  The modeling and uncertainty quantification of closed curves is an important
problem in the field of shape analysis, and can have significant ramifications
for subsequent statistical tasks. Many of these tasks involve collections of
closed curves, which often exhibit structural similarities at multiple levels.
Modeling multiple closed curves in a way that efficiently incorporates such
between-curve dependence remains a challenging problem. In this work, we
propose and investigate a multiple-output (a.k.a. multi-output),
multi-dimensional Gaussian process modeling framework. We illustrate the
proposed methodological advances, and demonstrate the utility of meaningful
uncertainty quantification, on several curve and shape-related tasks. This
model-based approach not only addresses the problem of inference on closed
curves (and their shapes) with kernel constructions, but also opens doors to
nonparametric modeling of multi-level dependence for functional objects in
general.
