  Clustering algorithms partition a dataset into groups of similar points. The
clustering problem is very general, and different partitions of the same
dataset could be considered correct and useful. To fully understand such data,
it must be considered at a variety of scales, ranging from coarse to fine. We
introduce the Multiscale Environment for Learning by Diffusion (MELD) data
model, which is a family of clusterings parameterized by nonlinear diffusion on
the dataset. We show that the MELD data model precisely captures latent
multiscale structure in data and facilitates its analysis. To efficiently learn
the multiscale structure observed in many real datasets, we introduce the
Multiscale Learning by Unsupervised Nonlinear Diffusion (M-LUND) clustering
algorithm, which is derived from a diffusion process at a range of temporal
scales. We provide theoretical guarantees for the algorithm's performance and
establish its computational efficiency. Finally, we show that the M-LUND
clustering algorithm detects the latent structure in a range of synthetic and
real datasets.
