Diffusion Maps : Using the Semigroup Property for Parameter Tuning

TL;DR

This paper introduces a semigroup-based criterion for selecting the diffusion time parameter in diffusion maps, improving the robustness and effectiveness of the dimension reduction process for data on low-dimensional manifolds.

Contribution

It proposes a novel semigroup criterion for tuning the diffusion time parameter, addressing a key challenge in applying diffusion maps.

Findings

The semigroup criterion effectively selects diffusion time t.

The approach is robust across different datasets.

Experiments demonstrate improved dimension reduction quality.

Abstract

Diffusion maps (DM) constitute a classic dimension reduction technique, for data lying on or close to a (relatively) low-dimensional manifold embedded in a much larger dimensional space. The DM procedure consists in constructing a spectral parametrization for the manifold from simulated random walks or diffusion paths on the data set. However, DM is hard to tune in practice. In particular, the task to set a diffusion time t when constructing the diffusion kernel matrix is critical. We address this problem by using the semigroup property of the diffusion operator. We propose a semigroup criterion for picking t. Experiments show that this principled approach is effective and robust.