# Riemannian Functional Map Synchronization for Probabilistic Partial Correspondence in Shape Networks

arXiv: https://arxiv.org/abs/2111.14762 · HTML: https://mlsanity.com/paper/2111.14762/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 51 references that are in the map.
- First posted on arXiv 29 Nov 2021.

## Builds on (most cited first, 18 of 18)

- [Deep Functional Maps: Structured Prediction for Dense Shape Correspondence](https://mlsanity.com/paper/1704.08686/) (2017)
- [Deep Geometric Functional Maps: Robust Feature Learning for Shape Correspondence](https://mlsanity.com/paper/2003.14286/) (2020)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Learning multiview 3D point cloud registration](https://mlsanity.com/paper/2001.05119/) (2020)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [SyncSpecCNN: Synchronized Spectral CNN for 3D Shape Segmentation](https://mlsanity.com/paper/1612.00606/) (2016)
- [MultiBodySync: Multi-Body Segmentation and Motion Estimation via 3D Scan Synchronization](https://mlsanity.com/paper/2101.06605/) (2021)
- [Isometric Multi-Shape Matching](https://mlsanity.com/paper/2012.02689/) (2020)
- [Quantum Permutation Synchronization](https://mlsanity.com/paper/2101.07755/) (2021)
- [Multiway Non-rigid Point Cloud Registration via Learned Functional Map Synchronization](https://mlsanity.com/paper/2111.12878/) (2021)
- [Learning Transformation Synchronization](https://mlsanity.com/paper/1901.09458/) (2019)
- [FARM: Functional Automatic Registration Method for 3D Human Bodies](https://mlsanity.com/paper/1807.10517/) (2018)
- [Probabilistic Permutation Synchronization using the Riemannian Structure of the Birkhoff Polytope](https://mlsanity.com/paper/1904.05814/) (2019)
- [Bayesian Pose Graph Optimization via Bingham Distributions and Tempered Geodesic MCMC](https://mlsanity.com/paper/1805.12279/) (2018)
- [Synchronizing Probability Measures on Rotations via Optimal Transport](https://mlsanity.com/paper/2004.00663/) (2020)
- [Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev](https://mlsanity.com/paper/2010.05263/) (2020)
- [Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry](https://mlsanity.com/paper/1802.02628/) (2018)
- [Functional Maps Representation on Product Manifolds](https://mlsanity.com/paper/1809.10940/) (2018)

## Built on by (most cited first, 1 of 1)

- [G-MSM: Unsupervised Multi-Shape Matching with Graph-based Affinity Priors](https://mlsanity.com/paper/2212.02910/) (2022)

## Abstract (arXiv)

We consider the problem of graph-matching on a network of 3D shapes with uncertainty quantification. We assume that the pairwise shape correspondences are efficiently represented as functional maps, that match real-valued functions defined over pairs of shapes. By modeling functional maps between nearly isometric shapes as elements of the Lie group SO(n), we employ synchronization to enforce cycle consistency of the collection of functional maps over the graph, hereby enhancing the accuracy of the individual maps. We further introduce a tempered Bayesian probabilistic inference framework on SO(n). Our framework enables: (i) synchronization of functional maps as maximum-a-posteriori estimation on the Riemannian manifold of functional maps, (ii) sampling the solution space in our energy based model so as to quantify uncertainty in the synchronization problem. We dub the latter Riemannian Langevin Functional Map (RLFM) Sampler. Our experiments demonstrate that constraining the synchronization on the Riemannian manifold SO(n) improves the estimation of the functional maps, while our RLFM sampler provides for the first time an uncertainty quantification of the results.