Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
Authors: Shikang Liu
Chuyang Wei
Xiren Zhou
Huanhuan Chen
Venue: ICML 2025 poster (2025)
Keywords: Reservoir Computing; Spectral Awareness; Classification; Time Series;
Primary area: general_machine_learning->sequential_network_and_time_series_modeling
TL;DR: This paper incorporates spectral insights into Reservoir Computing, offering a multi-scale analysis for fast and accurate time series classification.

Abstract: Analyzing inherent temporal dynamics is a critical pathway for time series classification, where Reservoir Computing (RC) exhibits effectiveness and high efficiency. However, typical RC considers recursive updates from adjacent states, struggling with long-term dependencies. In response, this paper proposes a Spectral-Aware Reservoir Computing framework (SARC), incorporating spectral insights to enhance long-term dependency modeling. Prominent frequencies are initially extracted to reveal explicit or implicit cyclical patterns. For each prominent frequency, SARC further integrates a Frequency-informed Reservoir Network (FreqRes) to adequately capture both sequential and cyclical dynamics, thereby deriving effective dynamic features. Synthesizing these features across various frequencies, SARC offers a multi-scale analysis of temporal dynamics and improves the modeling of long-term dependencies. Experiments on public datasets demonstrate that SARC achieves state-of-the-art results, while maintaining high efficiency compared to existing methods.

Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
Shikang Liu! Chuyang Wei!
Xiren Zhou
Huanhuan Chen
Abstract
Unlike other forms of sequential data, time series data typi- cally record only few numerical values each time step, leading to insufficient information at individual time points (Wu et al., 2021). Consequently, researchers have focused on inherent temporal dynamics of time series for classifi- cation (Wang et al., 2022). This stems from the observa- tion that each point in time series is influenced by both adjacent point and several past ones, real-world data of- ten possesses cyclical patterns at multiple frequency scales (Wen et al., 2021). For example, weather data exhibits daily and seasonal cycles, while electrocardiograms display short-term repeats and circadian rhythms. Identifying such explicit or implicit cyclic patterns through spectral analysis and mining for prominent frequencies enhances the mod- eling and classification of time series data (Yi al., 2024; Wu et al., 2023).
Analyzing inherent temporal dynamics is a criti- cal pathway for time series classification, where Reservoir Computing (RC) exhibits effectiveness and high efficiency. However, typical RC con- siders recursive updates from adjacent states, struggling with long-term dependencies. In re- sponse, this paper proposes a Spectral-Aware Reservoir Computing framework (SARC), incor- porating spectral insights to enhance long-term dependency modeling. Prominent frequencies are initially extracted reveal explicit or im- plicit cyclical patterns. For each prominent fre- quency, SARC further integrates Frequency- informed Reservoir Network (FreqRes) to ade- quately capture both sequential and cyclical dy- namics, thereby deriving effective dynamic fea- tures. Synthesizing these features across various frequencies, SARC offers a multi-scale analysis of temporal dynamics and improves the model- ing of long-tern dependencies. Experiments on public datasets demonstrate that SARC achieves state-of-the-an results, while maintaining high efficiency compared to existing methods.
Following the widespread use of Deep Learning (DL), deep neural networks, particularly Convolutional Neural Net- works (CNNs) and Recurrent Neural Networks (RNNs), have been applied to TSC tasks (Ismail Fawaz et al., 2019). CNN-based methods mainly slide convolutional kernels along the temporal dimension capture changing patterns in time series (Franceschi et al., 2019; Tang et al., 2022). However, the locality of CNN kernels limits their ability to handle long-term dependencies. RNNs generate hidden states by combining the current input with past states (Shen et al., 2020). Nevertheless, due to issues like gradient vanish- ing or exploding. RNNs may struggle with memory decay, thereby hindering the capture of long-term dependencies. Recent studies (Wu et al., 2023; Yang et al., 2024) have also introduced spectral information into DL to enhance time se- ries modeling. Yet, such methods remain essource-intensin and time-consuming, referring to the reliance on gradient descent for optimization.
1. Introduction
Time series classification (TSC) plays significant role in numerous applications, such as diagnosing medical condi- tions (Malik et al., 2022), identifying financial fraud (Al- ghofaili al., 2020), detecting anomalies in industrial pro- cesses (Wang al., 2024), and classifying music genres (Chen al., 2024b). This broad applicability has sparked considerable research interest in developing efficient and robust TSC methods (Iwana & Uchida, 2021).
As a subset of RNNs, Reservoir Computing (RC) demon- strates effectiveness and high efficiency in processing time series data (Chen et al., 2015). Typically, RC uses fixed, randomly initialized recurrent cells (the reservoir) to pro- duce rich set of high high-dimensional states, facilitating the derivation of valuable dynamic features (Yan et al., 2024). A key characteristic of RC is that only the output layer is trainable, thus eliminating the need for backpropagation through time. Although such simplification significantly
'School of Computer Science and Technology, University of Science and Technology of China, Hefei, Anhui, China. Corre- spondence to: Xiren Zhou <zhou0612@uset udw...du..e......... Huanhuan Chen <hehen @ udt...du.cn...
Proceedings of the 4nnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
Input
Identifying Prominent Frequencies
Learning Dynamic Features with FregRes
u()
(())
Waveld decompoostt
Fu
Prominent frequencies
FreqRes Module
x(n)
Next-step prediction task
F
Classifier
HPFFilte
Concat
FFT
tanh
Detrended residual
W
W
W
fin!
x(x-p) p)
x(1-D)
(
u(L)
Cyclical components
Dynamic features
Figure 1. Overview of the SARC framework. Input series is decomposed into multi-scale cyclical components and analyzed spectrally to extract prominent frequencies (Section 4.1). An ensemble of FregRes then captures temporal dynamics at local levels and across various cyclical intervals, deriving effective dynamic features for classification (Section 4.2).
reduces computational costs, single and standard reservoir falls short in modeling long-term dependencies and cyclical patterns due to its recursive updates from adjacent states (Liu et al., 2025). Moreover, directly applying gating mech- anisms (e.g.. LSTM GRU) to the reservoir also yields unsatisfactory results, evident in experimental study.
this, an autoregressive task is applied to derive a "readout model" for each FreqRes. This model maps past states to fu- ture inputs, reflecting the dependencies between time steps. Hence, it encapsulates the captured dynamic information and serves as dynamic feature. Synthesizing these fea- tures across various frequencies, SARC offers multi-scale representation of time series and improves the modeling of long-term dependencies. As result, even simple classifier could be used for effective classification.
Addressing the above issues, we propose novel Spectral- Aware Reservoir Computing (SARC) framework, which introduces spectral insights into RC, comprehensively cap- turing temporal dynamics within time series. As illustrated in Figure 1, SARC consists of two stages: Extracting multi- scale prominent frequencies to reveal explicit implicit eyclical patterns, and Capturing dynamic information within each cyclical pattern for dynamic feature derivation. The first stage begins with detrending, which removes a long- term trend and preserves the cyclical residual. Wavelet decomposition then separates the residual into multi-scale components, representing multiple cyclical patterns. These components are analyzed through Fast Fourier Transform (FFT) to select frequencies with the highest amplitudes. Af- terward, to enrich analytical scales, extend the selected frequencies to their sub-harmont harmonics. These comprise set of multi-scale prominent frequencies, allowing SARC to capture broader range of cyclical dynamics.
Summarily, our main contributions are as follows?:
SARC innovatively introduces spectral insights to RC, capturing temporal dynamics both within neighbor- hoods and across long-span cycles. This provides multi-scale analysis for time series, effectively model- ing long- and short-term dependencies.
Our framework is highly lightweight, requiring gra- dient backpropagation. FreqRes independently oper- ates on each sequence and directly solves for dynamic features using ridge regression. These allow for excep- tional efficiency compared baselines.
SARC adapts to different RC' implementations. Exper- imental studies on public datasets validate this flexi- bility, and also demonstrate that SARC achieves con- sistently superior performance when equipped with various RC implementations.
Subsequently, we integrate a modular component, namely the Frequency-infrmme Reservoir Network (FreqRes). for each extracted frequeney. Unlike standard RC models, Fre- qRes features two reservoirs in its hidden layer, each repre- senting connections between adjacent time steps and across cycles at the given frequency. When fitting a time series, FreqRes considers both of the two aspects to update the cur- rent state, enabling it to simultaneously capture sequential dynamics and long-span cyclical dynamics'. Building upon
2. Related Work
We briefly review state-of-the-an TSC methods, followed by introduction reservoir computing and its applications.
State-of-the-art TSC methods can be broadly divided into
'Henceforth, "cyclical dynamics" " refers to the dynamic infor- mation within a specific cyclical pattern or at a certain frequency.
Code available https: htgggghu.ccm////e
SARC.
2

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SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
four categories: dictionary-based, shapelet-based, ensem- ble, and deep learning methods. As a latest dictionary- based method, Hydra (Dempster et al., 2023) transforms time series using random convolutional kernels arranged into groups, counting the kernel with the maximum and minimum response to represent key patterns, In contrast, Shapelet Transform (ST; Hills et al. 2014) identifies the most discriminative subsequences, or shapelets, from time series, and then converts them into features for classification. Regarding ensemble methods, Collection of Transformation Ensembles (COTE; Lines et al. 2018) integrates multiple classifiers, such as Time Series Forest (Deng et al., 2013), Bag-of-SFA- SFA-Smbbls (Schäfer, 2015), and Elastic Ensem- ble (Lines & Bagnall, 2015), to enhance TSC performance.
RC models can sometimes limit their accuracy, suggesting need for more complex or nuanced RC architectures.
3. Preliminaries
Our framework adapts to various RC implementations. For a clear illustration of its core improvements, we formalize the simplest form of RC. namely the ESN, which underpins the subsequent methodological explanation.
An ESN consists of three main components: fixed, ran- domly initialized linear input layer and reservoir, along with a readout layer. During iteration, its hidden states are up- dated based on the previous state and the current input. The readout layer then converts these states into final outputs. Formally, the iterative and output formulas are as follows:
On the other hand, deep learning methods leverage neural networks to automatically extract features from time series data, achieving remarkable results in various TSC tasks (Ismail Fawaz et al., 2019). For example, Inception'ime (Ismail Fawaz et al., 2020) employs an ensemble of Incep- tion modules to capture complex temporal patterns. LSTNet (Lai et al., 2018) combines CNNs and RNNs to extract lo- cal dependencies and incorporates a recurrent-skip layer to handle long-term dependencies. Another notable method is TimesNet (Wu et al., 2023), which is more related to our approach as it also introduces periodicity. TimesNet uses the periods obtained from Fourier Transform to segment and transform time series into 2D formats, then employs vision backbones for classification. Besides, transformers (Wu et al., 2021; Zhou et al., 2022) have also gained attention for their ability to handle long-term dependencies. Despite the effectiveness of these methods, they are all constrained by the high computational and time costs.
x(n) = -[Wxx(n - 1) + Winu(n)),
(1)
y(n) = W (nn)(()),
(2)
where u(n). x(n). and y(n) denote the input, hidden state, and output at time step n; Win and W... are the input and readout weight matrices; W represents the reservoir weight matrix; and f is an activation function, typically tanh. According to Equation (1), a typical ESN only con- siders recursion between adjacent hidden states, struggling to capture long-term dependencies within time series.
4. Methodology
The proposed SARC framework comprises two main parts: 1) Extracting multi-scale prominent frequencies to identify potential cyclical patterns; 2) Capturing cyclical dynamics via FreqRes and deriving dynamic features. These parts are illustrated in Figure 1 and further elaborated below.
Reservoir Computing uses reservoirs with randomly inter- connected neurons to adeptly capture temporal dynamics from time series (Yan et al., 2024). Unlike traditional neural networks, it offers an efficiency advantage by maintaining fixed connection weights. The foundational concepts of RC were introduced via two primary models: Echo State Networks (ESNs; Jaeger 2001) and Liquid State Machines (LSMs; Maass et al. 2002). Following these, researchers have proposed various alternatives such as the Echo State Gaussian Process (Chatzis & Demiris, 2011), Delayed Feed- back Reservoir (Appeltant et al., 2011), DeepESN (Gal- licchio et al., 2018), and 2D-ESN (Chen et al., 2024a). Recently, RC has been increasingly applied to time series classification, demonstrating its efficacy in many tasks (Liu et al., 2024). For instance, COnvMESN (Ma et al., 2019) incorporates reservoirs with varying skips to compute multi- timescale echo states, followed by a ID convolutional layer for feature extraction. Additionally, rmESN (Bianchi et al., 2021) fits a linear transformation between adjacent echo states, using the weights as a representation of temporal dynamics. However, simplifying the reservoir in standard
4.1. Identifying Potential Cyclical Patterns
To identify cyclical patterns and capture long-span cycli- cal dynamics, is crucial to extract prominent frequencies from time series data. A straightforward approach involves selecting the top-k amplitudes from FFT (Wu et al., 2023). However, this strategy risks neglecting higher frequencies (Duhamel & Vetterli, 1990; Zhou et al., 2022). Additionally, if the time series shows a clear non-cyclical trend, the high- est amplitudes tend to concentrate near zero frequency, ren- dering the extraction less meaningful. In response, SARC detrends the time series and decomposes the residual into multi-scale cyclical components, after which the prominent frequencies are extracted from each scale.
4.1.1. DETRENDING AND CYCLICAL DECOMPOSITION
Given a time series u = {u(n) of length L with C variables, detrending is initially conducted to reduce non- cyclical influences. To this end, we apply the Hodrick-
3

