HyperTime: Implicit Neural Representations for Time Series Generation
Authors: Elizabeth Fons
Alejandro Sztrajman
Yousef El-Laham
Alexandros Iosifidis
Svitlana Vyetrenko
Venue: Submitted to ICLR 2023 (2023)
Keywords: Time Series
Implicit Neural Representations
Time Series Generation

Abstract: Implicit neural representations (INRs) have recently emerged as a powerful tool that provides an accurate and resolution-independent encoding of data. Their robustness as general approximators has been shown in a wide variety of data sources, with applications on image, sound, and 3D scene representation. However, little attention has been given to leveraging these architectures for the representation and analysis of time series data. In this paper, we propose a new INR architecture for time series (iSIREN) designed to perform an accurate reconstruction of univariate and multivariate data, while also providing an interpretable encoding of the signal. We compare our architecture against SIREN and INRs with different activations, in terms of training convergence, and the reconstruction accuracy of both the signal and its spectral distribution. 
To achieve generalization, we propose a hypernetwork architecture (HyperTime) that leverages iSIRENs to learn a latent representation of an entire time series dataset. In addition to the traditional reconstruction loss, we introduce an FFT-based loss that guides the training by enforcing a good match of the ground truth spectral distribution. We show how these architectures can be used for time series generation, and evaluate our method through fidelity metrics, presenting results that exceed the performance of state-of-the-art techniques. Finally, we propose an alternative hypernetwork architecture (iHyperTime) that incorporates interpretability into the latent representation, enabling the introduction of prior knowledge by imposing constraints into the generation process.

Under review as a conference paper at ICLR 2023
HYPERTIME: IMPLICIT NEURAL REPRESENTATIONS FOR TIME SERIES GENERATION
Anonymous authors Paper under double-blind review
ABSTRACT
Implicit neural representations (INRs) have recently emerged powerful tool that provides an accurate and resolution-independe bustmess as general approximators has been shown in a wide variety of data sources, with applications on image, sound, and 3D scene representation. However, little attention has been given to leveraging these architectures for the representation and analysis of time series data. In this paper, we propose a new INR architecture for time series (iSIREN) designed to perform an accurate reconstruction of univariate and multivariate data, while also providing an interpretable encoding of the signal. We compare our architecture against SIREN and INRs with different activations, in terms of training convergence, and the reconstruction accuracy of both the signal and its spectral distribution. To achieve generalization, we propose a hypernetwork architecture (HyperTime)) that leverages iSIRENs learn a latent representation of an entire time series dataset. In addition to the traditional reconstruction loss, we introduce an FFT-based loss that guides the training by enforeing a good match of the ground truth spectral distribution. We show how these architectures can be used for time series generation, and evaluate our method through fidelity met- rics, presenting results that exceed the performance of state-of sfate-of-the-art techniques. Finally, we propose an alternative hypernetwork architecture (iHyper that incorporates interpretability into the latent representation, enabling the introduction prior knowledge by imposing constraints into the generation process.
encoding of data. Their ro-
INTRODUCTION
Modeling time series data has been a key topic of research for many years, constituting a crucial component in a wide variety of areas such as climate modeling, medicine, biology, retail and finance (Lim Zohren 202T). Traditional methods for time series modeling have relied on parametric models informed by expert knowledge. However, the development of modern machine learning methods has provided purely data-driven techniques to learn temporal relationships. In particular, neural network-based methods have gained popularity in recent times, with applications to wide range of tasks, such as time series classification (Ismail Fawaz et al.) 2020). clustering (Ma et al.) 2019
Alqahtani al. 2021), segmentation (Perslev al. 2019] Zeng et al. 2022), anomaly detection Choi
et al. Xu et al. 2018 Hundman et al. 2018). upsampling (Oh et al. 2020 Bellos et al.) 2019).
imputation Liu 2018 Luo al. 2018 Cao al. 2018). forecasting (Lim & Zohren [021] Torres
ct al. 2021 and synthesis (Alaa et al. 2021 Yoon et al. 2019b Jordon et al. 20119) In particular, generation of synthetic time series has recentl gained atlention due the large number of potential
applications in medical and financial fields, where data cannot be shared, either due to privacy reasons or proprietary restrictions (Jordon et al. 202T 2019 Assefa et al.) (020). Moreover, synthetic time series can be used to augment training datasets to improve model generalization on downstream tasks, such as classification (Fons al. 2021). forecasting and anomaly detection.
In recent years, implicit neural representations (INRs) have gained popularity as an accurate and flexible method to parameterize signals from diverse sources, such as images, video, audio and 3D scene data (Sitzmann et al.] 2020b) Mildenhall et al. 2020). Conventional methods for data encoding often rely on discrete representations, such as data grids, which are limited by their spatial resolution and present inherent discretization artifacts. In contrast, INRs encode data in terms of continuous functional relationships between signals, and thus are uncoupled to spatial resolution. In practical terms, INRs provide new data representation framework that is resolution-indeeenden

