Learning on hexagonal structures and Monge-Amp\`ere operators

TL;DR

This paper explores the geometric structure of dually flat statistical manifolds, showing they are Monge-Ampère manifolds, and introduces a novel hexagonal framework for understanding optimal learning processes.

Contribution

It establishes that dually flat statistical manifolds are Monge-Ampère manifolds and introduces a hexagonal structure perspective for learning processes.

Findings

Dually flat statistical manifolds are Monge-Ampère manifolds.

Monge-Ampère operators govern learning in Boltzmann machines.

Learning can be modeled on hexagonal structures under certain topological axioms.

Abstract

Dually flat statistical manifolds provide a rich toolbox for investigations around the learning process. We prove that such manifolds are Monge-Amp\`ere manifolds. Examples of such manifolds include the space of exponential probability distributions on finite sets and the Boltzmann manifolds. Our investigations of Boltzmann manifolds lead us to prove that Monge-Amp\`ere operators control learning methods for Boltzmann machines. Using local trivial fibrations (webs) we demonstrate that on such manifolds the webs are parallelizable and can be constructed using a generalisation of Ceva's theorem. Assuming that our domain satisfies certain axioms of 2D topological quantum field theory we show that locally the learning can be defined on hexagonal structures. This brings a new geometric perspective for defining the optimal learning process.