Tight convex relaxations for sparse matrix factorization
This work addresses sparse matrix factorization for applications in machine learning and data analysis, but it is incremental as it builds on existing convex relaxation methods without a provably polynomial-time algorithm.
The authors tackled the problem of sparse matrix factorization by proposing a new convex formulation based on an atomic norm, which applies to tasks like sparse PCA and subspace clustering. They showed that this formulation has a statistical dimension an order of magnitude smaller than existing norms, and demonstrated promising numerical results with an active set algorithm.
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.