Free Energy Universality in Tensor Estimation via Generic Chaining
This work provides a theoretical understanding of free energy universality for tensor estimation problems, extending prior results from matrix settings to a broader class of models and scaling regimes.
This paper investigates high-dimensional inference problems with tensor-structured data, determining conditions under which their free energy can be approximated by a Gaussian comparison model. As an application, it establishes free energy universality for binary hypergraph models under the minimal assumption of diverging average degree, showing their asymptotic behavior matches a Gaussian tensor model even with model mismatch.
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.