.5++0.22
Fscooet 263
MSRCV
54.62:5.72
42.61=182
55.4613.0 1.80
47.45=3.39
60.4554.008
50.0522.19
56.36+ 102
66. 1840.17
70.77=1.23
WebKB
76.31=5.86
78.51=0.73
62.75=2.14
79 40:0000
71 82=0.00
82.28+0.11
69.02:0001
64. 6840000.01
76.7600000
Wiki
35.0520.98
36.0020.100
38.6520.13
17.57.0.18
22 22.2920.200
36.1920.02
34.9200000 09
23.5910.10
37.0510.00
Hdigit
44.48=3.29
52.52+2.50
45.21=1.68
53.62261
75.85=4.62
48.75e0.30
52 02.9600.05
63. 10+0.00
69. 19+0.02
YIF10
64.40=0.04
77.0++1
77 27+0000000
77.82=1.03
YTF20
57.42&1.17
69. 69.55a1 86
72.3810.36
70.9712 35
MNIST
44.1480.20
51.75+0.01
50.56+00.02
$8.8620.00
MSRCV
WebkB
Wiki
DAIMC
ACC
ACC
UEAF
BEIVIC
0.6
06
0.6
missing ratio
missing ratio YTF10
missing ratio
V-H
Hdigit
MNIST
IMVC-OBO
ACC
ACC
ACC
SCBGL
DVBAI
Proposed
missing ratio
missing ratio
missing ratio
Figure 2: The curves for ACC across different datasets as the missing rate varies.
The comparison results between the above variants and our method are shown in Table Both mod- ules in AIMC-CVR are essential, the absence of either one leads to decreased clustering performance as shown in AIMC-CVR-vI and AIMC-CVR-v2. Compared with AIMC-CVR-V3 appropriate con- straints on the sparsity of (P)) can prevent excessively divergent values. AIMC-CVR-Và uses initial anchors learned from single-view data with missing samples, leading to anchor-shif problem, while our method effectively mitigates it and enhances clustering results. Compared AIMC-CV IMC-CVR-v5, our method achieves better performance by reconstructing missing samples with affine combination of anchors, which leads more accurate learning of anchors and anchor graphs.
4.4 Convergence Study
the objective value of our proposed algorithm against the number of iterations on four datasets. The algorithm shows rapid convergence within the first iterations for all datasets. The objective value decreases significantly in the initial iterations and gradually stabilizes, indicating efficient attainment of an optimal or near-optimal solution. This rapid convergence especially notable on the MSRCV and WebKB datasets, where the objective value platcaus before 50 iterations.
Fig. Fig.Billustran Billsttrates

---

The consistent convergence patterns across different datasets validate the robustness and efficiency of our algorithm in handling incomplete multi-view clustering tasks.
Table 3: Ablation studies of AIMC-CVR with different variants.
Methods
MSRCV
WehKB
Wiki
Hdigit
YTFI0
YTF20
MNIST
ACC(%)
AIMC-CVR-VI
67.80=0.92
72.77=0.00
45.38+0. 10
54.01
80.31=1221
62.64+1.75
59.80#0.01
ALMC-CVR-v2
71.95=0.75
72.18=0.00
46.16+0.09
67.61=0.02
78. 78.8630.53
75.01+2.06
68.35=0.01
ATMC-CVR-x3
44.41=1.89
87.47=0.00
28.67+0.28
29.53=0.47
63.47=0.75
65.65+2.08
57.10=0.01
AIMC-CVR-v4
66,23+1.50
82.82+0.00
46.23+0.15
27.66+0.46
69.97=0.44 40
71.84&1.99
55.95+0.35
AIMC-CVR-v5
71.92+0.73
72.15=0.00
46.16+0.10
67.63=0.03
78. 78.5677
75.1242.08
68.35+0.01
AIMC-CVR
83.52+1.00
91.37=0.00
48.64:0.15
82.29=0.01
83.73+1.29
76.98+2.35
69.42=0.01
910
850
2400
20
$00
830
19-
730
Osjechve
15
820
030
9000
17
215
630
1900
18
850
550
1500
Numbber
Danstion Number
(a) MSRCV
(b) WebKB
(c) Wiki
(d) Hdigit
Figure 3: Variation of the objective value with increasing iteration number on four datasets.
