Algorithm 1 An iterative algorithm for optimization in FKMR.

1.1Perform FPCA (e.g., the R package fdapace) to extract the functional component features for the p functional predictors, and store them in a grand vector for each individual subject ${\overrightarrow{\mathbf{z}}}_i=[{\left({\mathbf{z}}_i^1\right)}^{\top },\dots ,{\left({\mathbf{z}}_i^p\right)}^{\top }){]}^{\top }$, $i=1,···,n$; 1.2Initialize $\mathit{\gamma }$ to be a vector of ones. which translates to mapping the original component scores to itself. Set up a grid of possible tuning parameters for ${\lambda }_1$ and ${\lambda }_2$, respectively. Set the kernel bandwidth parameter, which may depend on ${\lambda }_1$. For each pair of $({\lambda }_1,{\lambda }_2)$ from our grid, perform Steps 2.1–2.3 and 3.1 below. 2.1At the $(r+1)$-th step in the algorithm, first solve the LSKM problem with fixed (${\mathit{\gamma }}^{\left(r\right)},{\lambda }_1$) (based on a closed-form solution) to update ${\mathit{\beta }}^{(r+1)}$ and ${\mathit{\alpha }}^{(r+1)}$. 2.2Solve the group regularity problem (8) with fixed $\tilde{\mathit{\gamma }}={\mathit{\gamma }}^{\left(r\right)}$ and fixed (${\mathit{\alpha }}^{(r+1)}$, ${\mathit{\beta }}^{(r+1)}$, ${\lambda }_1$, ${\lambda }_2$) using the $r+1$ updates from the previous iteration. At this step, the proximal Gauss–Newton algorithm produces an update ${\mathit{\gamma }}^{(r+1)}$ at convergence. 2.3Repeat Steps 2.1–2.2 until convergence. 3.1Perform cross-validation over all pairs of (${\lambda }_1,{\lambda }_2$) to determine the final $(\mathit{\alpha },\mathit{\beta },\mathit{\gamma })$.