Tight convex relaxations for sparse matrix factorization by Emile Richard, Guillaume Obozinski, Jean-Philippe Vert
Based on a new atomic norm, we propose a new convex formulation for sparse matrix factorization problems in which the number of nonzero elements of the factors is assumed fixed and known. The formulation counts sparse PCA with multiple factors, subspace clustering and low-rank sparse bilinear regression as potential applications. We compute slow rates and an upper bound on the statistical dimension of the suggested norm for rank 1 matrices, showing that its statistical dimension is an order of magnitude smaller than the usual ℓ1-norm, trace norm and their combinations. Even though our convex formulation is in theory hard and does not lead to provably polynomial time algorithmic schemes, we propose an active set algorithm leveraging the structure of the convex problem to solve it and show promising numerical results.
The implementation of the Sparse PCA using various methods and some sparsely factorized matrices using the (k,q)-trace norm as penalty are on Emile Richard's software page. They will be added to the Advanced Matrix Factorization Jungle page shortly.