Learning by Unsupervised Nonlinear Diffusion

This paper proposes and analyzes a novel clustering algorithm that combines graph-based diffusion geometry with density 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 to introduce time of a data-adapted diffusion process as a scale parameter that is different from the local spatial scale parameter used in many clustering and learning 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. This analysis is leveraged to prove sufficient conditions guaranteeing the accuracy of the proposed learning by unsupervised nonlinear diffusion (LUND) algorithm. We implement the LUND algorithm numerically and confirm its theoretical properties on illustrative datasets, showing that the proposed method enjoys both theoretical and empirical advantages over current spectral clustering and density-based clustering techniques.