Statistical Inference in Classification of High-dimensional Gaussian Mixture

Hanwen Huang; Peng Zeng

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arXiv · 2410.19950

Statistical Inference in Classification of High-dimensional Gaussian Mixture

Hanwen Huang·Peng Zeng

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Abstract

We consider the classification problem of a high-dimensional mixture of two Gaussians with general covariance matrices. Using the replica method from statistical physics, we investigate the asymptotic behavior of a general class of regularized convex classifiers in the high-dimensional limit, where both the sample size $n$ and the dimension $p$ approach infinity while their ratio $\alpha=n/p$ remains fixed. Our focus is on the generalization error and variable selection properties of the estimators. Specifically, based on the distributional limit of the classifier, we construct a de-biased estimator to perform variable selection through an appropriate hypothesis testing procedure. Using $L_1$-regularized logistic regression as an example, we conducted extensive computational experiments to confirm that our analytical findings are consistent with numerical simulations in finite-sized systems. We also explore the influence of the covariance structure on the performance of the de-biased estimator.

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Hanwen Huang, Peng Zeng. 2024-10-25. Statistical Inference in Classification of High-dimensional Gaussian Mixture. https://arxiv.org/abs/2410.19950

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