Spectral convergence of diffusion maps: improved error bounds and an alternative normalisation
This work addresses theoretical gaps in diffusion maps for researchers in manifold learning, offering incremental improvements in error analysis and a new normalization method.
The paper tackles the problem of weak theoretical error bounds for diffusion maps in manifold learning by improving them for distributions on a hypertorus, matching long-standing pointwise error bounds for spectral data and operator convergence. It also introduces an alternative normalization using Sinkhorn weights, proving better convergence on flat domains and providing an efficient algorithm.
Diffusion maps is a manifold learning algorithm widely used for dimensionality reduction. Using a sample from a distribution, it approximates the eigenvalues and eigenfunctions of associated Laplace-Beltrami operators. Theoretical bounds on the approximation error are however generally much weaker than the rates that are seen in practice. This paper uses new approaches to improve the error bounds in the model case where the distribution is supported on a hypertorus. For the data sampling (variance) component of the error we make spatially localised compact embedding estimates on certain Hardy spaces; we study the deterministic (bias) component as a perturbation of the Laplace-Beltrami operator's associated PDE, and apply relevant spectral stability results. Using these approaches, we match long-standing pointwise error bounds for both the spectral data and the norm convergence of the operator discretisation. We also introduce an alternative normalisation for diffusion maps based on Sinkhorn weights. This normalisation approximates a Langevin diffusion on the sample and yields a symmetric operator approximation. We prove that it has better convergence compared with the standard normalisation on flat domains, and present a highly efficient algorithm to compute the Sinkhorn weights.