Maximum Tsallis Entropy Distributions for Robust and Efficient Sparse Learning from Correlated Data

Abstract:This paper addresses the limitations of Gaussian distribution assumptions in statistical sparse learning, particularly in modeling correlated and heterogeneous data. Conventional Gaussian models often lack robustness towards outliers and underlying distribution assumptions. To overcome these limitations, we propose the use of the $q$Gaussian distribution, derived from Tsallis entropy maximization, as a robust alternative. This is notably relevant in biostatistics, where the presence of correlated observations and heterogeneity, such as in genetic and longitudinal studies, is prevalent. Our contributions include modeling of correlated data through the re-derived multivariate probability density function from Tsallis entropy maximization, thereby addressing the limitations inherent in conventional Gaussian models. Furthermore, we introduce a novel framework that adapts numerical methods designed to find equilibria in flows to tackle composite optimization problems prevalent in statistical sparse learning. Applying this framework to the Hager-Zhang conjugate gradient algorithm \cite{Hager2005}, we develop a numerically stable and efficient algorithm for sparse statistical learning. The $q$Gaussian distribution, informed by the principle of maximizing Tsallis entropy, presents a viable and flexible alternative to Gaussian-based methods. This paper not only contributes to the theoretical understanding of statistical distributions and optimization techniques, but also paves the way for practical data analysis.

Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Statistics Theory (math.ST); Computation (stat.CO); Machine Learning (stat.ML)

arXiv:2608.17244 [math.OC]

(or arXiv:2608.17244v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2608.17244 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

https://doi.org/10.82308/38780 Focus to learn more DOI(s) linking to related resources

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.