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We introduce the {\it diffusion $K$-means} clustering method on Riemanniansubmanifolds, which maximizes the within-cluster connectedness based on thediffusion distance. The diffusion $K$-means constructs a random walk on thesimilarity graph with vertices as data points randomly sampled on the manifoldsand edges as similarities given by a kernel that captures the local geometry ofmanifolds. The diffusion $K$-means is a multi-scale clustering tool that issuitable for data with non-linear and non-Euclidean geometric features in mixeddimensions. Given the number of clusters, we propose a polynomial-time convexrelaxation algorithm via the semidefinite programming (SDP) to solve thediffusion $K$-means. In addition, we also propose a nuclear norm regularizedSDP that is adaptive to the number of clusters. In both cases, we show thatexact recovery of the SDPs for diffusion $K$-means can be achieved undersuitable between-cluster separability and within-cluster connectedness of thesubmanifolds, which together quantify the hardness of the manifold clusteringproblem. We further propose the {\it localized diffusion $K$-means} by usingthe local adaptive bandwidth estimated from the nearest neighbors. We show thatexact recovery of the localized diffusion $K$-means is fully adaptive to thelocal probability density and geometric structures of the underlyingsubmanifolds.

Diffusion $K$-means clustering on manifolds: provable exact recovery via semidefinite relaxations