Tight convex relaxations for sparse matrix factorization

TL;DR

This paper introduces a new convex approach based on an atomic norm for sparse matrix factorization, demonstrating improved statistical efficiency and proposing an active set algorithm for practical solution despite theoretical complexity.

Contribution

It presents a novel convex formulation for sparse matrix factorization using a new atomic norm, with theoretical analysis and an active set algorithm for practical computation.

Findings

Statistical dimension of the new norm is significantly smaller than traditional norms.

Proposed active set algorithm shows promising numerical results.

The formulation applies to sparse PCA, subspace clustering, and low-rank bilinear regression.

Abstract

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 $\ell\_1$-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.