# Tensor Moments of Gaussian Mixture Models: Theory and Applications

arXiv: https://arxiv.org/abs/2202.06930 · HTML: https://mlsanity.com/paper/2202.06930/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 38 references that are in the map.
- First posted on arXiv 14 Feb 2022.

## Builds on (most cited first, 7 of 7)

- [Learning Mixtures of Gaussians in High Dimensions](https://mlsanity.com/paper/1503.00424/) (2015)
- [Robust Learning of Mixtures of Gaussians](https://mlsanity.com/paper/2007.05912/) (2020)
- [Estimation under group actions: recovering orbits from invariants](https://mlsanity.com/paper/1712.10163/) (2017)
- [A Tensor Approach to Learning Mixed Membership Community Models](https://mlsanity.com/paper/1302.2684/) (2013)
- [Method of moments for 3-D single particle ab initio modeling with non-uniform distribution of viewing angles](https://mlsanity.com/paper/1907.05377/) (2019)
- [Tensor decomposition for learning Gaussian mixtures from moments](https://mlsanity.com/paper/2106.00555/) (2021)
- [Estimating Gaussian mixtures using sparse polynomial moment systems](https://mlsanity.com/paper/2106.15675/) (2021)

## Built on by (most cited first, 7 of 7)

- [Unraveling the Smoothness Properties of Diffusion Models: A Gaussian Mixture Perspective](https://mlsanity.com/paper/2405.16418/) (2024)
- [Geometry and Optimization of Shallow Polynomial Networks](https://mlsanity.com/paper/2501.06074/) (2025)
- [Moment Estimation for Nonparametric Mixture Models Through Implicit Tensor Decomposition](https://mlsanity.com/paper/2210.14386/) (2022)
- [Diagonally-Weighted Generalized Method of Moments Estimation for Gaussian Mixture Modeling](https://mlsanity.com/paper/2507.20459/) (2025)
- [Gaussian Variational Schemes on Bounded and Unbounded Domains](https://mlsanity.com/paper/2410.06219/) (2024)
- [Mixtures Closest to a Given Measure: A Semidefinite Programming Approach](https://mlsanity.com/paper/2509.22879/) (2025)
- [Data Understanding Survey: Pursuing Improved Dataset Characterization Via Tensor-based Methods](https://mlsanity.com/paper/2510.14161/) (2025)

## Abstract (arXiv)

Gaussian mixture models (GMMs) are fundamental tools in statistical and data sciences. We study the moments of multivariate Gaussians and GMMs. The d-th moment of an n-dimensional random variable is a symmetric d-way tensor of size nd, so working with moments naively is assumed to be prohibitively expensive for d>2 and larger values of n. In this work, we develop theory and numerical methods for implicit computations with moment tensors of GMMs, reducing the computational and storage costs to O(n2) and O(n3), respectively, for general covariance matrices, and to O(n) and O(n), respectively, for diagonal ones. We derive concise analytic expressions for the moments in terms of symmetrized tensor products, relying on the correspondence between symmetric tensors and homogeneous polynomials, and combinatorial identities involving Bell polynomials. The primary application of this theory is to estimating GMM parameters (means and covariances) from a set of observations, when formulated as a moment-matching optimization problem. If there is a known and common covariance matrix, we also show it is possible to debias the data observations, in which case the problem of estimating the unknown means reduces to symmetric CP tensor decomposition. Numerical results validate and illustrate the numerical efficiency of our approaches. This work potentially opens the door to the competitiveness of the method of moments as compared to expectation maximization methods for parameter estimation of GMMs.