# Model Selection and Parameter Estimation for Multidimensional Gaussian Mixture Models with a Common Covariance Matrix

arXiv: https://arxiv.org/abs/2603.19657 · HTML: https://mlsanity.com/paper/2603.19657/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 23 references that are in the map.
- First posted on arXiv 20 Mar 2026.

## Builds on (most cited first, 2 of 2)

- [On Learning Mixtures of Well-Separated Gaussians](https://mlsanity.com/paper/1710.11592/) (2017)
- [Optimal estimation of high-dimensional location Gaussian mixtures](https://mlsanity.com/paper/2002.05818/) (2020)

## Abstract (arXiv)

We study model-order selection and component-mean estimation for multidimensional Gaussian mixture models with a known common covariance matrix. Using empirical characteristic-function measurements, we construct Fourier covariance matrices whose population counterparts have rank equal to the number of mixture components. We establish a minimax lower bound showing that distinguishing a separated k-component mixture from the class of (k-1)-component mixtures requires Omega(Delta-(4k-4)) samples. We then develop an oracle spectral-thresholding estimator with a sufficient sample size of order Delta-(8k-8) for fixed k, together with a practical singular-value-ratio estimator. Given the model order, we estimate the component means by score-initialized gradient descent on a MUSIC-type projection objective. Under an explicit sample-size condition, a qualifying sample initialization lies in a certified attraction region with high probability, after which the iterates converge linearly. For fixed positive component separation, the resulting mean estimates achieve the parametric rate Op(n-1/2). Numerical experiments demonstrate competitive accuracy and lower computational cost than expectation-maximization across a range of multidimensional settings.