# Exactly computable and continuous metrics on isometry classes of finite and 1-periodic sequences

arXiv: https://arxiv.org/abs/2205.04388 · HTML: https://mlsanity.com/paper/2205.04388/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 May 2022.

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

- [Smooth, exact rotational symmetrization for deep learning on point clouds](https://mlsanity.com/paper/2305.19302/) (2023)
- [Recognizing Rigid Patterns of Unlabeled Point Clouds by Complete and Continuous Isometry Invariants with no False Negatives and no False Positives](https://mlsanity.com/paper/2303.15385/) (2023)
- [Completeness of Atomic Structure Representations](https://mlsanity.com/paper/2302.14770/) (2023)

## Abstract (arXiv)

The inevitable noise in real measurements motivates the problem to continuously quantify the similarity between rigid objects such as periodic time series and proteins given by ordered points and considered up to isometry maintaining inter-point distances. The past work produced many Hausdorff-like distances that have slow or approximate algorithms due to minimizations over infinitely many isometries. For finite and 1-periodic sequences under isometry in any high-dimensional Euclidean space, we introduce continuous metrics with faster algorithms. The key novelty in the periodic case is the continuity of new metrics under perturbations that change the minimum period.