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SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
Detrending
nent is analyzed through FFT:
HPPiitter
A |FFT(1))), 1s = argmax (Az)-
(4)
Sep.....1.1
Time Series u
Derrended Series
Specifically, [FFT(-) calculates the magnitude of the FFT results. A, contains the non-normalized amplitudes of ry different frequencies. The most prominent frequency ʃg is determined by selecting the highest amplitude in Aj. This frequency accurately reflects the cyclical pattern, with a cycle length of pj = [y] as it is free from long-term trends and other prominent frequencies.
Wavelet Decomposition
FFT
Spectral Analysis
Prequency
Prequency f
Prequency
To fully capture the cyclical dynamics at these frequencies, design to model the multi-scale dependencies spanning multiple cycles. Concretely, the selected frequencies extended to their sub-harmonics, comprising a set of multi- scale prominent frequencies F:
KCUS))
Multi-scalc Prominent Frequency Set F
Figure 2.
An illustration of prominent frequency extraction: The series u is first detrended and decomposed into cyclical compo- nents (r, 1= Each component then undergoes FFT to extract the highest amplitude and the corresponding frequency fy.
J F F= - HH)(()))
- 5=0
I <n-. <p, (5)
1=0
Here, Hr denotes the transformation from frequency to its sub-harmonics with a threshold parameter s. Each sub- harmonic f ///k characterizes a dependency spanning k cy- cles of length pj. constrained by K. Therefore, k defines the maximum span of dependencies that needs to be modeled. Typically, a larger к implies more meticulous analysis but also leads higher computational complexity.
Prescott filter (Hodrick & Prescott, 1997) featuring high efficiency and effectiveness in estimating long-term trends from time series data (Wen et al., 2021). This involves minimizing the following objective function:
+n)) = arg min
Σ n=1
(u(n) (n)))
N-1
4.2. Frequency-informee Reservoir Network (FreqRes)
+ A Σ
[[T(n -
+ (n 1n) 1)1²
For each frequency f E F, SARC integrates a FreqRes module. Unlike standard reservoir with only adjacent con- nections, FreqRes features additional cycle connection, which updates the hidden state by combining states from both previous steps and cycles. This improvement simul- taneously captures the sequential dynamics and long-span cyclical dynamics for more comprehensive modeling of time series. Moreover, given the diversity of cycle lengths, inte- grating multiple Freq modules facilitates multi-scale analysis of the temporal dynamics, thus effectively handling both short- and long-term dependencies. Incorporating such cycle connection into any existing RC model results in a different FreqRes implementation. For clarity, we now pro- pose the iteration of FreqRes and the derivation dynamic features based on an ESN model introduced Section 3.
n=2
where + (f(m)) is the estimated trend of u; À is the smoothing parameter: smaller À retains more short- term fluctuations in f, while larger À makes it smoother. Typically, A is set to 1600, and this problem can be solved using dynamic programming.
Through the above, the long-term trend is removed from the original time series, yielding a cyclical residual u +. This residual represents mixture of multiple cyclical influ- ences within the data. Decoupling the mixture, we employ wavelet decomposition and separate into J + cyclical components at different scales, including one approxima- tion component r2 and J/ levels of detail components rd.j. j E (1. J}. Each of these components reflects a po- tential cyclical pattern, aligned with a prominent frequency to be extracted. Note that, for simplicity and uniformity in notation, we will henceforth adopt the following conven- tions: Tu- = Ta. and r Td.j.
1.2.1. 1. ITERATION OF FREQRES
As shown in Figure 3, FreqRes iterates in 2D grid-like pattern. Given an input time series E L and specific frequency f. the process starts with an initial state x(0) O. Step by step, the raw inputs are weighted through a linear input layer and fed into a hidden layer containing two independent reservoirs. Each reservoir is composed
4.1.2. PROMINENT FREQUENCY EXTRACTION
Once wavelet decomposition is complete, we extract the corresponding prominent frequencies. Each cyclical compo-

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SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
Cyclical Connection
Sequential Connection
Algorithm Spectral-Aware Reservoir Computing
Time series u E R smoothing parameter A, thresh- old parameter K, regularization parameter <. Initialize: F <- 0. x(0) <- 0, Wi,, W., W. NO0, 1)². Output: Dynamic feature F.
Input:
5
$ HPFilter A), r u
|/ Detrending.
2: (ry (tz) +- WaveDec(r).
// Wavelet decomposition.
3: for 0
4: A, -- [FFT(r)))
5: f, + argmax (Ay).
SE(1.... 1.
Min
6: F F-FU liket
kp < k..
7: end for
Extract prominent frequencies.
8: for ʃ F do
9:
p [+]-
10:
for n
11:
Update state x(n) according to Eq. (1) and (6).
Figure 3. An example of the FreqRes iteration with a cycle 5.
12:
x(n) + [x(n - 1);x(n p)).
Hidden states arranged in 2D layout, each containing
13:
end for
p contiguous time steps. Hidden state update considers the state from the previous time step (left) and cycle (below). W, and W. represent the influence of the two independent reservoirs.
14:
U.X +- ulp L 1]. x/p app: L L+ 1].
15:
W + UXX" +
// Ridge regression.
16: F, f flatten W o..)
17: end for
Derive dynamic features from p.
18: F concatenate(P, for all F).
of randomly connected neurons, reflecting the influence of historical data on present states. One of the reservoirs accounts for variations between adjacent time steps, while the other focuses dynamics across cycles with span of [T/ During the first steps (n S p). hidden states of the previous cycle are not available; hence, FreqRes adheres to the base iteration formula in Equation (I). Then, for p. the hidden states are updated as follows:
prediction task" is implemented. It requires the output (n) to approximate input u(n), enabling the output layer to carry the transformation from previous states (both the pre- vious time step and cycle) to current sequence values. This approach models the recursive relationships within the in- put time series. Consequently, the derived readout model, y W out is considered to encapsulate the temporal dy- namies across both sequential and cyclical dimensions, and thus serve the dynamic features from frequency f. In practice, this readout model is represented by W which can be solved using ridge regression:
x(n) f (WW.f((WWWXl- 1)+ Wex(n pp) Wmu(n)). (6)
Here, Win in is the input weight matrix. W, and W. the reservoir weight matrices that describe the interconnections among neurons within each reservoir, modulating the influ- ence of states from the previous time step and cycle. Similar to the original ESN, these three matrices are randomly ini- tialized, remaining fixed and untrainable.
W out UX XX + +())
(8)
where U = (u(p), u( + 1), n u([.)] is the target matrix; X = {p)), x(p+1),,........ x(L) is the combined state matrix; is a regularization parameter; and I the identity matrix. By flattening and concatenating the Wowr derived from all frequencies, final dynamic features F are obtained. For further insights into the overall procedure, we detail framework in Algorithm 1.
In alignment with the added cyclical connection, the output formula of FreqRes has been refined accordingly. Rather than mapping x(n) directly, the output layer now uses combined state x(n) [x(n x); - p)] to determine the target output yn)):
x((n= 1)] x(n - p)
y(n) = W (()) [ out.s W Wotte]
(7)
5. Experiments
Here, W... is partitioned into W outs and W to map the contributions from x(n 1) and x(n p), respectively.
In this section, we evaluate the SARC on public benchmarks. As afforementioned. the FreqRes not confined to specific RC model. By default, we implement FreqRes based on Bidirectional ESN (BiESN), which is demonstrated to be optimal across four conventional RC models in our ablation
4.2.2. DERIVATION OF DYNAMIC FEATURES
Leveraging the rich hidden states from each FreqRes, we derive a representative "readout model" to serve as the dy- namic features. Concretely, an autoregressive "next-step
For brevity, input weight scaling for W. and spectral radius adjustments for W, and W, are omitted in Algorithm
5

