Neumann eigenmaps for landmark embedding

TL;DR

Neumann eigenmaps (NeuMaps) improve landmark-based diffusion map embeddings by using a Neumann Laplacian approach, offering computational efficiency and better stability in applications like digit classification and molecular dynamics.

Contribution

Introduction of NeuMaps, a novel landmark embedding method utilizing Neumann Laplacian eigendecomposition for improved efficiency and stability over existing techniques.

Findings

NeuMaps accurately recover diffusion distances on subgraphs.

NeuMaps enhance stability of embeddings when removing significant points.

NeuMaps outperform existing landmark-based diffusion map methods.

Abstract

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.