A Generalised Differential Framework for Measuring Signal Sparsity

The notion of signal sparsity has been gaining increasing interest in information theory and signal processing communities. Recent advances in fields like signal compression, sampling and analysis have accentuated the crucial role of sparse representations of signals. As a consequence, there is a strong need to measure sparsity and towards this end, a plethora of metrics has been presented in the literature. The appropriateness of these metrics is typically evaluated against a set of objective criteria that has been proposed for assessing the credibility of any sparsity metric. In this paper, we propose a Generalised Differential Sparsity (GDS) framework for generating novel sparsity metrics whose functionality is based on the concept that sparsity is encoded in the differences among the signal coefficients. We rigorously prove that every metric generated using GDS satisfies all the aforementioned criteria and we provide a computationally efficient formula that makes GDS suitable for high-dimensional signals. The great advantage of GDS is its flexibility to offer sparsity metrics that can be well-tailored to certain requirements stemming from the nature of the data and the problem to be solved. This is in contrast to current state-ofthe-art sparsity metrics like Gini Index (GI), which is actually proven to be only a specific instance of GDS, demonstrating the generalisation power of our framework. In verifying our claims, we have incorporated GDS in a stochastic signal recovery algorithm and experimentally investigated its efficacy in reconstructing randomly projected sparse signals. As a result, it is proven that GDS, in comparison to GI, both loosens the bounds of the assumed sparsity of the original signals and reduces the minimum number of projected dimensions, required to guarantee an almost perfect reconstruction of heavily compressed signals. The superiority of GDS over GI in conjunction with the fact that the latter is considered as a standard in numerous scientific domains, prove the great potential of GDS as a general purpose framework for measuring sparsity.