Linear Discriminant Analysis with Gradient Optimization

TL;DR

This paper introduces LDA-GO, a scalable gradient-based method for linear discriminant analysis that adapts to data structure and performs well in high-dimensional, sparse-signal settings.

Contribution

LDA-GO is a novel gradient optimization approach that learns a low-rank precision matrix, automatically selects loss functions, and has proven theoretical properties.

Findings

LDA-GO outperforms other LDA variants in high-dimensional simulations.

The method maintains linear per-iteration cost, suitable for large-scale data.

Theoretical guarantees include convexity, Bayes-optimality, and finite-sample error bounds.

Abstract

Linear discriminant analysis (LDA) is a fundamental classification and dimension reduction method that achieves Bayes optimality under Gaussian mixture, but often struggles in high-dimensional settings where the covariance matrix cannot be reliably estimated. We propose LDA with gradient optimization (LDA-GO), which learns a low-rank precision matrix via scalable gradient-based optimization. The method automatically selects between a Gaussian likelihood and a cross-entropy loss using data-driven structural diagnostics, adapting to the signal structure without user tuning. The gradient computation avoids any quadratic-sized intermediate matrix, keeping the per-iteration cost linear in the number of dimensions. Theoretically, we prove several properties of the method, including the convexity of the objective functions, Bayes-optimality of the method, and a finite-sample bound of the excess error. Numerically, we conducted a variety of simulations and real data experiments to show that LDA-GO wins a majority of settings among other LDA variants, particularly in sparse-signal high-dimensional regimes.