## 1 Introduction  
Brain-computer interface (BCI) is a technology that allows users and computers to interact with each other through brain activity. An electroencephalogram (EEG) is used to record the brain activity under certain BCI experimental task [1]. For example, users can control the mouse on the screen to move left and right by imagining their left and right hand movements, respectively [2]. Therefore, BCI has a wide range of uses in patients with disabilities, such as patients with severe neuromuscular disease or interlocking symptoms [3, 4]. Many different types of EEG signals can be used in BCI field, such as steady state visual evoked potential (SSVEP) [5], motor imagery (MI) [6], and P300 [7]. In this article, we are interested in the P300-EEG signal, which is based on event-related potentials. The P300-EEG signal is a natural response of our brain to a specific external stimulus; in response, the EEG signal will have a positive peak after about 0.3 ms of the stimulation [8]. One of the main reasons hindering the widespread use of BCI systems is the variability of EEG signals [1; 9]. Due to the variability, the feature space distribution of the EEG signal collected from different subjects or different sessions are inconsistent [10]. In addition, the BCI system requires a long calibration phase before each time it is used because, to achieve a good performance, every subject's BCI system needs to be trained by their own EEG signals and cannot use other's EEG signals [11]. One of the potential solutions to reduce or even eliminate the calibration phase is a transfer learning algorithm. In this article, we mainly study the offline transfer learning of the P300-EEG signal. In the field of machine learning, transfer learning is defined as the ability to use the knowledge learned in a previous task or domain in a new task or domain [12]. Transfer learning in the BCI field has received extensive attention in improving the generalization performance of the classifier. [PERSON] et al. [13] proposed combining a Bayesian model and learning from label proportion (LLP). [PERSON] et al. [14] completed a cross-session transfer of P300 data using a nonlinear transform obtained by solving an optimal transport problem and reached the highest AUC score of 0.835 for one particular subject. [PERSON] et al. [15] proposed an adaptive classification method. The initialization of the classifier is subject-independent, and, after several minutes of online adaptation, the accuracy converges to that of a fully trained supervised subject-specific model. [PERSON] et al. [16] proposed to learn a dictionary of spatial filters. Some other transfer learning methods are semi-supervised learning [17] and using uniform local binary patterns [18] or artificial data generation [11]. However, one of the methods with the most potential is Riemannian Geometry method [19; 20; 21]. The Riemannian Geometry classifier is a promising and new classification method in BCI field. The main idea of Riemannian Geometry is to represent the data in the form of symmetric positive definite (SPD) covariance matrix, and then directly map the SPD covariance matrix on the Riemannian manifold. The data on the Riemannian manifold can be manipulated directly, including direct classification using the Riemannian distance. We further study the potential of this classifier in this article. Although Riemannian Geometry method has achieved many good results in the BCI field, it still has shortcomings. If the data dimension is too large, the Riemannian Geometry method will perform many calculations, which is time-consuming and causes statistical deviations [22]. Therefore, the use of Riemannian Geometry method needs to be combined with dimensionality reduction algorithms. Thus, we introduce XDAWN spatial filters. XDAWN spatial filters, specially design for Event Related Potentials (ERP), were proposed by [PERSON] [23]. They can enhance the P300 component and reduce the data dimensions, which is very suitable to our needs.