cs.LGOct 19, 2026

Sector-Ring Linear Reservoirs: Sector-Wise Decay-Timescale Allocation With State Reuse Across Candidate Settings

Authors: Yudai Ebato, Sou Nobukawa, Yusuke Sakemi, Takashi Kanamaru, Nina Sviridova, Kazuyuki Aihara

Organizations: Graduate School of Information and Computer Science, Chiba Institute of Technology, Narashino, Chiba 275-0016, Japan · Department of Computer Science, Chiba Institute of Technology, Narashino, Chiba 275-0016, Japan · Research Center for Mathematical Engineering, Chiba Institute of Technology, Narashino, Chiba 275-0016, Japan · Department of Preventive Intervention for Psychiatric Disorders, National Institute of Mental Health, National Center of Neurology and Psychiatry, Kodaira, Tokyo 187-8551, Japan · 

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Abstract

In reservoir computing, performance depends on whether the fixed dynamics of the reservoir suit the task, and the appropriate memory length and frequency response differ across tasks. The reservoir settings are found by a search on each task, and driving the reservoir with the input and collecting its states for every candidate setting dominates the cost of the search. In a reservoir with a linear recurrence, the dynamics decompose into modes, one per eigenvalue, and the eigenvalue distribution becomes a design variable. We propose the Sector-Ring Linear Reservoir (SRLR), in which the eigenvalue angles are fixed at evenly spaced positions around a circle and one decay timescale, chosen from a few values, is assigned to each sector of consecutive angles. To keep the scale of each mode's state equal across candidates, the input weights are scaled using the power spectrum of the training input. To reduce the cost of the search, the states of the modes for each decay-timescale value are computed once and reused across candidates. On prediction and reconstruction benchmarks built from chaotic systems, the SRLR achieved, in all eight cases, a lower mean test error than a reference model in which all modes share one decay timescale, and was competitive with nonlinear echo state network and leaky-integrator echo state network baselines. The selected decay-timescale profiles changed with the prediction target and horizon. The state reuse reduced the number of computed mode trajectories in the search by 99.8% relative to per-candidate recomputation.

