# Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signals

arXiv: https://arxiv.org/abs/2303.05798 · HTML: https://mlsanity.com/paper/2303.05798/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 85 references that are in the map.
- First posted on arXiv 10 Mar 2023.

## Builds on (most cited first, 22 of 22)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Learning with minibatch Wasserstein : asymptotic and gradient properties](https://mlsanity.com/paper/1910.04091/) (2019)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [Sliced Wasserstein Kernels for Probability Distributions](https://mlsanity.com/paper/1511.03198/) (2015)
- [HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections](https://mlsanity.com/paper/2106.03306/) (2021)
- [Asymptotic Guarantees for Learning Generative Models with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1906.04516/) (2019)
- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods](https://mlsanity.com/paper/1605.06182/) (2016)
- [Minimax Confidence Intervals for the Sliced Wasserstein Distance](https://mlsanity.com/paper/1909.07862/) (2019)
- [Distribution Regression with Sliced Wasserstein Kernels](https://mlsanity.com/paper/2202.03926/) (2022)
- [Manifold-regression to predict from MEG/EEG brain signals without source modeling](https://mlsanity.com/paper/1906.02687/) (2019)
- [Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections](https://mlsanity.com/paper/2211.10066/) (2022)
- [Unsupervised Hierarchy Matching with Optimal Transport over Hyperbolic Spaces](https://mlsanity.com/paper/1911.02536/) (2019)
- [Intrinsic Sliced Wasserstein Distances for Comparing Collections of Probability Distributions on Manifolds and Graphs](https://mlsanity.com/paper/2010.15285/) (2020)
- [Statistical and Topological Properties of Gaussian Smoothed Sliced Probability Divergences](https://mlsanity.com/paper/2110.10524/) (2021)
- [Fast and Accurate Multiclass Inference for MI-BCIs Using Large Multiscale Temporal and Spectral Features](https://mlsanity.com/paper/1806.06823/) (2018)

## Built on by (most cited first, 19 of 19)

- [Sliced Wasserstein with Random-Path Projecting Directions](https://mlsanity.com/paper/2401.15889/) (2024)
- [Geodesic Optimization for Predictive Shift Adaptation on EEG data](https://mlsanity.com/paper/2407.03878/) (2024)
- [Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions](https://mlsanity.com/paper/2404.15378/) (2024)
- [SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges](https://mlsanity.com/paper/2504.18882/) (2025)
- [Neural Networks on Symmetric Spaces of Noncompact Type](https://mlsanity.com/paper/2601.01097/) (2026)
- [Fast Estimation of Wasserstein Distances via Regression on Sliced Wasserstein Distances](https://mlsanity.com/paper/2509.20508/) (2025)
- [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025)
- [Summarizing Bayesian Nonparametric Mixture Posterior -- Sliced Optimal Transport Metrics for Gaussian Mixtures](https://mlsanity.com/paper/2411.14674/) (2024)
- [Schur's Positive-Definite Network: Deep Learning in the SPD cone with structure](https://mlsanity.com/paper/2406.09023/) (2024)
- [Learning to Embed Distributions via Maximum Kernel Entropy](https://mlsanity.com/paper/2408.00549/) (2024)
- [Weakly supervised covariance matrices alignment through Stiefel matrices estimation for MEG applications](https://mlsanity.com/paper/2402.03345/) (2024)
- [EEG-MACS: Manifold Attention and Confidence Stratification for EEG-based Cross-Center Brain Disease Diagnosis under Unreliable Annotations](https://mlsanity.com/paper/2405.00734/) (2024)
- [EEG-ReMinD: Enhancing Neurodegenerative EEG Decoding through Self-Supervised State Reconstruction-Primed Riemannian Dynamics](https://mlsanity.com/paper/2501.08139/) (2025)
- [Brain Age Revisited: Investigating the State vs. Trait Hypotheses of EEG-derived Brain-Age Dynamics with Deep Learning](https://mlsanity.com/paper/2310.07029/) (2023)
- [Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing](https://mlsanity.com/paper/2510.04579/) (2025)
- [Revisiting the Sliced Wasserstein Kernel for persistence diagrams: a Figalli-Gigli approach](https://mlsanity.com/paper/2602.06539/) (2026)
- [Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty](https://mlsanity.com/paper/2603.17271/) (2026)
- [A Sliced-Wasserstein Framework on Correlation Matrices for EEG Decoding](https://mlsanity.com/paper/2606.06104/) (2026)
- [Geometric Feature Learning for Functional Data Valued on the Symmetric Positive Definite Manifold](https://mlsanity.com/paper/2609.30487/) (2026)

## Abstract (arXiv)

When dealing with electro or magnetoencephalography records, many supervised prediction tasks are solved by working with covariance matrices to summarize the signals. Learning with these matrices requires using Riemanian geometry to account for their structure. In this paper, we propose a new method to deal with distributions of covariance matrices and demonstrate its computational efficiency on M/EEG multivariate time series. More specifically, we define a Sliced-Wasserstein distance between measures of symmetric positive definite matrices that comes with strong theoretical guarantees. Then, we take advantage of its properties and kernel methods to apply this distance to brain-age prediction from MEG data and compare it to state-of-the-art algorithms based on Riemannian geometry. Finally, we show that it is an efficient surrogate to the Wasserstein distance in domain adaptation for Brain Computer Interface applications.