Exact Partitioning of High-order Planted Models with a Tensor Nuclear Norm Constraint

TL;DR

This paper introduces a convex optimization approach with a tensor nuclear norm constraint for the exact partitioning of hypergraphs generated by high-order planted models, under certain conditions with high probability.

Contribution

It presents the first efficient convex method for exact partitioning of high-order hypergraph models, extending spectral and combinatorial techniques to tensor-based formulations.

Findings

Exact partitioning achievable via convex optimization under specific conditions

High probability of success in recovering true cluster structures

Applicable to various high-order hypergraph models such as hypercliques and stochastic block models

Abstract

We study the problem of efficient exact partitioning of the hypergraphs generated by high-order planted models. A high-order planted model assumes some underlying cluster structures, and simulates high-order interactions by placing hyperedges among nodes. Example models include the disjoint hypercliques, the densest subhypergraphs, and the hypergraph stochastic block models. We show that exact partitioning of high-order planted models (a NP-hard problem in general) is achievable through solving a computationally efficient convex optimization problem with a tensor nuclear norm constraint. Our analysis provides the conditions for our approach to succeed on recovering the true underlying cluster structures, with high probability.