# Multiple Kernel Stein Spatial Patterns for the Multiclass Discrimination of Motor Imagery Tasks
## 5 Discussion and Concluding Remarks  
This work presents a methodology for the classification of SPD matrices, termed MKSSP and outlined in Figure 2. MKSSP performs a supervised nonlinear dimension reduction with the following benefits for the discrimination of multichannel EEG trials: firstly, the projection of PSD matrices into a smaller-dimensional Riemannian manifold allows preserving the geometry of the channel space. Secondly, the introduced CKA-based supervised learning optimizes the linear mapping between Riemannian manifolds to exploit the nonlinear relationships that discriminate classes. In this regard, we benefit from the Stein kernel parameterized by the Jensen-Bregman LogDet divergence, which is devoted to comparing SPD matrices. Thirdly, the spectral decomposition highlights discriminative patterns hidden by large-power frequencies. Since each band yields a single kernel, the MKL scheme gathering several kernel representations augmented the class separability. Regarding the performance evaluation, we considered the four-class EEG signals from the Dataset IIa of the BCI Competition IV. According to Figure 3, illustrating the kappa score across the number of components, the dimension reduction carried out by MKSSP outperforms the kappa score of the input channel-space at all subjects, so avoiding the curse of dimensionality. Moreover, Table 1 evidences a significant improvement in class discrimination in comparison with state-of-the-art approaches. Specifically, MKSSP considerably increases the kappa score of usually low-performing subjects, such as S05, indicating that nonlinear relationships between channels hold discriminant MI information. While spatial filtering approaches, that map into euclidean spaces, hinder the nonlinearity, MKSSP takes decode it into a single kernel to enhance class separability. Concerning the noise sensitivity, the classification results in Figure 4 guarantee reliable performances for SNR levels better than 30 dB. SNR level within 20 to 30 dB compromises the class-wise TPR. In fact, the proposed MKSSP reaches the best TPR on Right and Left-hand classes, possibly due to their contralateral nature. Despite the performance of around 80% TPR, the identification of the Feet movement remains challenging compared to other movements. For the model interpretability, Figures 5 and 6 illustrate the kernel resulting from Equation (19) in a 2D KPCA-based projection for subjects S07 and S02 (the best and worst-performing, respectively). Note that MKSSP finds a space suitably separating classes for the subject S07. Despite the lowest performance of subject S02 and mixed Left and Right trials, the proposed methodology enhances  
Figure 9: First four spatial patterns of the best performing (S07), worst performing (S02), and most improved (S05) subjects computed from MKSSP. For each subject, the top and bottom row hold the least and most weighted frequency band, respectively. the discrimination of three groups, namely, Foot, Tongue, and both hands. Such a result proves the capability of MKSSP to distinguish mental tasks despite the spatially close activity sources. Furthermore, Figure 7, comparing four feature sets for subject S05, indicates that CSP-based approaches lack separability as they miss geometric nonlinear channel relationships. Although including spectral decomposition, MKSSP without dimension reduction tangles the trials, due to the high dimension in the covariance matrices hinders discriminative patterns. On the contrary, MKSSP with the optimal number of components enhances class separation, agreeing with the test classification results, because the manifold information benefits the MI task identification. In the spectral relevance, Figure 8 depicts the band-wise MKL weights per subject. The weight distribution highlights the between-subject variability in terms of spectral relevance. For the best performing subjects, MKSSP concentrates the kernel weights in fewer bands than for the worst ones.