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

arXiv:2603.19657 · stat.ML, cs.LG · Submitted 2026-03-20 · Read on arXiv

stat.ML, cs.LG

Submitted: 2026-03-20

Updated: 2026-08-31

License: http://creativecommons.org/licenses/by/4.0/

The gist: We study model-order selection and component-mean estimation for multidimensional Gaussian mixture models with a known common covariance matrix.

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Abstract

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 Ω(Δ-(4k-4)) samples. We then develop an oracle spectral-thresholding estimator with a sufficient sample size of order Δ-(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 O p(n-1/2). Numerical experiments demonstrate competitive accuracy and lower computational cost than expectation-maximization across a range of multidimensional settings.

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