# The Vector Heat Method

arXiv: https://arxiv.org/abs/1805.09170 · HTML: https://mlsanity.com/paper/1805.09170/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2018.

## Built on by (most cited first, 30 of 31)

- [Deep Orientation-Aware Functional Maps: Tackling Symmetry Issues in Shape Matching](https://mlsanity.com/paper/2204.13453/) (2022)
- [BimArt: A Unified Approach for the Synthesis of 3D Bimanual Interaction with Articulated Objects](https://mlsanity.com/paper/2412.05066/) (2024)
- [Mesh Neural Networks for SE(3)-Equivariant Hemodynamics Estimation on the Artery Wall](https://mlsanity.com/paper/2212.05023/) (2022)
- [Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2303.11323/) (2023)
- [PICO: Reconstructing 3D People In Contact with Objects](https://mlsanity.com/paper/2504.17695/) (2025)
- [Intrinsic Neural Fields: Learning Functions on Manifolds](https://mlsanity.com/paper/2203.07967/) (2022)
- [Complex Functional Maps : a Conformal Link Between Tangent Bundles](https://mlsanity.com/paper/2112.09546/) (2021)
- [Hodge-Compositional Edge Gaussian Processes](https://mlsanity.com/paper/2310.19450/) (2023)
- [Field Convolutions for Surface CNNs](https://mlsanity.com/paper/2104.03916/) (2021)
- [Physics-informed neural networks to learn cardiac fiber orientation from multiple electroanatomical maps](https://mlsanity.com/paper/2201.12362/) (2022)
- [Tangent Bundle Filters and Neural Networks: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2210.15058/) (2022)
- [DA Wand: Distortion-Aware Selection using Neural Mesh Parameterization](https://mlsanity.com/paper/2212.06344/) (2022)
- [Learning atrial fiber orientations and conductivity tensors from intracardiac maps using physics-informed neural networks](https://mlsanity.com/paper/2102.10863/) (2021)
- [Intrinsic Gaussian Vector Fields on Manifolds](https://mlsanity.com/paper/2310.18824/) (2023)
- [HotSpot: Signed Distance Function Optimization with an Asymptotically Sufficient Condition](https://mlsanity.com/paper/2411.14628/) (2024)
- [A Heat Diffusion Perspective on Geodesic Preserving Dimensionality Reduction](https://mlsanity.com/paper/2305.19043/) (2023)
- [Implicit Gaussian process representation of vector fields over arbitrary latent manifolds](https://mlsanity.com/paper/2309.16746/) (2023)
- [Supervised Training of Dense Object Nets using Optimal Descriptors for Industrial Robotic Applications](https://mlsanity.com/paper/2102.08096/) (2021)
- [Learning the Structure of Connection Graphs](https://mlsanity.com/paper/2510.11245/) (2025)
- [An Intrinsic Vector Heat Network](https://mlsanity.com/paper/2406.09648/) (2024)
- [NeRF-Texture: Synthesizing Neural Radiance Field Textures](https://mlsanity.com/paper/2412.10004/) (2024)
- [Wall Shear Stress Estimation in Abdominal Aortic Aneurysms: Towards Generalisable Neural Surrogate Models](https://mlsanity.com/paper/2507.22817/) (2025)
- [Voronoi-Assisted Optimization for Diffusing Unsigned Distance Fields from Unoriented Points](https://mlsanity.com/paper/2510.12524/) (2025)
- [Deep Feature Deformation Weights](https://mlsanity.com/paper/2601.12527/) (2026)
- [From Extrinsic to Intrinsic: Geodesic-Guided Representation Learning for 3D Geometric Data](https://mlsanity.com/paper/2606.02268/) (2026)
- [Heat Kernel Textures: the Geodesic Gaussians That Do Not Splat](https://mlsanity.com/paper/2609.07557/) (2026)
- [Sinkhorn Normalization of Diffusion Kernels](https://mlsanity.com/paper/2507.06161/) (2025)
- [Region of Interest Segmentation and Morphological Analysis for Membranes in Cryo-Electron Tomography](https://mlsanity.com/paper/2602.21195/) (2026)
- [Medial Axis Aware Learning of Signed Distance Functions](https://mlsanity.com/paper/2604.16512/) (2026)
- [Points as Tori: Fast Pointwise Signed Distance for Point Clouds](https://mlsanity.com/paper/2607.16946/) (2026)

## Abstract (arXiv)

This paper describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Fr'echet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations (iDT) so that they can be used in the context of tangent vector field processing.