## 2 Materials and Methods
### Conformal Bayesian Optimization with Gaussian Process Priors  
To optimize the hyperparameters \(\theta\) controlling the deformation model, we employ Bayesian optimization with Gaussian Process (GP) priors. The objective is to minimize the cost function \(f(\theta)=\mathcal{L}(\mathcal{P},\mathcal{Q},\mathcal{T}(\theta))\) over the bounded domain \(\Theta\). We model \(f(\theta)\) as a sample from a GP prior, specified by the mean function \(m(\theta)\) (often zero) and the covariance function \(k(\theta,\theta^{\prime})\). The GP provides a probabilistic model of the function, allowing us to predict \(f(\theta)\) at unobserved points. The acquisition function \(a(\theta)\) guides the selection of the next hyperparameters to evaluate by balancing exploration and exploitation. We use Conformal Bayesian Optimization [16], which provides robustness against model misspecification and covariate shift, offering coverage guarantees. Table 1 presents the hyperparameters optimized within the proposed Bayesian framework for deformable point cloud registration. Each hyperparameter plays a crucial role in balancing the accuracy and computational feasibility of the registration process. Number of clusters (\(c\)): This parameter determines the granularity of local deformations. A higher number of clusters allows for finer transformations; however, excessively high values may lead to overfitting or increase computational complexity. The range [2; 20] ensures adaptability without compromising efficiency. Regularization weight (\(\lambda_{r}\)): This controls the trade-off between alignment precision and deformation smoothness. Higher values promote smoother transformations, which are crucial for maintaining anatomical plausibility, particularly in sensitive brain structures. The number of neighboring rings (\(r\)): This parameter defines the local neighborhood around each point and influences how adjacent points contribute to the transformation. Smaller values encourage localized deformations, while higher values capture broader contextual relationships. Influence factor (\(\xi_{ij}\)): This modulates the importance of local energy contributions from each neighboring ring. This ensures that the algorithm appropriately balances the impact of different regions on the final deformation, which is essential for aligning structures with varying geometric properties. Threshold for global energy correction (\(d\)): This parameter determines when global transformations, such as translation and rotation, are applied. A lower threshold triggers more frequent global adjustments, ensuring accurate alignment across large-scale transformations. By integrating anatomical plausibility through regularization and optimizing hyperparameters via Conformal Bayesian Optimization, our method aims to achieve accurate deformable registrations of point clouds. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Hyperparameter** & **Symbol** & **Range** \\ \hline
**Number of clusters** & \(c\) & \([2,20]\) \\
**Regularization weight** & \(\lambda_{r}\) & \([0.001,1.0]\) \\
**Number of neighboring rings** & \(r\) & \([1,5]\) \\
**Influence factors for local energy** & \(\xi_{ij}\) & \([0.1,10]\) \\
**Threshold for global energy correction** & \(d\) & \([0.01,1.0]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Hyperparameters for deformable point cloud registration. Figure 1 shows the proposed framework for the deformable registration of point clouds, using Bayesian optimization to enhance their alignment. The process starts with two input point clouds: a fixed cloud \(\mathcal{P}\) and a moving cloud \(\mathcal{Q}\). The transformation function \(\mathcal{T}\) estimates the non-rigid transformations needed to align \(\mathcal{Q}\) with \(\mathcal{P}\). In addition, the alignment is driven by a cost function, \(\mathcal{L}(\mathcal{P},\mathcal{Q},\mathcal{T})\), combining geometric distance with a regularization term to ensure smooth and anatomically plausible deformations.