Restricted Quasiconvexity Isometry Property for Symmetric $\alpha$-Stable Random Matrices

TL;DR

This paper introduces the Restricted Quasiconvexity Isometry Property (RQIP) for symmetric alpha-stable random matrices, providing bounds on the number of measurements needed for sparse recovery, highlighting limitations of traditional RIP in this context.

Contribution

It generalizes the RIP to RQIP for alpha-stable matrices and derives a lower bound on the sample size needed for RQIP to hold, using concentration inequalities and covering number bounds.

Findings

Derived a lower bound on the number of rows for RQIP to hold.

Showed the sample complexity depends on the polynomial tail behavior of alpha-stable distributions.

Indicated that traditional RIP may need replacement in sparse recovery frameworks.

Abstract

We formulate a generalization of the Restricted Isometry Property (RIP) referred to as the Restricted Quasiconvexity Isometry Property (RQIP) for alpha stable random projections with $0<\alpha<1$. A lower bound on the number of rows for RQIP to hold for random matrices whose entries are drawn from a symmetric $\alpha$-stable ($S\alpha S$) distribution is derived. The proof leverages two key components: a concentration inequality for empirical fractional moments of $S\alpha S$ variables and a covering number bound for sparse $\ell_\alpha$ balls. The resulting sample complexity reflects the polynomial tail behavior of the concentration and reinforces an observation made in the literature that the RIP framework may have to be replaced with other sparse recovery formulations in practice, such as those based on the null space property.