Neumann eigenmaps for landmark embedding
This work addresses computational efficiency and stability issues in manifold learning for researchers in data science and applied mathematics, representing an incremental improvement over existing landmark-based methods.
The authors tackled the problem of enhancing diffusion map embeddings using landmarks by proposing Neumann eigenmaps (NeuMaps), which provide a computationally efficient embedding that accurately recovers diffusion distances and improves stability to point removal, as demonstrated in digit classification and molecular dynamics examples.
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 Nyström 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.