A note on exponential varieties, statistical manifolds and Frobenius structures

TL;DR

This paper explores the deep connections between algebraic geometry, information theory, and Topological Field Theory by showing that statistical manifolds of noisy data models form algebraic varieties with rich geometric and algebraic structures.

Contribution

It explicitly demonstrates that statistical pre-Frobenius manifolds are algebraic varieties over 5, revealing new algebraic and geometric properties of these statistical models.

Findings

Statistical manifolds form 5-toric varieties.

Web symmetries are characterized as Commutative Moufang Loops.

Web structures are hexagonal and isoclinic, affecting data geometry.

Abstract

New relations between algebraic geometry, information theory and Topological Field Theory are developed. One considers models of databases subject to noise i.e. probability distributions on finite sets, related to exponential families. We prove explicitly that these manifolds have the structure of a pre-Frobenius manifold, being a pre-structure appearing in the process of axiomatisation of Topological Field Theory. On one hand, this allows us to develop relations to algebraic geometry, by proving explicitly that a statistical pre-Frobenius manifold forms an algebraic variety over $\mathbb{Q}$ (i.e. $\mathbb{Q}$-toric variety). On the other hand, this allows further developments of recent results concerning the hidden symmetries of those objects. Using classical web theory, it has been shown that those symmetries have the structure of Commutative Moufang Loops. Our result allows to develop more algebraically this statement, in a two-fold way. First, from an algebraic point of view it follows that statistical pre-Frobenius manifolds are equipped with algebraizable webs. Secondly, from the differential geometry point of view, it follows that these webs are hexagonal and isoclinic. This statement is important since it directly impacts the geometric properties of the {\it statistical data}, which are tightly related to the webs. Hence, this allows deeper connections to the branch of algebraic statistics, which is concerned with the development of techniques in algebraic geometry, commutative algebra, to address problems in statistics and its applications. Examples are provided and discussed.