Ordinary Differential Equation-based Sparse Signal Recovery

TL;DR

This paper introduces an ODE-based method for sparse signal recovery, analyzing its convergence properties and proposing a variational optimization to enhance performance, validated through numerical experiments.

Contribution

It presents a novel continuous-time dynamical system approach for sparse recovery, including convergence analysis and an optimized regularization scheme using deep unfolding.

Findings

The ODE-based method converges to the sparse solution.

Linear approximation provides insights into local convergence behavior.

Deep unfolded optimization improves convergence speed and solution quality.

Abstract

This study investigates the use of continuous-time dynamical systems for sparse signal recovery. The proposed dynamical system is in the form of a nonlinear ordinary differential equation (ODE) derived from the gradient flow of the Lasso objective function. The sparse signal recovery process of this ODE-based approach is demonstrated by numerical simulations using the Euler method. The state of the continuous-time dynamical system eventually converges to the equilibrium point corresponding to the minimum of the objective function. To gain insight into the local convergence properties of the system, a linear approximation around the equilibrium point is applied, yielding a closed-form error evolution ODE. This analysis shows the behavior of convergence to the equilibrium point. In addition, a variational optimization problem is proposed to optimize a time-dependent regularization parameter in order to improve both convergence speed and solution quality. The deep unfolded-variational optimization method is introduced as a means of solving this optimization problem, and its effectiveness is validated through numerical experiments.