# The Parametric Stability of Well-separated Spherical Gaussian Mixtures

arXiv: https://arxiv.org/abs/2302.00242 · HTML: https://mlsanity.com/paper/2302.00242/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 29 references that are in the map.
- First posted on arXiv 1 Feb 2023.

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

- [Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixtures](https://mlsanity.com/paper/1611.03473/) (2016)
- [Local Maxima in the Likelihood of Gaussian Mixture Models: Structural Results and Algorithmic Consequences](https://mlsanity.com/paper/1609.00978/) (2016)
- [Settling the Robust Learnability of Mixtures of Gaussians](https://mlsanity.com/paper/2011.03622/) (2020)
- [On Learning Mixtures of Well-Separated Gaussians](https://mlsanity.com/paper/1710.11592/) (2017)
- [A Bayesian information criterion for singular models](https://mlsanity.com/paper/1309.0911/) (2013)
- [The More, the Merrier: the Blessing of Dimensionality for Learning Large Gaussian Mixtures](https://mlsanity.com/paper/1311.2891/) (2013)
- [Model Selection for Gaussian Mixture Models](https://mlsanity.com/paper/1301.3558/) (2013)

## Abstract (arXiv)

We quantify the parameter stability of a spherical Gaussian Mixture Model (sGMM) under small perturbations in distribution space. Namely, we derive the first explicit bound to show that for a mixture of spherical Gaussian P (sGMM) in a pre-defined model class, all other sGMM close to P in this model class in total variation distance has a small parameter distance to P. Further, this upper bound only depends on P. The motivation for this work lies in providing guarantees for fitting Gaussian mixtures; with this aim in mind, all the constants involved are well defined and distribution free conditions for fitting mixtures of spherical Gaussians. Our results tighten considerably the existing computable bounds, and asymptotically match the known sharp thresholds for this problem.