Logistic lasso regression with nearest neighbors for gradient-based dimension reduction

TL;DR

This paper introduces a novel gradient estimation method using localized nearest-neighbor logistic regression with l1-penalty for high-dimensional binary classification, enabling effective dimension reduction and outperforming existing methods.

Contribution

It proposes a new gradient estimation technique with theoretical optimal convergence rates and a practical dimension reduction approach using cross-validation.

Findings

Achieves optimal convergence rate for gradient estimation.

Effectively estimates the central subspace for dimension reduction.

Outperforms existing methods in synthetic and real data experiments.

Abstract

This paper investigates a new approach to estimate the gradient of the conditional probability given the covariates in the binary classification framework. The proposed approach consists in fitting a localized nearest-neighbor logistic model with $\ell_1$-penalty in order to cope with possibly high-dimensional covariates. Our theoretical analysis shows that the pointwise convergence rate of the gradient estimator is optimal under very mild conditions. Moreover, using an outer product of such gradient estimates at several points in the covariate space, we establish the rate of convergence for estimating the so-called central subspace, a well-known object allowing to carry out dimension reduction within the covariate space. Our implementation uses cross-validation on the misclassification rate to estimate the dimension of this subspace. We find that the proposed approach outperforms existing competitors in synthetic and real data applications.