处于稳定性边缘的线性简单循环储层对输入驱动信号执行傅里叶分解
Linear Simple Cycle Reservoirs at the edge of stability perform Fourier decomposition of the input driving signals
AI总结：
本文研究稳定性边缘处线性 SCR 的状态表征，通过逆向工程时间序列核并分析度量张量特征空间，证明其特征向量对齐傅里叶基，并有数值实验验证。
摘要：
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征。通过赋予状态空间标准点积，我们对在原始时间序列空间中起作用的相应核（内积）进行“逆向工程”。该时间序列核的作用完全由相应度量张量的特征空间刻画。我们证明，当线性 SCR 构建于稳定性边缘时，时间序列核的特征向量与傅里叶基对齐。这一理论洞见得到了数值实验的支持。
英文摘要：
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.
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