(a) MSRCV
(b) WebKB
(c) Wiki
(d) Hdigit
Figure 4: Sensitivity analysis of 3 and on four datasets.
4.5 Parameter Analysis
AIMC-CVR has two main hyperparameters: 8 controls the sparsity of the anchor graph and À controls the weight of the cross-view learning module. We test the performance of AIMC-CVR with different combinations of these hyperparameters, across datasets, but values of /3 and À close to generally perform well. Experimental results shows that the cross-view learning module is crucial for constructing complete anchor graphs, while ensures the graph to be not too sparse. Proper parameter settings lead to good clustering results.
shown in Fig. The impact of these parameters varies
5 Conclusion
In this paper, we introduce an AIMC-CVR method to alleviate the anchor-shift problem in anchor- based incomplete multi-view clustering. Additionally, we explore the blind spots in sample recon- struction with affine combination. Experiments and theoretical analysis validate the effectiveness of the proposed AIMC-CVR method.
10

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Acknowledgment
This work supported by National Science and Technology Innovation 2030 Major Project under Grant No. 2022ZD0209100 103, the National Science Fund for Distinguished Young Scholars of China (No. 62325604), and the National Natural Science Foundation of China (No. 62276271, 62476281 and 62406329).
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A Appendix
A.1 Limitation
The primary limitation of AIMC-CVR lies in its difficulty in handling high-dimensional data. Al- though the designed symmetric cross-view projection mechanism help preserve high-dimensional features, it also introduces additional time complexity of (d2 and space complexity of O(d,dy). direction for future research is the design of a well-structured unified cross-view metric space address high-dimensional data. Moreover, different levels of confidence should be assigned to relationships between different views in cross-view learning. Incorporating the interrelations among views into cross-view learning help improve AIMC-CVR.
A.2 Proof of Theorem
To prove Theorem |
we first present two lemmas along with their corresponding proofs. Lemma 1. the set of anchors A = {a1..... 4m contains distinct points and for any anchor a, a, not a vertex of the convex hull of A. i.e., O₁ E conv(A)) {a:}) where conv(-) denotes the convex hull, then A must be empty set.
Proof. When j = 1, we have a₁ E cono(A) {ai)), then
conv(A(((1)) conw (A)(a)))
2 (A
conv(conul
(18)
commi(A).
Since A\fa1} S A, then
cono(AA((1)) C conv(A).
(19)
Along with Eq. (88
we have
conv (on=(A)(a1)) = cA)
(20)
Assuming that for j < m, we have
conv(A) U {a,)) conv(A).
(21)
Based on as+1 E conv((A) (M()))))))))))))) }), ajt1 not a vertex of conv(A). Therefore, aj+1 is not a vertex of conv(A)) U._1 {a:)). Then we have
aj-1 conv( (A) U (aiHW(at1))
(22)
-com(AA) U!t (ar)).
Therefore,
conv(A) U {a;}) ) = come(cone) (A) U (ait))
conv (A) 1 {ait) (a++i))
2 comv((A) U1+11 (ay+zl)
(23)
conv((A) Up {a:)))
= conv(A).
In summary, we have
conv(A) U {a:}) = conw(A),
conv Ø ) conv(A),
(24)
A = Ø.
This completes the proof.
14

---

Lemma 2. If the set of anchors A = fai...., am I contains distinct anchors and A & Ø, then there exists an anchor a, such that a is a vertex of the convex hull conw(A). other words, & (o.)(A)(a....
