# Riemannian Gradient Descent for Gaussian Mixture Models with unknown diagonal covariances

arXiv: https://arxiv.org/abs/2609.30220 · HTML: https://mlsanity.com/paper/2609.30220/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 30 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 12 of 12)

- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [High-dimensional limit theorems for SGD: Effective dynamics and critical scaling](https://mlsanity.com/paper/2206.04030/) (2022)
- [Sparse Optimization on Measures with Over-parameterized Gradient Descent](https://mlsanity.com/paper/1907.10300/) (2019)
- [Exact Support Recovery for Sparse Spikes Deconvolution](https://mlsanity.com/paper/1306.6909/) (2013)
- [The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy](https://mlsanity.com/paper/1811.06416/) (2018)
- [FastPart: Over-Parameterized Stochastic Gradient Descent for Sparse optimisation on Measures](https://mlsanity.com/paper/2312.05993/) (2023)
- [Gaussian Mixture Model with unknown diagonal covariances via continuous sparse regularization](https://mlsanity.com/paper/2509.12889/) (2025)
- [Fast Spawn&Prune (FS&P): Global convergence of stochastic conic particle gradient descent via birth/death process](https://mlsanity.com/paper/2605.19784/) (2026)
- [Fenchel-Young Duality Gaps: Certified Early Stopping for Regularized Inverse Problems](https://mlsanity.com/paper/2609.17629/) (2026)

## Abstract (arXiv)

This paper investigates the numerical resolution of the Beurling-LASSO (BLASSO), a convex optimization framework that promotes sparsity in the space of measures. We consider its application to the estimation of Gaussian mixture models (GMMs) with an unknown number of components and unknown diagonal covariance matrices. Our approach combines the Conic Particle Gradient Descent (CPGD) principle with Riemannian gradient descent, to account for the underlying Fisher-Rao geometry of Gaussian distributions. Our contributions are twofold. First, we provide theoretical guarantees for the convergence of our algorithm. In particular, we establish exponential local convergence under a non-degeneracy condition on the solution and relate this assumption to a separation condition on the underlying statistical target. Second, we address practical implementation aspects of CPGD and present numerical experiments illustrating its performance. On the test cases considered, these experiments suggest that CPGD is more robust to overspecification of the number of components than the EM algorithm. We also investigate the impact of component separation on recovery accuracy.