# Advancing Brain Tumor Segmentation with Spectral-Spatial Graph Neural Networks
## 2 Methods and Material
### Hyperparameters Study  
In this study, we utilized Bayesian Optimization [29] for hyperparameter tuning due to its superior efficiency over Grid Search and Random Search. Bayesian Optimization intelligently navigates the hyperparameter space, significantly reducing computational costs and time by focusing on promising areas based on a probabilistic model. This approach allows for a more efficient search and often yields better results faster than traditional methods.  
Multiple parameter sensitivity experiments were performed to test the influence of the selected crucial hyperparameters. It is crucial to consider the impact of different learning rates on the training process of machine learning models (See Figure 5). According to [PERSON] (2017) [30], a high learning rate, such as 0.1, can lead to rapid convergence of the model but may introduce instability during training. This instability is characterized by significant fluctuations in accuracy from one epoch to another, as the model may overshoot optimal points in the loss landscape. Consequently, training curves associated with higher learning rates are likely to display more noise due to these fluctuations.  
Conversely, a medium learning rate, exemplified by 0.01, offers a balance between convergence speed and stability [31]; while models trained with a medium learning rate may still exhibit some fluctuations in accuracy, these variations are typically less pronounced compared to those observed with high learning rates. The noise level in the training process is moderate when using a medium learning rate.  
On the other hand, a low learning rate, such as 0.005 or lower, results in slower convergence but promotes a more stable training process with minimal fluctuations in accuracy [30]. This stability is achieved because the model makes smaller weight updates, leading to a smoother progression of accuracy over epochs and reduced noise in the training curves associated with lower learning rates.  
Given the objective of comparing the efficacy of Spatial, Spectral, and Spectral-Spatial GNN models for brain MRI segmentation, the hyperparameter selection was critically strategized. As demonstrated in Table 3, a uniform learning rate of 0.005 was chosen across models to ensure a fair ground for performance comparison, negating any bias introduced by learning dynamics. This rate strikes a balance between adequate model training and the prevention of rapid convergence to local minima. The consistency in epochs, layer sizes, and dropout rates further standardizes the experimental setup, allowing direct attribution of performance differences to the architectural distinctions of each model. Such a methodical approach enables a comprehensive analysis of how spatial and spectral data processing capabilities of each GNN variant contribute to the segmentation accuracy, providing valuable insights into their applicability in medical imaging analysis. Table 3 provides a comparative overview of the configuration settings used for three types of Graph Neural Networks (GNNs): Spatial GNN, Spectral GNN, and Spectral-Spatial GNN. In this study, we used a consistent setup across all three models for many parameters, including the number of training epochs (200), number of node features (20), number of output classes (4), learning rate (0.005), learning rate decay (0.95), weight decay (5 \(\times\) 10\({}^{-4}\)), layer sizes (four layers of 256 units each), dropout rate (0.5), activation function (ReLU), and the optimizer (Adam). Unique to the Spectral GNN and the Spectral-Spatial GNN is the use of a Chebyshev filter size of 2. Additionally, the Spatial GNN and the Spectral-Spatial GNN utilize an LSTM aggregator, which is absent in the Spectral GNN, due to the difference in how node information is integrated in these models.