# Diffusion Maps for Group-Invariant Manifolds

arXiv: https://arxiv.org/abs/2303.16169 · HTML: https://mlsanity.com/paper/2303.16169/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 21 references that are in the map.
- First posted on arXiv 28 Mar 2023.

## Builds on (most cited first, 4 of 4)

- [Earthmover-based manifold learning for analyzing molecular conformation spaces](https://mlsanity.com/paper/1911.06107/) (2019)
- [Minimax-optimal semi-supervised regression on unknown manifolds](https://mlsanity.com/paper/1611.02221/) (2016)
- [The steerable graph Laplacian and its application to filtering image data-sets](https://mlsanity.com/paper/1802.01894/) (2018)
- [Approximation of Functions over Manifolds: A Moving Least-Squares Approach](https://mlsanity.com/paper/1711.00765/) (2017)

## Built on by (most cited first, 4 of 4)

- [Sharp error estimates for target measure diffusion maps with applications to the committor problem](https://mlsanity.com/paper/2312.14418/) (2023)
- [The G-invariant graph Laplacian](https://mlsanity.com/paper/2303.17001/) (2023)
- [Neumann eigenmaps for landmark embedding](https://mlsanity.com/paper/2502.06689/) (2025)
- [Group-invariant Coresets for Data-efficient Active Learning](https://mlsanity.com/paper/2607.01089/) (2026)

## Abstract (arXiv)

In this article, we consider the manifold learning problem when the data set is invariant under the action of a compact Lie group K. Our approach consists in augmenting the data-induced graph Laplacian by integrating over the K-orbits of the existing data points, which yields a K-invariant graph Laplacian L. We prove that L can be diagonalized by using the unitary irreducible representation matrices of K, and we provide an explicit formula for computing its eigenvalues and eigenfunctions. In addition, we show that the normalized Laplacian operator LN converges to the Laplace-Beltrami operator of the data manifold with an improved convergence rate, where the improvement grows with the dimension of the symmetry group K. This work extends the steerable graph Laplacian framework of Landa and Shkolnisky from the case of operatornameSO(2) to arbitrary compact Lie groups.