A Homotopy Coordinate Descent Optimization Method for $l_0$-Norm Regularized Least Square Problem

TL;DR

This paper introduces a homotopy coordinate descent method for efficiently solving the $l_0$-norm regularized least squares problem in compressed sensing, improving convergence speed and accuracy.

Contribution

It combines homotopy techniques with a novel coordinate descent approach, incorporating strategies like warm start, active set updating, and strong rules for faster convergence.

Findings

Effective in reconstructing sparse signals accurately

Works well with noisy and noise-free observations

Demonstrates improved computational efficiency

Abstract

This paper proposes a homotopy coordinate descent (HCD) method to solve the $l_0$-norm regularized least square ($l_0$-LS) problem for compressed sensing, which combine the homotopy technique with a variant of coordinate descent method. Differs from the classical coordinate descent algorithms, HCD provides three strategies to speed up the convergence: warm start initialization, active set updating, and strong rule for active set initialization. The active set is pre-selected using a strong rule, then the coordinates of the active set are updated while those of inactive set are unchanged. The homotopy strategy provides a set of warm start initial solutions for a sequence of decreasing values of the regularization factor, which ensures all iterations along the homotopy solution path are sparse. Computational experiments on simulate signals and natural signals demonstrate effectiveness of the proposed algorithm, in accurately and efficiently reconstructing sparse solutions of the $l_0$-LS problem, whether the observation is noisy or not.