## 1 Introduction
(2023) applied Koopman embeddings of individual signals using Multilayer Perceptrons (MLPs) and classified them with Convolutional Neural Networks (CNN). While demonstrating strong classification performance, the study focused solely on accuracy and receiver-operating characteristic (ROC) plots, leaving the discriminant features unexplained [17]. **Temporal and latent structure learning**--Integrated Dynamic Mode Decomposition (DMD) and Singular Value Decomposition (SVD) were employed in work by [PERSON] et al. (2021) [18] to capture temporal structures for anomaly detection. Although the delay embedding approach was promising, the study lacked interpretability and robustness analysis. [PERSON] et al. (2024) [19] used reduced-order Diffusion Maps (DMs) and Geometric Harmonics to predict the long-term dynamics of Functional magnetic resonance imaging (fMRI) signals. While DMs captured brain dynamics effectively, they were not optimised for EEG data. The spatio-temporal Koopman decomposition described in [20] learnt quasiperiodic dynamics by combining higher-order SVD and higher-order DMD. Alternative embeddings, such as Koopman spectral operators, remain unexplored in this context. **Graph-based representations--**[PERSON] (2019) [8] proposed a Koopman dynamic learning framework that combined Koopman operators with linear dynamical systems for classification using Support Vector Machine (SVM). However, the approach did not address delay characteristics or feature explainability. [PERSON] (2022) [21] applied Koopman embeddings in autoencoders for real-time electrical neuromodulation, demonstrating the feasibility of Koopman-based frameworks. However, the application of such embeddings to EEG datasets was not explored. **Graph kernels and spectral Koopman operators--**Graph kernels, particularly the Weisfeiler-Lehman (WL) kernel, Random Walk kernel, and Graphlet kernel, have shown promise in capturing graph structural dynamics through spectral decomposition. The WL kernel, in particular, is computationally efficient and robust, making it suitable for EEG connectivity graphs where computational constraints are critical. Studies like Refs. [5; 7] highlight graph kernels' ability to encode dynamic systems' spectral properties, enabling the identification of metastable structures and connectivity dynamics. However, the robustness of these kernels in learning latent metastructures from EEG data remains underexplored [5]. **Cost-sensitive learning (CSL)--**A common issue in EEG datasets, class imbalance leads to biased models that overlook critical minority classes. CSL addresses this by adjusting the loss function to penalise misclassifications of minority classes more heavily. For example, Scikit-learn's class_weight = 'balanced' option automatically assigns weights based on class frequencies. CSL has been shown to improve classifier performance without altering dataset distributions, making it particularly effective for EEG-based classification tasks [5; 7]. **Gaps in the literature--**Despite promising developments, several critical gaps persist. The robustness of graph kernels in estimating latent metastructures of connectivity has not been systematically evaluated. While Koopman operators are potent tools for spectral analysis, their application to EEG connectivity graphs is underexplored. Existing methods rarely focus on interpretable features or discriminant structures in brain connectivity. Graph kernels' robustness and comparative efficacy for capturing latent metastable patterns in EEG remain unexplored. While Koopman spectral operators have been applied to other domains, their integration with graph kernels for EEG data analysis is a novel direction requiring rigorous investigation. The study reported in our earlier work [22] described a novel workflow that performed optimal selection of the graph kernel for Koopman embeddings (GKKE) with ensemble classifiers for EEG-based classification.