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.

翻译：在处理脑电或脑磁图记录时，许多有监督预测任务通过协方差矩阵总结信号来解决。学习这些矩阵需要利用黎曼几何来考虑其结构。本文提出了一种处理协方差矩阵分布的新方法，并在M/EEG多元时间序列上展示了其计算效率。具体而言，我们定义了具有强理论保证的对称正定矩阵测度之间的切片-瓦瑟斯坦距离。然后，利用该距离的性质和核方法，将其应用于基于MEG数据的脑年龄预测，并与基于黎曼几何的最新算法进行比较。最后，我们证明了该距离在脑机接口应用的域适应中可作为瓦瑟斯坦距离的高效替代方法。