Parallel Transport on the Cone Manifold of SPD Matrices for Domain Adaptation

Abstract

The problem of domain adaptation has become central in many applications from a broad range of fields. Recently, it was proposed to use Optimal Transport (OT) to solve it. In this paper, we model the difference between the two domains by a diffeomorphism and use the polar factorization theorem to claim that OT is indeed optimal for domain adaptation 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.Preprint. Under review.

Riemannian Procrustes Analysis: Transfer Learning for Brain–Computer Interfaces

“…Another geometrical approach [9] matches the statistical distributions from the source and target sessions by means of a re-centering of their data points to the origin of the SPD space, the identity matrix. During the process of review of this paper, we came across the work in [10], which also proposes a Transfer Learning procedure for SPD matrices in the spirit of [9], but with the main difference that the data points are re-centered to the midpoint between the geometric means of the source and target datasets Besides of distribution matching based on geometrical transformations, there has been mainly two other kinds of proposals for doing Transfer Learning in the BCI literature [2]. One is based on the concept of ensemble classifiers [3], [11]- [13], where the information from multiple source datasets are combined into a "global" classifier, which is then used to label the trials from any other target dataset.…”

“…RPA can be seen as an evolution of the aforementioned procedures [9] and [10], with the re-centering step corresponding to the first of a series of geometrical transformations. Furthermore, the procedures in [9] and [10] are completely unsupervised, since they do not use any information from the labels of the data points, whereas RPA benefits from the labels in the source session (which are all known in advance) as well as from (at least part of) the labels that become sequentially available in the target session trial after trial.…”

Manifold learning for organizing unstructured sets of process observations

“…Our approach can then be useful in a domain-adaptation context, by constructing meaningful realizations of the input and output ("source" and "target") domains. It would be interesting to compare the Mahalanobis-like metric driven observation fusion with other registration approaches developed in the domain-adaptation literature [42,43]. There is a conceptual similarity to the Dynamic Laplacian [44], in that, to gather covariance information, we have to start at several nearby initial trial points and then perform the trial associated with each one of them.…”

Cross-Dataset Variability Problem in EEG Decoding With Deep Learning

“…Recently, many Transfer Learning approaches have been introduced into BCIs to reduce cross-subject variability (Zanini et al, 2018;Rodrigues et al, 2019;Yair et al, 2019). An approach named Riemannian Procrustes Analysis (RPA) was proposed by Rodrigues et al (2019).…”