  We introduce the {\it diffusion $K$-means} clustering method on Riemannian
submanifolds, which maximizes the within-cluster connectedness based on the
diffusion distance. The diffusion $K$-means constructs a random walk on the
similarity graph with vertices as data points randomly sampled on the manifolds
and edges as similarities given by a kernel that captures the local geometry of
manifolds. The diffusion $K$-means is a multi-scale clustering tool that is
suitable for data with non-linear and non-Euclidean geometric features in mixed
dimensions. Given the number of clusters, we propose a polynomial-time convex
relaxation algorithm via the semidefinite programming (SDP) to solve the
diffusion $K$-means. In addition, we also propose a nuclear norm regularized
SDP that is adaptive to the number of clusters. In both cases, we show that
exact recovery of the SDPs for diffusion $K$-means can be achieved under
suitable between-cluster separability and within-cluster connectedness of the
submanifolds, which together quantify the hardness of the manifold clustering
problem. We further propose the {\it localized diffusion $K$-means} by using
the local adaptive bandwidth estimated from the nearest neighbors. We show that
exact recovery of the localized diffusion $K$-means is fully adaptive to the
local probability density and geometric structures of the underlying
submanifolds.
