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Life sciences · Preprint
arXiv · September 9, 2026
Raises a question worth testing. It does not answer one.
This preprint proposes a diagonal saturation principle predicting that optimal Tikhonov regularization in modal inverse problems follows a closed-form power law independent of domain, supported by FEM acoustic simulations and comparisons of learned architectures. The work is theoretical and computational, identifying a mathematical structure in regularization rather than demonstrating a practical advantage over existing methods.
Theoretical analysis with computational validation via finite-element simulation. FEM-simulated acoustic rooms and heat diffusion systems; no real experimental systems or human subjects.. Intervention: Closed-form diagonal Tikhonov regularizer with power-law shape Gamma_k proportional to lambda_k^|s|, and three learned diagonal architectures trained on FEM-simulated acoustic data.. Compared with: Per-room oracle tuning; learned Iterative Ridge (which exploits cross-mode coupling and crosses the diagonal boundary)..
Three diagonal architectures trained on the same data matched the closed-form reconstruction error within 1 percentage point despite learning qualitatively different spectra. On FEM-simulated acoustic rooms, the closed form is near-optimal relative to per-room oracle tuning across observation windows. The closed-form Bayes-optimal Tikhonov shape is a power law Gamma_k proportional to lambda_k^|s|, determined by the prior alone and independent of domain.
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This is a theoretical and computational study proposing a mathematical principle (diagonal saturation) about regularization in inverse problems, supported by simulation and limited empirical validation but lacking clinical or real-world outcome data.
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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 the closed form. On FEM-simulated acoustic rooms, the closed form is near-optimal relative to per-room oracle tuning across observation windows, and three diagonal architectures trained on the same data match its reconstruction error within 1 pp despite learning qualitatively different spectra. The framework extends to heat diffusion via a known exponential Green's function correction with no new free parameters. Saturation is restricted to the diagonal family: Learned Iterative Ridge crosses the boundary by exploiting cross-mode coupling, locating where learning starts to help.
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