Proof. Assume that for any anchor d;, we have d; ccon((A)aa) According to Lemma 1 we have A= 2, which contradicts the given condition. Therefore, the assumption is false. There exists an anchor such that a; is a vertex of the convex hull cAn In other words, & (on This completes the proof.
Based on Lemma and Lemma
2 we provide the proof of Theorem ! as follows:
Proof. Based on Lemma there exists an anchor a, such that a, a vertex of the convex hull comv(AA). Assume au: C comv(A). Since is vertex of como(A), we have & conv(cono(A)) (A))))) Given Xx, S conv(AA, have:
Na, Mai} S como(A)) fast. conv Mai}) cono(cono (A) (a,)),))
(25)
Then
0: conv Ma;t),
(26)
which contradicts the fact that a; is the cluster center of Xaç- Thus, the assumption does not hold, and Xa; I conw(A). There exists Xa; C X' such that
2
m
min
Σ
faa
S, 1= 0.
(27)
This completes the proof.
A.3 Proof of Theorem
Proof. We denote f* and g" as the optimal values of f and in Eq. theorem, have
According to the Pythagorean
Σ
9. a
J. a4
fi 03
Σ f(a:
(28)
In the projection space aff (A) where aff (.) denotes the affine hull, Theorem ! still holds. Therefore, we have
Σ
02
S
ai
> 0.
(29)
1=1
1=1
2
Then
m
m
Σ
g
04
f 04
(30)
i=1
This completes the proof.
A.4 Proof of Doubly Stochastic Matrix S
Lemma 3. Let Z be x mairix where the sum of each column is 1. Define Q as a diagonal matrix where the i-th diagonal element the sum of the elements the i-th row 7. Then the matrix S Z Q 'Z is doubly stochastic, meaning that each and each column of sums to l.
15

---

Proof. Denote Sig to be the j-th element in the i-th row of S, qi to be i-th diagonal element of Q. we can derive that
Sij
ZB
Zy
qi
j=1
j=1
1=1
Z11
Zi
P
1=1
18
(31)
Z
qI
Zui
1=1
1.
12
Similarly, it can be proven that Σ S.j = This completes the proof.
A.5 Convergence Study on More Datasets
In Fig. 5 we further plot the curve of the objective function values against the number of iterations for the proposed algorithm the YTF10, YTF20, and MNIST datasets. It can be observed that the objective values monotonically decrease with the increasing number of iterations on these three datasets, gradually approaching stability, thus confirming the convergence of AIMC-CVR.
122
1.18
115
7.0
134
Iloration Numroe"
Eorazion Number
lloration Number
(a) YTFI0
(b) YTF20
(c) MNIST
Figure 5: Variation of the objective value with increasing iteration number other three datasets.
A.6 Parameter Analysis on More Datasets
In Fig. 6. further plot the variation of clustering performance of the proposed algorithm the YTF10, YTF20, and MNIST datasets with different values of two hyperparameters, B and X. can be observed that AIMC-CVR exhibits less sensitivity to parameters However, on the MNIST dataset, higher values of À lead to better clustering performance.
the YTF10 and YTF20 datasets.
(a) YTF10
(b) YTF220
(c) MNIST
Figure Sensitivity analysis of / and on other three datasets.
16

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A.7 Ablation Study on More Metrics
To further demonstrate the effectiveness our approach, we compared our algorithm with five variants and three other clustering metrics in Table Compared to the other variants, AIMC-CVR exhibits better NMI, Purity, and Fscore on all datasets, validating the effectiveness of our method.
Table 4: Ablation studies of AIMC-CVR with different variants on Other Metrics.
