Discrete uncertainty principles and sparse signal processing

TL;DR

This paper introduces new discrete uncertainty principles based on numerical sparsity, a continuous measure that better captures near-sparse signals, and explores their implications in sparse signal processing.

Contribution

It develops novel uncertainty principles using numerical sparsity and analyzes their impact on sparse signal processing applications.

Findings

Numerical sparsity provides a continuous alternative to 0-norm.

The principles accommodate nearly sparse functions.

Implications for sparse signal processing are identified.

Abstract

We develop new discrete uncertainty principles in terms of numerical sparsity, which is a continuous proxy for the 0-norm. Unlike traditional sparsity, the continuity of numerical sparsity naturally accommodates functions which are nearly sparse. After studying these principles and the functions that achieve exact or near equality in them, we identify certain consequences in a number of sparse signal processing applications.