Innovation Capacity of Dynamical Learning Systems

TL;DR

This paper introduces the innovation capacity in dynamical learning systems, explaining the missing capacity in noisy reservoirs and providing a theoretical framework for understanding the tradeoff between predictable and innovation components.

Contribution

It defines the innovation capacity, proves a conservation law with predictable capacity, and offers an explicit tradeoff and geometric interpretation in linear-Gaussian regimes.

Findings

Conservation law: predictable plus innovation capacity equals the rank of the covariance matrix.

Explicit tradeoff between temperature and predictable capacity in Gaussian regimes.

Innovation capacity relates to the complexity of distinguishable histories and sample complexity.

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 $C_{\mathrm{ip}}+C_{\mathrm{i}}=\mathrm{rank}(\Sigma_{XX})\le d$, so predictable and innovation capacities exactly partition the rank of the observable readout dimension covariance $\Sigma_{XX}\in \mathbb{R}^{\rm d\times d}$. In linear-Gaussian Johnson-Nyquist regimes, $\Sigma_{XX}(T)=S+T N_0$, 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.