Domain Adaptation with Optimal Transport on the Manifold of SPD matrices
This addresses domain adaptation for applications like Brain-Computer Interfaces, but it is incremental as it builds on existing optimal transport methods.
The paper tackles domain adaptation by modeling domain differences with a diffeomorphism and using optimal transport on Riemannian manifolds, specifically SPD matrices, resulting in state-of-the-art performance on two BCI datasets.
In this paper, we address the problem of Domain Adaptation (DA) using Optimal Transport (OT) on Riemannian manifolds. We model the difference between two domains by a diffeomorphism and use the polar factorization theorem to claim that OT is indeed optimal for DA in a well-defined sense, up to a volume preserving map. We then focus on the manifold of Symmetric and Positive-Definite (SPD) matrices, whose structure provided a useful context in recent applications. We demonstrate the polar factorization theorem on this manifold. Due to the uniqueness of the weighted Riemannian mean, and by exploiting existing regularized OT algorithms, we formulate a simple algorithm that maps the source domain to the target domain. We test our algorithm on two Brain-Computer Interface (BCI) data sets and observe state of the art performance.