  This paper proposes and analyzes a novel clustering algorithm that combines
graph-based diffusion geometry with techniques based on density and mode
estimation. The proposed method is suitable for data generated from mixtures of
distributions with densities that are both multimodal and have nonlinear
shapes. A crucial aspect of this algorithm is the use of time of a data-adapted
diffusion process as a scale parameter that is different from the local spatial
scale parameter used in many clustering algorithms. We prove estimates for the
behavior of diffusion distances with respect to this time parameter under a
flexible nonparametric data model, identifying a range of times in which the
mesoscopic equilibria of the underlying process are revealed, corresponding to
a gap between within-cluster and between-cluster diffusion distances. These
structures can be missed by the top eigenvectors of the graph Laplacian,
commonly used in spectral clustering. This analysis is leveraged to prove
sufficient conditions guaranteeing the accuracy of the proposed \emph{learning
by unsupervised nonlinear diffusion (LUND)} procedure. We implement LUND and
confirm its theoretical properties on illustrative datasets, demonstrating the
theoretical and empirical advantages over both spectral clustering and
density-based clustering techniques.
