Centrality-Based Pruning for Efficient Echo State Networks

Echo State Networks (ESNs) offer an efficient framework for temporal prediction, but their randomly initialized reservoirs are often over-parameterized and dynamically redundant. Existing pruning methods largely rely on static connectivity or activation statistics, which may overlook neurons that shape input-driven state transitions. We propose Dynamical Mode Pruning (DMP), a reservoir pruning method that ranks neurons by their contribution to dominant transition modes obtained from a trajectory-averaged Jacobian Gramian. DMP removes low-impact units and retrains only the readout. Experiments on chaotic and real-world time-series benchmarks show that DMP improves or preserves forecasting accuracy while reducing redundant reservoir components. Our results suggest that dynamical influence is a useful criterion for reservoir refinement beyond static structural importance alone.

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.

Echo State Networks (ESNs), as lightweight implementations of reservoir computing, provide efficient solutions for temporal data processing in resource-constrained environments. While standard ESNs offer computational efficiency, their dense readout layers inflate the stored model and the per-sample inference cost. This paper introduces a Smooth-L1 regularized ESN (SL1ESN) that promotes sparsity in the readout layer while preserving differentiability for gradient-based optimization. The method employs a twice-differentiable Smooth-L1 approximation compatible with Newton's method, which we exploit both for centralized training and for a second-order (Newton-step) federated aggregation scheme. Under a single fixed shrinkage threshold shared by every dataset and both settings, experiments on seven time-series classification benchmarks show that SL1ESN learns readouts that are $77$--$95\%$ sparse in the centralized setting and $75$--$95\%$ sparse in the federated one, while staying within $0.7$ accuracy points of the strongest dense baseline on five of the seven benchmarks in the centralized setting and within $1.6$ points on the same five in the federated one, and the second-order federated update converges in only a few rounds. The resulting sparsity compresses the stored model, the per-sample inference cost, and the server-to-client broadcast by a comparable factor. These results demonstrate the practical value of sparse readouts for edge computing and IoT deployments.