Complex normalizing flows can almost be information K\"ahler-Ricci flows

TL;DR

This paper explores the deep connection between complex normalizing flows used in data modeling and nonlinear K"ahler-Ricci flows, revealing geometric insights into their structure and behavior.

Contribution

It establishes a theoretical link between complex normalizing flows and K"ahler-Ricci flows, bridging statistical and geometric perspectives.

Findings

The log determinant in normalizing flows matches Ricci curvature terms.

The log density relates to a Fisher information metric under an augmented Jacobian.

The framework recovers a K"ahler-Einstein flow from the normalizing flow dynamics.

Abstract

We develop interconnections between the complex normalizing flow for data drawn from Borel probability measures on the twofold realification of the complex manifold and a nonlinear flow nearly K\"ahler-Ricci. The complex normalizing flow relates the initial and target realified densities under the complex change of variables, necessitating the log determinant of the ensemble of Wirtinger Jacobians. The Ricci curvature of a K\"ahler manifold is the second order mixed Wirtinger partial derivative of the log of the local density of the volume form. Therefore, we reconcile these two facts by drawing forth the connection that the log determinant used in the complex normalizing flow matches a Ricci curvature term under differentiation and conditions. The log density under the normalizing flow is kindred to a spatial Fisher information metric under an augmented Jacobian and a Bayesian perspective to the parameter, thus under the continuum limit the log likelihood matches a Fisher metric, recovering a K\"ahler-Ricci flow variation up to a time derivative and expectation, or an average-valued K\"ahler-Einstein flow. Using this framework, we establish other relevant results, attempting to bridge the statistical and ordinary behaviors of the complex normalizing flow to the geometric features of our derived K\"ahler flow.