Comparing Measures of Sparsity (0811.4706v2)

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.

• The paper compares fifteen sparsity measures using six criteria to assess their effectiveness in representing sparsity.
• Analytical proofs and numerical experiments reveal that the Gini Index uniquely satisfies all six sparsity criteria.
• The findings provide actionable insights for selecting robust sparsity measures in signal processing and machine learning applications.

Essay on "Comparing Measures of Sparsity"

This article, "Comparing Measures of Sparsity" by Niall Hurley and Scott Rickard, explores the comparative analysis of various sparsity measures employed in signal processing and machine learning. Sparsity, defined as a representation where a small number of coefficients account for a large proportion of the energy, has become integral in applications ranging from blind source separation to compression and denoising. This paper critically assesses fifteen commonly used sparsity measures against six intuitive criteria to determine their adequacy in capturing sparsity effectively.

The six criteria, namely Robin Hood (D1), Scaling (D2), Rising Tide (D3), Cloning (D4), Bill Gates (P1), and Babies (P2), are meticulously derived from both financial inequity measures and proposed sparsity properties. These criteria ensure that a good sparsity measure should penalize redistribution of energy (D1), remain invariant under scaling (D2), reduce sparsity when a constant is added (D3), exhibit invariance under duplication of coefficients (D4), increase sharply with the addition of outliers (P1), and increase with the addition of zeros (P2).

The paper proceeds to apply these criteria to fifteen measures including the classical $\ell^0$ norm, $\ell^1$ norm, $\ell^p$ norms (for $0$H_S$) and Gaussian entropy ($H_G$), amongst others. The authors employ both analytical proofs and numerical comparisons to systematically evaluate each measure.

Presented in the paper is a compelling argument for the Gini Index as the only measure that meets all six criteria. The Gini Index, originally used in economics to measure income inequality, finds a novel application here as an effective sparsity measure. It not only conforms to the desired properties but also provides a normalized value between 0 and 1, making it intuitive and easy to interpret.

In contrast, other measures exhibit varying degrees of compliance with these criteria. For instance, while the Hoyer measure ($\frac{\sqrt{N}-\frac{\sum_j c_j}{\sqrt{\sum_j c_j^2)}}{\sqrt{N}-1}$) fares well with most criteria, it fails to satisfy Cloning (D4). Similarly, measures like $\kappa_4$ and $\ell^p$ for negative $p$ (e.g., $\ell^p_{-}$) have limitations in specific criteria, thus rendering them less universally applicable compared to the Gini Index.

The theoretical discussion is complemented by numerical experiments that visualize the behavior of these measures against distributions with controllable sparsity parameters. For example, when evaluating sets drawn from Poisson and Bernoulli distributions, the Gini Index and the Hoyer measure demonstrate superior performance in reflecting sparsity accurately.

From a practical standpoint, the findings of this paper underscore the importance of selecting an appropriate sparsity measure based on the application context. Measures like the Gini Index are particularly advocated for when the number of coefficients is variable and a normalized, consistent sparsity assessment is desired. This can significantly enhance tasks such as feature extraction, signal separation, and data compression.

In conclusion, while sparsity is a widely utilized concept, this paper’s in-depth comparative analysis highlights the nuances in different measures, providing valuable insights for researchers. The advocacy for the Gini Index as a robust measure reaffirms its potential utility beyond its traditional economic applications. Future research can build on these findings to further refine sparsity measures or explore their implications in novel signal processing and machine learning frameworks.