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.

翻译：本文探讨了在稳定性边缘运行的线性简单循环储层（SCR）的表征结构。我们将SCR视为在其状态空间中提供输入驱动时间序列的特征表示。通过赋予状态空间标准点积，我们“逆向工程”出在原始时间序列空间中运行的相应核（内积）。该时间序列核的作用完全由相应度量张量的特征空间所表征。我们证明，当线性SCR在稳定性边缘构建时，时间序列核的特征向量与傅里叶基对齐。这一理论见解得到了数值实验的支持。