  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.