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SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
Table Comparison of accuracy and Fi-score between the proposed SARC and haselines across 128 datasets. Bold values represent the maximum scores for each metric. P-values > 0.05 are underlined. indicating no statistically significant difference. "Wins/iies/Losses count datasets where each baseline performed better, equally, worse than SARC, respectively.
Accuracy
Fl-score
Methods
Avg. Acc.
Avg. Rank
P-value
Wins/Ties/Losses
Avg. Fl
Avg. Rank
P-value
Wins/Ties/Losses
ImESN
0.7689
6.7266
1.60E-20
9/5/114
0.7560
6.7265
7,08E-20
7/5/116
ConvMESN
0,8108
5,8164
4.01E-16
20/3/105
0.7929
5,9487
3.15E-15
18/2/108
TimesNet
0.6867
8.3984
1.76E-22
1/1/126
0.6714
8.1624
8.41E-23
2/3/123
Inception Time
0.8352
4.5234
1.60E-03
42/10/76
0.8313
4.4188
4.03E-03
45/6/77
COTE
0.8484
3.7539
4.03E-01
59/8/61
0.8405
3.9017
5.48E-01
56/8/64
Hydra
0,8423
4,2695
3.78E-03
41/10/77
0.8377
4.3120
7.52E-03
45/9/74
Rocket 0,8450
3.8008
2.21E-01
56/12/60
0.8410
3.7991
1.73E-01
53/14/61
MiniRocket
0.8411
4.3711
1.81E-03
42/10/76
0.8368
4.3761
2.50E-03
44/10/74
SARC
0.8508
3.3398
0.8449
3.3547
studies (Section 5.3)*. All experiments are conducted using Python 3.11 on desktop with an Intel Core i7-14700KF CPU, and an NVIDIA GeForce RTX 4090D GPU.
TimesNer
MESS 6725
3:754
COTE
5.1. Experimental Settings
ConvMESN 5816
3.901
Rockst
Incoption Time
Hlyrr
5.1.1. DATASETS
MinRoooket
We use the full UCR Time Series Archive (Dau al., 2019) with 128 datasets spanning various applications such activity recognition, health monitoring, and spectrum anal- ysis. These datasets encompass time series lengths from 15 to 2844, classes ranging from 2 to 60, and training sets from 16 to 8926 samples. Such diverse characteristics and complexity enable a thorough evaluation of our method.
Figure 4. Critical Difference diagram for accuracy on 128 datascts.
lengths or missing values, we first interpolate middle NaNs (i.e., those surrounded by real values), then align real values to the right end of the tensor, and finally fill leading NaNs with zeros. Key hyperparameters are determined through a five-fold cross-validation on the training set, selecting input scaling from (0.5, 1, 2, 4), spectral radii from {0.4, 0.6, 0.8), regularization C from (0.5, 1}. and leaky rates ranging from O to 0.8 in 0.2 increments. The reservoir size is set to 10, the connectivity is 1. and the threshold к is set to 100. For classification, features with the max-pooled hidden states and feed them to default Ridge classifier. The experiments are repeated five times using different random seeds. Regarding the baselines, we use official implementations or configurations from the original papers. More detailed information provided in Section A. of the Appendix.
5.1.2. COMPARATIVE METHODS
The proposed SARC is evaluated against 8 state-of-the-art methods, which have been introduced in the Related Work. These include two RC-based methods: rmESN (Bianchi et al., 2021) and COnYMESN (Ma et al., 2019); two deep neural networks: Inception Time (Ismail Fawaz et al., 2020) and TimesNet (Wu et al., 2023); and four advanced methods on the UCR Archive, namely COTE (Lines et al., 2018), Rocket (Dempster al., 2020), MiniRocket (Dempster al., 2021), and Hydra (Dempster al., 2023). Similar to approach, TimesNet identifies cyclical patterns within time series for classification. Inception Time and COTE both demonstrate strong performance. Additionally, Rocket, MiniRocket, and Hydra are relatively lightweight, making them suitable for assessing SARC's efficiency.
concatenate the derived dynamic
5.2. Classification Performance Evaluation
5.2.1. ACCURACY AND F1 SCORE
We analyze accuracy and Fl scores across 128 datasets from multiple perspectives, including average metrics, rankings, the Wilcoxon Signed-Rand Test, and counts of wins/ties/losses. From the results in Table 1, SARC achieves the overall best performance, surpassing state-of-the-art methods like COTE, Rocket, and Hydra. It not only boasts the highest average accuracy of 0.8508 and the highest aver- age F1 score of 0. 0.8449, but also maintains the top average
5.1.3. IMPLEMENTATION DETAILS
In our experiments, all datasets standardized to maintain uniformity across evaluations. For datasets with variable
*FregRes based on BiESN involves: performing forward and backward iterations as Eq. (6). concatenating the bidirectional hidden states and calculating dynamic features using Eq. (8).
6

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SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
10'
Tocal Triining Time
Tosal Testing Time
10'
27716
LSTM
BIESN
Time (0)
10'
GRU
4.529
LSakyESN
10'
ESN
3
DeepESN
Varying RC Models
10'
mFSN
SARC MinilRadett
1yym
Kndert
Cone
Timas
Inog
CDT
Final State Avg. Pooling
Figure 5. Runtime comparison (in log scale) on 128 UCR datasets.
2,706
Max Pooling
Varying Feature Utilization
S
ranking. Statistically, SARC exhibits significant differences from most comparative methods and consistently wins more in pairwise comparisons, Compared to rmESN and Con- vMESN, which are also RC-based, SARC demonstrates significant performance advantages, with average accuracy improvements of 8.% and 4.00%, respectively, highlight- ing the benefits of incorporating spectral awareness.
KNN 3.554
2,500 Ridge
LR 2
SVM
RF 2:766
(c) Varying Classifiers
Figure 6. Critical Difference diagrams for ablation study on 128 datasets, based on accuracy rankings.
5.2.2. COMPLEXITY AND TIME
The main factors affecting SARC's complexity include sam- ple size M, series length L, threshold k, and reservoir size S. According to experiments, SARC's runtime is primarily spent on iteration of RreqRes modules and computation of dynamic features, with a complexity of Ox&M (S2 8*)).
and 1.61% % in Fl score. Even using the weakest ESN, SARC remains competitive with the state-of-the-att art Inception Time, highlighting its superiority and flexibility. This demon- strates SARC's potential to accommodate advancements in the field of RC and leverage its strengths. On the other hand, a notable accuracy decline is observed when implementing SARC based on LSTM and GRU, which confirms that di- rectly applying gating mechanisms in RC without adequate training often leads to unsatisfactory results.
This means that under specific settings, the runtime nearly linear with the data complexity (characterized by M and L). showcasing exceptional efficiency.
Figure 5 compares the runtime of various methods on 128 datasets. Clearly. SARC is faster than most baseline meth- ods. Compared to CESN and Inception Time, which require gradient descent, SARC tens hundreds of times faster in training. Considering relatively lightweight ap- proaches like MiniRocket, SARC's training is still several times faster, and the advantage is even more pronounced in the testing phase. This exceptional efficiency stems from the absence of gradient backpropagation and the high paral- lelizability among FreqRes modules. The only method that slightly exceeds SARC in runtime is mEEN; however, its accuracy lags substantially, rendering less comparable.
5.3.2. IMPACT OF FEATURE UTILIZATION
To validate the performance improvement from incorporat- ing the derived dynamic features, we compare the utilized features with three commonly used RNN feature types: Max Pooling and Average Pooling of hidden states, and the Fi- nal State From the results in Figure 6b and Table using only Max Pooling shows modest performance decline, while the others lead to significant decreases in average ac- curacy by 5.97% and 10.81%. This is because the derived dynamic feature effectively integrates past and present in- formation, encapsulating the inherent temporal dynamics, which leads to improved performance.
5.3. Ablation Study
5. IMPACT OF RC MODELS
We evaluate SARC based on four conventional RC models: ESN, LeakyESN, BiESN, and DeepESN, and two gated RNN models: LSTM and GRU, with implementation de- tails provided in Section A.2 of the Appendix. From Figure 6a and Table 2, we find no significant differences when us- ing the four RC models, while LSTM and GRU lag behind. Compared to BiESN, other RC models show only minor declines, with a maximum reduction of 1. 1.55% accuracy
5.3.3. IMPACT OF CLASSIFIERS
We also assess the impact of different classifiers, including Logistic Regression (LRR), Random Forest (RF), K-Nearest Neighbors (KNN), Support Vector Machines (SVM), and Ridge Regression (Ridge) classifier. Figure 6c shows that Ridge achieves the best performance, which is why we use it by default. RF and SVM are relatively weaker, yet not statis-
7

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SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
0.45
9.50
Avenag Nccuracy
0.81
Total Rumtime (min)
Avcuae Acountcy
Raitine (miin)
0.93
Ata Arc
0.83
Dec
0.664
Sri.
0.52
Avrmps
Hartinee
Rantize Time Ski
0.01
0.70
50
0.5
20
0.73
03
04 05
0:00
25
100
125
200
0.50
70
scaling
Lenky mate
Throshold
Roserveir sine
(a) Impact of input scaling
(b) Impact of leaky rate
(c) Impact of threshold x
(d) Impact of reservoir size
Figure Impact input scaling, leaky rate, threshold x reservoir size on average accuracy and total runtime across datasets.
0.86
0.550
Aoc.
0.8
Wil
Sppertn Reciooo W.
60 04 60 00 07 00 69
6838
0.85
0.8454
0.8661
0.6459
0.8110
0:2130
0.7
Loose
Accumazy
0,80
0.6
Average
Sod. Dev.
0.5
-0015
0,535
0.4
0.50
0.3
002
04
0300
0.80
0.5
0.525
BIESN
Spectral Raduus
Connectivity
Regularization
(a) SARC vs. BiESN
(b) Impact of spectral radii
(a) Impact of connectivity
(b) Impact regularization Ç
Figure 8. (a) Accuracy comparison between SARC and BiESN, and (b) the impact of spectral radius on average accuracy.
Figure 9. Impact of varying reservoir connectivity and regulariza- tion on average accuracy across 128 datasets.
tically significant, indicating that the derived features adapt well various classifiers. However, both LR and KNN yield inferior outcomes. Results in Table also confirm this lag, suggesting their difficulties in integrating features across multi-scale prominent frequencies.
proves threshold increases. This confirms that extend- ing prominent frequencies provides richer scale insights and effectively enhances TSC performance. Unexpectedly, reser- voir size has minimal impact on accuracy but significantly elevates runtimes, as depicted in Figure 7d. Thus, we have fixed it at 10 for efficiency. From Figure 8b, Spectral radii markedly influence SARC's performance, with optimal ac- curacy near spectral radius of 0.8 for W, and 0.6 for W.. This implies a weighting between sequential and cyclical dynamics: the former should slightly outweigh the latter.
5.3.4. COMPARISON WITH BIESN
To further investigate the contribution spectral awareness in the SARC framework, we compare with a variant lack- ing this mechanism, namely a single BiESN. As illustrated in Figure 8a, SARC consistently surpasses the BiESN in accuracy across 119 datasets, achieving over 10% im- provement in 65 cases. This highlights that incorporating spectral insights into RC models enhances the capture of temporal dynamics within time series data, leading to more discriminative dynamic features for classification.
From Figure 9a, accuracy remains stable as connectivity increases, with a dip around 0.5 and peaks near 0.1 or 1.0. Conversely, the standard deviation is lowest 0.5, and high- est at 0.1, highlighting a trade-off between accuracy and stability. Even so, the standard deviation stays below 0.5%, demonstrating the overall stability of SARC. Figure 9b fur- ther illustrates that average accuracy is highest around Ç = 0.5 to 1.0. Low ( values can cause overfitting, leading to the capture of excessive details and reducing generalizability, High < values may result underfitting, as the regulariza- tion suppresses informative temporal dynamics, diminishing the eategory-diserininbblly of the derived features.
5.4. Hyperparameter Study
We now turn to the impact of key hyperparameters SARC. Figure 7a shows a notable rise accuracy as input scaling increases about 2.8, beyond which the changes become minimal, suggesting that larger input scaling might be preferable. In contrast, Figure 7b indicates that average accuracy slightly improves with a leaky rate increase from () to 0.2, but diminishes with further increases, recommending a smaller leaky rate close to zero.
6. Conclusion
This paper proposes SARC, a novel Reservoir Computing framework with spectral awareness, integrating multiple FreqRes modules corresponding multi-scale prominent
Figure also shows that average accuracy markedly im-
8

