# Classification of Event-Related Potentials with Regularized Spatiotemporal LCMV Beamforming
## 1 Introduction  
Brain-computer interfaces (BCIs) establish a direct communication pathway between the brain and an external device [1]. Severely disabled patients with impaired or absent communication capabilities can benefit from BCIs to restore normal functioning [2, 3]. BCIs can be implemented in multiple ways, using non-invasive recording techniques such as electroencephalography (EEG) [4], magnetoencephalography (MEG) [5], functional near-infrared spectroscopy (fNIRS) [6], and optically pumped magnetometers (OPM MEG) [7], or semi-invasive and invasive methods such as electrocorticography (ECoG) [8] or microelectrode arrays [9], which require surgery to implant a recording device. Although invasive BCIs yield the highest information transfer rate [10], non-invasive BCIs are preferable for short-term use since they are not susceptible to the risks that come with surgery. Among the non-invasive options, EEG is the most cost-effective and practical as it is not limited to the same controlled settings as MEG and OPM MEG.  
In addition to the recording method, BCIs differ in the communication paradigms used for communication [4]. A popular class of BCI paradigms relies on the evocation of event-related potentials (ERPs) in the brain in response to visual, auditory, or tactilestimulation, given their low decoding cost and generally short calibration time before usage [11; 12]. In this study we focused on the visual P300 oddball ERP in response to a rare but attended visual stimulus. The decoder detects whether this ERP is present to determine which stimulus the user attended. The P300 paradigm has been used extensively in BCI development and is easy to set up [13; 14; 15; 16].  
There are multiple state-of-the-art P300 classification methods, such as support vector machines (SVMs) [17], deep learning models [18; 19], and Riemannian geometry classifiers [15]. Although these models often return a high classification accuracy, there is a need for lightweight models, as lightweight models lead to fast off-line analyses and can be transferred to consumer-grade hardware. When moving towards plug-and-play solutions, BCI calibration sessions should be short and model training times should be low. The spatiotemporal beamformer [20; 21] belongs to this class of ERP-decoding models as it achieves state-of-the-art performance and is fast to train. Earlier work has shown that it is possible to apply the spatiotemporal beamformer to multiple time-locked visual BCI paradigms, including the P300 oddball paradigm, steady-state visually evoked potentials (SSVEPs), code-modulated visually evoked potentials (cVEPs) [22], and motion-onset visually evoked potentials (mVEPs) [23].  
This work shows that the original spatiotemporal beamformer [21] can fall short in its performance when BCI calibration data are restricted. We also show that the spatiotemporal beamformer does not scale well for higher spatial and temporal resolution cases. As a response to these issues, we introduce a regularization method that exploits prior knowledge about the spatiotemporal nature of the EEG signal to improve the accuracy for settings with low data availability and to speed up the classifier training time, thereby considerably reducing memory usage. Similar structured regularization approaches have been applied to other linear ERP classifiers [24; 25] and have shown significant increases in performance. Additionally, we show that regularization results in an interpretable classification model, which can aid in analyzing and developing spatiotemporal beamformer-based classifiers.