Reducing the Dimensionality of SPD Matrices with Neural Networks in BCI (2023)
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1 Abstract
In brain–computer interface (BCI)-based motor imagery, the symmetric positive definite (SPD) covariance matrices of electroencephalogram (EEG) signals with discriminative information features lie on a Riemannian manifold, which is currently attracting increasing attention. Under a Riemannian manifold perspective, we propose a non-linear dimensionality reduction algorithm based on neural networks to construct a more discriminative low-dimensional SPD manifold. To this end, we design a novel non-linear shrinkage layer to modify the extreme eigenvalues of the SPD matrix properly, then combine the traditional bilinear mapping to non-linearly reduce the dimensionality of SPD matrices from manifold to manifold. Further, we build the SPD manifold network on a Siamese architecture which can learn the similarity metric from the data. Subsequently, the effective signal classification method named minimum distance to Riemannian mean (MDRM) can be implemented directly on the low-dimensional manifold. Finally, a regularization layer is proposed to perform subject-to-subject transfer by exploiting the geometric relationships of multi-subject. Numerical experiments for synthetic data and EEG signal datasets indicate the effectiveness of the proposed manifold network.
2 NOTES
In this paper the authors propose a Siamese Neural Network, with two Deep Riemannian Network (DRN) > SPD-Mani-Net whose outputs are used to calculate an Riemannian distance and contrastive loss. This is used so that similar samples (from same class) are closer and vice versa. From this the authors are able to construct new SPD matrices with reduced dimensionality that are still containing the relevant information, but they also should be learning information about the correct class. The second factor is probably why when used Riemannian Minimum Distance to Mean (RMDM) to classify the data it achieved the best accuracy overall. They also introduce two layers: Bilinear map (same as SPDNet > 1.1 BiMap) and Shrinkage Layer. This second one used the idea of shrinkage for better estimation of the covariance matrix, shrinking and stretching the eigenvalues, and appear to be more effective then the ReEig layer, since in their experiments it made it achieve better accuracy within fewer epochs. I like it, it has nice ideas and the authors show how they obtained all the derivatives, the only problem is that the code is in matlab.