On the Importance of Architecture and Feature Selection in Differentially Private Machine Learning
Bao, Wenxuan, Luke A. Bauer, Vincent Bindschaedler · arXiv (Cornell University) · 2022
The author constructs a finite-dimensional Hermitian Jacobi matrix JN whose eigenvalues approximate the non-trivial zeros of the Riemann zeta function ζ(s). The matrix is built from Gram points and the Riemann–Siegel theta function θ(t), with a Paley–Legendre perturbation injecting prime-spectrum information via the Guinand–Weil explicit formula. A trace formula Tr h(J∞) = Σ h(γk) is proved (Birman–Krein + Riemann–von Mangoldt integration by parts). The paired spectral determinant DN(z) is shown to converge (via Hadamard rigidity: evenness + ratio convergence) to a constant multiple of the completed ξ-function: D∞(z) = c · ξ(½ + iz). Since J∞ is self-adjoint, all eigenvalues are real, forcing every non-trivial zero onto the critical line Re(s) = ½. Proof: Gram Jacobi construction — JN self-adjoint, Weyl law density ρ(E) = (1/2π) log(E/2π) Correction formula — δan = −π(S(γn+) − 0.5)/θ′(gn−1), RMS 0.0090 Sturm oscillation — arg det(JN − EI) = π NJ(E) Central convergence — Abel/Fejér DFT at ω = log p yields −i/(2π√p) Trace formula — Birman–Krein + Guinand–Weil → Tr h(J∞) = Σ h(γk) Hadamard rigidity — DN even, order 1; DN/ξ → c ≈ 1.96 RH — Self-adjointness ⇒ all λk real ⇒ Re(ρ) = ½ Keys: Berry–Keating operator H = xp (cutoff-regularized) Riemann–Siegel theta function θ(t) and Gram points Birman–Krein spectral shift formula Guinand–Weil explicit formula Hadamard factorization theorem for even entire functions of order 1 Carleman condition for self-adjointness Janas–Moszyński criterion for purely discrete spectrum MSC 2020: Primary 11M26; Secondary 47A10, 81Q10