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

arXiv:2603.19657v2 Announce Type: replace-cross 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 O_p(n^{-1/2}). Numerical experiments demonstrate competitive accuracy and lower computational cost than expectation-maximization across a range of multidimensional settings.

Source: arXiv cs.LG | 2026-09-01