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

TL;DR

This paper introduces a convex tensor cone relaxation method for exact partitioning of high-order polynomial models, enabling precise group assignment recovery with theoretical guarantees.

Contribution

It defines a novel convex tensor cone and demonstrates its effectiveness for exact partitioning of high-order models, extending prior approaches.

Findings

Convexity of the Carathéodory symmetric tensor cone established.

Primal-dual certificate guarantees exact recovery of true partition.

Provides statistical bounds for the success of the method.

Abstract

In this paper we propose an algorithm for exact partitioning of high-order models. 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 high-order combinatorial problem to a convex conic form problem. To this end, we carefully define the Carath\'eodory symmetric 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) and to analyze the statistical upper bound of exact partitioning.