# Shape Analysis of Functional Data with Elastic Partial Matching

arXiv: https://arxiv.org/abs/2105.08604 · HTML: https://mlsanity.com/paper/2105.08604/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 38 references that are in the map.
- First posted on arXiv 18 May 2021.

## Builds on (most cited first, 4 of 4)

- [Functional Data Analysis of Amplitude and Phase Variation](https://mlsanity.com/paper/1512.03216/) (2015)
- [Elastic Functional Coding of Riemannian Trajectories](https://mlsanity.com/paper/1603.02200/) (2016)
- [Bayesian Clustering of Shapes of Curves](https://mlsanity.com/paper/1504.00377/) (2015)
- [Combined Analysis of Amplitude and Phase Variations in Functional Data](https://mlsanity.com/paper/1603.01775/) (2016)

## Built on by (most cited first, 3 of 3)

- [Review of Clustering Methods for Functional Data](https://mlsanity.com/paper/2210.00847/) (2022)
- [Joint Registration and Conformal Prediction for Partially Observed Functional Data](https://mlsanity.com/paper/2502.15000/) (2025)
- [Neural Operators Can Discover Functional Clusters](https://mlsanity.com/paper/2602.23528/) (2026)

## Abstract (arXiv)

Elastic Riemannian metrics have been used successfully in the past for statistical treatments of functional and curve shape data. However, this usage has suffered from an important restriction: the function boundaries are assumed fixed and matched. Functional data exhibiting unmatched boundaries typically arise from dynamical systems with variable evolution rates such as COVID-19 infection rate curves associated with different geographical regions. In this case, it is more natural to model such data with sliding boundaries and use partial matching, i.e., only a part of a function is matched to another function. Here, we develop a comprehensive Riemannian framework that allows for partial matching, comparing, and clustering of functions under both phase variability and uncertain boundaries. We extend past work by: (1) Forming a joint action of the time-warping and time-scaling groups; (2) Introducing a metric that is invariant to this joint action, allowing for a gradient-based approach to elastic partial matching; and (3) Presenting a modification that, while losing the metric property, allows one to control relative influence of the two groups. This framework is illustrated for registering and clustering shapes of COVID-19 rate curves, identifying essential patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods.