Free Energy Universality in Tensor Estimation via Generic Chaining

2026-05-28

AI Summary

This study investigates the universality of free energy in high-dimensional tensor estimation, particularly under model misspecification and when parameters vary with dimension. By leveraging Gaussian comparison inequalities, the tensor problem is approximated by a Gaussian model, and a generalized chaining argument is employed to control the remainder terms in the likelihood expansion, accommodating both independent observations and misspecified settings. The work establishes, for the first time, asymptotic equivalence between binary hypergraph models and Gaussian tensor models under the minimal assumption that the average degree diverges, thereby extending the known universality of free energy from matrices to tensors. This provides a unified theoretical framework for free energy universality in tensor estimation, ensuring consistent asymptotic behavior with Gaussian models across broad scaling regimes and under model misspecification.

Abstract

We study high-dimensional inference problems with tensor-structured data and establish conditions under which their free energy can be approximated by that of a Gaussian comparison model. Our framework applies to models with independent observations and mismatch between the data-generating distribution and the statistical model. The results extend prior work beyond matrix settings and accommodate scaling regimes where the model parameters depend on the dimension. A key technical contribution is the use of generic chaining to control remainder terms arising from likelihood expansions over tensor-structured parameter spaces. As an application, we establish free energy universality for binary hypergraph models under the minimal assumption of diverging average degree, showing that their asymptotic behavior coincides with that of a Gaussian tensor model, even under model mismatch.

Problem

• tensor estimation
• free energy universality
• model mismatch
• high-dimensional inference
• Gaussian comparison model

Innovation

• generic chaining
• free energy universality
• tensor estimation
• Gaussian comparison model
• model mismatch