Frobenius statistical manifolds & geometric invariants

TL;DR

This paper establishes that statistical manifolds related to exponential families possess Frobenius manifold structures and introduces Gromov--Witten-like invariants for these manifolds, linking geometry, topology, and learning processes.

Contribution

It proves the existence of Frobenius structures on certain statistical manifolds and defines analogous Gromov--Witten invariants with applications to learning theory.

Findings

Statistical manifolds have Frobenius manifold structures.

Existence of Gromov--Witten-like invariants for these manifolds.

Invariants relate to intersection points of para-holomorphic curves and learning success.

Abstract

In this paper, we explicitly prove that statistical manifolds, related to exponential families and with flat structure connection have a Frobenius manifold structure. This latter object, at the interplay of beautiful interactions between topology and quantum field theory, raises natural questions, concerning the existence of Gromov--Witten invariants for those statistical manifolds. We prove that an analog of Gromov--Witten invariants for those statistical manifolds (GWS) exists. Similarly to its original version, these new invariants have a geometric interpretation concerning intersection points of para-holomorphic curves. However, it also plays an important role in the learning process, since it determines whether a system has succeeded in learning or failed.