## 2 Theory
### Lower-Dimensional Remapping: PCA Analysis
Figure 1: (a) Section example of a 3D Random Smooth Scalar Field with periodic boundary conditions in the \(x\) and \(y\) axis. Arbitrary units. [PERSON]’s \(I\) = 0.5702. (b) Cumulative Explained Variance of 5184 Principal Components of a 10,000 (20 \(\times\) 24 \(\times\) 12 voxels) 3D Random Smooth Scalar Field Samples set. The 99.5% of the variance is represented by dashed lines with the number of principal components required. Since the spatial autocorrelation of the set is positive, we can find a sufficient number of principal components significantly lower than the original number of 3D variables, as shown in Figure 1b. Each element represents a smooth scalar field from an optimally selected set that, along with the other principal components, can reconstruct the original space by linear combinations. That set of maximally mutually linearly independent smooth scalar fields will constitute our lower dimensional base, drastically reducing the computational costs of our optimisation problem. Here, we can see that considering between 350 and 400 components, 99.5% of the total variance is fully defined. The choice of using a 99.5% cumulative variance threshold is empirically justified, as capturing a higher percentage of variance would require a significantly larger number of principal components, as shown in Figure 1b, which would undermine the goal of dimensionality reduction. Conversely, selecting fewer components reduces significantly the explained variance which risks capturing the structure of the solution space in an imprecise way, leading to reduced optimization accuracy. The resolution or the remapping ability to capture more abrupt changes in the field distribution of the vector directors \(\theta\) and \(\phi\) will depend on the initial dataset properties with which the PCA is conducted. An initial dataset with a stronger smoothness will allow a more substantial dimensional reduction to an even lower dimensional remapping space. However, the system will likely struggle to represent sharper local variations accurately. Furthermore, a larger dataset will facilitate the PCA procedure to better capture and reconstruct the entire 3d space, achieving a higher fidelity and avoiding possible missing features or inhomogeneities. Thus, an appropriate preparation of the procedure parameters is required, analogous to the crucial hyperparameter tuning of any deep learning model. The method described here can be generalised to include other forms of linear interdependencies beyond spatial autocorrelation, which can similarly be exploited to reduce the dimensionality of an optimisations problem. By constructing a synthetic dataset where these linear relationships are present, PCA can be applied to extract the principal components that capture the majority of the variance within the data and form a reduced set of variables in the same fashion as the described in the present work. This allows the method to be applied to a broader range of scenarios, enabling efficient and robust optimisation. The novelty of this work lies in its innovative application of PCA specifically through the construction of an artificial dataset with engineered linear interdependencies, while PCA is widely recognised for its effectiveness in reducing dimensionality in naturally occurring datasets, the deliberate creation and use of synthetic datasets exhibiting controlled linear dependencies to enhance complex optimisation systems represents a novel approach. A thorough review of the existing literature reveals that this specific methodology has not been widely documented or applied.