## 2 Related Work
While we consider a direct clustering of EEG background waves by age groups to be an intuitive approach, it must be acknowledged that direct clustering without considering the multi-channeled property of EEG signals overlooks the inherently spatial connections among EEG channels, thereby preventing the best performance that clustering can achieve. Graph AutoEncoders (GAEs) and their variants, such as Variational Graph AutoEncoders (VGAEs) [15], have demonstrated effectiveness in enhancing the performance of EEG-related tasks [16; 17; 18] and sleep staging [19; 20] by modeling the interrelationships between EEG channels as graph structures and analyzing them through graph embeddings. Although there are some works related to graph clustering, such as Adaptive Graph Clustering (AGC) [21; 22; 23; 24], their application in clustering EEG signals by age groups has not been explored or applied, partly due to the challenges posed by the high dimensionality and complexity of EEG data. Nevertheless, leveraging GAE to understand the interconnectivity between channels and apply it to age-based clustering analysis remains a promising approach. We currently face three main challenges in the task of clustering EEG signals using GAE. Firstly, as an unsupervised learning model, GAE cannot learn from true labels, which increases the complexity of results validating and optimizing, making it more difficult to ensure the model's accuracy and generalizability compared to supervised learning tasks. The next challenge relates to the issue of the separateness of the encoder's training process. Traditionally, the training of the encoder often occurs separately from downstream tasks (such as prediction, clustering, or classification). This separated training strategy prevents the encoder from fully utilizing task-specific information, thus reducing the model's accuracy in accomplishing specific tasks. Lastly, existing methods have not effectively leveraged the inherent spectral properties of graph structures. In EEG signal analysis, the spectral characteristics of graph structures are crucial for identifying similar wave patterns and functional regions, yet conventional graph encoder approaches often overlook this aspect. To address these challenges, this study proposes the following three innovative solutions. First, by integrating graph encoder methods with semi-supervised learning [25], we are able to identify representative cluster centers from the clustering outcomes and feed these centers' label information back to the graph attention encoder. This allows for the guidance of the encoder in generating graph embeddings with distinct separability, without the need for explicit labels. Secondly, we employ a multi-objective optimization strategy that jointly optimizes the graph reconstruction loss of the decoder and the clustering loss, guided by pseudo-labels. This joint optimization approach facilitates the encoder in learning feature representations more relevant to the downstream tasks and provides a self-supervised validation mechanism. Lastly, by integrating Laplacian filters into the graph attention encoder, we optimize the graph embedding process, enhancing the local consistency and maintenance of global information among nodes. This helps improve the precision and interpretability of clustering, especially in the domain of spectral clustering and EEG data processing.