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

Abstract

We study the problem of efficient exact partitioning of the hypergraphs generated by highorder 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. IntroductionOn a higher level, a planted model simulates interactions among various groups of entities in a network. Typical planted models assume that nodes are grouped into a number of clusters, and each pair of nodes is connected randomly with some probability related to the cluster membership. The generative and non-deterministic nature makes planted models of both theoretical and practical interests in the field of community detection, data mining, engineering, biology, among others. Various classical planted models have been studied extensively in recent years. This includes, for instance, the stochastic block models (SBMs) [

Information Theoretic Limits of Exact Recovery in Sub-hypergraph Models for Community Detection

“…The aforementioned works only focus on the stochastic block model in hypergraphs. [13] provided an algorithm for community detection for the densest sub-hypergraph model but provided an upper bound that is not tight. In practice, the obtained hypergraph data usually contains a large number of small groups consisting of few nodes, which makes the community detection problem harder.…”