Innovation Capacity of Dynamical Learning Systems

Abstract: In noisy physical reservoirs, the classical information-processing capacity $C_{\mathrm{ip}}$ quantifies how well a linear readout can realize tasks measurable from the input history, yet $C_{\mathrm{ip}}$ can be far smaller than the observed rank of the readout covariance. We explain this ``missing capacity'' by introducing the innovation capacity $C_{\mathrm{i}}$, the total capacity allocated to readout components orthogonal to the input filtration (Doob innovations, including input-noise mixing). Using a basis-free Hilbert-space formulation of the predictable/innovation decomposition, we prove the conservation law $C_{\mathrm{ip}}+C_{\mathrm{i}}=\mathrm{rank}(Σ{XX})\le d$, so predictable and innovation capacities exactly partition the rank of the observable readout dimension covariance $Σ{XX}\in \mathbb{R}^{\rm d\times d}$. In linear-Gaussian Johnson-Nyquist regimes, $Σ{XX}(T)=S+T N_0$, the split becomes a generalized-eigenvalue shrinkage rule and gives an explicit monotone tradeoff between temperature and predictable capacity. Geometrically, in whitened coordinates the predictable and innovation components correspond to complementary covariance ellipsoids, making $C{\mathrm{i}}$ a trace-controlled innovation budget. A large $C_{\mathrm{i}}$ forces a high-dimensional innovation subspace with a variance floor and under mild mixing and anti-concentration assumptions this yields extensive innovation-block differential entropy and exponentially many distinguishable histories. Finally, we give an information-theoretic lower bound showing that learning the induced innovation-block law in total variation requires a number of samples that scales with the effective innovation dimension, supporting the generative utility of noisy physical reservoirs.