Why Adaptive Spectral Shrinkage Resists a Hilbert-Pólya Bridge to the Riemann Hypothesis: An Honest Assessment

Abstract

This essay examines whether the adaptive spectral shrinkage mechanism developed in the EigenBayes algorithm for overfitted factor models can be mapped to the spectral theory of the Riemann zeta function under the Hilbert-Pólya conjecture.

Overview

This essay explores a potential bridge between high-dimensional statistical factor analysis and the Riemann Hypothesis (RH). The source paper introduces EigenBayes, a method for matrix factorization that handles "overfitted" models by adaptively shrinking superfluous latent components to zero using spectral empirical Bayes techniques.

The Structural Analogy

The central temptation is to view the Hilbert-Pólya conjecture—which posits that the non-trivial zeros of ζ(s) correspond to eigenvalues of a self-adjoint operator—as an "overfitted" spectral problem. In this metaphor, hypothetical zeros off the critical line Re(s) = 1/2 would represent "superfluous components" that a suitable adaptive shrinkage procedure ought to eliminate, leaving only the true spectrum on the line.

Assessment of the Bridge

We rate this correspondence as a suggestive metaphor only. The EigenBayes mechanism relies on finite sample sizes n and dimensions p, Gaussian error structures, and Bayesian conjugacy that have no direct counterparts in the analytic theory of the zeta function. The infinite sequence of zeta zeros lacks the finite-rank "signal plus diagonal noise" decomposition that makes the source shrinkage effective.

Proposed Computational Tests

Despite the weak formal analogy, we propose experiments applying the EigenBayes shrinkage calibration to eigenvalues of large random matrices from the Circular Unitary Ensemble (CUE). These matrices model the local statistics of zeta zeros (GUE statistics). The experiment would test whether the adaptive shrinkage hyperparameters ψ_l exhibit universal decay patterns matching the Montgomery-Odlyzko law, potentially revealing a shadow of the analogy in random matrix theory.

This is clearly-labeled exploratory speculation; no proof of RH is claimed.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.