@polakaRiemannianGeometryPreservingVariational2026

Riemannian Geometry-Preserving Variational Autoencoder for MI-BCI Data Augmentation (2026)

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1 Abstract

This paper addresses the challenge of generating synthetic electroencephalogram (EEG) covariance matrices for motor imagery brain-computer interface (MI-BCI) applications. Objective. We aim to develop a generative model capable of producing highfidelity synthetic covariance matrices while preserving their symmetric positive-definite nature. Approach. We propose a Riemannian geometry-preserving variational autoencoder (RGP-VAE) integrating geometric mappings with a composite loss function combining Riemannian distance, tangent space reconstruction accuracy and generative diversity. Results. The model generates valid, representative EEG covariance matrices, while learning a subject-invariant latent space. Synthetic data proves practically useful for MI-BCI, with its impact depending on the paired classifier. Contribution. This work introduces and validates the RGP-VAE as a geometry preserving generative model for EEG covariance matrices, highlighting its potential for signal privacy, scalability and data augmentation.

2 NOTES

In this paper the authors propose a VAE that receives as input the tangent vector obtained from a covariance matrix. This is then fed to an encoder, that consists of five sequential blocks (linear → batch normalization → LeakyReLU) with dimensions $D_{spd}\to 32\to 64\to 16\to 32\to 64$, followed by two separate linear projections to produce ${\mu }_i,log{\sigma }_i^2\in {\mathbb{R}}^{D_{lat}}$ where $D_{lat}=64$. At the end the reconstructed vector is converted again into a covariance matrix and can be classified by something like RMDM. There are a couple of things to keep an eye here that are interesting:

• For projection into and back from the manifold, the authors use the reference matrix as the karcher’s mean of the training class;
• They train, for each fold, class-specific RGP-VAEs, which explains how they can sample new vectors without worrying about the label;
• They define four losses: $L_{total}=(L_{manifold}+L_{tangent})+\beta L_{KL}+\gamma L_{diversity}$, which are $L_{manifold}=\frac{1}{B}{\sum }_{i-1}^B{d}_r(X_i,\widehat{X_i})$ $L_{tangent}=\frac{1}{B}{\sum }_{i=1}^B\frac{{\sum }_j(h_{\text{decoded},i,j}-h_{\text{tangent},i,j}{)}^2}{{\sum }_j{h}_{\text{tangent},i,j}^2+ϵ}$ where $ϵ=10^{-6}$. $L_{KL}=\frac{1}{B}{\sum }_{i=1}^B\left[-0.5{\sum }_{k=1}^{D_{lat}}(1+\log {\sigma }_{i,k}^2-{\mu }_{i,k}^2-\exp (\log {\sigma }_{i,k}^2))\right]$ $L_{diversity}=-\log \det (\text{Cov}(H_{\text{decoded}}^T)+{ϵ}_{\text{cov}}I)$ where ${ϵ}_{\text{cov}}=10^{-6}$ for numerical stability, $\gamma =0.035$ and $\beta$ linearly increases from $0.0001$ to $0.2$ during training.

These losses are very interesting because the VAE does not care at all if the data is SPD, unlike the SPDNet-VAE, or anything like that. Instead, they rely on defining losses that will force the network to consider it, which probably has a lot of influence of $L_{manifold}$. I would have loved to see an ablation where they remove each loss and see their impact. Still, I like it very much.