We introduce a diagonalization-based optimization for Linear Echo State Networks (ESNs) that reduces the per-step computational complexity of reservoir state updates from quadratic to linear. By reformulating reservoir dynamics in the eigenbasis of the recurrent matrix, the recurrent update becomes a set of independent element-wise operations, eliminating the matrix multiplication. We further propose three methods to use our optimization depending on the situation: (i) Eigenbasis Weight Transformation (EWT), which preserves the dynamics of standard and trained Linear ESNs, (ii) End-to-End Eigenbasis Training (EET), which directly optimizes readout weights in the transformed space and (iii) Direct Parameter Generation (DPG), that bypasses matrix diagonalization by directly sampling eigenvalues and eigenvectors, achieving comparable performance to standard Linear ESNs. Across all experiments, both our methods preserve predictive accuracy while offering significant computational speedups, making them a replacement for standard Linear ESNs computations and training, and suggesting a shift of paradigm in linear ESN towards the direct selection of eigenvalues.

翻译：我们提出了一种基于对角化的线性回声状态网络（ESN）优化方法，该方法将储层状态更新的每步计算复杂度从二次降至线性。通过在特征基下重新表述储层动力学，循环更新变为一组独立的逐元素运算，从而消除了矩阵乘法。我们进一步提出了三种根据具体场景使用该优化方法的技术：（i）特征基权重变换（EWT），该方法能保持标准及已训练线性ESN的动力学特性；（ii）端到端特征基训练（EET），该方法在变换空间中直接优化读出权重；（iii）直接参数生成（DPG），通过直接采样特征值与特征向量绕过矩阵对角化过程，其性能可与标准线性ESN相媲美。在所有实验中，我们的方法在保持预测精度的同时实现了显著的计算加速，可替代标准线性ESN的计算与训练过程，并揭示了线性ESN向直接选择特征值的范式转变趋势。