Riemannian Optimization on Tensor-Train Manifolds and the Riemann Hypothesis: An Exploratory Negative

Abstract

This essay examines the structural patterns in kernel regression with tensor trains and Hadamard overparameterization (KReTTaH) for potential analogies to the Riemann Hypothesis.

Overview

This essay analyzes the paper Kernel Regression with Tensor Trains and Hadamard Overparameterization (arXiv:2607.17390v1) through the lens of the Riemann Hypothesis (RH). The source paper presents KReTTaH, a method for imputing missing data in multi-way arrays using kernel regression constrained to tensor-train (TT) manifolds.

The Source Framework

The core mathematical structure involves optimization on a Riemannian product manifold: the Cartesian product of the manifold of fixed-rank TT tensors and the manifold of positive-definite covariance matrices. The method employs Hadamard overparameterization (entry-wise products of parameter tensors) to promote sparsity and uses reproducing kernel Hilbert spaces (RKHS) to capture nonlinear dependencies.

The Attempted Analogy

The terminology invites speculation: "Riemannian" geometry echoes the study of Riemann surfaces and automorphic forms; "Hadamard" products recall Hadamard's factorization of entire functions; "kernel" methods suggest the spectral kernels of the Hilbert–Pólya approach. We explore whether the optimization landscape on these manifolds could mirror the distribution of zeta zeros or the extremal principles underlying the explicit formula.

Assessment

The analogy is rated a structural mismatch. The source paper deals with finite-dimensional approximation manifolds for data imputation, lacking the functional equation, analytic continuation, and infinite-dimensional spectral theory central to RH. The "Hadamard" operation is an algebraic trick for sparsity, not the infinite product over zeros. The essay provides an honest negative assessment, detailing precisely where the correspondence breaks down.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.