Gaussian mixture models in Hilbert spaces via kernel methods

TL;DR

This paper introduces a Gaussian mixture modeling approach for Hilbert-space-valued data using kernel mean embeddings, addressing challenges in modeling infinite-dimensional data like functional data and graphs.

Contribution

It develops a novel Gaussian mixture framework tailored for Hilbert spaces, with efficient algorithms and theoretical guarantees for approximation quality.

Findings

Algorithm is well-defined and converges in infinite-dimensional spaces.

Framework effectively models diverse data structures including functional data and graphs.

Experimental results demonstrate applicability across various data geometries.

Abstract

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 $L^2$-functional data and random graphs in Laplacian spaces arising in modern medical applications.