Fast and Robust Sparsity-Aware Block Diagonal Representation

• Published in: IEEE transactions on signal processing Vol. 72; pp. 305 - 320
• Main Authors: Tastan, Aylin, Muma, Michael, Zoubir, Abdelhak M.
• Format: Journal Article
• Language: English
• Published: New York IEEE 2024; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Affinity; affinity matrix; Block diagonal representation; Cluster analysis; Clustering; Data analysis; Data models; eigenvalues; Eigenvalues and eigenfunctions; Enumeration; Indexes; Laplace equations; Linear functions; Mathematical analysis; Matrices (mathematics); Outliers (statistics); Representations; Robustness; Robustness (mathematics); similarity matrix; Sparse matrices; Sparsity; subspace clustering; Symmetric matrices
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2023.3343565

The block diagonal structure of an affinity matrix is a commonly desired property in cluster analysis because it represents clusters of feature vectors by non-zero coefficients that are concentrated in blocks. However, recovering a block diagonal affinity matrix is challenging in real-world applications, in which the data may be subject to outliers and heavy-tailed noise that obscure the hidden cluster structure. To address this issue, we first analyze the effect of different fundamental outlier types in graph-based cluster analysis. A key idea that simplifies the analysis is to introduce a vector that represents a block diagonal matrix as a piece-wise linear function of the similarity coefficients that form the affinity matrix. We reformulate the problem as a robust piece-wise linear fitting problem and propose a Fast and Robust Sparsity-Aware Block Diagonal Representation (FRS-BDR) method, which jointly estimates cluster memberships and the number of blocks. Comprehensive experiments on a variety of real-world applications demonstrate the effectiveness of FRS-BDR in terms of clustering accuracy, robustness against corrupted features, computation time and cluster enumeration performance.