Restricted Isometry Property for General p-Norms

TL;DR

This paper extends the understanding of the Restricted Isometry Property (RIP) for general p-norms, providing bounds on the number of rows needed for RIP and revealing a unique behavior at p=2, with implications for sparse recovery.

Contribution

The paper derives almost tight bounds on the minimum number of rows for RIP in p-norms for all p, highlighting the special case p=2 and exploring implications for sparse recovery.

Findings

Optimal row bounds for RIP vary with p, especially at p=2.

RIP matrices have bounds on column sparsity.

Results impact stable sparse recovery methods.

Abstract

The Restricted Isometry Property (RIP) is a fundamental property of a matrix which enables sparse recovery. Informally, an $m \times n$ matrix satisfies RIP of order $k$ for the $\ell_p$ norm, if $\|Ax\|_p \approx \|x\|_p$ for every vector $x$ with at most $k$ non-zero coordinates. For every $1 \leq p < \infty$ we obtain almost tight bounds on the minimum number of rows $m$ necessary for the RIP property to hold. Prior to this work, only the cases $p = 1$, $1 + 1 / \log k$, and $2$ were studied. Interestingly, our results show that the case $p = 2$ is a "singularity" point: the optimal number of rows $m$ is $\widetilde{\Theta}(k^{p})$ for all $p\in [1,\infty)\setminus \{2\}$, as opposed to $\widetilde{\Theta}(k)$ for $k=2$. We also obtain almost tight bounds for the column sparsity of RIP matrices and discuss implications of our results for the Stable Sparse Recovery problem.