# Moment Estimation for Nonparametric Mixture Models Through Implicit Tensor Decomposition

arXiv: https://arxiv.org/abs/2210.14386 · HTML: https://mlsanity.com/paper/2210.14386/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 73 references that are in the map.
- First posted on arXiv 25 Oct 2022.

## Builds on (most cited first, 9 of 9)

- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Pomegranate: fast and flexible probabilistic modeling in python](https://mlsanity.com/paper/1711.00137/) (2017)
- [Beyond the Low-Degree Algorithm: Mixtures of Subcubes and Their Applications](https://mlsanity.com/paper/1803.06521/) (2018)
- [Learning Mixtures of Smooth Product Distributions: Identifiability and Algorithm](https://mlsanity.com/paper/1904.01156/) (2019)
- [Learning Mixtures of Discrete Product Distributions using Spectral Decompositions](https://mlsanity.com/paper/1311.2972/) (2013)
- [Tensor Moments of Gaussian Mixture Models: Theory and Applications](https://mlsanity.com/paper/2202.06930/) (2022)
- [Tensor decomposition for learning Gaussian mixtures from moments](https://mlsanity.com/paper/2106.00555/) (2021)
- [Mahalanobis Distance for Class Averaging of Cryo-EM Images](https://mlsanity.com/paper/1611.03193/) (2016)
- [Estimation of the Number of Components of Non-Parametric Multivariate Finite Mixture Models](https://mlsanity.com/paper/1908.03656/) (2019)

## Built on by (most cited first, 4 of 4)

- [Geometry and Optimization of Shallow Polynomial Networks](https://mlsanity.com/paper/2501.06074/) (2025)
- [Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications](https://mlsanity.com/paper/2311.05116/) (2023)
- [Optimal Bias-variance Tradeoff in Matrix and Tensor Estimation](https://mlsanity.com/paper/2509.17382/) (2025)
- [Data Understanding Survey: Pursuing Improved Dataset Characterization Via Tensor-based Methods](https://mlsanity.com/paper/2510.14161/) (2025)

## Abstract (arXiv)

We present an alternating least squares type numerical optimization scheme to estimate conditionally-independent mixture models in ℝn, without parameterizing the distributions. Following the method of moments, we tackle an incomplete tensor decomposition problem to learn the mixing weights and componentwise means. Then we compute the cumulative distribution functions, higher moments and other statistics of the component distributions through linear solves. Crucially for computations in high dimensions, the steep costs associated with high-order tensors are evaded, via the development of efficient tensor-free operations. Numerical experiments demonstrate the competitive performance of the algorithm, and its applicability to many models and applications. Furthermore we provide theoretical analyses, establishing identifiability from low-order moments of the mixture and guaranteeing local linear convergence of the ALS algorithm.