Kernelized Diffusion maps

TL;DR

This paper introduces a kernelized approach to diffusion maps that improves Laplacian estimation in high-dimensional data, overcoming the curse of dimensionality and enabling efficient computation.

Contribution

It proposes a novel RKHS-based Laplacian estimator that adapts to data regularity and provides non-asymptotic rates, with techniques for computational efficiency.

Findings

Kernel estimator circumvents curse of dimensionality.

Non-asymptotic statistical rates established.

Techniques like Nyström and Fourier features reduce computational cost.

Abstract

Spectral clustering and diffusion maps are celebrated dimensionality reduction algorithms built on eigen-elements related to the diffusive structure of the data. The core of these procedures is the approximation of a Laplacian through a graph kernel approach, however this local average construction is known to be cursed by the high-dimension d. In this article, we build a different estimator of the Laplacian, via a reproducing kernel Hilbert space method, which adapts naturally to the regularity of the problem. We provide non-asymptotic statistical rates proving that the kernel estimator we build can circumvent the curse of dimensionality. Finally we discuss techniques (Nystr\"om subsampling, Fourier features) that enable to reduce the computational cost of the estimator while not degrading its overall performance.