# Gaussian mixture models in Hilbert spaces via kernel methods

arXiv: https://arxiv.org/abs/2605.05996 · HTML: https://mlsanity.com/paper/2605.05996/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 69 references that are in the map.
- First posted on arXiv 7 May 2026.

## Builds on (most cited first, 15 of 15)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Statistical Inference for Generative Models with Maximum Mean Discrepancy](https://mlsanity.com/paper/1906.05944/) (2019)
- [Fast and Eager k-Medoids Clustering: O(k) Runtime Improvement of the PAM, CLARA, and CLARANS Algorithms](https://mlsanity.com/paper/2008.05171/) (2020)
- [A Kernel Two-Sample Test for Functional Data](https://mlsanity.com/paper/2008.11095/) (2020)
- [Finite sample properties of parametric MMD estimation: robustness to misspecification and dependence](https://mlsanity.com/paper/1912.05737/) (2019)
- [scikit-fda: A Python Package for Functional Data Analysis](https://mlsanity.com/paper/2211.02566/) (2022)
- [Functional Models for Time-Varying Random Objects](https://mlsanity.com/paper/1907.10829/) (2019)
- [Kernel k-Groups via Hartigan's Method](https://mlsanity.com/paper/1710.09859/) (2017)
- [A Dictionary of Closed-Form Kernel Mean Embeddings](https://mlsanity.com/paper/2504.18830/) (2025)
- [Modeling Time-Varying Random Objects and Dynamic Networks](https://mlsanity.com/paper/2104.04628/) (2021)
- [Kernel Biclustering algorithm in Hilbert Spaces](https://mlsanity.com/paper/2208.03675/) (2022)
- [Continuous-Time Learning of Probability Distributions: A Case Study in a Digital Trial of Young Children with Type 1 Diabetes](https://mlsanity.com/paper/2603.24427/) (2026)

## Abstract (arXiv)

Modern datasets across many disciplines increasingly consist of time-evolving, potentially infinite-dimensional random objects, such as dynamic functional data, which are naturally modeled in Hilbert spaces. In these settings, characterizing probability measures, for example, through densities, can be ill-defined or technically challenging. Motivated by clustering applications, we propose a Gaussian mixture framework for Hilbert-space-valued data based on kernel mean embeddings and develop efficient optimization algorithms for estimation. We establish theoretical guarantees showing that the proposed algorithm is well defined and that the model yields a dense class of approximations in infinite-dimensional spaces. We evaluate the framework through extensive experiments on diverse structures and data geometries, including L2-functional data and random graphs in Laplacian spaces arising in modern medical applications.