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Why Learning Rediscovers the Closed-Form Diagonal Regularizer

arXiv · AI, language, vision and robotics · article · Sep 9, 2026 · UTC

We identify a diagonal saturation principle in modal inverse problems: when truncation noise is isotropic, the Bayes-optimal Tikhonov shape is a closed-form power law Gamma_k proportional to lambda_k^|s| set by the prior alone, independent of the domain. Berry's random-wave conjecture decorrelates the truncation noise across modes, and Weyl's eigenvalue counting law supplies enough modes for the conclusion to survive empirical Berry violations. Together they predict an approximately flat loss landscape across the per-mode family, leaving narrow scope for a diagonal regularizer to robustly beat

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