# Local Minima Structures in Gaussian Mixture Models

arXiv: https://arxiv.org/abs/2009.13040 · HTML: https://mlsanity.com/paper/2009.13040/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2020.

## Built on by (most cited first, 9 of 9)

- [A theoretical perspective on mode collapse in variational inference](https://mlsanity.com/paper/2410.13300/) (2024)
- [Unveiling the Cycloid Trajectory of EM Iterations in Mixed Linear Regression](https://mlsanity.com/paper/2405.18237/) (2024)
- [Jigsaw Game: Federated Clustering](https://mlsanity.com/paper/2407.12764/) (2024)
- [Scalable Clustering: Large Scale Unsupervised Learning of Gaussian Mixture Models with Outliers](https://mlsanity.com/paper/2302.14599/) (2023)
- [Learning Overspecified Gaussian Mixtures Exponentially Fast with the EM Algorithm](https://mlsanity.com/paper/2506.11850/) (2025)
- [Overspecified Mixture Discriminant Analysis: Exponential Convergence, Statistical Guarantees, and Remote Sensing Applications](https://mlsanity.com/paper/2510.27056/) (2025)
- [Learning Mixture Density via Natural Gradient Expectation Maximization](https://mlsanity.com/paper/2602.10602/) (2026)
- [The elbow statistic: Multiscale clustering statistical significance](https://mlsanity.com/paper/2603.03235/) (2026)
- [Local linear convergence of gradient methods for overparameterized Gaussian mixtures](https://mlsanity.com/paper/2605.30936/) (2026)

## Abstract (arXiv)

We investigate the landscape of the negative log-likelihood function of Gaussian Mixture Models (GMMs) with a general number of components in the population limit. As the objective function is non-convex, there can be multiple local minima that are not globally optimal, even for well-separated mixture models. Our study reveals that all local minima share a common structure that partially identifies the cluster centers (i.e., means of the Gaussian components) of the true location mixture. Specifically, each local minimum can be represented as a non-overlapping combination of two types of sub-configurations: fitting a single mean estimate to multiple Gaussian components or fitting multiple estimates to a single true component. These results apply to settings where the true mixture components satisfy a certain separation condition, and are valid even when the number of components is over- or under-specified. We also present a more fine-grained analysis for the setting of one-dimensional GMMs with three components, which provide sharper approximation error bounds with improved dependence on the separation.