---

Under review as a conference paper at ICLR 2023
with many potential applications to time series, where missing values and irregularly sampled data common occurrences (Fang & Wang 2020). While there have been a few recent works exploring the application of INRs to time series data, there is virtually work leveraging these architectures for generating synthetic time series or producing interpretable time series representations (Jeong & Shin 2022] Woo et al. (202)) Applications with regulatory focus such finance often require transparency and interpretability of proposed machine learning solutions as well as injection of expert knowledge as constraints into the training process guide learning. For instance, explainable construction of trading agents is preferred when actions of trading agents need to be explicitly attributed to market signals (Vyetrenko & Xu 2019).
In this paper, we propose a novel methodology that utilizes INRs to encode and generate time series data based on interpretable latent representations. To the best of our knowledge, we are the first to incorporate an interpretable decomposition into the generation of time-series. Our contributions are as follows:
We introduce iSIREN, INR archi- tecture for multivariate time-series representation which provides an interpretable trend-seasonally decomposition of the data. We show that interpretability does not lead loss of reconstruction accuracy, and in some cases increases the spectral reconstruction quality. Moreover, we leverage hypernetwork for time-series generation via interpolation of learned embeddings.
Representation and Generation of time-series using INRs:
Spectral Loss:
To improve the training of the hypernetwork, we introduce a novel spectral loss that enforces the correct reconstruction of the signal's 's spectral distribution. We show that for some datasets this loss plays a crucial role in the learning process.
Interpretable time-series generation: We propose iHyper'Times hypernetwork architecture for time-series generation that learns a disentangled seasonal-trent representation time series, enabling the introduction of expert knowledge into the synthesis process. We compare iHyper'Time against current state-of-the-an methods for time-series generation, showing improved results in terms of standard fidelity metrics.
2 RELATED WORK
Implicit Neural Representations Implicit Neural Representations (INRs) provide continuous representation of multidimensional data, by encoding a functional relationship between input co- ordinates and signal values, avoiding possible discretization artifacts. They have recently gained popularity in visual computing (Mescheder development of positional encodings (Tancik et al. 2020) and periodic activations (SIREN (Sitzmann)
al. 2019 Mildenhall et al., 2020) due the key
(ct al. (020b)), which have proven to be critical for the learning of INRs have been shown to produce accurate reconstructions in wide variety of data sources, such as video, images and audio (Sitzmann et al., 2020b Chen et al. 2021 Rott Shaham et al.
details. Whilst
2021), few
works have leveraged them for time series representation (Jeong & Shin 2022 Woo al.
2022 and
none have focused on interpretability and generation.
Hypernetworks Hypernetworks neural network architectures that are trained to predict the parameters of secondary networks, referred to as Hyponetworks (Ha et al., 2017]: Sitzzann et al.) 2020a). In the last few years, some works have leveraged different hypernetwork architectures for the prediction of INR weights, in order to learn priors over image data (Skorokhodov et al., 2021) and 3D scene data (Littwin & Woll 2019) Sitzmann et al., 2019; Sztrajman et al. (022]). Sitzmann et al.) (2020b) leverage a set encoder and a hypernetwork decoder to learn a prior over SIRENs encoding image data, and apply for image in in-painting. Our HyperTime and iHyperTims architectures detailed in Section 3 Juee similar encoder-decoder structure, however they learn latent representation over our interpretable SIREN INRs (iSIREN), which encode time series data. Moreover, we apply these architectures for time series generation via interpolation of learned embeddings.
Time Series Generation Synthesis of time series data using deep generative models has been previously studied in the literature. Examples include the TimeGAN architecture (Yoon et al. 2019a), as well as QuantGAN (Wiese et al., 2020). More recently. Desai et al. (2021) proposed Time VAE variational autoencoder alternative to GAN-based time series generation. Alaa et al.) (202I) introduced Fourier Flows, a normalizing flow model for time series data that leverages the frequency domain representation, which is currently considered together with TimeGAN state-
2