Methods
MSRCV
WebKB
Wiki
Hdigit
YTFI0
YTF20
MNIST
NMI(%)
AIMC-CVR-VI
59.26=1.14
20.65=0.00
35.47+0.13
47.33=0.05
81.06=0.84
68 68.43+0.91
52,40=0.01
AIMC-CVR-v2
61.66+0.83
18.29=0.01
35.70=0.08
56.50=0.02
79.89=0.39
79.9940.8. 86
60.94+0.01
AIMC-CVR-v3
30.12=1.67
46.1620.02
11.8620.19
14.18
64. 43.9620.53
70.6440.1
57.51=0.00
AIMC-CVR-vd
53.33+1.65
39.77=0.00
35.50+0.17
12.60=0.18
67 4420.26
75.9240.62
50.73+0.15
AIMC-CVR-v5
61.57=0.81
18.32=0.00
35.59+0.08
56.51=0.03
79.96=2.33
10.10047.74
60.94=0.01
AIMC-CVR
72.17=1.20
51.65=0.00
39.7820.17
67.05=0.02
81811=0.99
83.20=0.97
61.68=0.00
(urity(S%)
AIMC-CVR-vI
70.03+0.84
79.46+0.00
47.94#0.09
58.56=0.07
83.29=0.94
68.3241.33
61,43#0.01
AIMC-CVR-v2
73.52=0.75
79.12=0.00
48.71+0.11
71.17=0.02
82.49=0.49
80.37+1.51
71.10=0.01
AIMC-CVR-V3
45.78+1.8
89.48=0.00
31.31+0.28
31.06=0.45
68. 68.78=0.66
70.66+1.35
62.07=0.01
ALMC-CVR-v4
66.85=1.43
87 0700000
48 14.
29.06=0.43
73.34=0.42
76.06+14
59.22=0.17
AIMC-CVR-v5
73.49+0.72
19.1200.00
48.81+0.10
11.18+0.03
82.58+0.41
80.53+
71.10#0.01
AIMC-CVR
83,52=1.00
91.37=0.00
518820.08
82.29=0.01
84.43+1.21
81.27#181
71888+0.0
Fscore %+)
AIMC-CVR-vI
55.37=1.01
67.17=0.00
34.6120.09
42.93=0.05
74.79
54.01#1.49
48.46:0.01
AIMC-CVR-V
58.25+0.90
66.74+0.00
36.1640.08
55.37=0.03
13.1040.58
69.53a
57.90+0.01
AIMC-CVR-V3
31.02+1.31
83.99=0.01
17.58+0.12
18.1840.17
57.5440.68
1..25
50.1540.00
AIMC-CVR-vd
51.70=1.60
79.80=0.00
35.43+0.12
17.9420.15
59.96#0.46
63.34+143
46.26:0.08
AIMC-CVR-v5
58.1620.87
66.72=0.00
35.0620.08
55.3920.03
73.27=0.50
69.6741.73
57.90=0.01
AIMC-CVR
70,77+1.22
87.7620.00
37.05+0.08
69.19a0.02
77.82+1.03
70.9742.35
58.8610.00
17

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18

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Justification:
Please refer to Section A.2
A.3 and A.4
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Reviews:
[{"review_id": "3ES2gf44OO", "reviewer": "Reviewer_nWqT", "summary": "This paper proposes a novel anchor-based IMVC method called AIMC-CVR to address the anchor-shift caused by missing data. AIMC-CVR consists of two modules: cross-view anchor learning and affine combination-based reconstruction. The former helps in learning a complete anchor graph, while the latter aims to recover the missing data. Carefully designed experiments validate the effectiveness of AIMC-CVR in clustering tasks.", "questions": "-What do the gray areas in Figure 1(c) represent? There seems to be a deviation between the reconstructed points in Figure 1(d) and the real points in Figure 1(a). Does this deviation affect subsequent results?\n-Why do different datasets show different convergence states in Figure 3? I observed that some converge earlier while others converge later. Can you provide further explanation?", "limitations": "NA", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "-The concept of anchor shift is novel and introduced for the first time in this paper.\n-Experiments demonstrate the effectiveness of the proposed method in clustering performance.", "weaknesses": "-The paper introduces two modules: cross-view anchor learning and affine combination-based reconstruction. However, it lacks a description of the relationship between these two modules. Are both modules necessary in the proposed method?\n-The description of the experimental setup is unclear, particularly the settings in Figure 1. Clear descriptions of the data sources and the anchor generation method are crucial to support the motivation of this paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. The necessity of the proposed modules:**\n\nThanks for the comment. In AIMC-CVR, both modules are essential for alleviating the anchor-shift problem caused by incomplete data. The cross-view anchor learning module mitigates such problem by leveraging available data across views to learn complete anchor graphs and more accurate anchors. Meanwhile, the affine combination-based reconstruction module focuses on reconstructing missing samples based on the anchors, thus preventing the anchors from being affected by the missing data. These modules cooperate towards the same goal: the former utilizes cross-view complementary information, and the latter employs imputation techniques to alleviate anchor-shift. However, in practice, the reconstruction module depends on the initial anchors learned by the cross-view anchor learning module. Ablation experiments show significant performance drops in versions V1 and V2, where each of these modules is removed, underscoring their necessity.