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SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
frequencies. These modules simultaneously capture both sequential and cyclical dynamic information within time series, providing comprehensive analysis of temporal dy- namics and deriving effective dynamic features for TSC. Ex- periments demonstrate that SARC achieves state-of-the-art accuracy while maintaining exceptional efficiency. Future work could delve into the mechanisms by which spectral information influences TSC performance or explore appli- cations in other tasks such as time series anomaly detection.
ference on Artificial Intelligence, pp. 3387-3394. AAAI Press, 2015.
Chen, I.,, Zhou, X., and Chen, H. Audio scanning network: Bridging time and frequency domains for audio classifica- tion. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pp. 11355- 10355-1163,,00 2024b.
Dau, H. A., Bagnall, A., Kamgar, K., Ych, C.-C. Zhu, Y., Gharghabi, S., Ratanamahatana, C. A., and Keogh, The UCR time series archive. IEEE/CAA Journal of Automatica Sinica, 6(6): 1663:129333355 2019.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (No. 62206261, 62137002, 62176245), in part by the Fundamental Research Funds for the Central Universities (No. WK21501 10039).
Dempster, A., Petitjean, F., and Webb, G. I. ROCKET: Exceptionally fast and accurate time series classification using random convolutional kernels. Data Mining and Knowledge Discovery, 14(5):1454-10 1495, 2020.
Dempster, A., Schmidt, D. F., and Webb, G. I. Minirocket: A very fast (almost) deterministic transform for time series classification. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pp. 248 257, 2021.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
Dempster, A., Schmidt, D. F., and Webb, G. I. Hydra: Competing convolutional kernels for fast and accurate time series classification. Data Mining and Knowledge Discovery, 37(5):1779-1805 2023.
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10

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SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
A. Implementation Details
Table 2. Ablation results on 128 UCR datasets.
A.1. Comparative Methods
Avg. Acc.
Avg. Fl-score
The eight advanced baselines are sourced from official im-
Varying RC Models
plementations to ensure reproducibility and consistency. Specifically, TimesNet from the TSLib package'. In-
BIESN
0.8508
0.8449
- ESN
0,8353 (-1.55%)
0.8288 (-1.61%) %)
ceptionTiime, COTE, Rocket, and MiniRocket are imple-
- LeakyESN
0,8416 (-0.92%)
0,8350 (-0.99%)
DeepESN
0,8394 (-1.14%)
0.8325 (-1.24%)
mented using the skt imme package Additionally, Con- vMESN and Hydra are available from their respective repositories while rmESN follows the hyperparameter settings from its original paper.
LSTM
0.7896 12%)
0.7852 (-5.97%)
- GRU
0.7926 (-5.82%)
0.7874 (-5.75%)
Varying Feature Utilization
Utilized Feature
+ Max Pooling
0.8365 (-1 (.43)
0.8301 (-1,48%)
A.2. SARC Variants
Avg. Pooling
0.7911 1 (-5. -5.97%
0.7835 14%)
Final Staste
0.7427 (- (-10.81%)
0.7343 (-11.06%)
The variants of SARC in Table 2, achieved by employing different RC models or RNNs, are detailed as follows:
Varying Classifiers
Ridge
+ -R
0,8253 (-2.55%)
0.8069 (-3.80%)
- RF
0,8391 (-1.17%)
0.8309 (-1.40%)
SARC with ESN:
Iterations and feature derivation of FreqRes modules conform to equations (6)-(8). The leaky rate is fixed at O. Other key hyperparameters not mentioned here follow the settings in Section 5.1.
KNN
0,8218 (-2.90%)
0.8178 (-2.71%) %)
SVM
0.8456 (-0.52%)
0.8307 (-1.42%)
For each sample and FreqRes module, the iteration includes an (S²) operation for updating hidden states at each step, culminating in O(S²L) operations for the en- tire series. Subsequent feature derivation, as delineated in Equation (8), requires S L operations for XX", O($)) op- erations for matrix inversion, and O + S²) operations for the remaining matrix multiplications. The combined complexity thus amounts to O T, + 5²). Given that each FreqRes corresponds to a cycle interval less than K, the num- ber of FreqRes modules is bounded by k. Hence, the overall complexity for all samples is (S L + S))).
Proof:
SARC with LeakyESN: A leaky integrator is incorpo- rated into equation (6). The update of x(n) includes leakage from x(n (mn--1p 1), weighted by the leaky rate. The leaky rate for all FreqRes is uniformly set and chosen from {0, 0.2, 0.4, 0.6, 0.8).
SARC with DeepESN: For a given layer, the hidden states derived from equation (6) serve inputs for the next layer, and those from the final layer are used for subsequent processing. The number of layers is selected from (2, 3), and the leakage is also adopted.
C. Supplementary Experiments
SARC with LSTM / GRU: For each extracted fre- quency, single-layer LSTM/ GRU is implemented, with its hidden dimension set to 10, matching the reser- voir size of the default SARC.
C.1. Impact of Sequence Length on Efficiency
For a deeper insight into the computational efficiency of the proposed SARC framework, we compare its time cost with four relatively lightweight baselines: rmESN, Hydra, Rocket, MiniRocket, and LSTNet, across datasets with varying sequence lengths. Specifically, we selected six datasets, including Crop(46), Electric Straw- be35y FordA(500), StarLightCurves(( (270 with sequence lengths noted in paren- theses. From the results shown in Figure 10, SARC exhibits the highest efficiency in scenarios with sequence lengths under 100, where training the classifier takes up most time. In this case, features derived by SARC have the lowest di- mensionality, thus the time spent the classifier is shorter. However, as the sequence length increases, SARC's run- time becomes slightly higher than that of rmESN, since the iterative process of the latter is simpler. Nevertheless, SARC remains faster than MiniRocket, Hydra, and Rocket, demonstrating its superior efficiency.
B. Complexity Analysis
In the SARC workflow, operations such as the HP filter, wavelet decomposition, FFT, and Ridge Classifier are effi- ciently handled by well-established libraries, ensuring rapid computation. The runtime is mainly spent the itera- tion of FreqRes modules and dynamic feature derivation. We demonstrate that the complexity of these processes is (s + 5²)), where M denotes the sample size, L. the series length, x the period limit, and S the reservoir size.
and
Time- Series
com/ skt inltimm///e i
"https://gt thtb. com com/qianl iaa ab/ConvMESN "https://gi thub. http://ggttuuu..oooongggg/////// 4 4/hydra

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SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
SARC
SABC
SARC
10
maFSS Misieede Tlytrs Racket
MicoRoker
Maillecies Hlydu
Rantonr penneppett Sumple.
Trainin Time pes Sample
Texting time per Sumpptee
Rocks
Roclict
(86)
FoetA1900 Sur(h3bl) (odl2789)
Stx,239g
Handayt0y Sarcll00 Baad(2766)
Coeetti
RadAA/M Shachor riedd (709
Datects
Datanco
(a) Runtime per sample
(b) Training time per sample
(c) Test time per sample
Figure Comparison time cost across five lightweight methods on datasets with varying sequence lengths: runtime per sample subfigure (a), training time per sample in subfigure (b), and test time per sample in subfigure (c). The time axis is on a logarithmic scale. Sequence lengths are indicated in parentheses the x-axis.
Table 3. Average accuracy ranks of different methods on the 128 UCR datasets, grouped by sequence length. Bold indicates the best rank, while underlined indicates the second-best.
Table 4. Pairwise accuracy comparison with and without Spectral Insights across different models. "Wins/Ties/Loss indicate the respective counts for the model with ST versus without ST.
Methods
1-200
201-500
501-1000
1000+
Avg. Acc. w/o S1
Avg. Ace. w/ SI
Models
Wins/Ties/L losses
P-value
#Datasets
41
45
20
ESN
0.6456
0.8353
122/1/5
6.44E-22
MESS
6.6098
7.3111
5,3750
6.9773
LeakyESN
0,7030
0,8416
117/0/11
3.79E-20
ConvMESN
5.5366
6.1222
5.6500
5.8636
BiESN
0.7055
0,8508
119/3/6
1.58E-21
TimesNet
8.1707
8,5667
8.0250
8.8182
DeepESN
0.6752
0.8394
117/2/9
1.05--20
Inception
4.5610
4.1667
4.8250
4.9091
LSTM
0.6212
0.7896
115/0/13
5,05E-20
сотЕ
3.6463
4.1333
4.7250
2.2955
GRU
0.6295
0,7926
114/0/14
2.18E-20
Hydra
4.4878
4.0333
4.5500
4.0909
Rocket
4.0000
3.3000
4.2000
4,0909
MiniRocket
4.3171
3.9556
5.1750
4.5909
SARC 3.6707
3.4111
2.4750
3.3636
C.2. Accuracy Comparison by Sequence Length
TimesNer 3.162
3355
SARC
IEESN 5.226
3:392
Racket
In Table 3, we group 128 datasets by sequence length and report the average accuracy rank of different methods across each group. The results show that SARC achieves the best average rank of 2.475 in the length range of 501-1000. In contrast, Rocket performs best in the 201-500 range, while COTE ranks first in the remaining two groups. Despite this, SARC attains at least the second-best performance across all cases, demonstrating consistent competitiveness.
ConVMESN
CUTE
Inceptian
Hydıa
Min:Rooket
Figure 11. Critical Difference diagrams for different methods on 128 datasets in terms of Fi-score.
D. Extension to Multivariate Case
The proposed SARC could be adapted to multivariate narios in straightforward manner: it extracts required fre- quencies from the mean series across variables, then applies FreqRes modules to iterate on the raw multivariate series. Another feasible extension involves separately extracting fre- quencies and modeling patterns for each individual variable before concatenating the derived features. While preserving variable-specifie information, it would require more sophis- ticated implementations to ensure parallelization efficiency. Moreover, inter-variable interactions are unaddressed in this design. Further improvements could focus on the modeling of inter-variable relationships during feature extraction.
C.3. Impact of Spectral Insights across Different Models
Table 4 provides a pairwise comparison of whether Spectral Insights (SI) are incorporated across various base models (in terms of accuracy). Specifically, Column reports the win/tie/loss counts of "w/ SI" version against "wwo SI", and the p-values between them from Wilcoxon tests. Results demonstrate that incorporating SI achieves average accuracy improvements ranging from 13.86% to 18.97% across all base models. Statistics further confirm the significance of these improvements. This validates the consistent effective- ness of our approach across broader range of models.
12

