Title: Alleviate Anchor-Shift: Explore Blind Spots with Cross-View Reconstruction for Incomplete Multi-View Clustering
Venue: NeurIPS 2024
Accepted: yes
Keywords: Incomplete Multi-View Clustering

Anchor-Based Multi-View Clustering
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-View Reconstruction for Incomplete Multi-View Clustering

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.

1 Introduction

In multi-view learning, data collected from different sensors or media often suffer from missing values [1, 2, 3, 4]. 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, 7]. To address this issue, incomplete multi-view learning methods have been developed to leverage the available data from all views to perform downstream tasks [8, 9, 10]. Among these, incomplete multi-view clustering (IMC) relies on view consistency and complementarity, effectively enabling the partitioning of data with missing values [11, 12, 13].

Existing IMC methods can be categorized into three types based on their approach: similarity-based, imputation-based, and matrix decomposition-based methods. Similarity-based methods recover a relationship matrix among all samples using available data in each view [14, 15, 16, 17, 18]. Imputation-based methods first fill in the missing parts, transforming the problem into a complete multi-view clustering problem [19, 20, 21]. Matrix decomposition-based methods map samples from all views into a common latent space to construct a unified representation for clustering [22, 23, 24, 24, 25]. However, these methods all require constructing an n × n similarity matrix,

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resulting in an O ( n 2 ) space complexity, which limits their application to large-scale scenarios [26, 27, 28].

In contrast, anchor-based IMC methods reduce space complexity to O ( 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. 1(a)(b), anchor learning can be misled by missing data, resulting in a 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 a novel Anchor-based Incomplete Multi-view Clustering with CrossView Reconstruction strategy, termed AIMC-CVR. AIMC-CVR encompasses two key modules designed to resolve the anchor-shift problem in anchor-based IMC. The first module, the cross-view anchor learning module, is dedicated to mitigating the anchor-shift problem by learning a complete anchor graph. Specifically, we designed a 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-based reconstruction module, iteratively updates the anchors with available and reconstructed data. Existing anchor-based IMC methods often use convex constraints to build relationships between anchors and samples [32, 33, 34, 35], leading to blind spots in sample reconstruction. As shown in Fig. 1(c), 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-based reconstruction strategy. This strategy broadens the scope of sample reconstruction, as depicted in Fig. 1(d), allowing for a more comprehensive and accurate representation of missing data.

Our main contributions are as follows:

By employing cross-view anchor learning and affine combination-based reconstruction, we effectively alleviate the anchor-shift 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 combinations further explore blind spots in sample reconstruction, as demonstrated by theoretical analysis and experimental results.

Comparative experiments with state-of-the-art IMC and anchor-based IMC methods validate the effectiveness and superiority of AIMC-CVR.

2 Related Work

Cross-View Learning. Cross-view learning is a specialized form of multi-view learning that leverages interactions between paired views to learn cross-view representations, enabling the exploration of 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 et 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 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, 42, 21]. Typically, existing methods perform sample-level imputation by leveraging multi-view information to learn complete similarity relationships between samples. For example, Yin et al. reconstruct missing data based on the decomposition matrix and sample similarity relationships [20]. Liu et al. integrate data imputation and clustering within a unified optimization framework [19]. However, these sample-level imputation methods typically rely on similarity matrices of size O ( n 2 ) , 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-based reconstruction module. We provide a theoretical analysis that demonstrate the merits of our proposed affine combination-based reconstruction strategy. Finally, we present the overall objective function of the algorithm and propose a four-step alternating iterative algorithm to solve the corresponding optimization problem.

3.1 Cross-View Anchor Learning Module

Given multi-view data { X ( p ) ∈ R d p × n } v p =1 , where d p and n denotes the dimension of the data and the total number of samples, respectively.

