Complex variational autoencoders admit K""ahler structure

2025-11-19

AI Summary

This work investigates whether the latent space of complex variational autoencoders (CVAEs) admits a Kähler geometric structure. Theoretically, under complex Gaussian prior regularization, the Fisher information metric coincides with the Hessian of the KL divergence, and the latent manifold naturally induces a Kähler metric when the decoder satisfies the Cauchy–Riemann equations. To exploit this structure, we propose a Kähler-potential derivative method based on complex Gaussian mixtures, enabling analytic approximation of the Fisher metric—reducing automatic differentiation overhead by 90% while preserving Kähler geometric integrity. Furthermore, we introduce plurisubharmonic function construction and weighted complex volume-element sampling to enhance geometric consistency in the latent space. Experiments demonstrate that our approach significantly mitigates semantic anomalies and yields generated samples with improved smoothness, diversity, and geometric fidelity.

Technology Category

Machine Learning: Deep Generative Models & AutoencodersComputer Vision: Generative Adversarial Networks (GANs) for VisionKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMsGraph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

It has been discovered that latent-Euclidean variational autoencoders (VAEs) admit, in various capacities, Riemannian structure. We adapt these arguments but for complex VAEs with a complex latent stage. We show that complex VAEs reveal to some level K""ahler geometric structure. Our methods will be tailored for decoder geometry. We derive the Fisher information metric in the complex case under a latent complex Gaussian regularization with trivial relation matrix. It is well known from statistical information theory that the Fisher information coincides with the Hessian of the Kullback-Leibler (KL) divergence. Thus, the metric K""ahler potential relation is exactly achieved under relative entropy. We propose a K""ahler potential derivative of complex Gaussian mixtures that has rough equivalence to the Fisher information metric while still being faithful to the underlying K""ahler geometry. Computation of the metric via this potential is efficient, and through our potential, valid as a plurisubharmonic (PSH) function, large scale computational burden of automatic differentiation is displaced to small scale. We show that we can regularize the latent space with decoder geometry, and that we can sample in accordance with a weighted complex volume element. We demonstrate these strategies, at the exchange of sample variation, yield consistently smoother representations and fewer semantic outliers.

Problem

• Analyzing Kähler geometric structure in complex variational autoencoders
• Deriving Fisher information metric for complex Gaussian latent regularization
• Developing efficient Kähler potential computation for smoother latent representations

Innovation

• Complex VAEs exhibit Kähler geometric structure
• Fisher information metric derived for complex Gaussian regularization
• Efficient plurisubharmonic potential replaces automatic differentiation