\n\n**2. Experimental setup in Fig.1:**\n\nThanks for the comment. When generating data, we initially uniformly select four points on a circle with a radius of 1 centered at the origin in a two-dimensional coordinate system as cluster centers. Subsequently, we randomly generate 40 two-dimensional vectors following the standard normal distribution. Each group of 10 vectors is added to the corresponding cluster center, resulting in four classes with a total of 40 data points. Notably, the two-dimensional vectors generated on the two views are distinct but equally important. In Fig. 1a, the true anchors are obtained by applying k-means on the complete data. Fig. 1b shows the initial anchors derived from k-means on the incomplete data. Fig. 1c presents the anchors based on AIMC-CVR-V5, where the affine constraint is replaced with a convex constraint. Fig. 1d displays the anchors learned using the proposed AIMC-CVR method. We will add the above setting in the final version.\n\n**3. Further description of Fig.1:**\n\nThanks for the comment. In Fig. 1c, the white area represents the convex hull formed by four anchors, indicating that all points within this area can be expressed as a convex combination of the anchors. The gray area represents the blind spots where points cannot be expressed by the convex combination of these anchors. The points in Fig. 1a are the original sample points, whereas the black points in Fig. 1d are reconstructed with our proposed method. Although there are some differences in representation, the reconstructed points in Fig. 1d are evidently closer to the true points compared to those restricted within the convex hull in Fig. 1c. Our objective is not to precisely recover the missing samples, but to mitigate the impact of missing samples on the anchors. Compared to the anchors generated in Fig. 1b, the anchors in Fig. 1d are closer to the true anchors. Thus, while our method can not completely recover the missing samples, it effectively alleviates the anchor-shift problem in incomplete multi-view clustering, achieving state-of-the-art clustering performance.\n\n**4. Difference in convergence states on different datasets:**\n\nThanks for the comment. We speculate that the differences may arise from the following three factors: Firstly, different types of data naturally have inherent differences, leading to varying numbers of iterations required to meet the convergence criteria during optimization. Secondly, when fusing multi-view data, the similarity between views varies across datasets. Data with more similar views converge faster during integration, while data with greater view differences take longer. Thirdly, the alternating optimization algorithm is highly influenced by the initial value settings, with different initial settings resulting in different numbers of iterations.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "naDjOzdeXL", "reviewer": "Reviewer_fq57", "summary": "This paper proposes an anchor-based incomplete multi-view clustering with cross-view reconstruction (AIMC-CVR). To tackle the anchor-shift induced by incomplete multi-view data, AIMC-CVR reconstructs missing samples with learned anchors. The traditional convex combination is replaced with affine combination for more reconstruction accuracy. Two theorems is proposed to demonstrate the advantages of affine combination.", "questions": "Why use v*v projection matrices? In fact, v matrices projecting data from each view to the same dimension should be sufficient to solve the dimensional inconsistency problem. Could the authors provide further explanation?\n\nIn 2021, Yin et al. studied the reconstruction of missing datap[1]. How does the reconstruction method in this paper compare to theirs in terms of advantages? \n[1] Yin J, Sun S. Incomplete multi-view clustering with reconstructed views. IEEE TKDE, 2021.", "limitations": "The authors have acknowledged some limitations of their work in the appendix and suggested potential solutions.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The motivation of this paper is clear, focusing on the anchor shift problem in missing data scenarios, and the corresponding solution is theoretically and experimentally validated.", "weaknesses": "The paper emphasizes the missing rates in the experimental data but lacks a detailed description of how the missing data is constructed.