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SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
Dataset
rmESN
Conv.
Times.
Incep.
COTE
Hydra
Rocket
Mini.
SARC
ACSFI
0,8300
0.8320
0.8200
0.8300
0.9360
0.8800
0,8700
0.8920
0.8500
Adiac
0.6859
0.7381
0.6609
0.7673
0.8128
0.8107
0.7754
0.8143
0.7683
W
0,4686
0.6714
0,4274
0.7600
0,6680
0.6926
0.7771
0.6951
0.7234
AllGesture Y
0,5826
0.6803
0,4997
0.7783
0,6831
0.7431
0.7549
0.7043
0.7583
0.6151
0.6060
0.4323
0.7797
0.6520
0.6966
0.7494
0,6809
0.7474
ArrowHead
0,8000
0.8400
0.7943
0.8480
0.8446
0.7943
0.8011
0,8091
0.8537
Beef
0.7667
0.8667
0.5667
0.6600
0.8867
0.8200
0,8000
0.8667
0.9133
BeetleFly
0,8500
0.9000
0,6500
0,8500
0.9500
0.9500
0,9000
0,9100
0.9500
BirdChicken
0,8000
0,9000
0,8500
0.8000
0.8900
0.9000
0,9000
0,8000
0.9400
BME
0,9067
0.9933
0.8280
0.9933
0.9000
1.0000
1.0000
1.0000
1.0000
Car
0.7667
0.8833
0.6500
0.9000
0.8833
0.9267
0.8833
0.9167
0.9267
CBF
0.9016
0.9727
0,9476
0.9956
0,9920
0.9778
0.9989
0.9989
0.9882
Chinatown
0.9708
0.9837
0.9598
0.9825
0.9825
0.9755
0.9796
0.9825
0.9808
ChlorineConcenntatioon
0,6534
0.7905
0.6766
0.8616
0.7424
0.7540
0.8061
0.7395
0.8672
CinCECGTorso
0.8061
0.8967
0.5817
0.8239
0.9981
0.9959
0.8275
0,8652
0,9468
Coffee
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
Computers
0.7880
0.7488
0.6328
0,7880
0.7320
0.7160
0.7720
0.7880
0.7936
CricketX
0,5708
0.7046
0,4626
0.7451
0.8046
0.7923
0.8164
0,8010
0.8036
CricketY
0,5759
0.7097
0.4492
0.8231
0.8364
0.8195
0.8421
0.8313
0.7887
CrickelZ
0.5785
0.7077
0.5005
0.8354
0.8067
0.8026
0.8467
0.8046
0.8097
Crop
0.5739
0.7528
0.6107
0.7608
0.7551
0.7205
0.7543
0.7427
0.7238
DiatomSiizRReduction
0,9510
0.9549
0.9281
0.9837
0.8856
0.9301
0.9673
0.9190
0.9843
DisaaPhallanxOOutiineggGGGCo
0.7482
0.7813
0.7281
0.7511
0.7626
0.7525
0.7410
0.7237
0.7612
DistalPhalanx Outilaannnnoninnnneeeeee
0.7681
0.7899
0.7326
0.7746
0.7855
0.7739
0.7645
0.7739
0.7746
DistalPhalanxTW
0,7079
0.7079
0.6763
0.6978
0.7036
0.7022
0.6777
0.6863
0.7396
DodgerLoopDay
0,4375
0.5750
0.3750
0.5175
0.5500
0.4375
0.5925
0,5900
0.5875
DodgerL.ooppaagg
0.7971
0.8130
0.4942
0.8551
0.9188
0.8333
0.8406
0.8725
0.9304
Dodgerl.oopWWeeenn
0.8768
0.9072
0.7333
0.9565
0.9710
0.9594
0.9710
0.9725
0.9826
Earthquakes
0.7597
0.7770
0.7482
0.7410
0.7482
0.7295
0.7424
0.7367
0.7568
ECG200
0,8900
0.9100
0.7900
0.9000
0.8800
0.8660
0.9100
0,9140
0.8980
ECG5000
0.9292
0,9412
0.9344
0.9398
0.9471
0.9472
0.9460
0.9437
0.9456
ECGFiveDays
0.9895
0.9965
0.5626
1.0000
1.0000
0.9981
1.0000
1.0000
1.0000
ElectrieDevices
0.7189
0.7480
0.6782
0.7189
0.7622
0.7325
0.7268
0.7388
0.7387
EOGHorizontaliignaal
0,5171
0,5945
0.4066
0.5663
0.6343
0.5713
0.6177
0,5707
0.6155
EOGVerticalSignal
0.3956
0.5459
0.3436
0.4762
0.5260
0.4718
0.5365
0,5110
0.5271
EthenolLeevel
0.4736
0.6420
0.3028
0.7700
0.7076
0.5668
0.5868
0.5728
0.6036
FaceAll
0,8214
0.7937
0.6112
0.8130
0.7934
0.8155
0.9462
0.8046
0.9140
FaceFour
0.6500
0.9227
0.5114
0.9659
1.0000
0.8932
0.9773
0.9773
0.9341
FaCesUCR
0.7999
0.9088
0.5637
0.9634
0.9565
0.9560
0.9611
0.9546
0.9410
Fifty Words
0,4677
0.7833
0,4958
0.8044
0.7820
0.8325
0.8290
0.8316
0.7776
Fish
0,8297
0.9097
0.8069
0.9851
0.9874
0.9851
0.9829
0.9623
0.9646
FordA
0.9117
0.9427
0.7211
0.9524
0.9465
0.9591
0.9353
0.9464
0.9561
FordB
0.7301
0.8032
0.6415
0.8432
0.8331
0.8175
0.7993
0.8123
0.8398
FreezoRRegularTrain
0.9790
0,9920
0.9830
0.9959
0.9979
0.9949
0.9973
0.9975
0.9959
FreezerSmallTain
0.9204
0,8575
0.7862
0.8406
0.9945
0.8640
0.9478
0.9648
0.9254
Fungi
0.9677
0.9366
0.8817
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
GestureMidAirD!!
0.4615
0.6769
0.5154
0.7385
0.6154
0.7585
0.7231
0.6938
0.7600
GestureMidAirD22
0.6308
0.6538
0.6000
0.6846
0.6154
0.6585
0.6692
0.6338
0.6877
GestureMidAArrD3
0.3477
0.3862
0.2154
0.3923
0.4077
0.4831
0.3938
0.3800
0.4615
GesturePebble211
0.7047
0.8791
0,5872
0.9291
0.9070
0.9012
0.9070
0.9093
0.8837
GesturePehble72
0.6342
0.8241
0.6203
0.8633
0.8861
0.8696
0.8241
0.8190
0.8266
GunPoint
0,9533
0.9867
0.7813
0.9933
1.0000
1.0000
1.0000
0.9960
1.0000
GunPointA Aeeann
0.9557
0.9842
0,9304
0.9937
0.9968
1.0000
0.9968
0.9949
0.9956
MaluPinntaaeeeessseeeelllll VersusFemale
0.9937
0,9968
0,9918
0.9968
0.9968
0.9968
0.9968
0.9968
1.0000
GunPointOld Versus Young
0.9905
0.9346
0.9752
0.9937
0.9968
0.9924
0.9879
0.9937
0.9981
Ham
0.6667
0.6876
0.6095
0.7810
0.6743
0.7067
0.7143
0.7086
0.7505
HandOutlines
0,8141
0.9351
0.8357
0.9508
0.9324
0.9416
0,9405
0,9427
0,9470
Haptics
0.4714
0.5266
0.4169
0.5519
0.5675
0.5078
0,5266
0,5500
0.5773
Herring
0.6250
0.6281
0.5938
0.7031
0.6344
0.7313
0.5938
0.6594
0.6188
HouseTwenty
0.9664
0.8756
0.7983
0.9496
0.9832
0.9580
0.9580
0.9445
0.9630
InatneSkate
0.4113
0.4658
0.2658
0.4818
0.4796
0.4873
0.4662
0.4345
0.5044
Table 5: Summary of Accuracy Results Across 128 Datasets.
13

---

SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
Dataset
rmESN
Conv.
Times.
Incep.
COTE
Hydra
Rocket
Mini.
SARC
InseotEPGRegularTraa
1.0000
1.0000
0.9960
1.0000
0.9992
1.0000
1.0000
0,9952
1.0000
InsactiPGSmallTraii
0.9197
0.9598
0.9044
0.8594
0.9671
0,9614
0.9799
0,9438
0.9831
InseetWigbbaatSound
0.3904
0.6563
0,4276
0.6157
0,6553
0.6503
0.6599
0.6663
0.6542
IaalyooweDDemandd
0,9592
0.9679
0.9372
0.9673
0.9670
0,9656
0.9689
0.9668
0.9668
LangeKicchenApplianess
0.8693
0.7520
0.6912
0.8805
0.7573
0.8635
0.8965
0.8491
0.8901
Lightning2
0.7869
0.7705
0.7377
0.8197
0.7639
0.7377
0.7738
0.7443
0.8197
Lightning?
0.7397
0.7671
0.7123
0.7452
0.7507
0.7753
0.8082
0.6849
0.8274
Mallat
0.9494
0.9441
0,9034
0.8936
0.9719
0,9485
0.9557
0.9357
0.9597
Meat
0,9500
0.9367
0.9167
0.9300
0.9000
0.8933
0.9500
0.9667
0.9800
Medicallmages
0.7961
0.7924
0.7221
0.7950
0.7587
0.7553
0.7911
0.8053
0.7511
MelhournePodestine
0,8731
0.8891
0,8304
0.9117
0.8921
0.8804
0,9031
0.8891
0.8815
MiddlcPhalanx Quttlai A
0,5455
0.5338
0.6234
0.5519
0,5740
0.5909
0,5532
0,5468
0.6273
MiddlePhal PhidllePhalanxOOuiinCCCCCCC Onnrert
0.8213
0.8014
0.7567
0.8275
0.8488
0.8275
0.8289
0.8419
0.8474
0,5831
0.5519
0.5974
0.5065
0.5818
0.5299
0.5403
0.5338
0.5805
MixedShapesRegularTrii
0,9089
0.9264
0.8268
0.9563
0.9742
0.9801
0.9676
0.9720
0,9494
MixedShapesSmallTTaa
0,8829
0.9037
0.7238
0.9039
0.9533
0.9620
0.9347
0.9492
0.8934
MoteStrain
0.6589
0,8922
0,7059
0.8778
0.9593
0,9462
0,9152
0.9321
0.8997
0,9077
0.9297
0,8031
0.9376
0.9296
0,9440
0.9524
0.9464
0.9539
DoalivasiveFeuallCCGTooaaaa
0,9223
0.9484
0.8494
0.9532
0.9469
0.9576
0.9656
0,9651
0,9595
OliveOil
0,8333
0.7400
0,8333
0.8333
0.8933
0.9333
0,9000
0,9333
0.9600
OSULeaf
0.8438
0.8372
0,4463
0.9215
0.9942
0.9876
0.9347
0.9603
0.8248
PhalangeSOutinesCCree
0.7858
0.7932
0.7744
0.8399
0.8317
0.8189
0.8275
0.8310
0.8235
Phoneme
0.3189
0.3315
0.1800
0.3204
0.3783
0.3205
0.2759
0.2897
0.3387
MickupGesureWinmoeeZ
0.7600
0.7600
0,5400
0.6160
0.7160
0.8280
0.8400
0.6800
0.8160
0,2308
0.1779
0.1356
0.5529
0.7788
0.7519
0.3913
0,8702
0.3750
PigArtPressuree
0,8712
0,8577
0.1779
0,9952
0.9856
0,9856
0.9519
0,9808
0.9837
PigCVP
0.7462
0.8817
0.1260
0.9173
0.9663
0.9423
0.9327
0.9096
0.9029
PLAID
0.8719
0.7568
0.7654
0.9128
0.8466
0.8648
0.8819
0.8879
0.9084
Plane
1.0000
0.9905
0.9714
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
PowerCons
0,9444
0.9467
0,8733
0.9444
0,9589
0.9511
0.9333
0.9733
0.9467
0,8654
0,8439
0,8449
0.8107
0.8439
0.8537
0.8449
0.8459
0.8634
ProximalPlalannOuuliCCCoo
0,8729
0.8873
0.8351
0.9107
0.8790
0.8942
0.9003
0.8880
0.9003
ProximalPhalanxTW
0,8000
0.8195
0.7912
0.7366
0.8195
0.8039
0.7912
0.7922
0.8117
RefrigerationDevices
0,5557
0.5253
0,5637
0.5387
0.5403
0.5013
0,5141
0,5152
0.5749
Rock
0,8200
0,8440
0.5000
0.8200
0.9720
0.8600
0.9000
0,8400
0.8560
ScreenType
0.5691
0.5104
0.4517
0.5653
0.4800
0.4736
0.4677
0.5035
0.5883
SemgHandGGenderCh2
0.8540
0.8763
0.7523
0.8687
0.9617
0.8587
0.9223
0.8870
0.8967
SemgHanddovemeentee
0,5680
0.5680
0.4280
0.4813
0.7840
0.4609
0.6298
0.5591
0.6036
SemgHHandSubeectCua
0.7698
0.7564
0.6938
0.7844
0.9356
0.8009
0.8676
0.8689
0.8884
ShakeGesture WiimoteZ
0,8400
0.7840
0.7600
0.8000
0.7720
0. 0.9000
O. 0.9000
0.8840
0.9040
ShapeletSim
0.9722
0.8144
0.6333
0.9889
1.0000
0.9833
1.0000
1.0000
1.0000
ShapesAll
0.6930
0.8120
0.6457
0.9300
0.9380
0.9273
0.9033
0.9203
0.8713
0.7707
0.7424
0.7205
0.7611
0.8133
0.8037
0.8091
0.8011
0.7675
SmoothSubspace
0.7733
0,9000
0,8200
0.9667
0.9773
0.9333
0.9773
0.9507
0.8880
SonyAAlBORobouSurfaeee
0.9408
0,8812
0.8106
0.9334
0.8439
0.9185
0.9158
0.8892
0.9464
SonyAlBBORobotSufacce
0.9412
0.8959
0.7142
0.9234
0.9383
0.9467
0.9196
0.9037
0.9324
StarLightCurres
0.9589
0.9189
0.9292
0.9777
0.9795
0.9814
0.9806
0,9812
0.9813
Strawberry
0,9514
0,9827
0,9200
0.9768
0.9751
0.9719
0.9811
0.9789
0.9757
S SwedishLeaf
0.9347
0.9274
0,8531
0.9616
0.9507
0.9798
0.9616
0.9648
0.9533
Symbols
0,8400
0.9254
0,8868
0.9739
0.9727
0.9791
0.9729
0.9783
0.9574
SyntheticControo
0.7033
0.9967
0.9800
0.9980
1.0000
0.9767
1.0000
0.9920
0.9993
ToeSegmentation
0.9211
0.8526
0.6754
0.9693
0.9737
0.9632
0.9518
0.9658
0.9518
ToeSegmentation2
0.9077
0.9154
0.8154
0.9308
0.9538
0.9108
0.9231
0.9077
0.9185
Trace
1.0000
0,9800
0,8900
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
TwoLeadECG
0.9991
0.9473
0.7034
0.9965
0.9939
0.9991
0.9991
0,9982
0.9991
TwoPatterns
0,5100
0.9958
0.8008
1.0000
0.9975
0.9858
1.0000
0.9961
0.9961
UMD
0,8403
0.9722
0.9861
0.9653
0.9931
0,9889
0.9931
0.9931
0.9944
U WaaeGestuurLLbbrayy
0.7085
0.9515
0.6289
0,9370
0,9674
0.9706
0.9739
0.9691
0.9523
U MaveGeestureLibrary
0.6602
0.7286
0.6640
0.8110
0.8130
0.8575
0.8557
0.8510
0.8211
"aaveGestureLibrrryy
0.6884
0.6304
0.6266
0.7499
0.7367
0.7931
0.7772
0.7740
0.7452
LWaverestureLiirraryy
0.6912
0.6940
0.6116
0.7552
0.7633
0.7992
0.7950
0.7968
0.7697
Table 5: Summary of Accuracy Results Across 128 Datasets.
14

