Free-Probability Kernels for Zero-Rollout Hyperparameter Selection in Reservoir Computing

Reservoir computing (RC) couples a fixed recurrent dynamical system with a trained lightweight readout, but this efficiency is partly lost during hyperparameter selection: the recurrent gain, input scale, and leakage rate determine the reservoir's stability and temporal processing regime and are usually tuned through many rollouts. We introduce a deterministic, pilot-informed selector for leaky linear reservoirs followed by coordinate-wise nonlinear features. Free probability yields cross-lag propagation coefficients that summarize how the reservoir mixes past inputs. In the large-width limit, these coefficients define a deterministic temporal kernel that approximates the finite-reservoir feature geometry. Kernel ridge regression on a short labelled pilot sequence therefore ranks candidate operating regimes without instantiating or rolling out a reservoir, and the selected configuration transfers across widths. Across ten synthetic temporal benchmarks, zero-rollout selection obtains a mean deployment score of $0.772$, compared with $0.774$ for exhaustive simulation-based search, while avoiding $156\,600$ selection rollouts. With a small rollout budget, the proposed ranking provides the strongest mean performance at every tested budget and reaches the exhaustive reference using $4.8\%$ of its rollout cost. On four public electricity-transformer-temperature (ETT) forecasting datasets, five retained candidates recover the exhaustive operating point on three datasets. On multivariate cellular-traffic forecasting, 15 rollouts per cell reach the 462-rollout exhaustive reference and outperform random search and Bayesian optimization at low budgets. These results position free-probability kernels as deterministic surrogates for selecting reservoir operating regimes when validation rollouts are scarce.

Geometric Dyson Brownian Motions and the Free Log-Normal Limit for a Non-Square Gaussian Matrix Product

We study the squared singular value spectrum of a non-square product of independent real Gaussian matrices, equivalently the feature covariance spectrum of a deep linear neural network at initialization. Starting from the fixed-$m$ covariance diffusion previously obtained in the proportional depth-width limit, we record an equivalent matrix realization, describe its affine invariance, and derive the interacting diffusion satisfied by its eigenvalues. We then take a second limit, sending $m\to\infty$ on the accelerated spectral clock $τ=mt$, which corresponds in this sequential construction to the relation $dm/n\to\barτ$. We establish convergence of the empirical spectral measure path to a deterministic mean-field limit and derive a closed Burgers equation for its $T$-transform. Together with the proportional depth-width limit, these results give a rigorous sequential route from the deep non-square Gaussian product to the free log-normal limit of its feature covariance spectrum; for more general initial laws, the transform yields a free multiplicative convolution form. We further analyze the support of the free log-normal law, give a fixed point iteration for numerical evaluation and a formal Marchenko--Pastur approximation at small time, and use the limiting spectrum to predict the risk in a toy random feature model.