Compressive Conjugate Directions: Linear Theory

TL;DR

This paper introduces a novel iterative algorithm that combines ADMM with conjugate directions to efficiently solve large-scale optimization problems involving L1/TV regularization, demonstrating fast convergence in imaging and inversion tasks.

Contribution

The paper proposes a new method that reuses conjugate search directions within ADMM, improving convergence speed for large-scale L1/TV regularized problems.

Findings

Achieves faster convergence in imaging applications

Reuses conjugate directions across iterations

Balances memory use with convergence speed

Abstract

We present a powerful and easy-to-implement iterative algorithm for solving large-scale optimization problems that involve $L_1$/total-variation (TV) regularization. The method is based on combining the Alternating Directions Method of Multipliers (ADMM) with a Conjugate Directions technique in a way that allows reusing conjugate search directions constructed by the algorithm across multiple iterations of the ADMM. The new method achieves fast convergence by trading off multiple applications of the modeling operator for the increased memory requirement of storing previous conjugate directions. We illustrate the new method with a series of imaging and inversion applications.