Linear Simple Cycle Reservoirs at the edge of stability perform Fourier decomposition of the input driving signals

This paper explores the representational structure of linear Simple Cycle Reservoirs (SCR) operating at the edge of stability. We view SCR as providing in their state space feature representations of the input-driving time series. By endowing the state space with the canonical dot-product, we ``reverse engineer" the corresponding kernel (inner product) operating in the original time series space. The action of this time-series kernel is fully characterized by the eigenspace of the corresponding metric tensor. We demonstrate that when linear SCRs are constructed at the edge of stability, the eigenvectors of the time-series kernel align with the Fourier basis. This theoretical insight is supported by numerical experiments.