---

Under review as a conference paper at ICLR 2023
of-the-art for time series generation. In the last few years, multiple methods have used INRs for data generation, with applications on image synthesis (Chan et al. 2021| | Skorokhodov et al. 2021).
super-resolution (Chen et al., 2021)) and panorama synthesis (Anokhin et al. 2021). However, there are currently no applications of INRs on the generation of time series data.
Interpretable Time Series Seasonal-trend decomposition is standard tool in time series analysis. The trend encapsulates the slow time-varying behavior of the signal, while seasonal components capture periodicity. These techniques introduce interpretability in time series, which plays an impor- tant role in downstream tasks such as forecasting and anomaly detection. The classic approaches for decomposition are the widely used STL algorithm (Cleveland al. 1990), and its variants (Wen et al., 2019; Bandara et al., (002)). Relevant to this work is the recent N BEATS architecture (Oreshkin al. 2020), deep learning-based model for univariate time series forecasting that provides inter- pretability capabilities. The model explicitly encodes seasonal-trend decomposition into the network by defining separate trend and seasonal blocks, which fit low degree polynomial and Fourier series. While N-BEATS provides interpretable decomposition, its applications do not cover multivariate time series and data synthesis. Desai al. (202I) recently proposed TimeVAE, combining VAE architecture with trend-seasonality decomposition structure allow for interpretable generation. However no results highlighting the advantages of this capability were demonstrated.
FORMULATION
In this Section, we describe the iSIREN network architecture for time series representation and interpretable decomposition, and the HyperTime and iHyperTime networks leveraged for prior learning and new data generation.
3.1 TIME SERIES REPRESENTATION
We consider time series signal encoded by a discrete sequence of N observations (y1, YN) where y, E R is the m--dimensional observation at time t,. This time series defines dataset D = {(4, ) of time coordinates associated with observations yi. We want to find J==1
continuous mapping f TR - I ~ ft) that parameterizes the discrete time series, so that f(4) y for i L...N. The function can be approximated by implicit neural representation (INR) architecture conditioned the training loss L -= Σ3 yi -(4)] The network is composed of fully-connected layers with sine activations (SIREN) as defined by Sitzmann et al. (20205):
xx(x.) sin(wo W W + b;)
where (-)-) corresponds to the jth layer of the network. general factor wo multiplying the network weights determines the order of magnitude of the frequencies that will be used encode the signal. Input and output of the INR are of dimensions and m, corresponding to the time coordinate and the prediction ft). After training, the network encodes continuous representation of the functional relationship ft) for a single time series.
3.1.1 INTERPRETABLE SIREN
Interpretable Hyponetwork (iSIREN)
Trend Block
Trend Black
Σ wiet
Rusonnaliyy Black
time (input)
Amplitude f(t)) (output)
Trend
SIREN
Figure I: Interpretable SIREN (iSIREN) architecture comprised of two blocks. Trend Blocl: FC layer that outputs polynomial fit of fixed degree. Seasonality Block: SIREN network.
3

