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

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