# Neumann eigenmaps for landmark embedding

arXiv: https://arxiv.org/abs/2502.06689 · HTML: https://mlsanity.com/paper/2502.06689/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 44 references that are in the map.
- First posted on arXiv 10 Feb 2025.

## Builds on (most cited first, 8 of 8)

- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling](https://mlsanity.com/paper/1711.06011/) (2017)
- [Chasing Collective Variables using Autoencoders and biased trajectories](https://mlsanity.com/paper/2104.11061/) (2021)
- [Landmark Diffusion Maps (L-dMaps): Accelerated manifold learning out-of-sample extension](https://mlsanity.com/paper/1706.09396/) (2017)
- [From latent dynamics to meaningful representations](https://mlsanity.com/paper/2209.00905/) (2022)
- [Diffusion Maps for Group-Invariant Manifolds](https://mlsanity.com/paper/2303.16169/) (2023)
- [Sharp error estimates for target measure diffusion maps with applications to the committor problem](https://mlsanity.com/paper/2312.14418/) (2023)
- [Transport Model for Feature Extraction](https://mlsanity.com/paper/1910.14543/) (2019)

## Abstract (arXiv)

We present Neumann eigenmaps (NeuMaps), a novel approach for enhancing the standard diffusion map embedding using landmarks, i.e distinguished samples within the dataset. By interpreting these landmarks as a subgraph of the larger data graph, NeuMaps are obtained via the eigendecomposition of a renormalized Neumann Laplacian. We show that NeuMaps offer two key advantages: (1) they provide a computationally efficient embedding that accurately recovers the diffusion distance associated with the reflecting random walk on the subgraph, and (2) they naturally incorporate the Nystr"om extension within the diffusion map framework through the discrete Neumann boundary condition. Through examples in digit classification and molecular dynamics, we demonstrate that NeuMaps not only improve upon existing landmark-based embedding methods but also enhance the stability of diffusion map embeddings to the removal of highly significant points.