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Geodesic Optimization for Predictive Shift Adaptation on EEG data

Apolline Mellot, Antoine Collas, Sylvain Chevallier, Alexandre Gramfort, Denis A. Engemann

TL;DR

This paper proposes a novel method termed Geodesic Optimization for Predictive Shift Adaptation (GOPSA), which exploits the geodesic structure of the Riemannian manifold to jointly learn a domain-specific re-centering operator representing site-specific intercepts and the regression model.

Abstract

Electroencephalography (EEG) data is often collected from diverse contexts involving different populations and EEG devices. This variability can induce distribution shifts in the data $X$ and in the biomedical variables of interest $y$, thus limiting the application of supervised machine learning (ML) algorithms. While domain adaptation (DA) methods have been developed to mitigate the impact of these shifts, such methods struggle when distribution shifts occur simultaneously in $X$ and $y$. As state-of-the-art ML models for EEG represent the data by spatial covariance matrices, which lie on the Riemannian manifold of Symmetric Positive Definite (SPD) matrices, it is appealing to study DA techniques operating on the SPD manifold. This paper proposes a novel method termed Geodesic Optimization for Predictive Shift Adaptation (GOPSA) to address test-time multi-source DA for situations in which source domains have distinct $y$ distributions. GOPSA exploits the geodesic structure of the Riemannian manifold to jointly learn a domain-specific re-centering operator representing site-specific intercepts and the regression model. We performed empirical benchmarks on the cross-site generalization of age-prediction models with resting-state EEG data from a large multi-national dataset (HarMNqEEG), which included $14$ recording sites and more than $1500$ human participants. Compared to state-of-the-art methods, our results showed that GOPSA achieved significantly higher performance on three regression metrics ($R^2$, MAE, and Spearman's $ρ$) for several source-target site combinations, highlighting its effectiveness in tackling multi-source DA with predictive shifts in EEG data analysis. Our method has the potential to combine the advantages of mixed-effects modeling with machine learning for biomedical applications of EEG, such as multicenter clinical trials.

Geodesic Optimization for Predictive Shift Adaptation on EEG data

TL;DR

This paper proposes a novel method termed Geodesic Optimization for Predictive Shift Adaptation (GOPSA), which exploits the geodesic structure of the Riemannian manifold to jointly learn a domain-specific re-centering operator representing site-specific intercepts and the regression model.

Abstract

Electroencephalography (EEG) data is often collected from diverse contexts involving different populations and EEG devices. This variability can induce distribution shifts in the data and in the biomedical variables of interest , thus limiting the application of supervised machine learning (ML) algorithms. While domain adaptation (DA) methods have been developed to mitigate the impact of these shifts, such methods struggle when distribution shifts occur simultaneously in and . As state-of-the-art ML models for EEG represent the data by spatial covariance matrices, which lie on the Riemannian manifold of Symmetric Positive Definite (SPD) matrices, it is appealing to study DA techniques operating on the SPD manifold. This paper proposes a novel method termed Geodesic Optimization for Predictive Shift Adaptation (GOPSA) to address test-time multi-source DA for situations in which source domains have distinct distributions. GOPSA exploits the geodesic structure of the Riemannian manifold to jointly learn a domain-specific re-centering operator representing site-specific intercepts and the regression model. We performed empirical benchmarks on the cross-site generalization of age-prediction models with resting-state EEG data from a large multi-national dataset (HarMNqEEG), which included recording sites and more than human participants. Compared to state-of-the-art methods, our results showed that GOPSA achieved significantly higher performance on three regression metrics (, MAE, and Spearman's ) for several source-target site combinations, highlighting its effectiveness in tackling multi-source DA with predictive shifts in EEG data analysis. Our method has the potential to combine the advantages of mixed-effects modeling with machine learning for biomedical applications of EEG, such as multicenter clinical trials.
Paper Structure (24 sections, 1 theorem, 37 equations, 11 figures, 1 table, 2 algorithms)

This paper contains 24 sections, 1 theorem, 37 equations, 11 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Notations
  5. Regression modeling from covariance matrices using Riemannian geometry
  6. Parallel transport
  7. Learning to recenter from highly shifted y distributions with GOPSA
  8. Train-time
  9. Test-time
  10. Empirical benchmarks
  11. Results
  12. Conclusion
  13. Appendix
  14. Matrix operations
  15. Proof of Lemma 2.1
  16. ...and 9 more sections

Key Result

Lemma 2.1

Given ${\bm{\Sigma}}, {\bm{\Sigma}}^\prime \in {\mathbb{S}_d^{++}}$, the parallel transport of ${\bm{\Sigma}}^\prime$ along the geodesic from ${\bm{\Sigma}}$ to the identity ${\bm{I}}_d$ at $\alpha \in [0, 1]$ is

Figures (11)

  • Figure 1: Joint shift in $X$ and $y$ distributions on the HarMNqEEG dataset li2022harmonized. Subset of mean PSDs (A) and age distributions (B) from three recording sites used for the empirical benchmarks.
  • Figure 2: $R^2$ scores $\uparrow$ for different methods on simulated data. Performance is measured across 5 source domains and 1 target domain, with shifts controlled by $\xi$ (0 to maximum). Data are generated 100 times, with 5 sensors and 300 covariance matrices per domain. The target domain is randomly selected between the 6 domains generated as presented in \ref{['subsec:simulated_data_gopsa']}, with the remaining domains used as sources. (A) A shift is applied on the covariance matrices following \ref{['eq:shift_data']}. (B) A shift is applied on the variances following \ref{['eq:shift_label']}. (C) Both shifts from \ref{['eq:shift_data']} and \ref{['eq:shift_label']} are applied simultaneously.
  • Figure 3: Normalized performance of the different methods on several source-target combinations for three metrics: Spearman's $\rho$$\uparrow$ (left), $R^{2}$ score $\uparrow$ (middle) and Mean Absolute Error $\downarrow$ (right). As a large variability in the score values was present between the site combinations, we applied a min-max normalization per combination to set the minimum score across all methods to 0 and the maximum score to 1. (A) Boxplot of the concatenated results for the three normalized scores. One point corresponds to one split of one site combination. (B) Boxplots of the difference between the normalized scores of GOPSA and DO Intercept. A row corresponds to one site combination, one point corresponds to one split. For each plot, the associated results of Nadeau's & Bengio's corrected t-test nadeau1999inference are displayed. A p-value lower than $0.05$ indicates a significant difference between the two methods. Ba: Barbados, Be: Bern, Chb: CHBMP (Cuba), Co: Columbia, Cho: Chongqing, Cu03: Cuba2003, Cu90: Cuba90, G: Germany, M: Malaysia, R: Russia, S: Switzerland
  • Figure 4: Model inspection of GOPSA versus No DA and Re-center. Power Spectral Densities (PSDs) and $\alpha$ values were computed on the source sites Barbados, Chongqing, Germany, and Switzerland. The remaining sites were used as target domains. (A) Mean PSDs computed across sensors for No DA, Recenter and GOPSA on two source (Barbados and Switzerland) and two target (New York and Columbia) sites. (B) $\alpha$ values versus site's mean age for Re-center and GOPSA. One point corresponds to one site. The coefficient of determination is reported for the GOPSA method.
  • Figure 5: Age distribution of the 14 sites of the HarMNqEEG dataset li2022harmonized. The distributions are represented with a kernel density estimate. The y-scales are not shared for visualization purpose.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Lemma 2.1: Parallel transport to the identity