Improved graph Laplacian via geometric self-consistency

TL;DR

This paper proposes a method to optimally set the kernel bandwidth in manifold learning by aligning the graph Laplacian with the underlying data geometry, leading to improved preservation of manifold structure.

Contribution

It introduces a geometry-based bandwidth selection method that enhances the construction of graph Laplacians in manifold learning.

Findings

The approach effectively preserves data geometry.

It demonstrates robustness across different datasets.

Experiments outperform traditional bandwidth selection methods.

Abstract

We address the problem of setting the kernel bandwidth used by Manifold Learning algorithms to construct the graph Laplacian. Exploiting the connection between manifold geometry, represented by the Riemannian metric, and the Laplace-Beltrami operator, we set the bandwidth by optimizing the Laplacian's ability to preserve the geometry of the data. Experiments show that this principled approach is effective and robust.