# Integrated Model Selection and Scalability in Functional Data Analysis Through Bayesian Learning
## 1 Introduction
Alternative node allocations may be better, such as Chebyshev nodes for superior accuracy [15] or sparse grids to reduce the combinatorial growth of the computational complexity [16]. Existing studies tend to rely on choosing the hyperparameters manually [6] or on cross-validation [3; 14; 17], which are known to be computationally prohibitive. Others employ approximated cross-validation [13] or marginal likelihood [8], but these still require the exhaustive testing of all candidate models. Methods with sparse Bayesian priors [7; 18] for model selection allow model selection with a single optimization. In [19; 20], the authors use shrinkage or sparse priors for data-adaptive basis selection to ensure minimal but effective sets of basis functions. Notably, [PERSON]. [21] proposed the Bayesian and Akaike information criteria, demonstrating state-of-the-art performance in simulation studies for sparse and dense functional data. In addition, probabilistic FDA emerges as a sophisticated adaptation of probabilistic methods tailored to incorporate the flexibility of latent variable models to manage functional data. A Bayesian latent factor regression model (LFRM) [18], for example, extends conventional regression to accommodate complex structures and dependencies in functional data, providing a robust framework to handle the complexities inherent in functional data. However, these Bayesian approaches are often limited by the computational demands of Monte Carlo methods in high dimensions [8]. To address increasingly high-dimensional FDA problems, recent efforts have emphasized scalability. For instance, [PERSON]. [6] introduced FDA for images with a fixed basis or grid. In [17], they further reduced the complexity in 2D fPCA via tensor product B-splines. Meanwhile, [PERSON]. [22] applied a Bayesian framework with basis expansion, adaptive regularization, and Gibbs sampling to 2D functional data in the form of EEG studies on children with autism. Furthermore, [PERSON]. [23] leveraged a parsimonious basis representation and variational Bayes to achieve computational efficiency, making it suitable for 3D brain imaging data. A Bayesian nonparametric model [24] leverages variational inference for efficient computation in high-dimensional functional time series and uses an Indian buffet process to automatically select latent factors. Nonetheless, it focuses on 1D functional observations with temporal dependencies and a common sampling grid. In parallel, the broader method of principal component analysis (PCA) remains a fundamental and effective tool. Classical PCA, rooted in eigendecomposition [25], effectively extracts dominant modes of variation in many settings but does not inherently accommodate the probabilistic nature of real-world data and their inherent uncertainties. Thus, [PERSON]. [26] introduced _probabilistic PCA_ (PPCA), which incorporates a probability distribution to manage these uncertainties more effectively. PCA has since evolved to address missing data [27], model selection [28], and complex data types [29]. In the context of functional data, these concepts motivate new approaches that unify probabilistic methodologies, latent factor models, and kernel expansions for continuous domains [1]. Within Bayesian machine learning, various priors have been proposed for sparse or robust formulations of PCA. Specifically, _sparse Bayesian learning_ (SBL) [30], with its mechanism automatic relevance determination (ARD) [31; 32], has proven adept in promoting parsimonious solutions [33]. SBL has emerged in Bayesian PCA [28], applying an iterative method to evaluate the relevance of each component and select the internal dimensionality by disregarding the redundant ones. In [34], the authors applied SBL to optimize the combination of base kernels to enhance model performance.