In the context of incomplete data scenarios, the data matrix X ( p ) for the p -th view can be partitioned into two distinct subsets, i.e. , [ X ( p ) o , X ( p ) m ] , where X ( p ) o ∈ R d p × n p representing the existing portion of the data, and X ( p ) m ∈ R d p × ( n -n p ) is the missing portion. The observed part is obtained by applying an index matrix G ( p ) ∈ { 0 , 1 } n × n p to the complete data matrix, such that X ( p ) o = X ( p ) G ( p ) . The index matrix G ( 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 X ( p ) o . Based on the samples present in each view, we can learn the anchors and their corresponding anchor graphs as follows:

The anchor graph Z ( p ) is incomplete, representing only the relationship between existing samples and anchors in the p -th view. While a complete anchor graph can 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 n p 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 a cross-view anchor learning module. This module constructs a 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 Z ( p ) in the p -th view, we update it with the following objective:

where W ( pq ) is the projection matrix, which projects the dimension-reduced data into a higherdimensional 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 d p > d q , W ( pq ) is the projection matrix that needs to be optimized; otherwise, W ( pq ) equals the identity matrix, as shown below:

By sequentially learning the similarity between the n q points present in the q -th view (where q = 1 , . . . , v ) and the m anchors in the p -th view, we can obtain an anchor graph with a size of m × n . Furthermore, the complete anchor graph can guide the learning of anchors, thereby implicitly alleviating the anchor-shift problem.

3.2 Affine Combination-based Reconstruction Module

The cross-view anchor learning module fills the missing columns of 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 cross-view representations is overly dependent on 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, we propose to 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 on the similarity among all samples, require an additional quadratic space complexity, making them impractical for large-scale problems. Thus, we propose a fast reconstruction module, as follows:

where X ( p ) o ∈ R d p × n p represents the existing samples in the view, and X ( p ) m ∈ R d p × n -n p represents the missing samples to be reconstructed. X ( p ) m is constructed from the anchors and their corresponding anchor graphs. The reconstructed X ( p ) m 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 ) m enables the learned anchors in the next iteration to be closer to the true global anchors, whereas inaccurate reconstruction exacerbates the anchor-shift problem. However, the reconstructed missing samples are constrained within the convex hull of the anchor set in Eq. (4). According to Theorem 1, there always exist samples that cannot be reconstructed by the convex combination of anchors.

Theorem 1. Suppose A = { a 1 , . . . , a m } is an anchor set composed of cluster centers from the dataset X = { x 1 , . . . , x n } . Then, there always exist a sample c belonging to X that lies outside the convex hull of the anchor set A . Mathematically,

When most missing samples are outside the convex hull of the anchors, the next iteration of anchor learning erroneously shifts inward, worsening anchor-shift. Therefore, we propose an affine-

combination based reconstruction strategy, which relaxes convex constraints on anchor graph rows to affine ones.

Theorem 2. For a sample c lying outside the convex hull of the anchor set A , the representation constructed by the affine combination of set A is always closer to c than that constructed by the convex combination of set A . Mathematically,

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 affine-combination based reconstruction module, the objective of AIMC-CVR is as follows:

where Φ = { X ( p ) m , W ( pq ) , A ( p ) , Z ( p ) } . The hyperparameter β helps to adjust the sparsity of the anchor graph, and λ is a hyperparameter balancing the influence of the two modules. Finally, we concatenate the anchor graph from each view to obtain the common one, Z = [ Z (1) ; . . . ; Z ( v ) ] , 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 a 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. (8), we propose a four step alternating iterative algorithm. When optimizing a 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 ( p ) m . Fixing W ( pq ) , A ( p ) , Z ( p ) and removing terms unrelated to X ( p ) m , we have the following optimization problem:

where E ( p ) ∈ { 0 , 1 } n × n p is the index matrix, E ( p ) ij = 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 X ( p ) m and setting it to zero, we obtain the closed-form solution for X ( p ) m as follows:

The solution for X ( p ) m shows that it incorporates the anchors and anchor graphs from the previous iteration, contributing to the update of anchors in the next iteration.

Step 2: Update W ( pq ) . Fixing X ( p ) m , A ( p ) , Z ( p ) and removing terms unrelated to W ( pq ) , we have the following optimization problem when d p < d q :

By further simplification, Eq. (11) can transform into the following form:

where B ( pq ) = X ( p ) o ( A ( q ) Z ( q ) G ( p ) ) ⊤ + A ( p ) Z ( p ) G ( q ) X ( p ) o ⊤ . According to reference [43], the optimal W ( pq ) is derived from the product of the left and right singular vectors of B ( pq ) .

