Neumann eigenmaps for landmark embedding
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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.
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References · 8
- 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 708
- 2017DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Samplingcited by 16
- 2021Chasing Collective Variables using Autoencoders and biased trajectoriescited by 13
- 2017Landmark Diffusion Maps (L-dMaps): Accelerated manifold learning out-of-sample extensioncited by 10
- 2022From latent dynamics to meaningful representationscited by 5
- 2023Diffusion Maps for Group-Invariant Manifoldscited by 4
- 2023Sharp error estimates for target measure diffusion maps with applications to the committor problemcited by 2
- 2019Transport Model for Feature Extractioncited by 1
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