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The notion of signal sparsity has been gaining increasing interest ininformation theory and signal processing communities. As a consequence, aplethora of sparsity metrics has been presented in the literature. Theappropriateness of these metrics is typically evaluated against a set ofobjective criteria that has been proposed for assessing the credibility of anysparsity metric. In this paper, we propose a Generalised Differential Sparsity(GDS) framework for generating novel sparsity metrics whose functionality isbased on the concept that sparsity is encoded in the differences among thesignal coefficients. We rigorously prove that every metric generated using GDSsatisfies all the aforementioned criteria and we provide a computationallyefficient formula that makes GDS suitable for high-dimensional signals. Thegreat advantage of GDS is its flexibility to offer sparsity metrics that can bewell-tailored to certain requirements stemming from the nature of the data andthe problem to be solved. This is in contrast to current state-of-the-artsparsity metrics like Gini Index (GI), which is actually proven to be only aspecific instance of GDS, demonstrating the generalisation power of ourframework. In verifying our claims, we have incorporated GDS in a stochasticsignal recovery algorithm and experimentally investigated its efficacy inreconstructing randomly projected sparse signals. As a result, it is proventhat GDS, in comparison to GI, both loosens the bounds of the assumed sparsityof the original signals and reduces the minimum number of projected dimensions,required to guarantee an almost perfect reconstruction of heavily compressedsignals. The superiority of GDS over GI in conjunction with the fact that thelatter is considered as a standard in numerous scientific domains, prove thegreat potential of GDS as a general purpose framework for measuring sparsity.

A Generalised Differential Framework for Measuring Signal Sparsity