Step 3: Update A ( p ) . Fixing X ( p ) m , W ( pq ) , Z ( p ) and removing terms unrelated to A ( p ) , we have the following optimization problem:

where C ( p ) = Z ( p ) Z ( p ) ⊤ + λ ∑ v q =1 Z ( p ) G ( q ) G ( q ) ⊤ Z ( p ) ⊤ , D ( p ) = X ( p ) Z ( p ) ⊤ + λ ∑ v q =1 W ( qp ) ⊤ W ( pq ) X ( q ) o G ( q ) ⊤ Z ( p ) ⊤ . By taking the derivative of Eq. (13) with respect to A ( p ) and setting it to zero, we obtain the closed-form solution for A ( p ) as follows:

Step4: Update Z ( p ) . Since each column of Z ( p ) is independent of others, we optimize the i -th column z ( p ) i of Z ( p ) while fixing X ( p ) m , W ( pq ) , A ( p ) as follows:

where H = ( λ ∑ v q =1 σ ( q ) i + 1) A ( p ) ⊤ A ( p ) + β I , t i = λ ∑ v q =1 σ ( q ) i X ( q ) : ,i ⊤ W ( pq ) ⊤ W ( qp ) A ( p ) + X ( p ) : ,i ⊤ A ( p ) . When i -th sample exists in the p -th view σ ( q ) i = 1 , else σ ( q ) i = 0 . We employ the Lagrange multiplier method to tackle the above problem. Firstly, the Lagrangian function for Eq. (15) is as follows:

where α i is the Lagrangian multiplier. The corresponding KTT conditions is

By substituting the first term into the second, we can get α i = 2 ( H -1 t i ) ⊤ 1 -1 1 ⊤ H -1 1 .Then we have z ( p ) i = H -1 ( t i -α 2 1 ) . The entire optimization procedure for AIMC-CVR is summarized in Algorithm 1.

Algorithm 1 The proposed AIMC-CVR

3.4 Complexity Analysis

The computational complexity of AIMC-CVR primarily consists of solving four variables. When updating X ( p ) m , the complexity of matrix multiplication is O ( nmd p ) . For updating W ( pq ) , the complexity of matrix multiplication is O ( nmd p + nd p 2 + d p 3 ) , and the SVD decomposition complexity is O ( d p 3 ) . When optimizing A ( p ) , the complexity of matrix multiplication is O ( nmd p + m 2 d p + nd p d q ) , and the complexity of inversion is O ( m 3 ) . When optimizing Z ( p ) , the complexity of matrix multiplication is O ( m 2 d p + nmd p + nd p d q ) . Therefore, in each iteration, AIMCCVR consumes a time complexity of O ( n ∑ v p =1 ( md p + d p 2 + d p ∑ v q =1 d q ) + ∑ v p =1 ( m 2 d p + d 3 p )) , which is linear with respect to n .

The space complexity of AIMC-CVR is O ( n ( m + ∑ v p =1 d p ) + m ∑ v p =1 d p + ∑ v p =1 ∑ v q =1 d p d q ) , 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 of 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 1, 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 AIMCCVR 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 a local minimum.

4 Experiments

4.1 Experimental Setup

Datasets description. Seven widely used multi-view datasets are employed to evaluate the performance of AIMC-CVR, including: MSRCV [44] is composed of images from seven categories. WebKB 2 contains text and citations collected from the website. Wiki [45] is a dual-view dataset with text-image pairs. Hdigit 3 is a dataset composed of handwritten digit images. YTF10 and YTF20

are two subsets extracted from the YouTubeFace 4 dataset. MNIST 5 is a subset extracted from a larger dataset supplied by NIST. Details of these datasets are list in the Table 1. Following [26], we 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 at least one view.

Compared methods. We compared our proposed method with the following eight state-of-the-art incomplete multi-view clustering algorithms: DAIMC[22], UEAF[23], EEIMVC[19], FLSD[24], V 3 H[15], IMVC-CBG[26], SCBGL[29], DVSAI[31]. The first five comparison methods are traditional similarity-based approaches, while the latter three are large-scale anchor-based 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, 2 k, 3 k ] , and the parameter λ and β are both searched in [0 . 001 , 0 . 01 , 0 . 1 , 1 , 10 , 100] . To evaluate the clustering performance, we employ accuracy (ACC), normalized mutual information (NMI), Purity, and Fscore for comparison. Each compared algorithm was tested across datasets with different missing rates, and the results for each missing rate were averaged to obtain the clustering results. Additionally, we 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 a desktop computer equipped with an Intel Core i9-10900X CPU, 64GB of RAM.

