Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signals

When dealing with electro or magnetoencephalography records, many supervised prediction tasks are solved by working with covariance matrices to summarize the signals. Learning with these matrices requires using Riemanian geometry to account for their structure. In this paper, we propose a new method to deal with distributions of covariance matrices and demonstrate its computational efficiency on M/EEG multivariate time series. More specifically, we define a Sliced-Wasserstein distance between measures of symmetric positive definite matrices that comes with strong theoretical guarantees. Then, we take advantage of its properties and kernel methods to apply this distance to brain-age prediction from MEG data and compare it to state-of-the-art algorithms based on Riemannian geometry. Finally, we show that it is an efficient surrogate to the Wasserstein distance in domain adaptation for Brain Computer Interface applications.

Dictionary and prior learning with unrolled algorithms for unsupervised inverse problems

Inverse problems consist in recovering a signal given noisy observations. One classical resolution approach is to leverage sparsity and integrate prior knowledge of the signal to the reconstruction algorithm to get a plausible solution. Still, this prior might not be sufficiently adapted to the data. In this work, we study Dictionary and Prior learning from degraded measurements as a bi-level problem, and we take advantage of unrolled algorithms to solve approximate formulations of Synthesis and Analysis. We provide an empirical and theoretical analysis of automatic differentiation for Dictionary Learning to understand better the pros and cons of unrolling in this context. We find that unrolled algorithms speed up the recovery process for a small number of iterations by improving the gradient estimation. Then we compare Analysis and Synthesis by evaluating the performance of unrolled algorithms for inverse problems, without access to any ground truth data for several classes of dictionaries and priors. While Analysis can achieve good results,Synthesis is more robust and performs better. Finally, we illustrate our method on pattern and structure learning tasks from degraded measurements.