Joint parameters estimation in cubic tensor model

Abstract

We study joint parameter estimation from a single observation in high-dimensional Gibbs measures with cubic tensor interactions, motivated by dense ERGMs, arithmetic-progression models, and inhomogeneous random hypergraphs. Focusing on the maximum pseudolikelihood estimator, we give checkable conditions for joint consistency and asymptotic ill-conditioning. For the edge-triangle ERGM, pseudolikelihood is ill-conditioned in the ferromagnetic regime with nonnegative field, but consistent in a sufficiently strong antiferromagnetic regime. For the edge-three-star ERGM, it is ill-conditioned for all inverse temperatures and external fields. We also study consistency for arithmetic-progression, and inhomogeneous hypergraph models. Our proofs develop nonlinear large-deviation and mean-field approximation tools for cubic tensor Gibbs measures, which have scope for broad applications.