# Multiple Kernel Stein Spatial Patterns for the Multiclass Discrimination of Motor Imagery Tasks
## 1 Introduction  
Brain-computer interfaces (BCI) are systems that aim to establish a direct connection between the human brain and a computer. These systems can convert specific patterns of brain electrical activity from electroencephalographic (EEG) signals into commands to control external devices, allowing direct interaction with the environment [1]. Thanks to the provided alternative means of mobility and communication, BCI systems become especially useful in people suffering from motor disabilities such as amyotrophic lateral sclerosis, cerebral strokes, and paralysis, [2, 3]. In particular, the motor imagery (MI) paradigm takes advantage of the mental simulation of actions without physical execution to capture electrical activity patterns from the sensory-motor cortex [4]. BCI systems translate the MI-associated task pattern from electroencephalography (EEG) into control commands. Despite the potential of EEG-based systems, the background noise, muscle artifacts, head movements, and blinks largely detriment its performance, which constrains its introduction into real-world applications [5, 6]. One way to deal with the above issues comprises the design of robust feature sets. Among the wide variety of feature extraction approaches, the Common Spatial Patterns (CSP) stands out for capturing the MI activity within the EEG. CSP projects signals from two classesinto a space with maximum variance for one class and minimum for the other, providing discriminant features [7]. In this sense, the relative power in the CSP space, computed from the channel-wise covariance matrix, characterizes a band-passed multichannel EEG trial. Nonetheless, the interclass variability within subjects hampers the system effectiveness for MI applications [8]. As an initial solution, multiple CSP-based variants optimize the spectral filtering for yielding discriminative features under varying conditions. For instance, the Common Spatio-Spectral Patterns (CSSP) approach enhances a finite impulse response (FIR) filter by incorporating a time delay for filtering to improve CSP performance proposed [9]. The Common Sparse Spectral-Spatial Patterns modified the CSSP by simultaneously tuning the FIR and CSP filters [10]. Later, [PERSON] et al. introduced the CSP features from multiple sub-bands for EEG classification with a score fusion strategy [11]. In the same approach, the Filter-Bank CSP (FBCSP) exploited the potential correlation between CSP characteristics extracted from different bands to improve the signal discrimination [8]. In general, filter-banked feature extraction approaches outperform conventional multichannel time-series representations, including traditional CSP, in supervised learning schemes [12]. Nonetheless, spectral variants of CSP hardly decode MI tasks that activate spatially close regions, no matter the frequency differences [13]. A different solution decodes the trial variability through Riemannian manifolds composed of Symmetric Positive Definite (SPD) matrices, i.e., trial covariance estimations. Due to preserving the geometric structure and behaving like a matrix Hilbert space, the Riemannian manifolds devote conventional pattern recognition machines to time-series classification [14]. However, the covariance matrices as features suffer from the curse of dimensionality, since the usual matrix dimensions compare to the number of training samples [15]. For coping with high-dimensionality, various approaches map data from the Riemannian manifold into lower-dimensional vector spaces. For instance, the extension of three nonlinear dimension reduction (DR) approaches, namely, Local Linear Embedding (LLE), Hessian LLE, and Laplacian Eigenmaps, to the Riemannian geometry allowed the motion and image segmentation from a clustering point of view [16]. Nonetheless, such nonlinear DR algorithms lack a parametric mapping to the low-dimensional space, depending on an interpolation stage [17].