# Recognizing Rigid Patterns of Unlabeled Point Clouds by Complete and Continuous Isometry Invariants with no False Negatives and no False Positives

arXiv: https://arxiv.org/abs/2303.15385 · HTML: https://mlsanity.com/paper/2303.15385/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 76 references that are in the map.
- First posted on arXiv 27 Mar 2023.

## Builds on (most cited first, 26 of 26)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Threat of Adversarial Attacks on Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/1801.00553/) (2018)
- [Deep Closest Point: Learning Representations for Point Cloud Registration](https://mlsanity.com/paper/1905.03304/) (2019)
- [Simple Black-box Adversarial Attacks](https://mlsanity.com/paper/1905.07121/) (2019)
- [Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipulation](https://mlsanity.com/paper/2112.05124/) (2021)
- [Functional Adversarial Attacks](https://mlsanity.com/paper/1906.00001/) (2019)
- [Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problem](https://mlsanity.com/paper/2101.08286/) (2021)
- [ROBIN: a Graph-Theoretic Approach to Reject Outliers in Robust Estimation using Invariants](https://mlsanity.com/paper/2011.03659/) (2020)
- [Equivariant representations for molecular Hamiltonians and N-center atomic-scale properties](https://mlsanity.com/paper/2109.12083/) (2021)
- [Learning an Effective Equivariant 3D Descriptor Without Supervision](https://mlsanity.com/paper/1909.06887/) (2019)
- [The Density Fingerprint of a Periodic Point Set](https://mlsanity.com/paper/2104.11046/) (2021)
- [The Gromov-Hausdorff distance between ultrametric spaces: its structure and computation](https://mlsanity.com/paper/2110.03136/) (2021)
- [Fast predictions of lattice energies by continuous isometry invariants of crystal structures](https://mlsanity.com/paper/2108.07233/) (2021)
- [The mergegram of a dendrogram and its stability](https://mlsanity.com/paper/2007.11278/) (2020)
- [Exactly computable and continuous metrics on isometry classes of finite and 1-periodic sequences](https://mlsanity.com/paper/2205.04388/) (2022)
- [Metricizing the Euclidean Space towards Desired Distance Relations in Point Clouds](https://mlsanity.com/paper/2211.03674/) (2022)
- [Voronoi-based similarity distances between arbitrary crystal lattices](https://mlsanity.com/paper/2002.11165/) (2020)
- [Introduction to Periodic Geometry and Topology](https://mlsanity.com/paper/2103.02749/) (2021)
- [Density functions of periodic sequences](https://mlsanity.com/paper/2205.02226/) (2022)
- [Complete invariants of atomic clouds under rigid motion with Lipschitz continuous metrics in a polynomial time](https://mlsanity.com/paper/2303.13486/) (2023)
- [Simplexwise Distance Distributions for finite spaces with metrics and measures](https://mlsanity.com/paper/2303.14161/) (2023)
- [Approximating Gromov-Hausdorff Distance in Euclidean Space](https://mlsanity.com/paper/1912.13008/) (2019)
- [Geographic-style maps for 2-dimensional lattices](https://mlsanity.com/paper/2109.10885/) (2021)
- [Isometry invariant shape recognition of projectively perturbed point clouds by the mergegram extending 0D persistence](https://mlsanity.com/paper/2111.04617/) (2021)
- [Density functions of periodic sequences of continuous events](https://mlsanity.com/paper/2301.05137/) (2023)

## Built on by (most cited first, 8 of 8)

- [RAPiD-Seg: Range-Aware Pointwise Distance Distribution Networks for 3D LiDAR Segmentation](https://mlsanity.com/paper/2407.10159/) (2024)
- [On the Completeness of Invariant Geometric Deep Learning Models](https://mlsanity.com/paper/2402.04836/) (2024)
- [Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds](https://mlsanity.com/paper/2307.09259/) (2023)
- [On the Expressive Power of Sparse Geometric MPNNs](https://mlsanity.com/paper/2407.02025/) (2024)
- [Universally Invariant Learning in Equivariant GNNs](https://mlsanity.com/paper/2510.13169/) (2025)
- [TraIL-Det: Transformation-Invariant Local Feature Networks for 3D LiDAR Object Detection with Unsupervised Pre-Training](https://mlsanity.com/paper/2408.13902/) (2024)
- [Toward bilipshiz geometric models](https://mlsanity.com/paper/2511.11735/) (2025)
- [LexiCan: Lexicographic Canonicalization of Point Clouds for Machine Learning Applications](https://mlsanity.com/paper/2610.15171/) (2026)

## Abstract (arXiv)

Rigid structures such as cars or any other solid objects are often represented by finite clouds of unlabeled points. The most natural equivalence on these point clouds is rigid motion or isometry maintaining all inter-point distances. Rigid patterns of point clouds can be reliably compared only by complete isometry invariants that can also be called equivariant descriptors without false negatives (isometric clouds having different descriptions) and without false positives (non-isometric clouds with the same description). Noise and motion in data motivate a search for invariants that are continuous under perturbations of points in a suitable metric. We propose the first continuous and complete invariant of unlabeled clouds in any Euclidean space. For a fixed dimension, the new metric for this invariant is computable in a polynomial time in the number of points.