# Diffusion Maps : Using the Semigroup Property for Parameter Tuning

arXiv: https://arxiv.org/abs/2203.02867 · HTML: https://mlsanity.com/paper/2203.02867/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 13 references that are in the map.
- First posted on arXiv 6 Mar 2022.

## Built on by (most cited first, 3 of 3)

- [Learning a distance measure from the information-estimation geometry of data](https://mlsanity.com/paper/2510.02514/) (2025)
- [Parametrized Power-Iteration Clustering for Directed Graphs](https://mlsanity.com/paper/2210.00310/) (2022)
- [Nonlinear Dimensionality Reduction with Diffusion Maps in Practice](https://mlsanity.com/paper/2601.20428/) (2026)

## Abstract (arXiv)

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.