On the Lagrangian Biduality of Sparsity Minimization Problems
This provides a theoretical framework for sparsity minimization, with incremental advances in optimization methods for compressive sensing and classification tasks.
The paper tackles the problem of solving underdetermined linear systems with sparsity regularization by analyzing Lagrangian biduality, showing that bidual relaxations like ℓ1-minimization derive from ℓ0-minimization and improve performance in robust face recognition.
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 $\ell_0$-minimization problem is the $\ell_1$-minimization problem; and the bidual of the $\ell_{0,1}$-minimization problem for enforcing group sparsity on structured data is the $\ell_{1,\infty}$-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.