Learning with Dually Flat Structure and Incidence Algebra

Statistical manifolds with dually flat structures, such as an exponential family, appear in various machine learning models. In this talk, I will introduce a close connection between dually flat manifolds and incidence algebras in order theory and present its application to machine learning. This approach allows us to flexibly design log-linear models equipped with partially ordered sample spaces, which include a number of machine learning problems such as learning of Boltzmann machines, tensor decomposition, and blind source separation. I will also talk about theoretical analysis of such models using Rissanen's stochastic complexity and draw the connection to the double descent phenomenon via model volumes.