\n\nThe descriptions of the five variants in the ablation study are not clear enough. Understanding these variants is crucial for validating the effectiveness of the proposed algorithm.\n\nThere are some symbol errors in the paper, such as \"KTT conditions\" which should be \"KKT conditions\".", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. Incomplete data construction**\n\nThanks for the comment. For the datasets mentioned in our method, we remove some instances on each view randomly to get their incomplete versions. Specifically, with the principle that each instance is present in at least one view, we generate missing datasets at missing rates in intervals of 0.1 from 0.1 to 0.9. \n\n**2. Descriptions of the five variants**\n\nThanks for the comment. The five variants of our proposed method are constructed as follows: (1) AIMC-CVR-v1 removes the cross-view anchor learning module by setting $\\lambda = 0$. (2) AIMC-CVR-v2 removes the affine combination-based reconstruction module. (3) AIMC-CVR-v3 removes the sparsity regularization term by setting $\\beta = 0$. (4) AIMC-CVR-v4 keeps the initialized $\\mathbf{A}^{(p)}$ fixed and does not update it during subsequent optimization. (5) AIMC-CVR-v5 replaces affine combination with convex combination by adding non-negative constraints to  $\\mathbf{Z}^{(p)}$.\n\n**3. Symbol errors**\n\nSorry for our mistake. We will correct the symbol errors in our final version.\n\n**4. Different projection strategy**\n\nIn fact, we only used $v(v-1)/2$ projection matrices because we project lower-dimensional data to higher-dimensional spaces when one view's data dimension is smaller than another's. Unlike previous methods that use $v$ projection matrices to project data into the same lower-dimensional space, our approach aims to preserve high-dimensional information and reduce information loss. To validate the effectiveness of our method, we replace the projection matrix $\\mathbf{W}^{(pq)}\\in \\mathbb{R}^{d_p \\times d_q}$ between the $p$-th and $q$-th view with $\\mathbf{W}^{(p)}\\in \\mathbb{R}^{k \\times d_p}$ to get the variant AIMC-CVR-V6. The clustering performance of our proposed AIMC-CVR and its variant AIMC-CVR-V6 is shown in Table 1 on the global rebuttal file, which demonstrates the superiority of our projection strategy.\n\n**5. Novelty of imputation strategy**\n\nThe imputation method proposed by Yin et al. is constructed based on an $n \\times n$ full graph, which incurs $O(n^2)$ space complexity and $O(n^2 log(n))$ time complexity. In contrast, our method achieves imputation based on the anchor graph, with both time and space complexity being linear with respect to $n$, offering a significant efficiency advantage, as demonstrated in Section A.6. Additionally, we expand the reconstruction space of the samples into the affine space of the anchors, which provides a larger search space compared to Yin's method.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ogf2iCZGIr", "reviewer": "Reviewer_4LaZ", "summary": "By employing cross-view anchor learning and affine combination-based reconstruction，the authors propose an incomplete multi-view clustering method to alleviate the anchor-shift problem. Besides, the authors theoretically analysis the advantages of affine combination-based reconstruction, which help to explore blind spots in sample reconstruction. Experimental results on several datasets validate the effectiveness of the proposed method.", "questions": "1.Can the authors provide a clearer explanation of Fig. 1, especially in its caption?\n\n2.Why do the last three datasets in Table 4 have identical dimensions? Is this an oversight by the authors?", "limitations": "The authors have acknowledged some limitations of their work but have not discussed future work directions.", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.The affine combination-based reconstruction module is a simple yet effective novel approach.\n\n2.The authors provide thorough theoretical and experimental validation of the proposed module's effectiveness.", "weaknesses": "1.The authors fail to provide precise definitions for some symbols used in the paper, such as n_p.