# Multiple Kernel Stein Spatial Patterns for the Multiclass Discrimination of Motor Imagery Tasks
## 1 Introduction
In addition, Principal Geodesic Analysis (PGA) emerges as a principal component analysis generalization for Riemannian manifolds by finding a tangent space with maximized variance [18]. Two subsequent tangent space approaches introduce kernel-based mappings [16; 19]. However, tangent space projections distort data structure, with larger distortions at regions far from the space origin [20]. Another DR alternative maps from high to low-dimensional manifolds, where the resulting output manifold serves as input for existing SPD-based algorithms [15]. As an example, a linear mapping takes advantage of provided labels to maximize the geodesic distance among samples from different classes while minimizing distances among equally-labeled samples [17]. Despite favoring the supervised tasks, the linear combination of distances as a cost function underperforms at inherently nonlinear distributed classes. For dealing with the above issues, this work proposes a methodology for the classification of SPD matrices with three main contributions: firstly, we introduce a dimension reduction for SPD matrices. Contrarily to other approaches, the approach uses the provided labels to learn the mapping that not only preserves the data structure but also favors the class discrimination thanks to a supervised learning scheme. Secondly, we propose two kernel mappings to decode the nonlinear relationships that enhance the separability of classes at the component level, namely, the Stein kernel devoted to SPD matrices, and the linear combination of kernel functions unifying the knowledge from different spectral bands. The methodology, termed Multi-Kernel Stein Spatial Patterns (MKSSP), includes a spectral decomposition, spatial filtering, and supervised kernel learning stages to enhance the discrimination of multichannel EEG signals. Firstly, a bank of bandpass filters decomposes the multichannel signals, from which MKSSP computes the covariance matrices. Then, the spatial filtering projects the band-wise covariances into smaller-dimensional Riemannian manifolds by maximizing a supervised kernel-basedcost function. The Jensen-Bregman LogDet Divergence, devoted to SPD matrices, parameterizes the considered kernel pair-wise comparing EEG trial at each band. Finally, a multiple-kernel yields a single similarity measure that feeds a classifier. The centered kernel alignment (CKA) works as the cost function for training the dimension reduction and the multikernel. We tested our methodology in the Dataset IIa of the BCI Competition IV, outperforming the state of the art in the four task MI classification for most of the subjects and achieving significant improvements for the ones with usual low-performance scores. Aiming to gain insight into the methodology benefits, we analyze the methodology from four perspectives. The data analysis evidences the class separation. The noise sensitivity validates the robustness and class unbiasing. The spectral and spatial points of view explain the subject variability and model interpretability, respectively. The paper outline is as follows. Section 2 describes the EEG decomposition, the time-series similarity through the Stein kernel, the proposed spatial filter optimization and multiple kernel learning based on CKA. Section 3 presents the evaluation setup for the proposed MKSSP. Section 4 describes the classification results, the comparison with other methods, and the model interpretability. Lastly, Section 5 concludes the manuscript discussing the main findings.