# Diffusion Means and Heat Kernel on Manifolds

arXiv: https://arxiv.org/abs/2103.00588 · HTML: https://mlsanity.com/paper/2103.00588/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2021.

## Built on by (most cited first, 2 of 2)

- [A Manifold-based Airfoil Geometric-feature Extraction and Discrepant Data Fusion Learning Method](https://mlsanity.com/paper/2206.12254/) (2022)
- [Second-order differential operators, stochastic differential equations and Brownian motions on embedded manifolds](https://mlsanity.com/paper/2406.02879/) (2024)

## Abstract (arXiv)

We introduce diffusion means as location statistics on manifold data spaces. A diffusion mean is defined as the starting point of an isotropic diffusion with a given diffusivity. They can therefore be defined on all spaces on which a Brownian motion can be defined and numerical calculation of sample diffusion means is possible on a variety of spaces using the heat kernel expansion. We present several classes of spaces, for which the heat kernel is known and sample diffusion means can therefore be calculated. As an example, we investigate a classic data set from directional statistics, for which the sample Fr'echet mean exhibits finite sample smeariness.