TMI: Thermodynamic inference of data manifolds

TL;DR

TMI introduces a thermodynamic inference method that approximates arbitrary data distributions by learning from features and descriptors, enabling interpretable dimensionality reduction and geometric analysis of data manifolds.

Contribution

It presents a novel thermodynamic approach to model complex distributions beyond Gibbs-Boltzmann form, incorporating geometric tools for data analysis.

Findings

Successfully applied to real and artificial datasets.

Provides a geometric framework for analyzing data manifolds.

Achieves interpretable dimensionality reduction.

Abstract

The Gibbs-Boltzmann distribution offers a physically interpretable way to massively reduce the dimensionality of high dimensional probability distributions where the extensive variables are `features' and the intensive variables are `descriptors'. However, not all probability distributions can be modeled using the Gibbs-Boltzmann form. Here, we present TMI: TMI, {\bf T}hermodynamic {\bf M}anifold {\bf I}nference; a thermodynamic approach to approximate a collection of arbitrary distributions. TMI simultaneously learns from data intensive and extensive variables and achieves dimensionality reduction through a multiplicative, positive valued, and interpretable decomposition of the data. Importantly, the reduced dimensional space of intensive parameters is not homogeneous. The Gibbs-Boltzmann distribution defines an analytically tractable Riemannian metric on the space of intensive variables allowing us to calculate geodesics and volume elements. We discuss the applications of TMI with multiple real and artificial data sets. Possible extensions are discussed as well.