Diffusion maps tailored to arbitrary non-degenerate Itô processes

Ralf Banisch*, Zofia Trstanova, Andreas Bittracher, Stefan Klus, Péter Koltai

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Itô diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling.

Original languageEnglish
Pages (from-to)242-265
Number of pages24
JournalApplied and Computational Harmonic Analysis
Issue number1
Publication statusPublished - Jan 2020


  • Diffusion maps
  • Dimensionality reduction
  • Itô process
  • Local kernel

ASJC Scopus subject areas

  • Applied Mathematics


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