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Partial observation is a pervasive obstacle in nonequilibrium physics: coarse graining may absorb hidden forcing into an apparently equilibrium-like reduced description, so a driven system can look reversible through the only variables one can measure. For scalar Gaussian observables of linear stochastic systems, no time-irreversibility statistic can detect the underlying drive. The Lucente--Crisanti ceiling constrains what one channel carries; what two channels carry is a different question, with a sharp closed-form answer. Two simultaneously observed channels retain an off-diagonal cross-spectral sector inaccessible to any scalar reduction; under channel-separable multiplicative structure the observed-channel response factors cancel identically, leaving a closed-form cross-spectral witness controlled only by the hidden spectrum, the loadings, and the innovation scales, strictly positive at every nonzero cross-coupling including at exact timescale coalescence where every scalar reduction is blind. Within general CSM this certifies shared hidden-sector drive; under the additional one-way coupling assumption the witness identifies the total entropy production rate at leading order with a square-root scaling.

We develop a theoretical framework for generalization in the interpolating regime of statistical learning. The central question is why highly overparameterized estimators can attain zero empirical risk while still achieving nontrivial predictive accuracy, and how to characterize the boundary between benign and destructive overfitting. We introduce a spectral-transport stability framework in which excess risk is controlled jointly by the spectral geometry of the data distribution, the sensitivity of the learning rule under single-sample replacement, and the alignment structure of label noise. This leads to a scale-dependent Fredriksson index that combines effective dimension, transport stability, and noise alignment into a single complexity parameter for interpolating estimators. We prove finite-sample risk bounds, establish a sharp benign-overfitting criterion through the vanishing of the index along admissible spectral scales, and derive explicit phase-transition rates under polynomial spectral decay. For a model-specific specialization, we obtain an explicit theorem for polynomial-spectrum linear interpolation, together with a proof of the resulting rate. The framework also clarifies implicit regularization by showing how optimization dynamics can select interpolating solutions of minimal spectral-transport energy. These results connect algorithmic stability, double descent, benign overfitting, operator-theoretic learning theory, and implicit bias within a unified structural account of modern interpolation.

Scaling laws for neural networks, in which the loss decays as a power-law in the number of parameters, data, and compute, depend fundamentally on the spectral structure of the data covariance, with power-law eigenvalue decay appearing ubiquitously in vision and language tasks. A central question is whether this spectral structure is preserved or destroyed when data passes through the basic building block of a neural network: a random linear projection followed by a nonlinear activation. We study this question for the random feature model: given data $x \sim N(0,H)\in \mathbb{R}^v$ where $H$ has $α$-power-law spectrum ($λ_j(H ) \asymp j^{-α}$, $α> 1$), a Gaussian sketch matrix $W \in \mathbb{R}^{v\times d}$, and an entrywise monomial $f(y) = y^{p}$, we characterize the eigenvalues of the population random-feature covariance $\mathbb{E}_{x }[\frac{1}{d}f(W^\top x )^{\otimes 2}]$. We prove matching upper and lower bounds: for all $1 \leq j \leq c_1 d \log^{-(p+1)}(d)$, the $j$-th eigenvalue is of order $\left(\log^{p-1}(j+1)/j\right)^α$. For $ c_1 d \log^{-(p+1)}(d)\leq j\leq d$, the $j$-th eigenvalue is of order $j^{-α}$ up to a polylog factor. That is, the power-law exponent $α$ is inherited exactly from the input covariance, modified only by a logarithmic correction that depends on the monomial degree $p$. The proof combines a dyadic head-tail decomposition with Wick chaos expansions for higher-order monomials and random matrix concentration inequalities.

An important factor contributing to the success of deep learning has been the remarkable ability to optimize large neural networks using simple first-order optimization algorithms like stochastic gradient descent. While the efficiency of such methods depends crucially on the local curvature of the loss surface, very little is actually known about how this geometry depends on network architecture and hyperparameters. In this work, we extend a recently-developed framework for studying spectra of nonlinear random matrices to characterize an important measure of curvature, namely the eigenvalues of the Fisher information matrix. We focus on a single-hidden-layer neural network with Gaussian data and weights and provide an exact expression for the spectrum in the limit of infinite width. We find that linear networks suffer worse conditioning than nonlinear networks and that nonlinear networks are generically non-degenerate. We also predict and demonstrate empirically that by adjusting the nonlinearity, the spectrum can be tuned so as to improve the efficiency of first-order optimization methods.

CNNs, MLPs, and quadratic problems, and find that SGD can perform on par with Adam on problems without block heterogeneity, but performs worse than Adam when the heterogeneity exists.

Spectroscopic techniques are essential tools for determining the structure of molecules.

Despite decades of progress in machine learning applications for predicting molecular structures from MS/MS spectra, the development of new methods is severely hindered by the lack of standard datasets and evaluation protocols. To address this problem, we propose MassSpecGym - the first comprehensive benchmark for the discovery and identification of molecules from MS/MS data.