---

SARC: Spectral-Aware Reservoir Computing for Fast and Accurate Time Series Classification
Dataset
rmESN
Conv.
Times.
Incep.
CОTЕ
Hydra
Rocket
Mini.
SARC
Wafer
0,9904
0.9972
0.9883
0.9978
0.9997
1.0000
0.9979
0,9993
0.9978
Wine
0.7778
0.7444
0.8148
0.7222
0.8481
0.8815
0.7815
0.8667
0.8667
WordSymonyms
0,5185
0.6890
0,4875
0.7132
0.6994
0.7382
0.7467
0.7251
0.7213
Worms
0.7403
0.6909
0.5974
0.7792
0.7117
0.7325
0.7273
0.6831
0.7896
WormsTwo
0,8052
0.7662
0.6234
0.7792
0.8182
0.7610
0.8052
0.7532
0.8286
Yoga
0.8451
0.8591
0.7899
0.9050
0.9203
0.9210
0,9066
0.9137
0.8793
Table 5: Summary of Accuracy Results Across 128 Datasets.
15


Reviews:
[{"review_id": "j7NM9awgjM", "reviewer": "Reviewer_wNEC", "summary": "The paper proposes SARC, a novel framework that combines spectral analysis with RNN-based models for time series classification. SARC identifies prominent frequencies and captures corresponding temporal dynamics. The framework analyzes time series on multiple scales, achieving state-of-the-art accuracy on the full UCR archive and showing exceptional speed.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer’s efforts and acknowledgement. Below, we provide detailed responses to your concerns:\n\n**1. Comparison by Sequence Length**\n\nIn the table below, we group 128 datasets by sequence length and report the average rank of different methods across each group. The results show that SARC achieves the best average rank of 2.475 in the (500, 1000] case. In contrast, Rocket performs best in the (200, 500] range, while COTE ranks first in the remaining two groups. Despite this, SARC attains at least the second-best performance across all cases, demonstrating consistent competitiveness.\n\n---\n| Length | Counts | rmESN | Conv. | Times. | Incep. | COTE | Hydra | Rocket | Mini. | SARC |\n|-----|-----|-------|----------|----------|---------------|---------|---------|--------|------------|--------|\n| (0, 200] | 41 | 6.6098 | 5.5366 | 8.1707 | 4.5610 | 3.6463 | 4.4878 | 4.0000 | 4.3171 | 3.6707 |\n| (200, 500] | 45 | 7.3111 | 6.1222 | 8.5667 | 4.1667 | 4.1333 | 4.0333 | 3.3000 | 3.9556 | 3.4111 |\n| (500, 1000] | 20 | 5.3750 | 5.6500 | 8.0250 | 4.8250 | 4.7250 | 4.5500 | 4.2000 | 5.1750 | 2.4750 |\n| (1000, 2844] | 22 | 6.9773 | 5.8636 | 8.8182 | 4.9091 | 2.2955 | 4.0909 | 4.0909 | 4.5909 | 3.3636 |\n---\n\n**2. SARC’s Iteration Based on BiESN**\n\nSARC's iteration based on BiESN has been introduced through Equations (6)-(8) and footnote 3 of the main paper. For further clarification, we formally describe the computation of FreqRes based on BiESN.\n\nGiven an input time series $\\mathbf{u}$ of length $L$, a specific frequency $f$, and $p = [L/f]$, this FreqRes module iterates bidirectionally as follows:\n\n- Forward hidden state update: \n$$ \\mathbf{x}^{\\text{f}}(n) = \\tanh \\left(\\mathbf{W}\\_{\\text{s}}^{\\text{f}}\\mathbf{x}^{\\text{f}}(n-1) + \\mathbf{W}\\_{\\text{c}}^{\\text{f}}\\mathbf{x}^{\\text{f}}(n-p) + \\mathbf{W}\\_{\\text{in}}^{\\text{f}}\\mathbf{u}(n) \\right),$$\n\n- Backward hidden state update: \n$$\\mathbf{x}^{\\text{b}}(n) = \\tanh \\left(\\mathbf{W}\\_{\\text{s}}^{\\text{b}}\\mathbf{x}^{\\text{b}}(n+1) + \\mathbf{W}\\_{\\text{c}}^{\\text{b}}\\mathbf{x}^{\\text{b}}(n+p) + \\mathbf{W}\\_{\\text{in}}^{\\text{b}}\\mathbf{u}(n) \\right),$$\n\n\nHere, superscripts f/b denote forward/backward directions; $\\mathbf{x}^{\\text{f}}(n) = 0$ for $n \\leq 0$; and $\\mathbf{x}^{\\text{b}}(n) = 0$ for $n > L$.\n\nAfter computing all hidden states, the states at the same time step are concatenated as: $\\mathbf{x}(n) = \\begin{bmatrix} \\mathbf{x}^{\\text{f}}(n) \\\\\\\\ \\mathbf{x}^{\\text{b}}(n) \\end{bmatrix}$, which is then used to compute dynamic features as described in Equations (7) and (8).\n\n**3. Discussion on SARC's Efficiency**\n\nWe attribute SARC’s high efficiency to three main factors:\n\n- Based on reservoir computing, SARC eliminates the need for iterative training and time-consuming gradient backpropagation.\n- Different FreqRes modules operate independently, enabling efficient parallel processing.\n- FreqRes maintains nearly linear complexity with respect to data size, ensuring scalability.", "claims_and_evidence": "Main claims of the paper, including the listed contributions, complexity analysis, and design tricks, are supported by reasonable evidence.", "methods_and_evaluation_criteria": "The proposed method uses a novel reservoir-based RNN network to analyze cyclical patterns at multiple scales, achieving high accuracy and very fast speed for TSC. The utilized UCR datasets provide diverse characteristics, and comparisons include highly efficient MiniRocket. Statistical tests (Wilcoxon Test, CD diagrams) further enhance evaluation reliability.", "theoretical_claims": "The theoretical claims involving complexity analysis and are proven correct.", "experimental_designs_or_analyses": "The experimental design is targeted and thorough, supporting main claims and showcasing the proposed method's effectiveness. It compares several relevant reservoir computing methods and state-of-the-art approaches. The analyses are detailed, evaluating accuracy, F1-score, and time across multiple dimensions.", "relation_to_broader_scientific_literature": "The paper adopts a multi-periodicity perspective to analyze time series, which may be influenced by prior work such as [1]. The difference is that it employs an ensemble of reservoir-based models to capture multi-scale temporal dynamics. This architecture is rare in the current literature and could offer new perspectives for time series modeling.\n[1] Wu et al. \"TimesNet: Temporal 2D-variation modeling for general time series analysis.\" ICLR 2023.", "essential_references_not_discussed": "I believe the current references are sufficient. Given the paper's emphasis on efficiency, excluding some large models like RDST and HC2 is acceptable.", "other_strengths_and_weaknesses": "Strengths:\n1. The algorithm's lightweight nature is a key highlight, especially for architectures comprising multiple recurrent structures. This encourages further exploration of reservoir computing.\n2. Technical details are clearly presented, making the reservoir model's properties easy to grasp.\n3. The proposed dynamic features break classification into forecasting + classification, suggesting a new pathway to leveraging forecasting models for classification.\n4. The architecture emphasizes accurate representation of time series, showing potential for broader applications like forecasting and imputation.\n\nWeaknesses:\n1. While described in footnotes and appendix, how the framework adapts to different RC models is not entirely intuitive.\n2. Analyzing performance on varying lengths could further enhance evaluation.\n\nDespite minor issues, I find this paper acceptable and capable of inspiring new ideas.", "other_comments_or_suggestions": "None.", "questions_for_authors": "1. Please provide formulas to further clarify SARC's iteration when based on BiESN, which is the primary model evaluated in the experiments.\n2. Since high efficiency is a core contribution, how does SARC achieve such performance?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1JF5FvudJ9", "reviewer": "Reviewer_uL8Q", "summary": "The paper enhances reservoir computing (RC) for TSC by incorporating spectral insights. It extracts prominent frequencies to identify cyclical patterns and refines RC module to capture cyclical dynamics. Features from next-step prediction tasks are used for classification. Extensive experiments on the UCR 128 archive show state-of-the-art performance with superior efficiency, highlighting the potential of RC in time series modeling.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are thankful for the reviewer’s recognition of the work. The processing of datasets with variable lengths or NaN values is clarified as follows:\n\n- For middle NaN values (i.e., real values exist on both sides of the NaN), we employ the interpolation for imputation.\n- Then, we shift real values in each time series sample to the rightmost positions of the tensor.\n- Finally, the rest leading NaNs are filled with zeros.\n\nShifting real values to the right is critical due to the recurrent nature of FreqRes. Specifically, this ensures that the initial hidden states start as zeros and transition to normal iteration when real values appear. In addition, the leading zero states do not affect subsequent ridge regression computations.", "claims_and_evidence": "The paper is rigorous and main claims are well-supported: \n1) Superior performance is shown by comparing with SOTAs across full UCR benchmarks.\n2) High efficiency is validated via time and complexity analysis.\n3) Ablation studies confirm the effectiveness of spectral insights and adaptability to various RC models.\n4) Other minor design and claims are supported by reasonable explanations and experiments.", "methods_and_evaluation_criteria": "The proposed method and evaluation criteria are suitable for TSC. Combining spectral analysis with RC is novel and addresses long-term dependencies effectively. The refined RC module, along with specially designed features, enables fast and accurate classification. The UCR 128 archive and strong baselines like mini rocket and cote ensure solid validation.", "theoretical_claims": "Time complexity analysis appears correct based on appendix proofs, showing linear complexity with respect to sample size and length, supporting the efficiency claim.", "experimental_designs_or_analyses": "The experimental design is sound and well-structured, with rich datasets and comprehensive baselines. Diverse metric indicators allow convincing comparisons. Additionally, extensive ablation studies and hyperparameter analyses support the claims, and the provided code aids reproducibility. Only a minor clarification is needed on data preparation (see Questions).", "relation_to_broader_scientific_literature": "RC techniques have been applied to TSC, but often with conventional structures. The paper advances RC by integrating spectral insights, aligning with trends in time series modeling, such as multi-scale and periodicity analysis. This design improves accuracy while maintaining high efficiency. Notably, the proposed method offers a generalizable extension to various RC models, representing an significant  advancement in the field.", "essential_references_not_discussed": "Essential works in both the RC and TSC fields are discussed.", "other_strengths_and_weaknesses": "Besides the above, there are two notable strengths:\n1) The derivation of dynamic features is interesting, using the predictive model as a sequence representation. This idea could inspire further cross-task research.