Figures & tables

Fig. 1: Concept of the Sector-Ring Linear Reservoir. The preprocessed input u~t\widetilde{u}_{t} enters through input weights scaled using the power spectral density (PSD) of the training input (Section III-A3c). Rotation–decay blocks assigned to sectors generate the states of the modes, which a fixed orthogonal mixing and a fixed tanh map transform into features for the ridge readout. Here q(ϕ)=[cos⁡ϕ,sin⁡ϕ]T\mathbf{q}(\phi)=[\cos\phi,\sin\phi]^{\mathsf{T}} denotes the unit direction of the input weights of one mode. In the pipeline and the bottom row, H\mathbf{H} denotes the states ht\mathbf{h}_{t} stacked in rows over time, so that the rows of HQ\mathbf{H}\mathbf{Q} are the mixed states QTht\mathbf{Q}^{\mathsf{T}}\mathbf{h}_{t} (Section III-B3). The upper-right callout summarizes the candidate-evaluation computation (Section III-B3): a state bank indexed by the half-life value (the per-sector decay timescale, Section III-A3b) reuses states across candidates. An associative scan is invoked only for a half-life value that the candidate requires and the bank does not yet hold.
Search configurationCoordinates and domains
ESNsin∈[0.1,3.0]s_{\mathrm{in}}\in[0.1,3.0] (lin), r∈[0.2,1.5]r\in[0.2,1.5] (lin), β∈[10−9,10−3]\beta\in[10^{-9},10^{-3}] (log)
LI-ESNESN coordinates plus αℓ∈[0.001,1]\alpha_{\ell}\in[0.001,1] (lin)
SRLR, PSD-scaled input weightsσ∗∈[0.25,4]\sigma_{*}\in[0.25,4] (lin), β\beta (log, as above), each ηk∈{0.25,0.5,1,2,4,8}\eta_{k}\in\{0.25,0.5,1,2,4,8\}
Common-radius referenceσ∗\sigma_{*} , β\beta as above, shared r∈[2−4,2−1/8]r\in[2^{-4},2^{-1/8}] (approximately [0.0625,0.9170][0.0625,0.9170] ; lin, continuous)
SRLR, γ\gamma -normalized input weightsσ∗\sigma_{*} , β\beta , η\bm{\eta} as above
SRLR, uniform random input weightssin∈[0.1,3.0]s_{\mathrm{in}}\in[0.1,3.0] (lin), β\beta , η\bm{\eta} as above
TABLE I: Search spaces. Realized evaluation counts are given per case in Appendix B . Here (lin) and (log) denote uniform sampling on a linear and a logarithmic scale, respectively.
Fig. 2: Test NRMSE of the SRLR and the common-radius reference on (a) the four Lorenz tasks and (b) the four Rössler tasks. For each system, the panels show, in order, x→xx\to x prediction at horizon 1, x→xx\to x prediction at horizon 12, x→zx\to z reconstruction at horizon 0, and x→zx\to z reconstruction at horizon 12. Bars show means over ten model seeds, and error bars show population SD.
TaskCommon-radius referenceSRLR
Lorenz
x→xx\to x , horizon 1(8.29±1.63)×10−6(8.29\pm 1.63)\times 10^{-6}(4.36±0.507)×10−6(\mathbf{4.36}\pm 0.507)\times 10^{-6}
x→xx\to x , horizon 120.0283±0.001900.0283\pm 0.001900.0209±0.00167\mathbf{0.0209}\pm 0.00167
x→zx\to z , horizon 00.132±0.007480.132\pm 0.007480.109±0.0112\mathbf{0.109}\pm 0.0112
x→zx\to z , horizon 120.162±0.008620.162\pm 0.008620.144±0.0121\mathbf{0.144}\pm 0.0121
Rössler
TABLE II: Test NRMSE for the common-radius reference and the SRLR: mean ±\pm population SD over ten model seeds. The lowest mean in each row is in bold.
Fig. 3: Test NRMSE of the ESN, the LI-ESN, and the SRLR on (a) the four Lorenz tasks and (b) the four Rössler tasks, in the task order of Fig. 2 . Bars show means over ten model seeds, and error bars show population SD.
TaskESNLI-ESNSRLR
Lorenz
x→xx\to x , horizon 1(19.5±3.09)×10−6(19.5\pm 3.09)\times 10^{-6}(19.7±3.33)×10−6(19.7\pm 3.33)\times 10^{-6}(4.36±0.507)×10−6(\mathbf{4.36}\pm 0.507)\times 10^{-6}
x→xx\to x , horizon 120.0536±0.004180.0536\pm 0.004180.0395±0.004560.0395\pm 0.004560.0209±0.00167\mathbf{0.0209}\pm 0.00167
x→zx\to z , horizon 00.0853±0.02000.0853\pm 0.02000.0328±0.0100\mathbf{0.0328}\pm 0.01000.109±0.01120.109\pm 0.0112
x→zx\to z , horizon 120.204±0.01670.204\pm 0.01670.126±0.0104\mathbf{0.126}\pm 0.01040.144±0.01210.144\pm 0.0121
Rössler
TABLE III: Test NRMSE for the ESN, the LI-ESN, and the SRLR: mean ±\pm population SD over ten model seeds. The lowest mean in each row is in bold.
Fig. 4: Half-life profiles selected by the validation searches for the SRLR on (a) the four Lorenz tasks and (b) the four Rössler tasks, in the task order of Fig. 2 . Each task panel is a heatmap with the six half-life values ηk∈{0.25,0.5,1,2,4,8}\eta_{k}\in\{0.25,0.5,1,2,4,8\} (in reservoir updates) as rows and the sectors 0–7 as columns (sector 0 contains the smallest angles θj\theta_{j} , near 0, and sector 7 the largest, near π\pi ; see ( 6 )). The color of a cell indicates the fraction of the ten model seeds that selected that half-life for that sector, and cells selected by at least five seeds are labeled with the seed count.
Fig. 5: Test NRMSE of the three input-weight designs of the SRLR (PSD-scaled, γ\gamma -normalized, and uniform random) on (a) the four Lorenz tasks and (b) the four Rössler tasks, in the task order of Fig. 2 , each design with the settings selected by its own search. Bars show means over ten model seeds, and error bars show population SD.
TaskPSD-scaledγ\gamma -normalizedUniform random
Lorenz
x→xx\to x , horizon 1(4.36±0.507)×10−6(\mathbf{4.36}\pm 0.507)\times 10^{-6}(5.56±0.691)×10−6(5.56\pm 0.691)\times 10^{-6}(4.59±0.424)×10−6(4.59\pm 0.424)\times 10^{-6}
x→xx\to x , horizon 120.0209±0.00167\mathbf{0.0209}\pm 0.001670.0245±0.002220.0245\pm 0.002220.0265±0.002860.0265\pm 0.00286
x→zx\to z , horizon 00.109±0.01120.109\pm 0.01120.0923±0.01130.0923\pm 0.01130.0864±0.00990\mathbf{0.0864}\pm 0.00990
x→zx\to z , horizon 120.144±0.0121\mathbf{0.144}\pm 0.01210.147±0.006140.147\pm 0.006140.154±0.008060.154\pm 0.00806
Rössler
TABLE IV: Test NRMSE for the three input-weight designs of the SRLR: mean ±\pm population SD over ten model seeds. The lowest mean in each row is in bold.
Fig. 6: Ratio of the 90th to the 10th percentile (P90/P10) of the 64 coordinate scales within each profile, where each scale is the population SD of one coordinate of the states of the modes, for the PSD-scaled, γ\gamma -normalized, and uniform random input-weight designs (Section III-A3c) on (a) the Lorenz tasks and (b) the Rössler tasks. Each thin line connects the values of one profile (40 profiles, one per task and model seed) across the three designs. Large outlined symbols mark medians, and vertical bars span the interquartile range. The dotted line marks a ratio of 1.
Fig. 7: Computational cost of the validation searches. (a) Full operational wall time and number of evaluations that returned a finite validation NRMSE for the SRLR, the ESN, the LI-ESN, and the common-radius reference (80 searches per search configuration). The searches of the four configurations ran sequentially, the ten seeds of each case ran concurrently, and other workloads on the machine were not controlled. (b) Time per evaluated candidate for the first 30 of the 128 replayed candidates (the first 128 candidates recorded in each search, re-evaluated in their recorded order) in each of 80 searches per search configuration, one per system, task, and model seed, measured with one worker process and one BLAS thread on one machine, using a fresh process for each search and, for the SRLR, a fresh state bank per search. Lines show medians across searches, and shading covers the middle 50% (25th–75th percentile range). For the SRLR, all six half-life values of the state bank had been computed by the third evaluation in all 80 searches, and a half-life value computed in an earlier evaluation could be reused before the bank was complete. (c) Mean evaluation-time components from the same measurements as panel (b), over all 128 replayed candidates (10,240 per search configuration). (d) Median time for Optuna to generate the next candidate before evaluation, recorded during the searches, over successive blocks of 1,024 candidates, separately for each case and for the four search configurations of panel (a) (10,240 measurements per point, 1,024 candidates ×\times ten seeds). Solid and dashed lines denote Lorenz and Rössler, respectively.
Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

