Convergence of LCA Flows to (C)LASSO Solutions

This paper establishes several convergence results about flows of the dynamical system LCA (Locally Competitive Algorithm) to the mixed $\ell_2$-$\ell_1$ minimization problem LASSO and the constrained version, called CLASSO here, where the parameters are required to be non-negative. (C)LASSO problems are closely related to various important applications including efficient coding, image recognition and image reconstruction. That the solution of (C)LASSO can be determined by LCA allows the former to be solved in novel ways such as through a physical realization of analog circuits or on non-von Neumann computers. As discussed in the paper, previous works that show convergence of LCA to LASSO are incomplete, and do not consider CLASSO. The main contributions of this paper are a particular generalization of LaSalle's invariance principle and its application to rigorously establish LCA's convergence to (C)LASSO.