Simple Bounds for Recovering Low-complexity Models

TL;DR

This paper provides a unified analysis of the recovery bounds for low-complexity models like sparse vectors and low-rank matrices from Gaussian measurements, matching or approaching the best known bounds.

Contribution

It introduces a simplified, unified analysis framework for recovery bounds of sparse vectors and low-rank matrices, including block sparse signals, using standard large deviation inequalities.

Findings

Recovery of s-sparse vectors from 2s log n measurements with high probability.

Recovery of rank r matrices from r(6n-5r) measurements with high probability.

Bounds are nearly optimal and robust to measurement matrix variations.

Abstract

This note presents a unified analysis of the recovery of simple objects from random linear measurements. When the linear functionals are Gaussian, we show that an s-sparse vector in R^n can be efficiently recovered from 2s log n measurements with high probability and a rank r, n by n matrix can be efficiently recovered from r(6n-5r) with high probability. For sparse vectors, this is within an additive factor of the best known nonasymptotic bounds. For low-rank matrices, this matches the best known bounds. We present a parallel analysis for block sparse vectors obtaining similarly tight bounds. In the case of sparse and block sparse signals, we additionally demonstrate that our bounds are only slightly weakened when the measurement map is a random sign matrix. Our results are based on analyzing a particular dual point which certifies optimality conditions of the respective convex programming problem. Our calculations rely only on standard large deviation inequalities and our analysis is self-contained.