Restricted $q$-Isometry Properties Adapted to Frames for Nonconvex $l_q$-Analysis

TL;DR

This paper introduces a generalized $q$-restricted isometry property (RIP) adapted to frames, providing new conditions for signal reconstruction from few measurements using $l_q$-analysis, especially effective for smaller $q$ values.

Contribution

It extends the standard $q$-RIP to a generalized version suited for frames, establishing new measurement bounds for approximate reconstruction in sparse and data separation problems.

Findings

Fewer measurements are needed for smaller $q$ in $l_q$-analysis.

The generalized $q$-RIP applies to signals sparse in frames and in data separation.

The approach improves reconstruction efficiency under mutual coherence conditions.

Abstract

This paper discusses reconstruction of signals from few measurements in the situation that signals are sparse or approximately sparse in terms of a general frame via the $l_q$-analysis optimization with $0<q\leq 1$. We first introduce a notion of restricted $q$-isometry property ($q$-RIP) adapted to a dictionary, which is a natural extension of the standard $q$-RIP, and establish a generalized $q$-RIP condition for approximate reconstruction of signals via the $l_q$-analysis optimization. We then determine how many random, Gaussian measurements are needed for the condition to hold with high probability. The resulting sufficient condition is met by fewer measurements for smaller $q$ than when $q=1$. The introduced generalized $q$-RIP is also useful in compressed data separation. In compressed data separation, one considers the problem of reconstruction of signals' distinct subcomponents, which are (approximately) sparse in morphologically different dictionaries, from few measurements. With the notion of generalized $q$-RIP, we show that under an usual assumption that the dictionaries satisfy a mutual coherence condition, the $l_q$ split analysis with $0<q\leq1 $ can approximately reconstruct the distinct components from fewer random Gaussian measurements with small $q$ than when $q=1$