Efficient Solvers for Sparse Subspace Clustering

Abstract Sparse subspace clustering (SSC) clusters n points that lie near a union of low-dimensional subspaces. The SSC model expresses each point as a linear or affine combination of the other points, using either l1 or l0 regularization. Using l1 regularization results in a convex problem but requires O ( n 2 ) storage, and is typically solved by the alternating direction method of multipliers which takes O ( n 3 ) flops. The l0 model is non-convex but only needs memory linear in n, and is solved via orthogonal matching pursuit and cannot handle the case of affine subspaces. This paper shows that a proximal gradient framework can solve SSC, covering both l1 and l0 models, and both linear and affine constraints. For both l1 and l0, algorithms to compute the proximity operator in the presence of affine constraints have not been presented in the SSC literature, so we derive an exact and efficient algorithm that solves the l1 case with just O ( n 2 ) flops. In the l0 case, our algorithm retains the low-memory overhead, and is the first algorithm to solve the SSC-l0 model with affine constraints. Experiments show our algorithms do not rely on sensitive regularization parameters, and they are less sensitive to sparsity misspecification and high noise.