\n\n2.Some inconsistencies are present in the experimental figures. For example, the x-axes of the first three subfigures in Fig. 3 are spaced at intervals of 20, whereas Fig. 3d uses intervals of 50. Additionally, Fig. 4c and 4d need to be rotated for better presentation.\n\n3.The writing quality needs improvement. The conclusion section is too brief and does not adequately summarize the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. Definition of symbols**\n\nThanks for the comment. $n_p$ represents the number of samples in the $p$-th view. In the global rebuttal file, we have provided a notation table that explains the main symbols used in the paper. Please refer to the Table 2 in the global rebuttal file.\n\n**2. Inconsistencies of figures**\n\nThanks for the comment. We have made corrections to the inconsistencies in Fig. 3 and the obscured parts in Fig. 4. Please refer to Figure 1 and Figure 2 in the global rebuttal file.\n\n**3. Rewrite conclusion**\n\nThanks for the comment. We have rewritten the conclusion as follows to better summarize: In this paper, we introduce an AIMC-CVR method to alleviate the anchor-shift problem in anchor-based incomplete multi-view clustering. Specifically, we introduce a cross-view learning module and a reconstruction module to fix the influence of incomplete data. Additionally, we explore the blind spots in sample reconstruction with affine combination. Experiments and theoretical analysis validate the effectiveness of the proposed AIMC-CVR method. We will replace the conclusion in our final version.\n\n\n**4. Clearer explanation of Fig. 1**\n\nWe further explain Fig.1 as follows: (a)Anchors learned in complete data: The true anchors are generated with k-means performed in the complete data. (b)Anchors initialized in incomplete data: The initial anchors are generated with k-means performed in the incomplete data, which is shift the origin point compared with the true anchors. (c)Data reconstructed with convex combination: The convex combination-based reconstructed data is restricted in the convex hull of anchors. (d)Data reconstructed with affine combination: The affine combination-based reconstructed data is breaking through the limitations of convex hull, which can represent any position in affine space.\n\n**5. Explanation of the dataset**\n\nAll three datasets consist of image data. We used four operators to extract features for each image, namely LBP (Local Binary Pattern), HOG (Histogram of Oriented Gradient), Gist, and Gabor. These operators provide four different views, each with the same dimensionality on different datasets.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5YCBSFClt7", "reviewer": "Reviewer_mjvb", "summary": "The paper proposes a cross-view reconstruction-based multi-view clustering algorithm to address the issue of anchor shift in missing data scenarios. Specifically, The method guides anchor learning by reconstructing the missing parts of the data. It uses an affine combination-based reconstruction strategy, rather than a convex combination, to avoid the negative impact of blind reconstruction areas. Theoretical analysis demonstrates the superiority of affine combination constraints over convex combination for this problem. Comparative experiments on multiple datasets with varying missing rates show the effective clustering capability of the proposed algorithm. The key innovations are the cross-view reconstruction approach, the use of affine combination constraints, and the theoretical justification for this choice. The experimental results validate the effectiveness of the proposed method for handling missing data in multi-view clustering tasks.", "questions": "Regarding Algorithm 1, the paper does not discuss the possibility of using spectral clustering instead of k-means in the final step. It's unclear how this substitution would affect the performance of the proposed algorithm.\n\nAdditionally, the paper does not clearly explain the relationship between the cross-view learning approach used in this work and the broader multi-view learning field. The connection between these concepts is not well-articulated.", "limitations": "See Weaknesses.", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "The anchor shift problem studied in this paper is a challenging issue that has been rarely addressed in the multi-view clustering field.\nThe proposed method effectively addresses the observed problems through the collaboration of two modules, yielding good experimental results.