Orthogonal Matching Pursuit: A Brownian Motion Analysis

TL;DR

This paper improves the theoretical understanding of orthogonal matching pursuit (OMP), showing it can recover sparse signals with fewer measurements than previously established, matching the performance of more complex methods like lasso.

Contribution

It demonstrates that OMP requires only 2 k log(n - k) measurements for asymptotic recovery, strengthening prior results and aligning its measurement bounds with those of lasso.

Findings

OMP can recover k-sparse signals with 2 k log(n - k) measurements.

This measurement bound also enables support detection with high SNR.

The results match the measurement requirements of lasso for similar recovery performance.

Abstract

A well-known analysis of Tropp and Gilbert shows that orthogonal matching pursuit (OMP) can recover a k-sparse n-dimensional real vector from 4 k log(n) noise-free linear measurements obtained through a random Gaussian measurement matrix with a probability that approaches one as n approaches infinity. This work strengthens this result by showing that a lower number of measurements, 2 k log(n - k), is in fact sufficient for asymptotic recovery. More generally, when the sparsity level satisfies kmin <= k <= kmax but is unknown, 2 kmax log(n - kmin) measurements is sufficient. Furthermore, this number of measurements is also sufficient for detection of the sparsity pattern (support) of the vector with measurement errors provided the signal-to-noise ratio (SNR) scales to infinity. The scaling 2 k log(n - k) exactly matches the number of measurements required by the more complex lasso method for signal recovery with a similar SNR scaling.