arXiv subjects

Sergio Cruces

Publications and source records attributed to Sergio Cruces.

4 recordsLinked to original sources

arXiv2025

The alpha-beta divergence for real and complex data

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.

cs.LG↗

arXiv2025

Mirror Descent and Novel Exponentiated Gradient Algorithms Using Trace-Form Entropies and Deformed Logarithms

This paper introduces a broad class of Mirror Descent (MD) and Generalized Exponentiated Gradient (GEG) algorithms derived from trace-form entropies defined via deformed logarithms. Leveraging these generalized entropies yields MD \& GEG algorithms with improved convergence behavior, robustness to vanishing and exploding gradients, and inherent adaptability to non-Euclidean geometries through mirror maps. We establish deep connections between these methods and Amari's natural gradient, revealing a unified geometric foundation for additive, multiplicative, and natural gradient updates. Focusing on the Tsallis, Kaniadakis, Sharma--Taneja--Mittal, and Kaniadakis--Lissia--Scarfone entropy families, we show that each entropy induces a distinct Riemannian metric on the parameter space, leading to GEG algorithms that preserve the natural statistical geometry. The tunable parameters of deformed logarithms enable adaptive geometric selection, providing enhanced robustness and convergence over classical Euclidean optimization. Overall, our framework unifies key first-order MD optimization methods under a single information-geometric perspective based on generalized Bregman divergences, where the choice of entropy determines the underlying metric and dual geometric structure.

cs.LG↗

arXiv2024

Generalized Exponentiated Gradient Algorithms and Their Application to On-Line Portfolio Selection

This paper introduces a novel family of generalized exponentiated gradient (EG) updates derived from an Alpha-Beta divergence regularization function. Collectively referred to as EGAB, the proposed updates belong to the category of multiplicative gradient algorithms for positive data and demonstrate considerable flexibility by controlling iteration behavior and performance through three hyperparameters: $\alpha$, $\beta$, and the learning rate $\eta$. To enforce a unit $l_1$ norm constraint for nonnegative weight vectors within generalized EGAB algorithms, we develop two slightly distinct approaches. One method exploits scale-invariant loss functions, while the other relies on gradient projections onto the feasible domain. As an illustration of their applicability, we evaluate the proposed updates in addressing the online portfolio selection problem (OLPS) using gradient-based methods. Here, they not only offer a unified perspective on the search directions of various OLPS algorithms (including the standard exponentiated gradient and diverse mean-reversion strategies), but also facilitate smooth interpolation and extension of these updates due to the flexibility in hyperparameter selection. Simulation results confirm that the adaptability of these generalized gradient updates can effectively enhance the performance for some portfolios, particularly in scenarios involving transaction costs.

cs.LG↗

arXiv2014

Log-Determinant Divergences Revisited: Alpha--Beta and Gamma Log-Det Divergences

In this paper, we review and extend a family of log-det divergences for symmetric positive definite (SPD) matrices and discuss their fundamental properties. We show how to generate from parameterized Alpha-Beta (AB) and Gamma Log-det divergences many well known divergences, for example, the Stein's loss, S-divergence, called also Jensen-Bregman LogDet (JBLD) divergence, the Logdet Zero (Bhattacharryya) divergence, Affine Invariant Riemannian Metric (AIRM) as well as some new divergences. Moreover, we establish links and correspondences among many log-det divergences and display them on alpha-beta plain for various set of parameters. Furthermore, this paper bridges these divergences and shows also their links to divergences of multivariate and multiway Gaussian distributions. Closed form formulas are derived for gamma divergences of two multivariate Gaussian densities including as special cases the Kullback-Leibler, Bhattacharryya, Rényi and Cauchy-Schwartz divergences. Symmetrized versions of the log-det divergences are also discussed and reviewed. A class of divergences is extended to multiway divergences for separable covariance (precision) matrices.

stat.CO↗