# Estimating Gaussian mixtures using sparse polynomial moment systems

arXiv: https://arxiv.org/abs/2106.15675 · HTML: https://mlsanity.com/paper/2106.15675/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jun 2021.

## Built on by (most cited first, 5 of 5)

- [Tensor Moments of Gaussian Mixture Models: Theory and Applications](https://mlsanity.com/paper/2202.06930/) (2022)
- [Gaussian Joint Embeddings For Self-Supervised Representation Learning](https://mlsanity.com/paper/2603.26799/) (2026)
- [Diagonally-Weighted Generalized Method of Moments Estimation for Gaussian Mixture Modeling](https://mlsanity.com/paper/2507.20459/) (2025)
- [Mixtures Closest to a Given Measure: A Semidefinite Programming Approach](https://mlsanity.com/paper/2509.22879/) (2025)
- [Identifying parameter couplings and uncertainties of mixed-noise stochastic systems via full-covariance Gaussian mixture network](https://mlsanity.com/paper/2608.15198/) (2026)

## Abstract (arXiv)

The method of moments is a classical statistical technique for density estimation that solves a system of moment equations to estimate the parameters of an unknown distribution. A fundamental question critical to understanding identifiability asks how many moment equations are needed to get finitely many solutions and how many solutions there are. We answer this question for classes of Gaussian mixture models using the tools of polyhedral geometry. In addition, we show that a generic Gaussian k-mixture model is identifiable from its first 3k+2 moments. Using these results, we present a homotopy algorithm that performs parameter recovery for high dimensional Gaussian mixture models where the number of paths tracked scales linearly in the dimension.