Comparing Measures of Sparsity

• Published in: IEEE transactions on information theory Vol. 55; no. 10; pp. 4723 - 4741
• Main Authors: Hurley, N., Rickard, S.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.10.2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptive signal processing; Applied sciences; Blind source separation; Blinds; Classification; Cloning; Compressing; Detection, estimation, filtering, equalization, prediction; Digital signal processors; Exact sciences and technology; Hoods; Image coding; Information; Information theory; Information, signal and communications theory; Machine learning; Measures of sparsity; measuring sparsity; Representations; Sampling; Sampling methods; Sampling, quantization; Sea measurements; Signal analysis; Signal and communications theory; Signal representation. Spectral analysis; Signal, noise; Source separation; sparse distribution; sparse representation; sparsity; Tables (data); Telecommunications and information theory; Tides; measuring sparsity; Signal processing; Sparse representation; Blind source separation; sparsity; Sampling; Signal analysis; Measures of sparsity; sparse distribution
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2009.2027527

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 commonly-used 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. Only two of these measures satisfy all six: the pq-mean with p les 1, q > 1 and the Gini index.