Convex Sparse Matrix Factorizations

We present a convex formulation of dictionary learning for sparse signal decomposition. Convexity is obtained by replacing the usual explicit upper bound on the dictionary size by a convex rank-reducing term similar to the trace norm. In particular, our formulation introduces an explicit trade-off between size and sparsity of the decomposition of rectangular matrices. Using a large set of synthetic examples, we compare the estimation abilities of the convex and nonconvex approaches, showing that while the convex formulation has a single local minimum, this may lead in some cases to performance which is inferior to the local minima of the non-convex formulation.

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: Francis Bach, Julien Mairal, Jean Ponce
• Keywords: Convex Formulation, Convex Rank-reducing Term, CORR 2008, Education, Explicit Upper Bound