  Data-dependent metrics are powerful tools for learning the underlying
structure of high-dimensional data. This article develops and analyzes a
data-dependent metric known as diffusion state distance (DSD), which compares
points using a data-driven diffusion process. Unlike related diffusion methods,
DSDs incorporate information across time scales, which allows for the intrinsic
data structure to be inferred in a parameter-free manner. This article develops
a theory for DSD based on the multitemporal emergence of mesoscopic equilibria
in the underlying diffusion process. New algorithms for denoising and dimension
reduction with DSD are also proposed and analyzed. These approaches are based
on a weighted spectral decomposition of the underlying diffusion process, and
experiments on synthetic datasets and real biological networks illustrate the
efficacy of the proposed algorithms in terms of both speed and accuracy.
Throughout, comparisons with related methods are made, in order to illustrate
the distinct advantages of DSD for datasets exhibiting multiscale structure.
