A Trivial Observation related to Sparse Recovery

TL;DR

This paper presents a simple modification to existing sparse recovery analysis, replacing the RIP with a potentially simpler RIP-type property, though its practical usefulness remains uncertain.

Contribution

It introduces a trivial change to the analysis of sparse recovery algorithms, questioning the necessity of the standard RIP and proposing an alternative RIP-type condition.

Findings

Proposes an alternative RIP-type property for sparse recovery analysis

Suggests the new property might be simpler or as complex as RIP

Highlights the uncertainty about the practical checkability of the new property

Abstract

We make a trivial modification to the elegant analysis of Garg and Khandekar (\emph{Gradient Descent with Sparsification} ICML 2009) that replaces the standard Restricted Isometry Property (RIP), with another RIP-type property (which could be simpler than the RIP, but we are not sure; it could be as hard as the RIP to check, thereby rendering this little writeup totally worthless).