• Diagonalize recurrent matrix to transform reservoir dynamics into eigenbasis, eliminating costly matrix multiplication. • Achieves per-step computational complexity reduction from O(N²) to O(N) for Linear Echo State Networks. • Introduces Eigenbasis Weight Transformation (EWT) to preserve standard Linear ESN dynamics during training. • End-to-End Eigenbasis Training (EET) optimizes readout weights directly in transformed space for faster learning. • Direct Parameter Generation (DPG) samples eigenvalues/vectors, bypassing diagonalization while matching performance. • Experiments confirm maintained predictive accuracy with significant speedups, advocating a paradigm shift toward eigenvalue selection.

Article Summaries:

  • A new study proposes a diagonalization‑based optimization for Linear Echo State Networks (ESNs) that cuts the per‑step computational cost from O(N²) to O(N). By expressing the recurrent dynamics in the eigenbasis of the weight matrix, the update becomes a set of independent element‑wise operations, eliminating costly matrix multiplications. The authors present three variants: Eigenbasis Weight Transformation (EWT), which preserves standard ESN dynamics; End‑to‑End Eigenbasis Training (EET), which trains readout weights directly in the transformed space; and Direct Parameter Generation (DPG), which samples eigenvalues and vectors without explicit diagonalization. Experiments show all methods maintain predictive accuracy while delivering significant speedups, indicating a potential shift toward eigenvalue‑directed design in linear ESNs.

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