---

Under review as a conference paper at ICLR 2023
We propose interpretable architecture to encode time series that reuses the INR with sine activation functions from the previous section. In particular, we assume that our INR follows a classic time series additive decomposition, i.c., f(t) = (t)(t) + fa(t),
(2)
where (t)(t) represents the trend component and f (t) represents the seasonal component of f(t), respectively. Note that this is standard assumption for time series decomposition techniques, such as STL and others. We elaborate on the structure of (t) and fa(t) in the following.
Trend block The trend component of a time series aims model slow-varying (and occasionally monotonic) behavior. Following the work by /Oreskki et al. (2020) (N-BEATS), for the trend component of the interpretable architecture, we consider polynomial regressor, i.e.,
[ir(()
Σ 1=0
(tr)
where p denotes the degree of the polynomial, and (tr) denotes the weight vector associated with the ith degree. In practice, p is chosen to be small (e.g., p = 2) to capture low frequency behavior.
Seasonality block The scasonal component of the time series ft) aims to capture the periodic behavior of the signal. In classical techniques, a pre-determined and fixed period (or set of periods) determines the resulting extracted seasonal components (c.g.. STL supports single seasonal compo- nent, whereas MSTL supports multiple). In this work, we propose to model the seasonal component of the time series SIREN with J/ layers (see Section[..]] Importantly, a single layer SIREN has been shown to be structurally similar to a Fourier mapped perception (FMP). An FMP h din + dout. a perceptron with identity activation and Fourier mapped input, i.e., for Rdd we have
h(x) WFS 7xx) b
7 (xx
(4)
sin 2
where WFS E dou X2M is a Fourier weight matrix, ES E Rdo and 7xx is a Fourier mapping of the input x, with B E Rddx the Fourier mapping matrix of M frequencies. For a particular choice of B and FS = 0, an FMP is equivalent to a Fourier series representation and the Fourier mapping (x) is equivalent
single layer SIREN with fixed Fourier mapping matrix (Benbarka et al. 2022). Fourier series is precisely what is used in the N-BEATS algorithm to model the seasonal component of the time series (Oreshkin al., 2020). The benefit of using SIREN is that the Fourier mapping becomes a trainable parameter - enabling the learning of the seasonal component of the time series in an unsupervised manner.
The training of iSIREN is performed in a supervised manner with an MSE reconstruction loss, as described for SIREN in Section 3.1 However, order stabilize the training, the process is performed in two stages: 1) we train the Trend Block for 100 epochs, computing the
Training of iSIREN
MSE loss between the ground truth time series and the output of the block: L1 = Ex || fir (t) This leads to a smooth approximation of the time series, which we use as initial guess for the second stage. 2) We then train the Trend and Seasonality blocks together, computing the MSE reconstruction
loss between the ground truth and the added output of both iSIREN blocks: L2 E, l1 f(t))ll.
3.2 TIME SERIES GENERATION WITH HYPERTIME
In Figure toop. we display a diagram of the HyperTime architecture, which allows us to leverage INRs learn priors over the space of time series. The Set Encoder (green network), composed of SIREN layers (Sitzmann et al. 2020b), takes as input a pair of values, corresponding to the time- coordinate and the time series signal (t). Each pair of values thus encoded into an embedding and evaluated by the HyperNet decoder (blue network), composed of fully-connected layers with ReLU activations (MLP). The output the HyperNet is one-dimensional vector that contains the network weights of an INR which encodes the time series data from the input. The INR architecture used within HyperTime is our interpretable SIREN (iSIREN), discussed in the previous section and illustrated in Figure [ However, in Section 4.2 will also consider the case of using SIREN as target INR. Following previous works (Sitzmann et al.) 2020a), in the context of hypemnetwrks we refer these predicted INRs as HypoNets in order to avoid ambiguities.
4

---

Under review as a conference paper at ICLR 2023
HyperTime
GT Tieze Strrie
mmv
Frod Timne Seriis
Baa
w
mMM
Ret Knrodar
Hyper Network
[t, ((0)
Speckeate
Interpretable HyperTime
Dieed Meighis
Z
Treed Mackk
M
w
MMM
fesacilty
Seamvaiity
fye
E(9))
desmcodity Weggls
Figure 2: Diagram of HyperTime network architecture. Each pair of coorciiae and time series f(t) is encoded as a 40-values embedding Z by the Set Encoder. The HyperNet decoder learns to predict HypoNet weights from the embeddings. During training, the output of the HyperNet is used to build HypoNet and evaluate it on in the input time-coordinates. The loss is computed as difference between [() and the output of the HypoNet f(1).
During the training of HyperTime, we use the weights predicted by the HyperNet decoder to instantiate a HypoNet and evaluate it on the input time-coorde coordinaod f, to produce the predicted time series value ftt). The entire chain of operations is implemented within the same differentiable pipeline, and hence the training loss can be computed the difference between the ground truth time series signal ft) and the value predicted by the HypoNet f(t).
The use of Set Encoder in the architecture introduces permutation invariance in the computation. This enables the encoding of data with missing values or irregularly sampled, which are common oc- currences in time series datasets. After training, the Set Encoder is able to generate latent embeddings to encode entire time series. Section 4.2] we show that these embeddings can be interpolated to produce new samples from the distribution learned by the network, enabling the synthesis of new time series from known ones.
Training Loss The training of HyperTimm is done by optimizing the following loss, which contains an MSE reconstruction term Lie and two regularization terms Lwiights and Ciaeens for the network weights and the latent embeddings, respectively:
L =
f(t:) ((ta))f(())))) +AI
+22
2 APPP
(5)
Eesigrts
Gyeen
Since we are working with time series, we want to ensure an accurate reconstruction not only of the signal but also of its spectral composition. Hence, we introduce an additional Fourier-based loss LFFT that penalizes differences in the distribution of frequencies of the INR-encoded time series with respect to the ground truth data:
CEPT
1 N
Σ FFT])(O) -FT((())))
(6)
In Section 4.2
we show that CFFT is crucial for the accurate reconstruction of some datasets that show large variations of amplitudes in the frequency domain.
3.3
INTERPRETABLE HYPERTIME
In the previous section, we proposed a hypernetwork architecture (HyperTime) to learn a prior over a dataset of time series, encoding each sample small embedding Z. Although the HT architecture
5

