Simple Cycle Reservoirs are Universal
Boyu Li, Robert Simon Fong, Peter Tiňo
TL;DR
Problem: determine whether highly constrained reservoir architectures can universally approximate time-invariant fading memory dynamics. Method: a constructive pipeline that first dilates a contracting state-coupling to a unitary form, then converts to permutation-based state coupling, yielding Complex SCR, Simple Multi-Cycle Reservoir, and Twin SCR that are ε-close to any linear reservoir with a continuous readout (preserving the norm via a cycle weight $\lambda$). Contributions: rigorous, constructive proofs that $C$-SCR, SMCR, and Twin SCR can approximate any linear reservoir with a continuous readout, hence universal for fading memory filters; the approach connects to existing universality results for polynomial readouts. Significance: provides universal, hardware-friendly reservoir designs with a single tunable parameter, enabling practical deployment in physical systems while maintaining theoretical guarantees.
Abstract
Reservoir computation models form a subclass of recurrent neural networks with fixed non-trainable input and dynamic coupling weights. Only the static readout from the state space (reservoir) is trainable, thus avoiding the known problems with propagation of gradient information backwards through time. Reservoir models have been successfully applied in a variety of tasks and were shown to be universal approximators of time-invariant fading memory dynamic filters under various settings. Simple cycle reservoirs (SCR) have been suggested as severely restricted reservoir architecture, with equal weight ring connectivity of the reservoir units and input-to-reservoir weights of binary nature with the same absolute value. Such architectures are well suited for hardware implementations without performance degradation in many practical tasks. In this contribution, we rigorously study the expressive power of SCR in the complex domain and show that they are capable of universal approximation of any unrestricted linear reservoir system (with continuous readout) and hence any time-invariant fading memory filter over uniformly bounded input streams.