# Complex variational autoencoders admit K"ahler structure

arXiv: https://arxiv.org/abs/2511.15172 · HTML: https://mlsanity.com/paper/2511.15172/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 41 references that are in the map.
- First posted on arXiv 19 Nov 2025.

## Builds on (most cited first, 19 of 19)

- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [EQ-VAE: Equivariance Regularized Latent Space for Improved Generative Image Modeling](https://mlsanity.com/paper/2502.09509/) (2025)
- [On Implicit Regularization in β-VAEs](https://mlsanity.com/paper/2002.00041/) (2020)
- [Learning Flat Latent Manifolds with VAEs](https://mlsanity.com/paper/2002.04881/) (2020)
- [A Geometric Perspective on Variational Autoencoders](https://mlsanity.com/paper/2209.07370/) (2022)
- [Robust Variational Autoencoders for Outlier Detection and Repair of Mixed-Type Data](https://mlsanity.com/paper/1907.06671/) (2019)
- [Hyperbolic VAE via Latent Gaussian Distributions](https://mlsanity.com/paper/2209.15217/) (2022)
- [Geometry-Aware Hamiltonian Variational Auto-Encoder](https://mlsanity.com/paper/2010.11518/) (2020)
- [Hessian Geometry of Latent Space in Generative Models](https://mlsanity.com/paper/2506.10632/) (2025)
- [Robust outlier detection by de-biasing VAE likelihoods](https://mlsanity.com/paper/2108.08760/) (2021)
- [Geometry-Preserving Encoder/Decoder in Latent Generative Models](https://mlsanity.com/paper/2501.09876/) (2025)
- [Training VAEs Under Structured Residuals](https://mlsanity.com/paper/1804.01050/) (2018)
- [Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation](https://mlsanity.com/paper/2507.11789/) (2025)
- [Probabilistic Autoencoder using Fisher Information](https://mlsanity.com/paper/2110.14947/) (2021)
- [Novel Applications for VAE-based Anomaly Detection Systems](https://mlsanity.com/paper/2204.12577/) (2022)
- [Ensuring Topological Data-Structure Preservation under Autoencoder Compression due to Latent Space Regularization in Gauss--Legendre nodes](https://mlsanity.com/paper/2309.08228/) (2023)
- [Complex Recurrent Variational Autoencoder with Application to Speech Enhancement](https://mlsanity.com/paper/2204.02195/) (2022)
- [A Hessian for Gaussian Mixture Likelihoods in Nonlinear Least Squares](https://mlsanity.com/paper/2404.05452/) (2024)
- [GMapLatent: Geometric Mapping in Latent Space](https://mlsanity.com/paper/2503.23407/) (2025)

## Abstract (arXiv)

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 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 acts as a rough proxy 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. Our methods leverage the law of total covariance to bridge behavior between our potential and the Fisher metric. 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.