Comparing Measures of Sparsity

TL;DR

This paper compares various sparsity measures used in signal processing, evaluating them against six intuitive properties, and identifies the Gini Index as the only measure satisfying all criteria.

Contribution

It provides proofs and a classification table for common sparsity measures based on six key properties, highlighting the Gini Index as the most comprehensive measure.

Findings

Gini Index satisfies all six sparsity properties.

Most measures fail to meet all criteria, except Gini Index.

The paper offers a systematic comparison and classification of sparsity measures.

Abstract

Sparsity of representations of signals has been shown to be a key concept of fundamental importance in fields such as blind source separation, compression, sampling and signal analysis. The aim of this paper is to compare several commonlyused sparsity measures based on intuitive attributes. Intuitively, a sparse representation is one in which a small number of coefficients contain a large proportion of the energy. In this paper six properties are discussed: (Robin Hood, Scaling, Rising Tide, Cloning, Bill Gates and Babies), each of which a sparsity measure should have. The main contributions of this paper are the proofs and the associated summary table which classify commonly-used sparsity measures based on whether or not they satisfy these six propositions and the corresponding proofs. Only one of these measures satisfies all six: The Gini Index. measures based on whether or not they satisfy these six propositions and the corresponding proofs. Only one of these measures satisfies all six: The Gini Index.