On the Lagrangian Biduality of Sparsity Minimization Problems

Recent results in Compressive Sensing have shown that, under certain conditions the solution to an underdetermined system of linear equations with sparsity-based regularization can be accurately recovered by solving convex relaxations of the original problem. In this work, we present a novel primal-dual analysis on a class of sparsity minimization problems. We show that the Lagrangian bidual i.e. the Lagrangian dual of the Lagrangian dual of the sparsity minimization problems can be used to derive interesting convex relaxations the bidual of the L0- minimization problem is the L1-minimization problem and the bidual of the L0,1- minimization problem for enforcing group sparsity on structured data is the l1,00- minimization problem. The analysis provides a means to compute per-instance non-trivial lower bounds on the group sparsity of the desired solutions. In a real-world application, the bidual relaxation improves the performance of a sparsity-based classification framework applied to robust face recognition.