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Gaussian Mixture Models in Hilbert Spaces via Kernel Methods

NeurIPS 2026 · Learning theory and algorithms › Kernels and regression theory · approach: Kernel regression under shift · 0 citations (OpenAlex)

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Fits Gaussian mixtures on separable Hilbert spaces by minimizing kernel-mean-embedding MMD with closed-form Gaussian/polynomial kernel expressions, finite-dimensional projection, and time-varying mixture weights applied to CGM and clustering benchmarks.

Problem

Clustering and density modeling of random objects in infinite-dimensional Hilbert spaces, where densities and likelihoods are ill-defined without a reference measure.

Approach

Fit finite Gaussian mixtures with Gaussian-measure components by minimizing MMD between the empirical measure and the mixture, using closed-form kernel expectations (Fredholm determinant formulas) and alternating a convex QP for weights with gradient steps on means and covariances; extend to time-varying weights via a neural ODE.

Key results

• MMD GMM (Gaussian) reaches ARI 0.692 on R^d benchmarks and 0.412 on L2 functional data, versus 0.683 for EM GMM on R^d.
• On the CGM dataset, MMD GMM (Gaussian) attains ARI 0.053 and SO(3) ARI 0.867, the best among compared methods on those datasets per the table.
• Treatment-arm total variation distance in intraday-curve mixture rises from about 0.11 at baseline to about 0.19 at end of trial.

Surprising findings

• Spectral clustering achieves the best R^d ARI (0.949), outperforming the proposed method on finite-dimensional benchmarks.

Limitations

• Scalability to large-scale biomedical studies is the main stated limitation.
• Joint optimization is nonconvex; alternating scheme gives no global guarantee.
• Uncertainty quantification for parameters and predictions is not addressed.

Experimental setup

Datasets: sklearn clustering datasets SO(3) synthetic rotations L2 functional data benchmarks CGM clinical trial data MUTAG

Benchmarks: sklearn synthetic datasets SO(3) synthetic L2 functional benchmarks CGM MUTAG

Metrics: Adjusted Rand Index MMD^2 total variation distance

Baselines: K-Medoids K-Means DBSCAN HDBSCAN Spectral clustering Funclust FunHDDC Gaussian Mixture (EM) Kernel k-Groups

Builds on: maximum mean discrepancy kernel k-groups Gaussian mixture models neural ODE

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