# Wasserstein Discriminant Analysis

arXiv: https://arxiv.org/abs/1608.08063 · HTML: https://mlsanity.com/paper/1608.08063/

## Key facts

- Cited by 25 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 29 Aug 2016.

## Built on by (most cited first, 25 of 25)

- [Differentiable Particle Filtering via Entropy-Regularized Optimal Transport](https://mlsanity.com/paper/2102.07850/) (2021)
- [A Unified Framework for Implicit Sinkhorn Differentiation](https://mlsanity.com/paper/2205.06688/) (2022)
- [Learning to Predict Graphs with Fused Gromov-Wasserstein Barycenters](https://mlsanity.com/paper/2202.03813/) (2022)
- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution](https://mlsanity.com/paper/2103.06263/) (2021)
- [Hilbert Curve Projection Distance for Distribution Comparison](https://mlsanity.com/paper/2205.15059/) (2022)
- [The quest for the GRAph Level autoEncoder (GRALE)](https://mlsanity.com/paper/2505.22109/) (2025)
- [Importance Sparsification for Sinkhorn Algorithm](https://mlsanity.com/paper/2306.06581/) (2023)
- [Unsupervised Ground Metric Learning using Wasserstein Singular Vectors](https://mlsanity.com/paper/2102.06278/) (2021)
- [Nearly Tight Convergence Bounds for Semi-discrete Entropic Optimal Transport](https://mlsanity.com/paper/2110.12678/) (2021)
- [Diffeomorphic Measure Matching with Kernels for Generative Modeling](https://mlsanity.com/paper/2402.08077/) (2024)
- [Wasserstein t-SNE](https://mlsanity.com/paper/2205.07531/) (2022)
- [Muon Dynamics as a Spectral Wasserstein Flow](https://mlsanity.com/paper/2604.04891/) (2026)
- [OTFace: Hard Samples Guided Optimal Transport Loss for Deep Face Representation](https://mlsanity.com/paper/2203.14461/) (2022)
- [Entropic Wasserstein Component Analysis](https://mlsanity.com/paper/2303.05119/) (2023)
- [Manifold learning in metric spaces](https://mlsanity.com/paper/2503.16187/) (2025)
- [SOTAlign: Semi-Supervised Alignment of Unimodal Vision and Language Models via Optimal Transport](https://mlsanity.com/paper/2602.23353/) (2026)
- [A Unified Joint Maximum Mean Discrepancy for Domain Adaptation](https://mlsanity.com/paper/2101.09979/) (2021)
- [On Understanding and Mitigating the Dimensional Collapse of Graph Contrastive Learning: a Non-Maximum Removal Approach](https://mlsanity.com/paper/2203.12821/) (2022)
- [Differentiable Cost-Parameterized Monge Map Estimators](https://mlsanity.com/paper/2406.08399/) (2024)
- [ReLATE: Resilient Learner Selection for Multivariate Time-Series Classification Against Adversarial Attacks](https://mlsanity.com/paper/2503.07882/) (2025)
- [A Push-Relabel Based Additive Approximation for Optimal Transport](https://mlsanity.com/paper/2203.03732/) (2022)
- [A Bi-level Nonlinear Eigenvector Algorithm for Wasserstein Discriminant Analysis](https://mlsanity.com/paper/2211.11891/) (2022)
- [Canonical Variates in Wasserstein Metric Space](https://mlsanity.com/paper/2405.15768/) (2024)
- [Supervised Quadratic Feature Analysis: Information Geometry Approach for Dimensionality Reduction](https://mlsanity.com/paper/2502.00168/) (2025)

## Abstract (arXiv)

Wasserstein Discriminant Analysis (WDA) is a new supervised method that can improve classification of high-dimensional data by computing a suitable linear map onto a lower dimensional subspace. Following the blueprint of classical Linear Discriminant Analysis (LDA), WDA selects the projection matrix that maximizes the ratio of two quantities: the dispersion of projected points coming from different classes, divided by the dispersion of projected points coming from the same class. To quantify dispersion, WDA uses regularized Wasserstein distances, rather than cross-variance measures which have been usually considered, notably in LDA. Thanks to the the underlying principles of optimal transport, WDA is able to capture both global (at distribution scale) and local (at samples scale) interactions between classes. Regularized Wasserstein distances can be computed using the Sinkhorn matrix scaling algorithm; We show that the optimization of WDA can be tackled using automatic differentiation of Sinkhorn iterations. Numerical experiments show promising results both in terms of prediction and visualization on toy examples and real life datasets such as MNIST and on deep features obtained from a subset of the Caltech dataset.