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Neumann eigenmaps for landmark embedding

Shashank Sule, Wojciech Czaja

TL;DR

NeuMaps address the scalability and robustness limitations of diffusion-map embeddings by embedding a landmark subset of the data through normalized Neumann eigenvectors of the landmark subgraph, yielding a diffusion distance for the reflecting random walk and a Nyström-compatible out-of-sample extension. The method reduces computation by solving a smaller eigenproblem while preserving geometric structure and exhibits improved stability to removal of highly significant points. Theoretical results establish an isometric embedding via the Neumann Laplacian and connect the extension to Nyström, with empirical validation on digit classification and molecular dynamics showing superior performance to Roseland and standard diffusion maps. Together, these contributions provide a boundary-aware, landmark-driven diffusion framework with practical impact for scalable, robust manifold learning.

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 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.

Neumann eigenmaps for landmark embedding

TL;DR

NeuMaps address the scalability and robustness limitations of diffusion-map embeddings by embedding a landmark subset of the data through normalized Neumann eigenvectors of the landmark subgraph, yielding a diffusion distance for the reflecting random walk and a Nyström-compatible out-of-sample extension. The method reduces computation by solving a smaller eigenproblem while preserving geometric structure and exhibits improved stability to removal of highly significant points. Theoretical results establish an isometric embedding via the Neumann Laplacian and connect the extension to Nyström, with empirical validation on digit classification and molecular dynamics showing superior performance to Roseland and standard diffusion maps. Together, these contributions provide a boundary-aware, landmark-driven diffusion framework with practical impact for scalable, robust manifold learning.

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 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.

Paper Structure

This paper contains 14 sections, 4 theorems, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions and organization
  4. Neumann spectra of subgraphs
  5. Neumann Maps
  6. Numerical experiments
  7. Digit classification and Roseland
  8. Learning collective variables in molecular dynamics
  9. Conclusion
  10. Proof of theorem \ref{['thm: Neummann eigenfunction main theorem']}
  11. Proof of proposition \ref{['prop: Neumann laplacian']}
  12. Proof of proposition \ref{['prop: Neumann random walk matrix is a random walk matrix']}
  13. Proof of \ref{['thm: Main theorem of Neumann diffusions']}
  14. Data and code availability

Key Result

Theorem 2.1

Let $V_S \subseteq V_G$ and $L_{V^*}$ the Laplacian of the graph generated by $V^*=V_S \cup \delta S$. The Neumann eigenfunction $f = f_{1}^{N}$ satisfies the following properties:

Figures (2)

  • Figure 1: Left: Embedding UCI digits with 25% random landmarking via Roseland. Right: Embedding UCI digits with Neumann maps.
  • Figure 2: Top to bottom: FP eigenfunction embeddings of data after removal of landmarks provided by subsampling uniformly in time, choosing metastable states, and uniformly subsampling in space via $\delta$-nets.

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.1: Lemma 8.1 in SGTbook
  • Remark 2.1
  • Definition 3.1
  • Definition 3.2
  • ...and 6 more