4.2 Clustering Performance Comparison

We compare AIMC-CVR with eight state-of-the-art algorithms across seven datasets, as shown in Table 2. Our algorithm demonstrates superior or competitive clustering performance across all datasets, highlighting its effectiveness. On smaller datasets such as MSRCV, WebKB, Wiki, and Hdigit, our 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 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-based methods (IMVC-CBG, SCBGL, DVSAI), we achieved a 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-v1 removes the cross-view anchor learning module by setting λ = 0 . (2) AIMC-CVR-v2 removes the affine combination-based reconstruction module. (3) AIMC-CVR-v3 removes the sparsity regularization term by setting β = 0 . (4) AIMC-CVR-v4 keeps the initialized 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 Z ( p ) .

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The comparison results between the above variants and our method are shown in Table 3. Both modules in AIMC-CVR are essential, the absence of either one leads to decreased clustering performance as shown in AIMC-CVR-v1 and AIMC-CVR-v2. Compared with AIMC-CVR-v3, appropriate constraints on the sparsity of Z ( p ) can prevent excessively divergent values. AIMC-CVR-v4 uses initial anchors learned from single-view data with missing samples, leading to anchor-shift problem, while our method effectively mitigates it and enhances clustering results. Compared to AIMC-CVR-v5, our method achieves better performance by reconstructing missing samples with affine combination of anchors, which leads to more accurate learning of anchors and anchor graphs.

4.4 Convergence Study

Fig. 3 illustrates the objective value of our proposed algorithm against the number of iterations on four datasets. The algorithm shows rapid convergence within the first 50 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 is especially notable on the MSRCV and WebKB datasets, where the objective value plateaus before 50 iterations.

The consistent convergence patterns across different datasets validate the robustness and efficiency of our algorithm in handling incomplete multi-view clustering tasks.

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4.5 Parameter Analysis

AIMC-CVR has two main hyperparameters: β controls the sparsity of the anchor graph and λ controls the weight of the cross-view learning module. We test the performance of AIMC-CVR with different combinations of these hyperparameters, as shown in Fig. 4. The impact of these parameters varies across datasets, but values of β and λ close to 1 generally perform well. Experimental results shows that the cross-view learning module is crucial for constructing complete anchor graphs, while β ensures the graph to be not too sparse. Proper parameter settings lead to good clustering results.

5 Conclusion

In this paper, we introduce an AIMC-CVR method to alleviate the anchor-shift problem in anchorbased incomplete multi-view clustering. 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.

Acknowledgment

This work is supported by National Science and Technology Innovation 2030 Major Project under Grant No. 2022ZD0209103, 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. Although the designed symmetric cross-view projection mechanism help to preserve high-dimensional features, it also introduces additional time complexity of O ( d p 3 ) and space complexity of O ( d p d q ) . A direction for future research is the design of a well-structured unified cross-view metric space to 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 can help to improve AIMC-CVR.

A.2 Proof of Theorem 1

To prove Theorem 1, we first present two lemmas along with their corresponding proofs.

Lemma 1. If the set of anchors A = { a 1 , . . . , a m } contains distinct points and for any anchor a i , a i is not a vertex of the convex hull of A , i.e., a i ∈ conv ( A\{ a i } ) where conv ( · ) denotes the convex hull, then A must be an empty set.

Proof. When j = 1 , we have a 1 ∈ conv ( A\{ a 1 } ) , then

Since A\{ a 1 } ⊆ A , then

Along with Eq. (18), we have

Therefore,

In summary, we have

This completes the proof.

Assuming that for 1 ≤ j < m , we have

Based on a j +1 ∈ conv ( A\{ a j +1 } ) , a j +1 is not a vertex of conv ( A ) . Therefore, a j +1 is not a vertex of conv ( A\ ∪ j i =1 { a i } ) . Then we have

Lemma 2. If the set of anchors A = { a 1 , . . . , a m } contains distinct anchors and A ̸ = ∅ , then there exists an anchor a i such that a i is a vertex of the convex hull conv ( A ) . In other words, a i / ∈ conv ( A\{ a i } ) .

