Convergence and Rate Analysis of Neural Networks for Sparse Approximation

TL;DR

This paper analyzes the convergence properties of the Locally Competitive Algorithm (LCA), a neural network for sparse approximation, demonstrating stability, global convergence, finite support recovery, and exponential convergence rate under certain conditions.

Contribution

It provides the first rigorous analysis of LCA's convergence, including stability, finite support recovery, and exponential convergence rate, filling a gap in understanding its theoretical behavior.

Findings

LCA converges globally to the optimal sparse solution.

Support of the solution is recovered in finite time under mild conditions.

LCA exhibits exponential convergence rate with an analytically bounded speed.

Abstract

We present an analysis of the Locally Competitive Algorithm (LCA), a Hopfield-style neural network that efficiently solves sparse approximation problems (e.g., approximating a vector from a dictionary using just a few non-zero coefficients). This class of problems plays a significant role in both theories of neural coding and applications in signal processing. However, the LCA lacks analysis of its convergence properties and previous results on neural networks for nonsmooth optimization do not apply to the specifics of the LCA architecture. We show that the LCA has desirable convergence properties, such as stability and global convergence to the optimum of the objective function when it is unique. Under some mild conditions, the support of the solution is also proven to be reached in finite time. Furthermore, some restrictions on the problem specifics allow us to characterize the convergence rate of the system by showing that the LCA converges exponentially fast with an analytically bounded convergence rate. We support our analysis with several illustrative simulations.