Complex normalizing flows can be information Kähler-Ricci flows

2026-04-20

AI Summary

This work establishes a theoretical equivalence between complex normalizing flows and the Kähler–Ricci flow, unifying probabilistic modeling with complex geometric evolution. By employing Wirtinger calculus, the logarithm of the Jacobian determinant under complex variable transformations is linked to the Ricci curvature of Kähler manifolds, thereby connecting density evolution with geometric structure. Within the Bayesian and information-geometric framework, the study reveals for the first time a profound correspondence between the log-density of complex normalizing flows and both the Fisher information metric and the geometry of the Kähler–Ricci flow. It further proves that, in the continuous limit, these two formulations are equivalent, and that the log-likelihood aligns with the Fisher metric in expectation, offering a novel complex-geometric interpretation of generative models.

Technology Category

Machine Learning: Learning with ManifoldsReasoning under Uncertainty: Relational Probabilistic ModelsKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMsWeb Mining and Content Analysis: Models for Web evolution

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 the Kähler-Ricci flow. The complex normalizing flow relates the initial and target realified densities under the complex change of variables, necessitating the log determinant of the Wirtinger Jacobian. The Ricci curvature of a Kähler 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 the Ricci curvature term under differentiation and conditions. The log density under the normalizing flow is kindred to a spatial Fisher information metric under a holomorphic pullback and a Bayesian perspective to the parameter, thus under the continuum limit the log likelihood matches a Fisher metric, recovering the Kähler-Ricci flow up to expectation. 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 the Kähler-Ricci flow.

Problem

• complex normalizing flow
• Kähler-Ricci flow
• Ricci curvature
• Wirtinger Jacobian
• Fisher information metric

Innovation

• complex normalizing flow
• Kähler-Ricci flow
• Wirtinger Jacobian
• Fisher information metric
• Borel probability measures