Proof. Assume that for any anchor a i , we have a i ∈ conv ( A\{ a i } ) . According to Lemma 1, we have A = ∅ , which contradicts the given condition. Therefore, the assumption is false. There exists an anchor a i such that a i is a vertex of the convex hull conv ( A ) . In other words, a i / ∈ conv ( A\{ a i } ) . This completes the proof.

Based on Lemma 1 and Lemma 2, we provide the proof of Theorem 1 as follows:

Proof. Based on Lemma 2, there exists an anchor a i such that a i is a vertex of the convex hull conv ( A ) . Assume X a i ⊆ conv ( A ) . Since a i is a vertex of conv ( A ) , we have a i / ∈ conv ( conv ( A\{ a i } )) . Given X a i ⊆ conv ( A ) , we have:

which contradicts the fact that a i is the cluster center of X a i . Thus, the assumption does not hold, and X a i ̸⊆ conv ( A ) . There exists c ∈ X a i ⊆ X such that

This completes the proof.

A.3 Proof of Theorem 2

Proof. We denote f ∗ and g ∗ as the optimal values of f and g in Eq. (7). According to the Pythagorean theorem, we have

In the projection space aff ( A ) where aff ( · ) denotes the affine hull, Theorem 1 still holds. Therefore, we have

This completes the proof.

A.4 Proof of Doubly Stochastic Matrix S

Lemma 3. Let Z be an m × n matrix where the sum of each column is 1. Define Q as a diagonal matrix where the i -th diagonal element is the sum of the elements in the i -th row of Z. Then the matrix S = Z ⊤ Q -1 Z is doubly stochastic, meaning that each row and each column of sums to 1.

Then

Then

Proof. Denote S ij to be the j -th element in the i -th row of S , q l to be i -th diagonal element of Q , we can derive that

Similarly, it can be proven that n ∑ i =1 S ij = 1 . 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 on 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.

0

20

40

60

80

100

Iteration Number

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

1.21

Objective Value

10 4

(a) YTF10

0

20

40

60

80

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Iteration Number

7.6

7.8

8

8.2

8.4

Objective Value

10 6

(b) YTF20

0

20

40

60

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Iteration Number

1.7

1.8

1.9

2

2.1

2.2

Objective Value

10 4

(c) MNIST

A.6 Parameter Analysis on More Datasets

In Fig. 6, we further plot the variation of clustering performance of the proposed algorithm on the YTF10, YTF20, and MNIST datasets with different values of two hyperparameters, β and λ . It can be observed that AIMC-CVR exhibits less sensitivity to parameters on the YTF10 and YTF20 datasets. However, on the MNIST dataset, higher values of λ lead to better clustering performance.

(a) YTF10

(b) YTF20

(c) MNIST

100

100

604

ACC (%)

ACC (%)

ACC (%)

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D

0

C-

-

C-

-2

1g(3)0

-1

1

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Ig(入)

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Ig(3)0

-1

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2

32

-1

Ig(入)

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1

-1

0

1

2

~7

Ig(入)

0

1

A.7 Ablation Study on More Metrics

To further demonstrate the effectiveness of our approach, we compared our algorithm with five variants and three other clustering metrics in Table 4. Compared to the other variants, AIMC-CVR exhibits better NMI, Purity, and Fscore on all datasets, validating the effectiveness of our method.

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\nHowever, 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.\n\nQUESTIONS:\nRegarding 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.\n\nLIMITATIONS:\nSee Weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-06"}

comments: 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.

STRENGTHS:
1.The affine combination-based reconstruction module is a simple yet effective novel approach.

2.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.

2.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.

3.The writing quality needs improvement. The conclusion section is too brief and does not adequately summarize the paper.

QUESTIONS:
1.Can the authors provide a clearer explanation of Fig. 1, especially in its caption?

2.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.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-09

comments: 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.

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.

The 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.

There are some symbol errors in the paper, such as "KTT conditions" which should be "KKT conditions".

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?

In 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? 
[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.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12

comments: 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.

STRENGTHS:
-The concept of anchor shift is novel and introduced for the first time in this paper.
-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?
-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.

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?
-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
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12