Innovation Capacity of Dynamical Learning Systems
Anthony M. Polloreno
TL;DR
The paper investigates why the classical information-processing capacity $C_{ m ip}$ can fall short of the readout covariance rank in noisy reservoirs. It introduces the innovation capacity $C_{ m i}$ via a basis-free Doob decomposition and proves the exact conservation law $C_{ m ip}+C_{ m i}= ext{rank}(oldsymbol{\Sigma}_{XX})\le d$, meaning missing input-focused capacity is reallocated to orthogonal innovation tasks. In linear-Gaussian Johnson-Nyquist settings the split reduces to generalized-eigenvalue shrinkage, yielding an explicit monotone tradeoff between temperature and $C_{ m ip}$; geometrically, the predictable and innovation parts form complementary ellipsoids in whitened space. The authors show that a large innovation budget drives a high-dimensional innovation subspace with a positive variance floor, leading to extensive block entropy and exponentially many distinguishable histories, and prove information-theoretic hardness for learning the induced innovation-law from data, underscoring the generative utility of noisy physical reservoirs for exploring rich histories.
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.