Gaussian Mixture Model with unknown diagonal covariances via continuous sparse regularization

2025-09-16 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

A novel approach extends the Beurling-LASSO (BLASSO) methodology to estimate Gaussian Mixture Models (GMMs) with unknown diagonal covariances from independent and identically distributed samples. This convex optimization framework simultaneously estimates the number of components and their parameters, offering greater flexibility than previous BLASSO applications that required known and identical covariances. The method establishes non-asymptotic recovery guarantees with nearly parametric convergence rates for component means, diagonal covariances, and weights, alongside density prediction. A critical theoretical contribution is an explicit separation condition on mixture components, enabling the construction of non-degenerate dual certificates, leveraging Fisher-Rao geometry and a novel semi-distance.

Key takeaway

For AI Scientists and Research Scientists developing GMM-based models who struggle with pre-specifying component numbers or assuming identical covariances, this extended BLASSO approach offers a robust, convex alternative to the EM algorithm. It simultaneously estimates component count and parameters, providing strong theoretical guarantees and nearly parametric convergence rates, even with unknown diagonal covariances. Consider its application for more flexible clustering or density estimation tasks, noting the explicit component separation conditions required for recovery.

Key insights

BLASSO is extended to GMMs with unknown diagonal covariances, offering simultaneous parameter and component number estimation.

Principles

• Component separation is crucial for statistical guarantees.
• Fisher-Rao geometry informs parameter space analysis.
• Reparametrization normalizes kernels for theoretical rigor.

Method

The method involves a convex optimization framework, reparametrizing measures with a weighting function W(x), and constructing non-degenerate dual certificates based on a novel semi-distance.

In practice

• Apply BLASSO for GMM estimation when component count is unknown.
• Use distinct covariance matrices for quadratic decision boundaries in clustering.

Topics

• Gaussian Mixture Models
• Beurling-LASSO
• Sparse Regularization
• Covariance Estimation
• Non-asymptotic Analysis
• Fisher-Rao Geometry

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.