Physics, Information and Inference: High-Dimensional Models under Structured Dependencies

Abstract

This thesis develops theory and methodology for high-dimensional models that exhibit complex global dependencies. It examines several fundamental problems in statistical physics, compressed sensing, and statistical inference using rotationally invariant random matrix ensembles—frameworks that more accurately capture global dependencies than traditional i.i.d. assumptions. The work is organized around three main topics: (i) an in-depth study of the classical Sherrington–Kirkpatrick model under rotationally invariant couplings, including a rigorous proof of the fundamental Thouless–Anderson–Palmer (TAP) equations for low-dimensional marginals of the high-dimensional Gibbs measure; (ii) the derivation of single-letter formulas characterizing key information-theoretic quantities in compressed sensing with right-rotationally invariant sensing matrices; and (iii) the development of data-driven corrections to one-step debiasing regularized estimators that model complex global dependencies in the covariates via rotationally invariant matrices.