# A diffusion-map-based algorithm for gradient computation on manifolds and applications

arXiv: https://arxiv.org/abs/2108.06988 · HTML: https://mlsanity.com/paper/2108.06988/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 74 references that are in the map.
- First posted on arXiv 16 Aug 2021.

## Builds on (most cited first, 7 of 7)

- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [A Theoretical and Empirical Comparison of Gradient Approximations in Derivative-Free Optimization](https://mlsanity.com/paper/1905.01332/) (2019)
- [Averaging Stochastic Gradient Descent on Riemannian Manifolds](https://mlsanity.com/paper/1802.09128/) (2018)
- [Optimization Techniques on Riemannian Manifolds](https://mlsanity.com/paper/1407.5965/) (2014)
- [DIRECTGO: A new DIRECT-type MATLAB toolbox for derivative-free global optimization](https://mlsanity.com/paper/2107.02205/) (2021)
- [Manifold Learning for Accelerating Coarse-Grained Optimization](https://mlsanity.com/paper/2001.03518/) (2020)
- [Damped Newton's Method on Riemannian Manifolds](https://mlsanity.com/paper/1803.05126/) (2018)

## Abstract (arXiv)

We recover the Riemannian gradient of a given function defined on interior points of a Riemannian submanifold in the Euclidean space based on a sample of function evaluations at points in the submanifold. This approach is based on the estimates of the Laplace-Beltrami operator proposed in the diffusion-maps theory. The Riemannian gradient estimates do not involve differential terms. Analytical convergence results of the Riemannian gradient expansion are proved. We apply the Riemannian gradient estimate in a gradient-based algorithm providing a derivative-free optimization method. We test and validate several applications, including tomographic reconstruction from an unknown random angle distribution, and the sphere packing problem in dimensions 2 and 3.