# The alpha-beta divergence for real and complex data

arXiv: https://arxiv.org/abs/2508.03272 · HTML: https://mlsanity.com/paper/2508.03272/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 36 references that are in the map.
- First posted on arXiv 5 Aug 2025.

## Builds on (most cited first, 6 of 6)

- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Alpha-Beta Divergence For Variational Inference](https://mlsanity.com/paper/1805.01045/) (2018)
- [Generalized Exponentiated Gradient Algorithms and Their Application to On-Line Portfolio Selection](https://mlsanity.com/paper/2406.00655/) (2024)
- [Computing Divergences between Discrete Decomposable Models](https://mlsanity.com/paper/2112.04583/) (2021)
- [Mirror Descent and Novel Exponentiated Gradient Algorithms Using Trace-Form Entropies and Deformed Logarithms](https://mlsanity.com/paper/2503.08748/) (2025)
- [Learning Log-Determinant Divergences for Positive Definite Matrices](https://mlsanity.com/paper/2104.06461/) (2021)

## Built on by (most cited first, 1 of 1)

- [No Unique Minimizer, No Problem: On the Consistency of Robust Neural Classifiers](https://mlsanity.com/paper/2608.08489/) (2026)

## Abstract (arXiv)

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.