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.

翻译：闭合曲线的建模与不确定性量化是形状分析领域的重要问题，对后续统计任务具有重要影响。许多这类任务涉及具有多层次结构相似性的闭合曲线集合。以有效纳入曲线间依赖关系的方式对多个闭合曲线进行建模仍具挑战性。本文提出并研究了一种多输出（亦称多变量）多维高斯过程建模框架。我们通过若干曲线及形状相关任务，阐述了所提出的方法学进展，并展示了有意义的不确定性量化的实用性。这种基于模型的方法不仅借助核构造解决了闭合曲线（及其形状）的推断问题，还为功能对象的非参数化多层次依赖建模开辟了普适性途径。