On the Optimality of Backward Regression: Sparse Recovery and Subset Selection

TL;DR

This paper analyzes the backward regression algorithm for sparse recovery, providing new polynomial-time bounds, proposing an improved two-stage method called SRR, and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces a polynomial-time residual bound for backward regression, proposes the SRR algorithm combining forward and backward steps, and empirically shows SRR's advantages in sparse recovery.

Findings

SRR maintains good performance on coherent dictionaries.

The polynomial-time residual bound extends the theoretical understanding.

SRR outperforms Subspace Pursuit in certain scenarios.

Abstract

Sparse recovery and subset selection are fundamental problems in varied communities, including signal processing, statistics and machine learning. Herein, we focus on an important greedy algorithm for these problems: Backward Stepwise Regression. We present novel guarantees for the algorithm, propose an efficient, numerically stable implementation, and put forth Stepwise Regression with Replacement (SRR), a new family of two-stage algorithms that employs both forward and backward steps for compressed sensing problems. Prior work on the backward algorithm has proven its optimality for the subset selection problem, provided the residual associated with the optimal solution is small enough. However, the existing bounds on the residual magnitude are NP-hard to compute. In contrast, our main theoretical result includes a bound that can be computed in polynomial time, depends chiefly on the smallest singular value of the matrix, and also extends to the method of magnitude pruning. In addition, we report numerical experiments highlighting crucial differences between forward and backward greedy algorithms and compare SRR against popular two-stage algorithms for compressed sensing. Remarkably, SRR algorithms generally maintain good sparse recovery performance on coherent dictionaries. Further, a particular SRR algorithm has an edge over Subspace Pursuit.