TaskESNLI-ESNSRLR (PSD-scaled)Common-radius referenceSRLR ( γ\gamma -normalized)SRLR (uniform random)
Lorenz
x→xx\to x , horizon 13,0723,0723,0723,0723,0724,096
x→xx\to x , horizon 123,0723,0723,0723,0723,0723,072
x→zx\to z , horizon 03,0723,0723,0723,0723,0723,072
x→zx\to z , horizon 123,0723,0723,0723,0723,0723,072
Rössler
Appendix
TABLE VI: Final evaluation counts of the validation searches for each search configuration and system–task case. Each entry is the count shared by the ten model seeds of that case. All 48 configuration–case pairs met the stopping rule of Section III-B2 before reaching the budget of 8,192 evaluations.
Fig. 8: Best validation NRMSE found so far versus the number of evaluated candidates for the six search configurations on the eight system–task cases ((a) Lorenz, (b) Rössler). Lines show the median across the ten model seeds, and bands span the interquartile range. Each curve ends at the evaluation count in Table VI . The vertical axis is truncated at the top so that the later part of the searches is visible, and the first evaluations of each curve lie above the plotted range.
Fig. 9: Best validation NRMSE found so far versus the number of evaluated candidates for the SRLR with the PSD-scaled design, shown separately for the ten model seeds ((a) Lorenz, (b) Rössler). Each line ends at the evaluation count of its case in Table VI , and a marker shows the final value of each line. The vertical axis is truncated at the top as in Fig. 8 .
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