mlsanityData to 7 Oct 2026

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

arXiv 2303.15385 ↗First posted 27 Mar 2023cs.CV cs.CG math.MGcited by 7

Not on the map. Why?

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.

Key facts. Cited by 7 papers in the map (data through 7 Oct 2026). Builds on 26 of its 76 references that are in the map. First posted on arXiv 27 Mar 2023.

Builds on

26 of 76 references.

References · 26

  1. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  2. 2016Geometric deep learning: going beyond Euclidean datacited by 1,042
  3. 2018Threat of Adversarial Attacks on Deep Learning in Computer Vision: A Surveycited by 459
  4. 2019Deep Closest Point: Learning Representations for Point Cloud Registrationcited by 318
  5. 2019Simple Black-box Adversarial Attackscited by 249
  6. 2021Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipulationcited by 204
  7. 2019Functional Adversarial Attackscited by 82
  8. 2021Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problemcited by 53
  9. 2020ROBIN: a Graph-Theoretic Approach to Reject Outliers in Robust Estimation using Invariantscited by 38
  10. 2021Equivariant representations for molecular Hamiltonians and N-center atomic-scale propertiescited by 19
  11. 2019Learning an Effective Equivariant 3D Descriptor Without Supervisioncited by 14
  12. 2021The Density Fingerprint of a Periodic Point Setcited by 5
14 more
  1. 2021The Gromov-Hausdorff distance between ultrametric spaces: its structure and computationcited by 5
  2. 2021Fast predictions of lattice energies by continuous isometry invariants of crystal structurescited by 4
  3. 2020The mergegram of a dendrogram and its stabilitycited by 3
  4. 2022Exactly computable and continuous metrics on isometry classes of finite and 1-periodic sequencescited by 3
  5. 2022Metricizing the Euclidean Space towards Desired Distance Relations in Point Cloudscited by 3
  6. 2020Voronoi-based similarity distances between arbitrary crystal latticescited by 2
  7. 2021Introduction to Periodic Geometry and Topologycited by 2
  8. 2022Density functions of periodic sequencescited by 2
  9. 2023Complete invariants of atomic clouds under rigid motion with Lipschitz continuous metrics in a polynomial timecited by 2
  10. 2019Approximating Gromov-Hausdorff Distance in Euclidean Spacecited by 1
  11. 2021Geographic-style maps for 2-dimensional latticescited by 1
  12. 2021Isometry invariant shape recognition of projectively perturbed point clouds by the mergegram extending 0D persistencecited by 1
  13. 2023Density functions of periodic sequences of continuous eventscited by 1
  14. 2023Simplexwise Distance Distributions for finite spaces with metrics and measurescited by 1

Built on by

Cited by year

Citing papers per year: 2023: 1; 2024: 4; 2025: 2.

7 papers.

  1. 2024RAPiD-Seg: Range-Aware Pointwise Distance Distribution Networks for 3D LiDAR Segmentationcited by 15
  2. 2024On the Completeness of Invariant Geometric Deep Learning Modelscited by 11
  3. 2023Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Cloudscited by 8
  4. 2024On the Expressive Power of Sparse Geometric MPNNscited by 7
  5. 2025Universally Invariant Learning in Equivariant GNNscited by 3
  6. 2024TraIL-Det: Transformation-Invariant Local Feature Networks for 3D LiDAR Object Detection with Unsupervised Pre-Trainingcited by 1
  7. 2025Toward bilipshiz geometric modelscited by 1