Entropy-Response Tuning for Recurrent Reservoirs

Use a tanh recurrent reservoir before a linear readout: h_(t+1) = (1-a)h_t + a tanh(g W h_t + U x_t + b) + sqrt(2D) xi_t, where a is the leak rate and g is the scanned control parameter. Freeze W and U, scan g over a fixed grid, and fit only a ridge-regression readout after selecting g. For every g, roll out identical initial states on the real input sequence and on an innate baseline with x_t equal to zero or with phase-shuffled inputs. Accumulate the mean squared hidden-state response R(g). Use automatic differentiation to obtain the hidden-state Jacobian at each step and accumulate the phase-space contraction term using a stable slogdet operation. For small reservoirs compute the determinant exactly; for large reservoirs estimate the log determinant with stochastic trace or Lanczos methods and record estimator variance. Calculate Delta_sigma and the normalized score S. As a cheap first experiment, use 100 tanh units on Lorenz-63 one-step prediction, scan g from 0.1 to 2.0 at 30 points, and compare against spectral-radius tuning and random gain selection. Repeat three seeds with identical data splits and ridge penalties. The quantitative prediction is that the gain maximizing S lies within one grid interval of the minimum validation error and that the Spearman correlation between S and negative validation error exceeds 0.6. The expected curve is low at weak driving, maximal at intermediate gain, and degraded near exploding hidden trajectories. If the score peak consistently disagrees with the accuracy peak, the transfer hypothesis is falsified.