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Deep Optimal Transport for Domain Adaptation on SPD Manifolds

Ce Ju, Cuntai Guan

TL;DR

Domain shifts in SPD covariance data for EEG threaten cross-session generalization. DOT integrates a deep SPDNet encoder with joint optimal-transport losses under the Log-Euclidean metric to align both marginal and conditional distributions while preserving SPD geometry. The approach extends OT-DA to SPD manifolds with Brenier-style transport, introduces $\mathcal{L}_{MDA}$ and $\mathcal{L}_{CDA}$ losses within a unified DOT objective, and validates on three cross-session EEG datasets, showing improved accuracy and geometry retention. This geometry-aware, data-efficient transfer method has practical impact for robust EEG motor imagery and broad applicability to SPD-valued biomedical signals.

Abstract

Recent progress in geometric deep learning has drawn increasing attention from the machine learning community toward domain adaptation on symmetric positive definite (SPD) manifolds, especially for neuroimaging data that often suffer from distribution shifts across sessions. These data, typically represented as covariance matrices of brain signals, inherently lie on SPD manifolds due to their symmetry and positive definiteness. However, conventional domain adaptation methods often overlook this geometric structure when applied directly to covariance matrices, which can result in suboptimal performance. To address this issue, we introduce a new geometric deep learning framework that combines optimal transport theory with the geometry of SPD manifolds. Our approach aligns data distributions while respecting the manifold structure, effectively reducing both marginal and conditional discrepancies. We validate our method on three cross-session brain computer interface datasets, KU, BNCI2014001, and BNCI2015001, where it consistently outperforms baseline approaches while maintaining the intrinsic geometry of the data. We also provide quantitative results and visualizations to better illustrate the behavior of the learned embeddings.

Deep Optimal Transport for Domain Adaptation on SPD Manifolds

TL;DR

Domain shifts in SPD covariance data for EEG threaten cross-session generalization. DOT integrates a deep SPDNet encoder with joint optimal-transport losses under the Log-Euclidean metric to align both marginal and conditional distributions while preserving SPD geometry. The approach extends OT-DA to SPD manifolds with Brenier-style transport, introduces and losses within a unified DOT objective, and validates on three cross-session EEG datasets, showing improved accuracy and geometry retention. This geometry-aware, data-efficient transfer method has practical impact for robust EEG motor imagery and broad applicability to SPD-valued biomedical signals.

Abstract

Recent progress in geometric deep learning has drawn increasing attention from the machine learning community toward domain adaptation on symmetric positive definite (SPD) manifolds, especially for neuroimaging data that often suffer from distribution shifts across sessions. These data, typically represented as covariance matrices of brain signals, inherently lie on SPD manifolds due to their symmetry and positive definiteness. However, conventional domain adaptation methods often overlook this geometric structure when applied directly to covariance matrices, which can result in suboptimal performance. To address this issue, we introduce a new geometric deep learning framework that combines optimal transport theory with the geometry of SPD manifolds. Our approach aligns data distributions while respecting the manifold structure, effectively reducing both marginal and conditional discrepancies. We validate our method on three cross-session brain computer interface datasets, KU, BNCI2014001, and BNCI2015001, where it consistently outperforms baseline approaches while maintaining the intrinsic geometry of the data. We also provide quantitative results and visualizations to better illustrate the behavior of the learned embeddings.
Paper Structure (24 sections, 7 theorems, 23 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 24 sections, 7 theorems, 23 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. PRELIMINARY
  3. Riemannian Metric for SPD Manifolds
  4. Discrete Optimal Transport
  5. Joint Distribution Optimal Transport
  6. EEG-Based Motor Imagery Classifier on SPD Manifolds
  7. Methodology
  8. OT-DA Framework on ($\mathcal{S}_{++}$, $g^{LEM}$)
  9. Deep Optimal Transport
  10. EXPERIMENTS
  11. Deep Optimal Transport-Based Motor Imagery Classifier
  12. Evaluation Dataset
  13. Experimental Settings
  14. Evaluated Baselines
  15. Main Results
  16. ...and 9 more sections

Key Result

Lemma 1

Let $(\mathcal{M}, g)$ be a connected, compact, and $C^3-$smooth Riemannian manifold without a boundary. Consider a compact subset $\mathcal{U} \subset \mathcal{M}$ and a fixed point $q \in \mathcal{U}$. Let $f(p) = \frac{1}{2} d_g^2(p, q)$ denote the squared distance function. Then, for $p$ not on

Figures (4)

  • Figure 1: Illustration of Deep Optimal Transport: Multi-channel signals are first transformed into covariance matrices, situating these matrices on the space of ($\mathcal{S}_{++}$, $g^{LEM}$). The SPD matrix-valued data is processed through the BiMap, ReEig, and LogEig layers, moving it to the tangent space at the identity. The loss function takes into account the cross-entropy loss, marginal distribution adaptation, and conditional distribution adaptation.
  • Figure 2: Illustration of DOT-Based Motor-Imagery Classifier: The proposed neural networks $\varphi$ simultaneously transports the Fréchet means $\overline{w}(\mathcal{B}_{S})$ and $\overline{w}(\mathcal{B}_{T}$ of each batch of source and target samples within each frequency band to a common subspace $Z$ associated with that frequency band. Assumption \ref{['a2']} guarantees that all frequency components are equally weighted in the classification process. Assumption \ref{['a3']} ensures that the transformations applied to the source and target domains are restricted to occur within corresponding EEG frequency bands.
  • Figure 3: Illustrations for experimental settings on three datasets: (a). KU; (b). BNCI2014001; (c). BNCI2015001
  • Figure 4: Comparison of OT and DOT on 2 by 2 SPD matrices: Each colored ball represents a 2 by 2 SPD matrix. The left diagram illustrates the OT method, where the source distribution (red) is mapped to the target distribution (green), with blue lines indicating the transport paths between each pair. The right diagram depicts the proposed DOT approach but only considers the marginal distribution adaptation, where both the source and target distributions are mapped onto the subspaces (brown) respectively.

Theorems & Definitions (14)

  • Lemma 1
  • proof : Proof
  • Lemma 2: Brenier's Polar Factorization mccann2001polar
  • Theorem 3
  • Remark
  • Remark
  • Remark
  • Corollary 3.1
  • proof
  • Theorem 4
  • ...and 4 more