Global convergence of the Locally Competitive Algorithm

The Locally Competitive Algorithm (LCA) is a continuous-time dynamical system designed to solve the problem of sparse approximation. This class of approximation problems plays an important role in producing state-of-the-art results in many signal processing and inverse problems, and implementing the LCA in analog VLSI may significantly improve the time and power necessary to solve these optimization programs. The goal of this paper is to analyze the dynamical behavior of the LCA system and guarantee its convergence and stability. We show that fixed points of the system are extrema of the sparse approximation objective function when designed for a certain class of sparsity-inducing cost penalty. We also show that, if the objective has a unique minimum, the LCA converges for any initial point. In addition, we prove that under certain conditions on the solution, the LCA converges in a finite number of switches (i.e., node threshold crossings).