# Nonparametric Multi-shape Modeling with Uncertainty Quantification

arXiv: https://arxiv.org/abs/2206.09127 · HTML: https://mlsanity.com/paper/2206.09127/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 87 references that are in the map.
- First posted on arXiv 18 Jun 2022.

## Builds on (most cited first, 20 of 20)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [GPflow: A Gaussian process library using TensorFlow](https://mlsanity.com/paper/1610.08733/) (2016)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [A Framework for Interdomain and Multioutput Gaussian Processes](https://mlsanity.com/paper/2003.01115/) (2020)
- [Generalized Spectral Clustering via Gromov-Wasserstein Learning](https://mlsanity.com/paper/2006.04163/) (2020)
- [Topological Optimization with Big Steps](https://mlsanity.com/paper/2203.16748/) (2022)
- [From Geometry to Topology: Inverse Theorems for Distributed Persistence](https://mlsanity.com/paper/2101.12288/) (2021)
- [Gaussian Process Landmarking on Manifolds](https://mlsanity.com/paper/1802.03479/) (2018)
- [Review: Nonstationary Spatial Modeling, with Emphasis on Process Convolution and Covariate-Driven Approaches](https://mlsanity.com/paper/1610.02447/) (2016)
- [Non-smooth Bayesian Optimization in Tuning Problems](https://mlsanity.com/paper/2109.07563/) (2021)
- [Numerical issues in maximum likelihood parameter estimation for Gaussian process interpolation](https://mlsanity.com/paper/2101.09747/) (2021)
- [Elastic analysis of irregularly or sparsely sampled curves](https://mlsanity.com/paper/2104.11039/) (2021)
- [Probabilistic Diffeomorphic Registration: Representing Uncertainty](https://mlsanity.com/paper/1701.03266/) (2017)
- [Simultaneous Registration and Clustering for Multi-dimensional Functional Data](https://mlsanity.com/paper/1711.04761/) (2017)
- [Generalized Penalty for Circular Coordinate Representation](https://mlsanity.com/paper/2006.02554/) (2020)
- [Bayesian registration of functions and curves](https://mlsanity.com/paper/1311.2105/) (2013)
- [Generalized Gaussian Process Regression Model for Non-Gaussian Functional Data](https://mlsanity.com/paper/1401.8189/) (2014)
- [Smoothing and mean-covariance estimation of functional data with a Bayesian hierarchical model](https://mlsanity.com/paper/1402.5723/) (2014)
- [Bayesian Detection of Image Boundaries](https://mlsanity.com/paper/1508.05847/) (2015)
- [Elastic k-means clustering of functional data for posterior exploration, with an application to inference on acute respiratory infection dynamics](https://mlsanity.com/paper/2011.12397/) (2020)

## Built on by (most cited first, 2 of 2)

- [A Unifying Perspective on Non-Stationary Kernels for Deeper Gaussian Processes](https://mlsanity.com/paper/2309.10068/) (2023)
- [Contrastive inverse regression for dimension reduction](https://mlsanity.com/paper/2305.12287/) (2023)

## Abstract (arXiv)

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.