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In this paper we propose an algorithm for exact partitioning of high-ordermodels. We define a general class of $m$-degree Homogeneous Polynomial Models,which subsumes several examples motivated from prior literature. Exactpartitioning can be formulated as a tensor optimization problem. We relax thishigh-order combinatorial problem to a convex conic form problem. To this end,we carefully define the Carath\'eodory symmetric tensor cone, and show itsconvexity, and the convexity of its dual cone. This allows us to construct aprimal-dual certificate to show that the solution of the convex relaxation iscorrect (equal to the unobserved true group assignment) and to analyze thestatistical upper bound of exact partitioning.

Exact Partitioning of High-order Models with a Novel Convex Tensor Cone Relaxation