Sparse Recovery by Non-convex Optimization -- Instance Optimality

TL;DR

This paper extends theoretical guarantees for non-convex $ ext{l}^p$ minimization decoders in compressed sensing, showing they are robust and instance optimal under weaker conditions than $ ext{l}^1$ methods.

Contribution

It generalizes existing results for $ ext{l}^1$ minimization to $ ext{l}^p$ with 0<p<1, establishing robustness and instance optimality under broader conditions.

Findings

$ ext{l}^p$ decoders are robust to noise.

They are (2,p) instance optimal for a large class of encoders.

They are (2,2) instance optimal in probability with appropriate measurement matrices.

Abstract

In this note, we address the theoretical properties of $\Delta_p$, a class of compressed sensing decoders that rely on $\ell^p$ minimization with 0<p<1 to recover estimates of sparse and compressible signals from incomplete and inaccurate measurements. In particular, we extend the results of Candes, Romberg and Tao, and Wojtaszczyk regarding the decoder $\Delta_1$, based on $\ell^1$ minimization, to $\Delta_p$ with 0<p<1. Our results are two-fold. First, we show that under certain sufficient conditions that are weaker than the analogous sufficient conditions for $\Delta_1$ the decoders $\Delta_p$ are robust to noise and stable in the sense that they are (2,p) instance optimal for a large class of encoders. Second, we extend the results of Wojtaszczyk to show that, like $\Delta_1$, the decoders $\Delta_p$ are (2,2) instance optimal in probability provided the measurement matrix is drawn from an appropriate distribution.