Finite‐Block Polynomial and Volterra Structure in Reservoir Computing

ABSTRACT Reservoir computing is analyzed through the algebraic structure generated by finite‐width reservoirs on fixed input blocks. For linear and quadratic logistic activations, the induced input‐to‐state map admits an exact finite Volterra representation, yielding an explicit characterization of the associated hypothesis class. It is shown that reservoir width imposes rank constraints on the admissible Volterra kernels, which in the quadratic case reduce to classical matrix‐rank limitations and lead to approximation estimates in terms of spectral decay and best low‐rank approximation error. A Volterra‐Barron complexity measure is introduced for finite‐block polynomial functionals, and corresponding approximation rates are established. Numerical experiments successfully validate the theoretical predictions.