Subspace Clustering by Block Diagonal Representation

TL;DR

This paper introduces a novel block diagonal matrix regularizer for subspace clustering, providing a unified theoretical framework and an effective algorithm that directly enforces block diagonal structure, improving clustering accuracy.

Contribution

It proposes the first direct block diagonal regularizer for subspace clustering and offers a unified theoretical guarantee for the block diagonal property.

Findings

The BDR method effectively clusters data from multiple subspaces.

The proposed solver converges reliably in experiments.

BDR outperforms existing methods on real datasets.

Abstract

This paper studies the subspace clustering problem. Given some data points approximately drawn from a union of subspaces, the goal is to group these data points into their underlying subspaces. Many subspace clustering methods have been proposed and among which sparse subspace clustering and low-rank representation are two representative ones. Despite the different motivations, we observe that many existing methods own the common block diagonal property, which possibly leads to correct clustering, yet with their proofs given case by case. In this work, we consider a general formulation and provide a unified theoretical guarantee of the block diagonal property. The block diagonal property of many existing methods falls into our special case. Second, we observe that many existing methods approximate the block diagonal representation matrix by using different structure priors, e.g., sparsity and low-rankness, which are indirect. We propose the first block diagonal matrix induced regularizer for directly pursuing the block diagonal matrix. With this regularizer, we solve the subspace clustering problem by Block Diagonal Representation (BDR), which uses the block diagonal structure prior. The BDR model is nonconvex and we propose an alternating minimization solver and prove its convergence. Experiments on real datasets demonstrate the effectiveness of BDR.