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.
Grokking shows that fitting the training data and learning the underlying signal can occur at very different stages. However, existing theories offer limited quantitative insight into how this delayed generalization depends on inductive bias and signal structure. Our work addresses the gap by developing a statistical t...
Gil Kur, Ileana Rugina, C. Dominé et al.· 0 citations
Whitening a foundation-model embedding and using its squared norm as a training-free likelihood surrogate is motivated by the observation that whitened coordinates often appear approximately standard normal. We show that this observation follows from the projection central limit theorem and therefore does not imply a G...
Local Gaussian models of constant-step learning predict output variability and expected losses, but weak convergence alone does not justify these moment predictions. We establish moment-accurate Gaussian mixtures by matching stationary energy with local Ornstein--Uhlenbeck limits, ruling out quadratic tail mass invisib...
As an alternative to the standard geometric analyses, we give an exact, information-theoretic analysis of stochastic gradient descent (SGD) and its variants. We show that a preconditioned SGD step is the posterior-mean update of a Gaussian Bayes model, and that its one-step regret splits into an intrinsic-time cost and...
Restricted eigenvalue (RE) bounds govern stable recovery by norm-regularized estimators. For isotropic sub-Gaussian measurements, the benchmark sample size is $1+w(A)^2$, where $w(A)$ is the Gaussian width of the normalized descent cone. The COLT 2015 open-problem note (Banerjee et al., 2015) asked whether the same law...
Shi Fu, Hui-Bo Xu, Qi-Xin Zhang et al.· 0 citations
It is proved that the extracted trend is invariant across representations, and a closed-form Bregman-type divergence quantifying their disagreement under a shared coefficient vector is obtained, and this divergence vanishes as $\lambda\to\infty$.
Learn how Google DeepMind and GovTech Singapore uses lightweight on-device Gemma models to intercept conversational scams in private chats with zero cloud data egress.