\nThe theoretical analysis provided in the paper intuitively demonstrates the superiority of the affine combination-based framework over previous convex combination approaches.", "weaknesses": "However, the paper contains some inaccurate descriptions, such as stating that matrix decomposition has n^2 complexity in the introduction, which is not entirely accurate.\nThe paper includes many symbols, but the lack of a symbol table reduces readability.\nThere is a contradiction regarding the final step of Algorithm 1, where it states k-means is directly applied to Z, while Section 3.2 describes applying k-means to the left singular vector matrix of Z.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. Inaccurate statements**\n\nThanks for the comment. Due to the additional introduction of regularization constraints on the entire graph, such as manifold constraints, most existing matrix decomposition methods exhibit an $O(n^2)$ complexity [1][2]. We will revise this statement in the final version.\n\n**2. Notation table**\n\nThanks for the comment. We have added a symbol table to explain the key symbols used in our paper in the global rebuttal file. This symbol table will also be added in the final version to enhance readability.\n\n**3. Contradiction in Algorithm 1**\n\nSorry for our mistake. In our method, the final clustering results are produced by applying $k$-means to the left singular vector matrix of $\\textbf{Z}$. We will correct it in our final version.\n\n**4. Final process of clustering**\n\nThanks for the comment. The anchor graph $\\textbf{Z}$ obtained from our method is $n \\times m$, which cannot be directly used for spectral clustering. According to [3], applying $k$-means to the left singular vectors of $\\textbf{Z}$ is equivalent to performing spectral clustering on the doubly stochastic matrix constructed from $\\textbf{Z}$. Our method differs from [3] in the constraints applied to $\\textbf{Z}$. Therefore, in Section A.4, we prove that the anchor graph $\\textbf{Z}$ obtained from our method can also be recovered as a doubly stochastic matrix.\n\n**5. Relationship of cross-view learning and multi-view learning**\n\nThanks for the comment. Cross-view learning is a subset of multi-view learning. Unlike most previous multi-view learning methods, which build relationships between samples within the single view, cross-view learning aims to explore the relationships between samples across different views. In multi-view alignment methods, cross-view learning is a key technique for aligning samples between different views. In our approach, based on the assumption that samples will always appear in at least one view under missing data scenarios, introducing cross-view learning helps mitigate the impact of missing samples within a single view on anchor point learning.\n\n**References**\n\n[1] Rai N, Negi S, Chaudhury S, et al. Partial multi-view clustering using graph regularized NMF. 2016 23rd International Conference on Pattern Recognition (ICPR). IEEE, 2016: 2192-2197.\n\n[2] Zhao H, Ding Z, Fu Y. Multi-view clustering via deep matrix factorization. Proceedings of the AAAI conference on artificial intelligence. 2017, 31(1).\n\n[3] Wang S, Liu X, Liu L, et al. Highly-efficient incomplete large-scale multi-view clustering with consensus bipartite graph. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2022: 9776-9785.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank the SAC, AC, and PCs for their efforts and constructive comments, which are helpful in further improving the quality of our manuscript. We respond to your questions carefully one by one carefully, and we hope our responses can address your concerns.

Note that there are two tables and two figure in the attached PDF, corresponding to RQ4 for Reviewer fq57, RQ1 and RQ2 for Reviewer 4LaZ, and RQ2 for Reviewer mjvb.

Meta-review: This paper proposes a novel anchor-based method for incomplete multi-view clustering. Specifically, the authors propose a cross-view reconstruction strategy to recover missing samples. The reviewers believe that the proposed method is simple yet efficient, and empirical studies demonstrate its superiority.  The reviewers have reached an agreement to accept this paper. Hence, it should be a clear acceptance. The authors are required to follow the reviewers' suggestions to polish the main paper.
Decision: Accept (poster)