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Interpretable Multi-View Clustering Based on Anchor Graph Tensor Factorization

Rui Wang, Jing Li, Quanxue Gao, Cheng Deng

TL;DR

Non-negative tensor factorization is used to decompose an anchor graph tensor that combines anchor graphs from multiple views to consider inter-view information comprehensively and enhances the interpretability of the factorization.

Abstract

The clustering method based on the anchor graph has gained significant attention due to its exceptional clustering performance and ability to process large-scale data. One common approach is to learn bipartite graphs with K-connected components, helping avoid the need for post-processing. However, this method has strict parameter requirements and may not always get K-connected components. To address this issue, an alternative approach is to directly obtain the cluster label matrix by performing non-negative matrix factorization (NMF) on the anchor graph. Nevertheless, existing multi-view clustering methods based on anchor graph factorization lack adequate cluster interpretability for the decomposed matrix and often overlook the inter-view information. We address this limitation by using non-negative tensor factorization to decompose an anchor graph tensor that combines anchor graphs from multiple views. This approach allows us to consider inter-view information comprehensively. The decomposed tensors, namely the sample indicator tensor and the anchor indicator tensor, enhance the interpretability of the factorization. Extensive experiments validate the effectiveness of this method.

Interpretable Multi-View Clustering Based on Anchor Graph Tensor Factorization

TL;DR

Non-negative tensor factorization is used to decompose an anchor graph tensor that combines anchor graphs from multiple views to consider inter-view information comprehensively and enhances the interpretability of the factorization.

Abstract

The clustering method based on the anchor graph has gained significant attention due to its exceptional clustering performance and ability to process large-scale data. One common approach is to learn bipartite graphs with K-connected components, helping avoid the need for post-processing. However, this method has strict parameter requirements and may not always get K-connected components. To address this issue, an alternative approach is to directly obtain the cluster label matrix by performing non-negative matrix factorization (NMF) on the anchor graph. Nevertheless, existing multi-view clustering methods based on anchor graph factorization lack adequate cluster interpretability for the decomposed matrix and often overlook the inter-view information. We address this limitation by using non-negative tensor factorization to decompose an anchor graph tensor that combines anchor graphs from multiple views. This approach allows us to consider inter-view information comprehensively. The decomposed tensors, namely the sample indicator tensor and the anchor indicator tensor, enhance the interpretability of the factorization. Extensive experiments validate the effectiveness of this method.
Paper Structure (21 sections, 4 theorems, 50 equations, 9 figures, 5 tables, 1 algorithm)

This paper contains 21 sections, 4 theorems, 50 equations, 9 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. MVC Based on Anchor Graph
  4. MVC Based on NMF
  5. Notations
  6. Methodology
  7. Motivation and Objective
  8. Optimization
  9. Convergence Analysis
  10. Experiments
  11. Clustering Performance
  12. Parametric Analysis
  13. Ablation Experiments
  14. Conclusion
  15. Appendix: Proof of Theorem \ref{['thm1']}
  16. ...and 6 more sections

Key Result

Theorem 1

Given two matrices $\mathbf{G}$ and $\mathbf{P}$, where $\mathbf{G} (\mathbf{G})^{\mathrm{T}}=\mathbf{I}$ and $\mathbf{P}$ has the singular value decomposition $\mathbf{P}=\mathbf{\Lambda} \mathbf{S}(\mathbf{V})^{\mathrm{T}}$, the optimal solution of is $\mathbf{G}^\ast=\mathbf{V}[\mathbf{I},\mathbf{0}](\mathbf{\Lambda})^{\mathrm{T}}$.

Figures (9)

  • Figure 1: Process flow of our model
  • Figure 2: Interpretation of the tensor Schatten $p$-norm
  • Figure 3: Clustering performance with different anchor rate on MSRC, HandWritten4 and Mnist4.
  • Figure 4: Running time (sec.) with different number of anchors on MSRC, HandWritten4 and Mnist4.
  • Figure 5: The influence of $p$ on clustering performance on MSRC, HandWritten4 and Mnist4.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Definition 1: t-product kilmer2011factorization
  • Definition 2
  • Remark 1: Explanation of the tensor Schatten $p$-norm
  • Theorem 1
  • Proof 1
  • Lemma 1
  • Theorem 2
  • Lemma 2: Proposition 6.2 of lewis2005nonsmooth