\n2) The writing is a strong point with high readability. Motivation is well-justified, and technical details are clear. Each step is explained with its purpose, ensuring rigor and logical flow.\nThe paper looks promising in many aspects, and I found no obvious weaknesses.", "other_comments_or_suggestions": "I have no other comments.", "questions_for_authors": "Data preparation: Some UCR datasets have variable lengths or NaN values. They can greatly affect the performance of RC-based and recurrent models. How were they processed in the experiments?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "d9xnfoIh6q", "reviewer": "Reviewer_X5Kk", "summary": "Typical reservoir computing (RC) considers recursive updates from adjacent states and has difficulty handling long-term dependencies. For this issue, this paper proposes a Spectrum-Aware Reservoir Computing framework (SARC) that incorporates spectral insights to enhance long-term dependency modeling. Prominent frequencies are first extracted to reveal explicit or implicit periodic patterns. For each prominent frequency, SARC further integrates a frequency-informed reservoir network (FreqRes) to fully capture the sequential and periodic dynamics, thereby deriving effective dynamic features. By synthesizing these features at different frequencies, SARC provides a multi-scale analysis of temporal dynamics and improves the modeling of long-term dependencies. Extensive experiments on pubic datasets demonstrate that SARC achieves state-of-the-art results.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s efforts and recognition. Our responses to the raised concerns are listed below:\n\n**1. Mathematical Explanation of FreqRes**\n\nWe formally define a general FreqRes, and then give some theoretical claims for ESN-based FreqRes, as described in Eq. (6).\n\n**(1) Definition of FreqRes**\n\nLet the following recursive process define a general RC model:\n$$\n\\begin{aligned}\n\\mathbf{\\hat{x}}(n) &= \\mathbf{W} \\mathbf{x}(n-1), \\\\\\\\\n\\mathbf{x}(n) &= \\sigma\\left(\\mathbf{\\hat{x}}(n), \\mathbf{u}(n)\\right),\n\\end{aligned}\n$$\nwhere $\\mathbf{x}(n)$ is the hidden state; $\\mathbf{u}(n)$ is the input; $ \\mathbf{W}$ is the reservoir weight matrix; and $ \\sigma(\\cdot) $ is the activation.\n\nThe corresponding FreqRes with a frequency $f$ redefines its recursive process as:\n$$\n\\begin{aligned}\n\\mathbf{\\hat{x}}(n) &= \\mathbf{W}\\_{\\text{s}} \\mathbf{x}(n-1) + \\mathbf{W}\\_{\\text{c}} \\mathbf{x}(n-p), \\\\\\\\\n\\mathbf{x}(n) &= \\sigma\\left(\\mathbf{\\hat{x}}(n), \\mathbf{u}(n)\\right).\n\\end{aligned}\n$$\nHere, $L$ is series length ; $p = [L/f]$ is cycle length; $ \\mathbf{W}\\_{\\text{s}}$ and $\\mathbf{W}\\_{\\text{c}}$ are weight matrices modulating sequential and cyclical dependencies. \n\n**(2) Lipschitz Continuity Analysis**\n\nTo analyze state stability, we treat the input influence $\\mathbf{W}\\_{\\text{in}} \\mathbf{u}(n)$ as fixed.\n\n**Claim 1:** The Lipschitz constant of state update function for the ESN-based FreqRes is bounded by $\\sqrt{ \\Vert\\mathbf{W}\\_{\\text{s}}\\Vert_2^2 + \\Vert\\mathbf{W}\\_{\\text{c}}\\Vert_2^2 }$. Thus, $\\Vert\\mathbf{W}\\_{\\text{s}}\\Vert_2^2 + \\Vert\\mathbf{W}\\_{\\text{c}}\\Vert_2^2 < 1$ ensures a contraction mapping and guarantees state stability.\n\nProof sketch：\n\nNoting that $\\tanh$ is 1-Lipschitz and applying the Cauchy-Schwarz inequality:\n$$\n\\begin{align}\n\\Vert \\mathbf{A} \\mathbf{x}\\_{1} + \\mathbf{B} \\mathbf{x}\\_{2} - \\mathbf{A} \\mathbf{x}\\_{1}' - \\mathbf{B} \\mathbf{x}\\_{2}'\\Vert\\_{2}\n& \\leq \\Vert \\mathbf{A}\\Vert\\_{2} \\Vert(\\mathbf{x}\\_{1} - \\mathbf{x}\\_{1}')\\Vert\\_{2} + \\Vert\\mathbf{B}\\Vert\\_{2} \\Vert(\\mathbf{x}\\_{2} - \\mathbf{x}\\_{2}')\\Vert\\_{2}\\\\\\\\\n& \\leq \\sqrt{\\Vert \\mathbf{A}\\Vert\\_{2}^{2} + \\Vert \\mathbf{B}\\Vert\\_{2}^{2}}\\sqrt{\\Vert\\mathbf{x}\\_{1} - \\mathbf{x}\\_{1}'\\Vert\\_{2}^{2} + \\Vert\\mathbf{x}\\_{2} - \\mathbf{x}\\_{2}'\\Vert\\_{2}^{2})}\n\\end{align}\n$$\n\n**(3) Discussion on Echo State Property**\n\n**Claim 2:** Given $p = [L/f]$, an ESN-based FreqRes with reservoir size $s$ satisfies the ESP  $\\iff$ a specially structured ESN with reservoir size $ps$ does.\n\nProof sketch：\n\nLet $\\mathbf{U}(n) = [\\mathbf{u}(n); \\mathbf{u}(n-1); \\dots; \\mathbf{u}(n-p+1)]$, and $\\mathbf{X}(n) = [\\mathbf{x}(n); \\dots; \\mathbf{x}(n-p+1)]$.\n\nThe state update could be rewritten as: $$\\mathbf{X}(n) = \\sigma\\left(\\begin{bmatrix}\n\\mathbf{W}\\_{\\text{s}} & \\mathbf{0} & \\mathbf{W}\\_{\\text{c}} \\\\\\\\\n\\mathbf{I}\\_{s} & \\mathbf{0} & \\mathbf{0}\\\\\\\\\n\\mathbf{0} & \\mathbf{I}\\_{(p-2)s} & \\mathbf{0}\\\\\\\\\n\\end{bmatrix}\\mathbf{X}(n-1) + \\begin{bmatrix}\n\\mathbf{W}\\_{\\text{in}} &  &  \\\\\\\\\n & \\mathbf{0}\\_{s} & \\\\\\\\\n &  & \\mathbf{0}\\_{(p-2)s}\\\\\\\\\n\\end{bmatrix}\\mathbf{U}(n)\\right)$$\n\nHere, $\\sigma$ is 1-Lipschitz, applying $\\tanh$ to the first $s$-dimensional part and identity function to the rest, which aligns with the common assumptions used in ESP discussions. Thus, existing theories can be applied to derive constraints on $\\mathbf{W}\\_{\\text{s}}$ and $\\mathbf{W}\\_{\\text{c}}$.\n\n**(4) Hyperparameter Analysis**\n\nDue to space constraints, we briefly discuss spectral radius (SR) and input scaling (IS) choices. Based on the above analysis, restricting SR $< 1$ is a necessary condition for state stability, justifying the range selection in Fig. 8b. For IS, Fig. 7a shows that overly small values markedly weaken input influence and degrade performance. With further validation, we find that excessively large IS also reduce performance, likely because states activated by $\\tanh$ saturate at $\\pm 1$, as weights and historical states remain bounded.\n\n**2. Comparison with Spectral-Based Methods**\n\nWe compare SARC with Fourier or wavelet-based methods in accuracy using results from their original papers. The table below shows SARC's consistent outperformance in pairwise comparisons, highlighting its competitiveness in spectrum-based TSC. \n\nThese methods include:\n- **AMSWR('21):** Learnable multi-scale wavelet decomposition via adaptive CNNs. DOI: 10.3390/info12060252\n- **SFCC('22):** Stratified frequency recombination for data augmentation to enhance ResNet. DOI: 10.1007/s11063-022-10965-9\n- **TF-Net('22):** Two-stage fusion of ResNets trained on raw series and wavelet representations. DOI: 10.1007/s10489-022-03485-5\n- **CoInception('24):** Noise-resilient wavelet contrastive views with cross-view alignment. DOI: 10.1109/ICDM59182.2024.00041\n\n---\n| | AMSWR| SFCC|TF-Net|CoIncep.|\n|--|--|--|--|--|\n|#Dataset|85|128|85|128|\n|Avg. Acc. $\\uparrow$|2.15%|1.8%|0.53%|0.82%|\n|Win/Tie/Loss|60/4/21|76/6/46|43/3/39|71/6/51|\n|P-value|1.17E-05|3.61E-03|6.01E-01|5.75E-02|\n---", "claims_and_evidence": "The claim that SARC provides multiscale analysis of temporal dynamics and improves modeling of long-range dependencies is supported by a large body of experimental evidence, including experimental results on multiple datasets. However, the theoretical analysis and description of the SARC framework could be clearer, especially in how long-term dependencies are captured.", "methods_and_evaluation_criteria": "The proposed SARC framework is suitable for handling long-term dependencies in RC. Besides, the comparison of SARC with existing RC methods helps confirm the effectiveness of SARC.", "theoretical_claims": "The theoretical claims about incorporating spectral insights to enhance the modeling of long-term dependencies are well-founded. However, the mathematical formulation of the frequency-informed reservoir network (FreqRes) could be more explicitly defined, especially about the convergence properties of its cyclical connections.", "experimental_designs_or_analyses": "The experiments are well designed and the benchmarks are clear. The comparison with eight state-of-the-art baselines and the use of statistical significance tests such as the Wilcoxon Signed-Rank Test strengthen the validity of the results.", "relation_to_broader_scientific_literature": "The paper positions itself within the broader context of time series classification and reservoir computing, comparing the Spectral-Aware Reservoir Computing (SARC) framework to traditional reservoir computing models and deep learning approaches. It highlights the advantages of integrating spectral analysis for improved long-term dependency modeling. However, a more comparison with other spectral-aware approaches would be appreciated.", "essential_references_not_discussed": "The paper cites key references, but would benefit from more discussion on the connection to other spectral analysis techniques in time series modeling, particularly those utilizing advanced wavelet transforms and adaptive frequency decomposition methods.", "other_strengths_and_weaknesses": "Strengths: \n1. This paper conducts comprehensive experiments on 128 UCR datasets to demonstrate the effectiveness and efficiency of the proposed SARC framework. Comparisons with multiple state-of-the-art methods further strengthen its empirical contribution. \n2. This paper clearly points out the limitations of existing reservoir computing methods in dealing with long-term dependencies and periodic patterns. The introduction of spectrum-aware modeling makes a meaningful contribution to time series classification.\nWeaknesses: \n1. Although this paper introduces the FreqRes component to enhance long-term dependency modeling, its mathematical formulation lacks sufficient clarity, especially in terms of theoretical guarantees.\n2. This paper does not fully compare its spectrum-aware approach with other frequency-domain methods. A more in-depth discussion will help position SARC in the broader field of spectrum-based machine learning techniques.", "other_comments_or_suggestions": "1. Provide a clearer definition of the frequency-informed reservoir network (FreqRes), especially its theoretical properties, stability, and impact on modeling long-term dependencies.\n2. Some hyperparameter choices (e.g., spectral radius, input scaling) appear to be empirically driven. A more systematic sensitivity analysis would enhance experimental validity.", "questions_for_authors": "1. Could you provide a more formal and detailed mathematical explanation of the frequency-informed reservoir network (FreqRes)? A clearer formulation would help strengthen the theoretical justification of the proposed method.\n2. How does SARC compare to other frequency-aware time series models, such as adaptive Fourier-based methods or alternative wavelet-based approaches? A more detailed discussion could better position SARC within the broader landscape of spectral-based time series classification.