Deep Linear Discriminant Analysis Revisited

TL;DR

This paper revisits deep linear discriminant analysis, identifying issues with maximum-likelihood training, and proposes the DNLL loss to improve class separation, calibration, and probabilistic interpretation.

Contribution

The paper introduces the Discriminative Negative Log-Likelihood (DNLL) loss, combining generative and discriminative training for deep LDA, improving calibration and class separation.

Findings

DNLL improves class separation and calibration.

Deep LDA with DNLL matches softmax accuracy on benchmarks.

DNLL yields more coherent probabilistic models.

Abstract

We show that for unconstrained Deep Linear Discriminant Analysis (LDA) classifiers, maximum-likelihood training admits pathological solutions in which class means drift together, covariances collapse, and the learned representation becomes almost non-discriminative. Conversely, cross-entropy training yields excellent accuracy but decouples the head from the underlying generative model, leading to highly inconsistent parameter estimates. To reconcile generative structure with discriminative performance, we introduce the \emph{Discriminative Negative Log-Likelihood} (DNLL) loss, which augments the LDA log-likelihood with a simple penalty on the mixture density. DNLL can be interpreted as standard LDA NLL plus a term that explicitly discourages regions where several classes are simultaneously likely. Deep LDA trained with DNLL produces clean, well-separated latent spaces, matches the test accuracy of softmax classifiers on synthetic data and standard image benchmarks, and yields substantially better calibrated predictive probabilities, restoring a coherent probabilistic interpretation to deep discriminant models.