  Magnetoencephalography and electroencephalography (M/EEG) can reveal neuronal
dynamics non-invasively in real-time and are therefore appreciated methods in
medicine and neuroscience. Recent advances in modeling brain-behavior
relationships have highlighted the effectiveness of Riemannian geometry for
summarizing the spatially correlated time-series from M/EEG in terms of their
covariance. However, after artefact-suppression, M/EEG data is often rank
deficient which limits the application of Riemannian concepts. In this article,
we focus on the task of regression with rank-reduced covariance matrices. We
study two Riemannian approaches that vectorize the M/EEG covariance
between-sensors through projection into a tangent space. The Wasserstein
distance readily applies to rank-reduced data but lacks affine-invariance. This
can be overcome by finding a common subspace in which the covariance matrices
are full rank, enabling the affine-invariant geometric distance. We
investigated the implications of these two approaches in synthetic generative
models, which allowed us to control estimation bias of a linear model for
prediction. We show that Wasserstein and geometric distances allow perfect
out-of-sample prediction on the generative models. We then evaluated the
methods on real data with regard to their effectiveness in predicting age from
M/EEG covariance matrices. The findings suggest that the data-driven Riemannian
methods outperform different sensor-space estimators and that they get close to
the performance of biophysics-driven source-localization model that requires
MRI acquisitions and tedious data processing. Our study suggests that the
proposed Riemannian methods can serve as fundamental building-blocks for
automated large-scale analysis of M/EEG.
