Statistical Inference in Classification of High-dimensional Gaussian Mixture

• URL: http://arxiv.org/abs/2410.19950v1
• Date: Fri, 25 Oct 2024 19:58:36 GMT
• Title: Statistical Inference in Classification of High-dimensional Gaussian Mixture
• Authors: Hanwen Huang, Peng Zeng,
• Abstract summary: We investigate the behavior of a general class of regularized convex classifiers in the high-dimensional limit. Our focus is on the generalization error and variable selection properties of the estimators.
• Score: 1.2354076490479515
• License:
• 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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