# Model Selection and Parameter Estimation of One-Dimensional Gaussian Mixture Models

arXiv: https://arxiv.org/abs/2404.12613 · HTML: https://mlsanity.com/paper/2404.12613/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 49 references that are in the map.
- First posted on arXiv 19 Apr 2024.

## Builds on (most cited first, 10 of 10)

- [Mixture Models, Robustness, and Sum of Squares Proofs](https://mlsanity.com/paper/1711.07454/) (2017)
- [List-Decodable Robust Mean Estimation and Learning Mixtures of Spherical Gaussians](https://mlsanity.com/paper/1711.07211/) (2017)
- [On Learning Mixtures of Well-Separated Gaussians](https://mlsanity.com/paper/1710.11592/) (2017)
- [Tight bounds for learning a mixture of two gaussians](https://mlsanity.com/paper/1404.4997/) (2014)
- [Clustering Mixtures with Almost Optimal Separation in Polynomial Time](https://mlsanity.com/paper/2112.00706/) (2021)
- [Cutoff for exact recovery of Gaussian mixture models](https://mlsanity.com/paper/2001.01194/) (2020)
- [Optimal estimation of high-dimensional location Gaussian mixtures](https://mlsanity.com/paper/2002.05818/) (2020)
- [Statistically Optimal K-means Clustering via Nonnegative Low-rank Semidefinite Programming](https://mlsanity.com/paper/2305.18436/) (2023)
- [A Fourier Approach to Mixture Learning](https://mlsanity.com/paper/2210.02415/) (2022)
- [Mathematical Theory of Computational Resolution Limit in Multi-dimensions](https://mlsanity.com/paper/2103.11632/) (2021)

## Abstract (arXiv)

In this paper, we study the problem of learning one-dimensional Gaussian mixture models (GMMs) with a specific focus on estimating both the model order and the mixing distribution from independent and identically distributed (i.i.d.) samples. This paper establishes the optimal sampling complexity for model order estimation in one-dimensional Gaussian mixture models. We prove a fundamental lower bound on the number of samples required to correctly identify the number of components with high probability, showing that this limit depends critically on the separation between component means and the total number of components. We then propose a Fourier-based approach to estimate both the model order and the mixing distribution. Our algorithm utilizes Fourier measurements constructed from the samples, and our analysis demonstrates that its sample complexity matches the established lower bound, thereby confirming its optimality. Numerical experiments further show that our method outperforms conventional techniques in terms of efficiency and accuracy.