aiqbench analysis
This research advances statistical learning theory by proposing efficient estimators for Gaussian mixture models using spectral methods and gradient descent, offering improved computational costs over traditional expectation-maximization algorithms. The work provides rigorous minimax bounds and convergence guarantees that are directly relevant to practitioners developing robust machine learning pipelines for high-dimensional data analysis. While the mathematical framework is foundational, it does not yet address specific AI applications or quantum computing architectures.
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 $\mathcal{O}_p(n^{-1/2})$. Numerical experiments demonstrate competitive accuracy and lower computational cost than expectation-maximization across a range of multidimensional settings.
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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Full text under the publisher's open licence · 253 words
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