  In statistical learning, high covariate dimensionality poses challenges for
robust prediction and inference. To address this challenge, supervised
dimension reduction is often performed, where dependence on the outcome is
maximized for a selected covariate subspace with smaller dimensionality.
Prevalent dimension reduction techniques assume data are $i.i.d.$, which is not
appropriate for longitudinal data comprising multiple subjects with repeated
measurements over time. In this paper, we derive a decomposition of the
Hilbert-Schmidt Independence Criterion as a supervised loss function for
longitudinal data, enabling dimension reduction between and within clusters
separately, and propose a dimensionality-reduction technique, $sklPCA$, that
performs this decomposed dimension reduction. We also show that this technique
yields superior model accuracy compared to the model it extends.
