Frobenius Statistical Manifolds and Geometric Invariants

Abstract

Using an algebraic approach, we 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. It also plays an important role in the learning process, since it determines whether a system has succeeded in learning or failed.