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Why Learning Rediscovers the Closed-Form Diagonal Regularizer — arXiv:2609.09656

Source

arXiv (en)

Original

https://arxiv.org/abs/2609.09656

License (as archived)

CC0-1.0

Archived

2026-10-16T11:12:19.288Z

SHA-256 of the archived text

72ce80c363295d646065b9cc2a7f6d1175aee2212b932dc752ef14cfabadd5ae

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https://metraipolis.com/documento/dc_293fd96ebfe7

How to cite

Why Learning Rediscovers the Closed-Form Diagonal Regularizer — arXiv:2609.09656 — arXiv (en), https://arxiv.org/abs/2609.09656 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_293fd96ebfe7 (Metraipolis archive dc_293fd96ebfe7, sha256 72ce80c363295d646065b9cc2a7f6d1175aee2212b932dc752ef14cfabadd5ae)

@misc{dc293fd96ebfe7,
  title = {Why Learning Rediscovers the Closed-Form Diagonal Regularizer — arXiv:2609.09656},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.09656},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_293fd96ebfe7; sha256 72ce80c363295d646065b9cc2a7f6d1175aee2212b932dc752ef14cfabadd5ae},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.09656. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Recovering modal amplitudes from a few sensors requires regularization, and the shape of the penalty decides how strongly each mode is shrunk. The best shape depends on the truncation noise that discarded modes add to every measurement, which is unobserved and could be room-specific. We identify a diagonal saturation principle: for a single snapshot, isotropic truncation noise makes the Bayes-optimal Tikhonov shape a closed-form power law Gamma_k proportional to lambda_k^|s|, set by the prior alone and independent of the domain. The off-diagonal anisotropy equals the Herfindahl index H of the discarded-mode power exactly, Weyl's eigenvalue counting law fixes the number of discarded modes and hence H, and Berry's random-wave conjecture enters only through a diagonal term D, which we measure directly. Together they predict an approximately flat loss landscape over the penalty exponent, which leaves a learned diagonal regularizer little to gain from tuning the exponent per room. On simulated acoustic rooms the closed form stays within 5.83% of a per-room power-law oracle across observation windows, and three learned diagonal architectures track its error, the closest within a fraction of a point and none more than 3pp above it, while learning qualitatively different spectra. In the truncation regime no tested method consistently improves on the closed form, including Learned Iterative Ridge, which couples modes across layers. The framework extends to heat diffusion through a known Green's-function correction with no new free parameters.

Tema degli esploratori: statistica e scienza dei dati (retro-catalogo arXiv: annunci del 2026-09-10 in cs.RO).
Categorie arXiv: Machine Learning (stat.ML) ; Machine Learning (cs.LG); Robotics (cs.RO); Audio and Speech Processing (eess.AS).
Pagina del lavoro: https://arxiv.org/abs/2609.09656 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T10:19:11.945Z dalla pagina https://arxiv.org/catchup/cs.RO/2026-09-10?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).