# Integrated Model Selection and Scalability in Functional Data Analysis Through Bayesian Learning
## 4 Faster Variant  
To enhance the computational efficiency of our primary algorithm, we introduce a faster variant, denoted as BSFDA Fast. This approach leverages conditional independence among the columns of \(W\), enabling separate updates and thereby reducing the computational complexity. Similar strategies have been described in [18; 28]. The model is defined with an introduced variable \(\zeta_{i}\) for the coefficient noise as follows:\[\theta_{i}=Z_{i}W+\zeta_{i}, \tag{37}\] \[\zeta_{ik}\sim\mathcal{N}(0,\zeta_{k}^{2}\theta_{k}^{-1}). \tag{38}\]  
Similarly to the above, we assign a conjugate Gamma prior to the precision:  
\[\zeta_{k}^{-2}\sim\Gamma(a_{0},b_{0}). \tag{39}\]  
This formulation ensures that the columns of \(W\) are conditionally independent, allowing the variational distribution to factorize as \(\mathcal{Q}_{\mathcal{W}}=\prod_{k}\mathcal{Q}_{W,k}\), thereby facilitating separate updates for each column. Consequently, the time complexity is reduced from \(\mathcal{O}({K^{(a)}}^{6})\) to \(\mathcal{O}({K^{(a)}}^{3})\).  
To align with the original model, it is necessary for \(\zeta\) and the associated variance parameters \(\varsigma\) to approach zero. Having \(\varsigma\) too high would allow the coefficient noise to corrupt the signal, biasing the model toward underestimating the true signal levels, particularly because this noise operates in the coefficient space where it introduces smooth, correlated variation (low entropy, like signals), which is harder to eliminate than high-frequency white noise (maximum entropy). Injecting the same amount of noise leads to the unbiased estimation of the signals but increases the estimation variance. Conversely, as \(\varsigma\) decreases, the columns of \(W\) become dependent, violating the independence assumption inherent in variational inference. This dependency degrades the approximation quality and slows down the optimization process. Such dependency issues are well documented in both the variational inference and MCMC literature--with recent efforts addressing them via structured VI [38] or blocked/collapsed Gibbs sampling [40]. Empirical validations of this noise impact are conducted with both BSFDAFast in Section 5.1 and with Bayesian PCA [28] in Appendix F.2 in the Appendix part.  
To balance the trade-off between the optimization speed and accuracy, we adopt a strategy of gradually decreasing the values of \(\varsigma_{k}\) during the optimization iterations. Specifically, we initialize \(\varsigma_{k}\) with a relatively large value and linearly decrease it from \(10^{-2}\) to \(10^{-5}\) over the first half of the iterations. After reaching \(10^{-5}\), \(\varsigma_{k}\) is fixed for the remaining iterations. This gradual reduction ensures that the algorithm initially maintains its efficiency, with benefits from minimizing the interdependency among the columns of \(W\) to accelerating convergence while later preserving the quality of the approximation by preventing the noise from obscuring the signal components. We unify the scales by scaling the basis functions so that \(Z_{i}\) is standard normal and \(W\) is an identity matrix in initialization. Empirical evaluations indicate that the strategy above is effective in most applications.  
By implementing these modifications, BSFDAFast offers a practical solution that substantially accelerates the algorithm without a significant loss in accuracy, making it well suited for large-scale, high-dimensional functional data analysis.