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 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 first consider a general formulation and provide a unified theoretical guarantee of the block diagonal property of the solutions. The block diagonal property of many existing subspace clustering methods falls into our special case. Second, we observe that many existing methods aim to approximate the block diagonal representation matrix by using different structure priors, e.g., sparsity and low-rankness, which are indirect methods. We propose the first block diagonal matrix induced regularizer for directly pursuing the block diagonal matrix as well as the 0- or 1-norm for pursuing sparsity and the rank function or nuclear norm for pursuing low-rankness. Based on this regularizer, we solve the subspace clustering problem by Block Diagonal Representation (BDR), which uses the block diagonal structure prior of the representation matrix in the ideal case. The BDR model is nonconvex and we propose to solve it by an alternating minimization algorithm and prove its convergence. Experiments on several real datasets demonstrate the effectiveness and high-efficiency of our BDR