# Learning Gaussian Mixtures Using the Wasserstein-Fisher-Rao Gradient Flow arXiv: https://arxiv.org/abs/2301.01766 ยท HTML: https://mlsanity.com/paper/2301.01766/ ## Key facts - Cited by 27 papers in the map (data through 14 Oct 2026). - Builds on 14 of its 36 references that are in the map. - First posted on arXiv 4 Jan 2023. ## Builds on (most cited first, 14 of 14) - [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018) - [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014) - [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (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) - [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021) - [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022) - [Accelerating Langevin Sampling with Birth-death](https://mlsanity.com/paper/1905.09863/) (2019) - [Local Maxima in the Likelihood of Gaussian Mixture Models: Structural Results and Algorithmic Consequences](https://mlsanity.com/paper/1609.00978/) (2016) - [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021) - [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022) - [On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising](https://mlsanity.com/paper/1712.02009/) (2017) - [Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Models](https://mlsanity.com/paper/2008.08244/) (2020) - [On Efficient and Scalable Computation of the Nonparametric Maximum Likelihood Estimator in Mixture Models](https://mlsanity.com/paper/2208.07514/) (2022) ## Built on by (most cited first, 27 of 27) - [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025) - [An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2302.12229/) (2023) - [Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities](https://mlsanity.com/paper/2407.15693/) (2024) - [Weighted quantization using MMD: From mean field to mean shift via gradient flows](https://mlsanity.com/paper/2502.10600/) (2025) - [Stein transport for Bayesian inference](https://mlsanity.com/paper/2409.01464/) (2024) - [Interaction-Force Transport Gradient Flows](https://mlsanity.com/paper/2405.17075/) (2024) - [Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2406.17263/) (2024) - [Wasserstein Gradient Flows for Optimizing Gaussian Mixture Policies](https://mlsanity.com/paper/2305.10411/) (2023) - [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023) - [Particle Semi-Implicit Variational Inference](https://mlsanity.com/paper/2407.00649/) (2024) - [Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals](https://mlsanity.com/paper/2501.17049/) (2025) - [Estimating the Rate-Distortion Function by Wasserstein Gradient Descent](https://mlsanity.com/paper/2310.18908/) (2023) - [Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2506.05905/) (2025) - [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023) - [Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamics](https://mlsanity.com/paper/2411.00214/) (2024) - [A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows](https://mlsanity.com/paper/2507.04330/) (2025) - [Multi-Objective Optimization via Wasserstein-Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2311.13159/) (2023) - [A Particle Algorithm for Mean-Field Variational Inference](https://mlsanity.com/paper/2412.20385/) (2024) - [Big Learning Expectation Maximization](https://mlsanity.com/paper/2312.11926/) (2023) - [Cramer Type Distances for Learning Gaussian Mixture Models by Gradient Descent](https://mlsanity.com/paper/2307.06753/) (2023) - [Geometry-Aware Instrumental Variable Regression](https://mlsanity.com/paper/2405.11633/) (2024) - [Computational and Statistical Asymptotic Analysis of the JKO Scheme for Iterative Algorithms to update distributions](https://mlsanity.com/paper/2501.06408/) (2025) - [On Model-Based Clustering With Entropic Optimal Transport](https://mlsanity.com/paper/2605.03240/) (2026) - [Model-free Estimation of Latent Structure via Multiscale Nonparametric Maximum Likelihood](https://mlsanity.com/paper/2410.22248/) (2024) - [Gaussian mixture layers for neural networks](https://mlsanity.com/paper/2508.04883/) (2025) - [Learning Relationship between Quantum Walks and Underdamped Langevin Dynamics](https://mlsanity.com/paper/2601.01589/) (2026) - [Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation](https://mlsanity.com/paper/2610.19588/) (2026) ## Abstract (arXiv) Gaussian mixture models form a flexible and expressive parametric family of distributions that has found applications in a wide variety of applications. Unfortunately, fitting these models to data is a notoriously hard problem from a computational perspective. Currently, only moment-based methods enjoy theoretical guarantees while likelihood-based methods are dominated by heuristics such as Expectation-Maximization that are known to fail in simple examples. In this work, we propose a new algorithm to compute the nonparametric maximum likelihood estimator (NPMLE) in a Gaussian mixture model. Our method is based on gradient descent over the space of probability measures equipped with the Wasserstein-Fisher-Rao geometry for which we establish convergence guarantees. In practice, it can be approximated using an interacting particle system where the weight and location of particles are updated alternately. We conduct extensive numerical experiments to confirm the effectiveness of the proposed algorithm compared not only to classical benchmarks but also to similar gradient descent algorithms with respect to simpler geometries. In particular, these simulations illustrate the benefit of updating both weight and location of the interacting particles.