发表机构
GIST（光州科学技术院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文发现模态逆问题中的对角饱和原理：各向同性噪声下闭式幂律正则化器接近最优，学习对角架构难以超越，但跨模态耦合可突破此界限。
AI 中文摘要
我们在模态逆问题中识别出一个对角饱和原理：当截断噪声是各向同性时，贝叶斯最优的Tikhonov形状是一个由先验单独决定的闭式幂律Gamma_k正比于lambda_k^|s|，与定义域无关。Berry随机波猜想使截断噪声在不同模态间去相关，而Weyl特征值计数定律提供了足够的模态，使得该结论在经验上违反Berry猜想时依然成立。两者共同预测了在逐模态族中损失景观近似平坦，留给对角正则化器稳健地超越闭式形式的空间很小。在有限元模拟的声学房间中，相对于逐房间的oracle调参，闭式形式在观测窗口上接近最优，并且三个在同一数据上训练的对角架构在重建误差上与其匹配，差异在1个百分点以内，尽管它们学习到的频谱在性质上不同。该框架通过已知的指数格林函数修正扩展到热扩散，无需新的自由参数。饱和仅限于对角族：学习迭代岭通过利用跨模态耦合跨越了这一边界，定位了学习开始发挥作用的点。
英文摘要
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.
Commentsmain paper: 9 pages, 3 figures appendix
