Simplex Deep Linear Discriminant Analysis

Maxat Tezekbayev, Arman Bolatov, Zhenisbek Assylbekov
Conference on Parsimony and Learning, PMLR 328:957-967, 2026.

Abstract

We revisit Deep Linear Discriminant Analysis (Deep LDA) from a likelihood-based perspective. While classical LDA is a simple Gaussian model with linear decision boundaries, attaching an LDA head to a neural encoder raises the question of how to train the resulting deep classifier by maximum likelihood estimation (MLE). We first show that end-to-end MLE training of an unconstrained Deep LDA model ignores discrimination: when both the LDA parameters and the encoder parameters are learned jointly, the likelihood admits a degenerate solution in which some of the class clusters may heavily overlap or even collapse, and classification performance deteriorates. Batchwise moment re-estimation of the LDA parameters does not remove this failure mode. We then propose a constrained Deep LDA formulation that fixes the class means to the vertices of a regular simplex in the latent space and restricts the shared covariance to be spherical, leaving only the priors and a single variance parameter to be learned along with the encoder. Under these geometric constraints, MLE becomes stable and yields well-separated class clusters in the latent space. On images (Fashion-MNIST, CIFAR-10, CIFAR-100) and texts (AG News, CLINC150), the resulting Deep LDA models achieve accuracy competitive with softmax baselines while offering a simple, interpretable latent geometry that is clearly visible in two-dimensional projections.

Cite this Paper

BibTeX
@InProceedings{pmlr-v328-tezekbayev26a, title = {Simplex Deep Linear Discriminant Analysis}, author = {Tezekbayev, Maxat and Bolatov, Arman and Assylbekov, Zhenisbek}, booktitle = {Conference on Parsimony and Learning}, pages = {957--967}, year = {2026}, editor = {Burkholz, Rebekka and Liu, Shiwei and Ravishankar, Saiprasad and Redman, William and Huang, Wei and Su, Weijie and Zhu, Zhihui}, volume = {328}, series = {Proceedings of Machine Learning Research}, month = {23--26 Mar}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v328/main/assets/tezekbayev26a/tezekbayev26a.pdf}, url = {https://proceedings.mlr.press/v328/tezekbayev26a.html}, abstract = {We revisit Deep Linear Discriminant Analysis (Deep LDA) from a likelihood-based perspective. While classical LDA is a simple Gaussian model with linear decision boundaries, attaching an LDA head to a neural encoder raises the question of how to train the resulting deep classifier by maximum likelihood estimation (MLE). We first show that end-to-end MLE training of an unconstrained Deep LDA model ignores discrimination: when both the LDA parameters and the encoder parameters are learned jointly, the likelihood admits a degenerate solution in which some of the class clusters may heavily overlap or even collapse, and classification performance deteriorates. Batchwise moment re-estimation of the LDA parameters does not remove this failure mode. We then propose a constrained Deep LDA formulation that fixes the class means to the vertices of a regular simplex in the latent space and restricts the shared covariance to be spherical, leaving only the priors and a single variance parameter to be learned along with the encoder. Under these geometric constraints, MLE becomes stable and yields well-separated class clusters in the latent space. On images (Fashion-MNIST, CIFAR-10, CIFAR-100) and texts (AG News, CLINC150), the resulting Deep LDA models achieve accuracy competitive with softmax baselines while offering a simple, interpretable latent geometry that is clearly visible in two-dimensional projections.} }
Endnote
%0 Conference Paper %T Simplex Deep Linear Discriminant Analysis %A Maxat Tezekbayev %A Arman Bolatov %A Zhenisbek Assylbekov %B Conference on Parsimony and Learning %C Proceedings of Machine Learning Research %D 2026 %E Rebekka Burkholz %E Shiwei Liu %E Saiprasad Ravishankar %E William Redman %E Wei Huang %E Weijie Su %E Zhihui Zhu %F pmlr-v328-tezekbayev26a %I PMLR %P 957--967 %U https://proceedings.mlr.press/v328/tezekbayev26a.html %V 328 %X We revisit Deep Linear Discriminant Analysis (Deep LDA) from a likelihood-based perspective. While classical LDA is a simple Gaussian model with linear decision boundaries, attaching an LDA head to a neural encoder raises the question of how to train the resulting deep classifier by maximum likelihood estimation (MLE). We first show that end-to-end MLE training of an unconstrained Deep LDA model ignores discrimination: when both the LDA parameters and the encoder parameters are learned jointly, the likelihood admits a degenerate solution in which some of the class clusters may heavily overlap or even collapse, and classification performance deteriorates. Batchwise moment re-estimation of the LDA parameters does not remove this failure mode. We then propose a constrained Deep LDA formulation that fixes the class means to the vertices of a regular simplex in the latent space and restricts the shared covariance to be spherical, leaving only the priors and a single variance parameter to be learned along with the encoder. Under these geometric constraints, MLE becomes stable and yields well-separated class clusters in the latent space. On images (Fashion-MNIST, CIFAR-10, CIFAR-100) and texts (AG News, CLINC150), the resulting Deep LDA models achieve accuracy competitive with softmax baselines while offering a simple, interpretable latent geometry that is clearly visible in two-dimensional projections.
APA
Tezekbayev, M., Bolatov, A. & Assylbekov, Z.. (2026). Simplex Deep Linear Discriminant Analysis. Conference on Parsimony and Learning, in Proceedings of Machine Learning Research 328:957-967 Available from https://proceedings.mlr.press/v328/tezekbayev26a.html.

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