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

In this paper we propose the first correct poly-time algorithm for exact partitioning of high-order models (a worst case NP-hard problem). We define a general class of $m$-degree Homogeneous Polynomial Models, which subsumes several examples motivated from prior literature. Exact partitioning can be formulated as a tensor optimization problem. We relax this NP-hard problem to a convex conic form problem (poly-time solvable by interior point methods). To this end, we carefully define the positive semidefinite tensor cone, and show its convexity, and the convexity of its dual cone. This allows us to construct a primal-dual certificate to show that the solution of the convex relaxation is correct (equal to the unobserved true group assignment) under some sample complexity conditions.