# Correspondence Learning via Linearly-invariant Embedding

arXiv: https://arxiv.org/abs/2010.13136 · HTML: https://mlsanity.com/paper/2010.13136/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Oct 2020.

## Built on by (most cited first, 30 of 48)

- [Unsupervised Learning of Robust Spectral Shape Matching](https://mlsanity.com/paper/2304.14419/) (2023)
- [Shape registration in the time of transformers](https://mlsanity.com/paper/2106.13679/) (2021)
- [Deep Orientation-Aware Functional Maps: Tackling Symmetry Issues in Shape Matching](https://mlsanity.com/paper/2204.13453/) (2022)
- [Multiway Non-rigid Point Cloud Registration via Learned Functional Map Synchronization](https://mlsanity.com/paper/2111.12878/) (2021)
- [Bending Graphs: Hierarchical Shape Matching using Gated Optimal Transport](https://mlsanity.com/paper/2202.01537/) (2022)
- [Self-Supervised Learning for Multimodal Non-Rigid 3D Shape Matching](https://mlsanity.com/paper/2303.10971/) (2023)
- [NICP: Neural ICP for 3D Human Registration at Scale](https://mlsanity.com/paper/2312.14024/) (2023)
- [Understanding and Improving Features Learned in Deep Functional Maps](https://mlsanity.com/paper/2303.16527/) (2023)
- [SE-ORNet: Self-Ensembling Orientation-aware Network for Unsupervised Point Cloud Shape Correspondence](https://mlsanity.com/paper/2304.05395/) (2023)
- [NCP: Neural Correspondence Prior for Effective Unsupervised Shape Matching](https://mlsanity.com/paper/2301.05839/) (2023)
- [Hybrid Functional Maps for Crease-Aware Non-Isometric Shape Matching](https://mlsanity.com/paper/2312.03678/) (2023)
- [Differentiable Surface Triangulation](https://mlsanity.com/paper/2109.10695/) (2021)
- [Point2SSM: Learning Morphological Variations of Anatomies from Point Cloud](https://mlsanity.com/paper/2305.14486/) (2023)
- [Denoising Functional Maps: Diffusion Models for Shape Correspondence](https://mlsanity.com/paper/2503.01845/) (2025)
- [Hierarchical Neural Semantic Representation for 3D Semantic Correspondence](https://mlsanity.com/paper/2509.17431/) (2025)
- [Implicit field supervision for robust non-rigid shape matching](https://mlsanity.com/paper/2203.07694/) (2022)
- [Non-Rigid Shape Registration via Deep Functional Maps Prior](https://mlsanity.com/paper/2311.04494/) (2023)
- [SRIF: Semantic Shape Registration Empowered by Diffusion-based Image Morphing and Flow Estimation](https://mlsanity.com/paper/2409.11682/) (2024)
- [Wormhole Loss for Partial Shape Matching](https://mlsanity.com/paper/2410.22899/) (2024)
- [On Unsupervised Partial Shape Correspondence](https://mlsanity.com/paper/2310.14692/) (2023)
- [DV-Matcher: Deformation-based Non-Rigid Point Cloud Matching Guided by Pre-trained Visual Features](https://mlsanity.com/paper/2408.08568/) (2024)
- [Neural Intrinsic Embedding for Non-rigid Point Cloud Matching](https://mlsanity.com/paper/2303.01038/) (2023)
- [Reduced Representation of Deformation Fields for Effective Non-rigid Shape Matching](https://mlsanity.com/paper/2211.14604/) (2022)
- [G-MSM: Unsupervised Multi-Shape Matching with Graph-based Affinity Priors](https://mlsanity.com/paper/2212.02910/) (2022)
- [Deep Virtual Markers for Articulated 3D Shapes](https://mlsanity.com/paper/2108.09000/) (2021)
- [DiscoMatch: Fast Discrete Optimisation for Geometrically Consistent 3D Shape Matching](https://mlsanity.com/paper/2310.08230/) (2023)
- [Shape Non-rigid Kinematics (SNK): A Zero-Shot Method for Non-Rigid Shape Matching via Unsupervised Functional Map Regularized Reconstruction](https://mlsanity.com/paper/2403.06804/) (2024)
- [Unsupervised Template-assisted Point Cloud Shape Correspondence Network](https://mlsanity.com/paper/2403.16412/) (2024)
- [Conjugate Product Graphs for Globally Optimal 2D-3D Shape Matching](https://mlsanity.com/paper/2211.11589/) (2022)
- [Kissing to Find a Match: Efficient Low-Rank Permutation Representation](https://mlsanity.com/paper/2308.13252/) (2023)

## Abstract (arXiv)

In this paper, we propose a fully differentiable pipeline for estimating accurate dense correspondences between 3D point clouds. The proposed pipeline is an extension and a generalization of the functional maps framework. However, instead of using the Laplace-Beltrami eigenfunctions as done in virtually all previous works in this domain, we demonstrate that learning the basis from data can both improve robustness and lead to better accuracy in challenging settings. We interpret the basis as a learned embedding into a higher dimensional space. Following the functional map paradigm the optimal transformation in this embedding space must be linear and we propose a separate architecture aimed at estimating the transformation by learning optimal descriptor functions. This leads to the first end-to-end trainable functional map-based correspondence approach in which both the basis and the descriptors are learned from data. Interestingly, we also observe that learning a canonical embedding leads to worse results, suggesting that leaving an extra linear degree of freedom to the embedding network gives it more robustness, thereby also shedding light onto the success of previous methods. Finally, we demonstrate that our approach achieves state-of-the-art results in challenging non-rigid 3D point cloud correspondence applications.