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：我们提出一种基于对角化的线性回声状态网络优化方法，将储备池状态更新的每步计算复杂度从二次降至线性。通过将储备池动力学重构到循环矩阵的特征基下，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。我们进一步根据应用场景提出三种利用该优化的方法：(i) 特征基权重转换法（EWT），可保留标准及训练后线性ESN的动力学特性；(ii) 端到端特征基训练法（EET），直接优化变换空间中的读出权重；(iii) 直接参数生成法（DPG），通过直接采样特征值与特征向量绕过矩阵对角化过程，达到与标准线性ESN相当的性能。在所有实验中，我们的方法在保持预测精度的同时显著提升计算速度，可作为标准线性ESN计算与训练的替代方案，这暗示了线性ESN研究范式向直接选择特征值的转变。