Fast rate of convergence in high dimensional linear discriminant analysis

TL;DR

This paper provides a theoretical analysis of high-dimensional linear discriminant analysis, highlighting limitations of standard methods and proposing new procedures with improved convergence rates under sparsity.

Contribution

It introduces two new discrimination procedures based on dimensionality reduction with proven convergence rates, improving understanding of high-dimensional Gaussian classification.

Findings

Standard procedures perform poorly when p > n

Proposed methods achieve O(log(p)/n) convergence under sparsity

Theoretical bounds relate excess risk to geometric estimation errors

Abstract

This paper gives a theoretical analysis of high dimensional linear discrimination of Gaussian data. We study the excess risk of linear discriminant rules. We emphasis on the poor performances of standard procedures in the case when dimension p is larger than sample size n. The corresponding theoretical results are non asymptotic lower bounds. On the other hand, we propose two discrimination procedures based on dimensionality reduction and provide associated rates of convergence which can be O(log(p)/n) under sparsity assumptions. Finally all our results rely on a theorem that provides simple sharp relations between the excess risk and an estimation error associated to the geometric parameters defining the used discrimination rule.