Truncated Sparse Approximation Property and Truncated $q$-Norm Minimization

TL;DR

This paper introduces the truncated sparse approximation property for robust recovery of approximately sparse signals and low-rank matrices, establishing its relation to the restricted isometry property and providing conditions for stable recovery.

Contribution

It defines the truncated sparse approximation property, explores its connection with RIP, and proves new conditions under which stable recovery is guaranteed.

Findings

Truncated sparse approximation property generalizes robust null space property.

Established the relationship between truncated sparse approximation property and RIP.

Derived conditions involving restricted isometry constants for stable recovery.

Abstract

This paper considers approximately sparse signal and low-rank matrix's recovery via truncated norm minimization $\min_{x}\|x_T\|_q$ and $\min_{X}\|X_T\|_{S_q}$ from noisy measurements. We first introduce truncated sparse approximation property, a more general robust null space property, and establish the stable recovery of signals and matrices under the truncated sparse approximation property. We also explore the relationship between the restricted isometry property and truncated sparse approximation property. And we also prove that if a measurement matrix $A$ or linear map $\mathcal{A}$ satisfies truncated sparse approximation property of order $k$, then the first inequality in restricted isometry property of order $k$ and of order $2k$ can hold for certain different constants $\delta_{k}$ and $\delta_{2k}$, respectively. Last, we show that if $\delta_{t(k+|T^c|)}<\sqrt{(t-1)/t}$ for some $t\geq 4/3$, then measurement matrix $A$ and linear map $\mathcal{A}$ satisfy truncated sparse approximation property of order $k$. Which should point out is that when $T^c=\emptyset$, our conclusion implies that sparse approximation property of order $k$ is weaker than restricted isometry property of order $tk$.