  Machine learning applications on signals such as computer vision or
biomedical data often face significant challenges due to the variability that
exists across hardware devices or session recordings. This variability poses a
Domain Adaptation (DA) problem, as training and testing data distributions
often differ. In this work, we propose Spatio-Temporal Monge Alignment (STMA)
to mitigate these variabilities. This Optimal Transport (OT) based method
adapts the cross-power spectrum density (cross-PSD) of multivariate signals by
mapping them to the Wasserstein barycenter of source domains (multi-source DA).
Predictions for new domains can be done with a filtering without the need for
retraining a model with source data (test-time DA). We also study and discuss
two special cases of the method, Temporal Monge Alignment (TMA) and Spatial
Monge Alignment (SMA). Non-asymptotic concentration bounds are derived for the
mappings estimation, which reveals a bias-plus-variance error structure with a
variance decay rate of $\mathcal{O}(n_\ell^{-1/2})$ with $n_\ell$ the signal
length. This theoretical guarantee demonstrates the efficiency of the proposed
computational schema. Numerical experiments on multivariate biosignals and
image data show that STMA leads to significant and consistent performance gains
between datasets acquired with very different settings. Notably, STMA is a
pre-processing step complementary to state-of-the-art deep learning methods.
