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

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 $\Omega(\Delta^{-(4k-4)})$ 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 $\mathcal{O}_p(n^{-1/2})$. Numerical experiments demonstrate competitive accuracy and lower computational cost than expectation-maximization across a range of multidimensional settings.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2603.19657 [stat.ML]

(or arXiv:2603.19657v2 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2603.19657 Focus to learn more arXiv-issued DOI via DataCite

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