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FNni1bGPvn", "reviewer": "Reviewer_CNcc", "summary": "This paper introduces a novel method for time series classification. The approach begins by decomposing the time series into multiple prominent frequencies, each representing different cyclic patterns in the data. For each extracted frequency, a reservoir computing model—called FreqRes—is applied to generate complex hidden states. These hidden states are then used to produce individual readouts, which are subsequently concatenated to form a final representation for classification. Experiments on the UCR dataset demonstrate that this method is effective and outperforms competing approaches.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We express our gratitude to the reviewer for your time and recognition. Detailed responses to the raised concerns are listed below. We agree with the suggestions mostly and will clarify those details.\n\n**1. Impact of Spectral Insights across Different Models**\n\nThe table below provides a comparison of whether Spectral Insights (SI) are incorporated across various base models (in terms of accuracy). Specifically, Row 4 reports the win/tie/loss counts of \"w/ SI\" version against \"w/o SI\", and the p-values between them are from Wilcoxon tests. \n\nResults demonstrate that incorporating SI achieves average accuracy improvements ranging from 13.86\\% to 18.97\\% across all models. Statistics further confirm the significance of these improvements. This validates the consistent effectiveness of our approach across a broader range of models.\n\n---\n|   | ESN | LeakyESN | BiESN | DeepESN | LSTM | GRU |\n|--|--|--|--|--|--|--|\n| Avg. Acc. w/o SI | 0.6456 | 0.7030 | 0.7055 | 0.6752 | 0.6212 | 0.6295 |\n| Avg. Acc. w/ SI | 0.8353 | 0.8416 | 0.8508 | 0.8394 | 0.7896 | 0.7926 |\n| Wins/Ties/Losses | 122/1/5 | 117/0/11 | 119/3/6 | 117/2/9 | 115/0/13 | 114/0/14 |\n| P-value | 6.44E-22 | 3.79E-20 | 1.58E-21 | 1.05E-20 | 5.05E-20 | 2.18E-20 |\n---\n\n**2. The Need for Reservoir Computing**\n\nAs Eq. (6) shows, the FreqRes modules iterate on raw time series and implicitly utilize frequency information. Thus, an explicit frequency component for direct classification could be unavailable. In response to the reviewer's expectation, the following features in our experiments enable comparison and demonstrate the contribution of reservoir computing (RC): \n\n- $O$: Original time series data\n- $F_1$: For each $f \\in \\mathcal{F}$ and $p = [L/f]$, compute a differenced series of the raw sequence and concatenate statistical features, including mean absolute value, standard deviation, kurtosis, zero-crossing rate, and first-order autocorrelation coefficient.\n- $F_2$: For each $f \\in \\mathcal{F}$, extract frequency band components (centered at $f$, 10% relative bandwidth) and concatenate features, including relative band energy ratio alongside the above five statistical features.\n- $F_3$: Feature vectors composed of FFT amplitude values at each $f \\in \\mathcal{F}$.\n\nThe average accuracies using Ridge and Random Forest classifiers with these features are presented below. This comparison highlights the necessity of RC and FreqRes modules, as they achieve significantly better performance by capturing temporal dependencies at multi-scale frequencies.\n\n---\n|   | $O$ | $F_1$ | $F_2$ | $F_3$ | $O+F_1$ | $O+F_2$ | $O+F_3$ | $O+F_{1,2,3}$ | SARC |\n|--|--|--|--|--|--|--|--|--|--|\n| Ridge | 0.6475 | 0.7112 | 0.6322 | 0.6178 | 0.7270 | 0.6961 | 0.7048 | 0.7422 | 0.8508 |\n| RF | 0.7209 | 0.7281 | 0.6920 | 0.6610 | 0.7712 | 0.7680 | 0.7430 | 0.7894 | 0.8391|\n---\n\n\n**3. Extension to Multivariate TSC**\n\nThe proposed SARC is currently adapted to multivariate scenarios in a straightforward manner: it extracts required frequencies from the mean series across variables, then applies FreqRes modules to iterate on the raw multivariate series. This approach proves effective, as evidenced by the submitted supplementary materials, which contain a demo achieving ~98% accuracy on the \"ArticularyWordRecognition\" dataset from the UEA archive.\n\nAnother feasible extension involves separately extracting frequencies and modeling patterns for each individual variable before concatenating the derived features. While preserving variable-specific information, it would require more sophisticated implementations to ensure parallelization efficiency. Moreover, inter-variable interactions are unaddressed in this design. Further improvements could focus on the modeling of inter-variable relationships during feature extraction, and we plan to explore this in future studies.\n\n**4. Impact of Frequency Quantity**\n\nAs described in Section 4.1.2, the extracted frequency set $\\mathcal{F}$ consists of several \"root\" frequencies and their sub-harmonics. Figure 7(c) in the main paper has visualized the impact of threshold $\\kappa$ (which controls the number of sub-harmonics) on average accuracy across 128 datasets. Here, we offer an analysis regarding the impact of \"root\" frequency quantity.\n\nThe table below shows SARC's average accuracy when selecting and extending only the top-$\\alpha$ smallest \"root\" frequencies. Similar to the trend with $\\kappa$, SARC achieves better performance as $\\alpha$ increases (i.e., more frequencies are included). Notably, the number of extracted \"root\" frequencies is less than 10 for most datasets, resulting in identical accuracy when $\\alpha=9$ and $10$ on these datasets. This explains why the average accuracy growth gradually plateaus when $\\alpha$ approaches 10. \n\n---\n| $\\alpha=$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |\n|--|--|--|--|--|--|--|--|--|--|--|\n| Avg. Acc.| 0.6863 | 0.7641 | 0.7937 | 0.8140 | 0.8279 | 0.8373 | 0.8454 | 0.8503 | 0.8507 | 0.8508 |\n---", "claims_and_evidence": "**Key Claims and Comments**\n\n- *Incorporating Spectral Information into Echo State Neural Networks:*\nAblation studies in Section 5.3 (particularly Figure 8a) demonstrate that integrating spectral insights into BiESN enhances performance across most datasets, providing strong evidence of its effectiveness. However, it would be valuable to see similar analyses for other architectures, such as ENS, LeakyESN, DeepESN, and LSTM/GRU, to determine whether spectral insights benefit all models.\n\n- *Computational Efficiency and Performance:*\nFigure 5 presents a computational cost analysis, highlighting the method’s low computational overhead alongside a time complexity evaluation. Additionally, comparisons of F1-score and accuracy against multiple competitors show a performance improvement, reinforcing the effectiveness of the approach.", "methods_and_evaluation_criteria": "*Evaluation Criteria:*\nThe authors evaluate their method based on computational cost and several widely accepted metrics in the community, including F1-score, accuracy, Win/Tie/Loss analysis, and statistical significance testing using p-values. Additionally, the benchmarks used are standard, ensuring a fair comparison. However, a limitation is that the method only applies to univariate time series, whereas the main challenges often lie in the multivariate setting.\n\n*Proposed Method:*\nIncorporating spectral insights into time series analysis is a logical approach, as frequency-based representations are a well-established and widely used method for capturing time series patterns.", "theoretical_claims": "1- No theoretical claims in the paper.", "experimental_designs_or_analyses": "The experimental design is well-executed, with a comprehensive set of benchmarks (128 UCR datasets) used for comparison. The chosen metrics are appropriate, and p-values are computed to assess statistical significance. However, the authors should place greater emphasis on evaluating the impact of spectral insights—the main contribution of the paper. Specifically, they should demonstrate that these insights not only enhance performance beyond BiESN but also apply to other model variants.", "relation_to_broader_scientific_literature": "To the best of my knowledge, this is the first work to incorporate spectral insights into Echo State Neural Networks (ESNs), despite the existence of several studies leveraging frequency information for classification. In the specific context of ESNs, this approach appears to be the first in this direction, making it the key innovation of the paper", "essential_references_not_discussed": "No essential references that I know.", "other_strengths_and_weaknesses": "**Strengths:**\n- Extensive experimental studies\n- Well-written article with a clear motivation\n\n**Weaknesses:**\n- The method is limited to univariate time series classification.\n- A direct comparison should be made between applying classifiers (e.g., Ridge, Random Forest) directly to the extracted frequency components and using reservoir computing. This would clarify whether the reservoir computing truly adds value, especially given that the reservoir size does not appear to be a key factor. If the reservoir computing is not significantly improving performance, it may be unnecessary, as a direct classification approach could be more computationally efficient.", "other_comments_or_suggestions": "1- Need for incorporating the baselines of using classifiers directly on the concatenation of raw extracted from each frequency.\n2 Need to comment on multivariate extensions\n\n\n**After rebutal**\nI increase my score to 4 given the answer of the authors", "questions_for_authors": "1- Could the authors explain at least experimentally the need for the reservoir computing?\n2- Can the authors discuss the extension to multivariate time series classification?\n3- Could the authors make some experiments regarding the number of frequencies extracted ?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper makes a significant contribution to time series modeling through a method that uses "spectral insights to enhance long-term dependency modeling". 

The reviewers appreciated the novelty of the method, its broad applicability, the clarity of the presentation and the strength of the results. The reviewers also determined that the claims of the paper are well supported by the experiments, in particular the ablation studies.

Questions raised by the reviewers involving some of the technical details of the method were successfully addressed in the rebuttal. The authors also provided a comparison of the performance of different methods for timeseries of different lengths, an experiment motivating the need for reservoir computing, and a study on incorporating "spectral insights" into different base models.

Overall, the paper appears ready for publication.
Decision: Accept (poster)