The alpha-beta divergence for real and complex data

TL;DR

This paper extends the alpha-beta divergence framework to complex data, providing a versatile tool for signal processing that generalizes classical distances and offers closed-form solutions for complex vector approximation.

Contribution

It introduces a novel formulation of alpha-beta divergences for complex vectors, generalizes classical distances, and derives closed-form solutions for complex data approximation.

Findings

Unified framework for real and complex data divergences

Closed-form expression for complex vector centroid

Versatile extension of classical divergences

Abstract

Divergences are fundamental to the information criteria that underpin most signal processing algorithms. The alpha-beta family of divergences, designed for non-negative data, offers a versatile framework that parameterizes and continuously interpolates several separable divergences found in existing literature. This work extends the definition of alpha-beta divergences to accommodate complex data, specifically when the arguments of the divergence are complex vectors. This novel formulation is designed in such a way that, by setting the divergence hyperparameters to unity, it particularizes to the well-known Euclidean and Mahalanobis squared distances. Other choices of hyperparameters yield practical separable and non-separable extensions of several classical divergences. In the context of the problem of approximating a complex random vector, the centroid obtained by optimizing the alpha-beta mean distortion has a closed-form expression, which interpretation sheds light on the distinct roles of the divergence hyperparameters. These contributions may have wide potential applicability, as there are many signal processing domains in which the underlying data are inherently complex.