---

Under review as a conference paper at ICLR 2023
uses interpretable SIRENs as HypoNetworks, the decomposition between trend and seasonality signals is limited to the predicted INR, and is not explicitly reflected in the generated embeddings.
In Figure bottom, display our interpretable HyperTime (iHTT) network, which builds over the previous HyperTime architecture to incorporate interpretability into the latent representation. Here the embedding Z is split in two parts Zr and Zg, corresponding to the encodings of the Trend and Seasonality signals. Each these two embeddings is decoded by a different HyperNetwork (Trend HyperNet and Seasonality HyperNet) and affects the weights of different block (Trend Block and Seasonality Block) within the predicted iSIREN. Only at the output of the iSIREN both signals are recomposed into predicted time series, which compared against the ground truth signal via the reconstruction and spectral losses Reeec and FFFT).
The training of iHyper is performed in two stages, in order to improve stability, as explained for iSIREN in Section 3.1.1 After training, the Set Encoder produces a decomposed encoding of time series into interpretable embeddings Zr and Zg. As with the HyperTime architecture, these embeddings can be interpolated to produce new unseen time series. However, as we will show in Section 4.2 we can now interpolate only one of the embeddings while leaving the other fixed. This way we can interpolate between time series with different trends while keeping the seasonality fixed and vice-versa allowing us to introduce fixed temporal structures to the data generation which can be used to inject domain expertise in cases where there is not sufficient data.
EXPERIMENTS
4.1 RECONSTRUCTION
We start by analyzing the reconstruction performance of our model iSIREN over univariate and multivari- ate time series datasets from the UCR archive (Bag- nall al. 2017). We selected datasets with vary- ing lengths and numbers of features, and sampled 100 time series from each of them, training sin- gle INR for each time series. We compare against other INRs with different activations functions (Tanh,
MSF (log smale)
SIR STREN
P.E.
10:-
10:1
Tonh
10:5
ReLU, Sine) and with ReLU positional encoding (P.F.) (Mildenhall et al. 2020) using INRs with equal size (3 hidden layer of 60 FC neurons). In addition, iSIREN presents 4 more trainable parameters per time series feature, due the Trend block.
20000
(000)
0000 Epochs
S000
100000
10000
100
SIREIN
10m*
RoLU
Teahs
In Figure B we display two measures of reconstruc- tion error averaged over 100 samples of dataset (FordA). Figure displays the MSE training loss of each model in log space as function of the train- ing epoch. We observe that both models based on sine activations converge to similarly low values of error. We can also see that iSIREN's convergence delayed during the first training iterations due to the sequential training of the Trend and Seasonality blocks, detailed Section 3.1.1
PSD
zon
10
0.4
Fooquency
Figure 3: Comparison of iSIREN with other INR encodings. Top: MSE loss. Bottom:
MAE of power spectral density.
In Figure 3 bottom, we focus on the accurate reconstruction of the GT spectral distribution. We illustrate the MAE of the power spectrum density (log) vs the frequency. Here we observe too that the lowest emrors correspond to SIREN and iSIREN for all frequencies, with a slight advantage of our method for low frequencies. The reconstruction across all datasets has been summarized in Table [ where we observe that iSIREN presents the best reconstruction performance, especially in terms of the spectral distribution. Moreover, for multivariate data iSIREN shows the lowest errors over all metrics and datasets, with particularly large advantage in the case of PhoremeSpectra.
Finally in Figure we illustrate the capacity of iSIREN to perform an interpretable decomposition of time series. In column (a) show the signal reconstruction, and in columns (b) and (c) the

---

Under review as a conference paper at ICLR 2023
corresponding outputs of the Trend and Seasonality blocks. As it would be expected, the Trend curve follows the overall shape of the signal, while the Seasonality prediction oscillates around zero. good agreement with analogous
For qualitative comparison, in columns (d) and (e) decomposition performed by a traditional method (STL. (Cleveland et al.) 1990)). In contrast with iSIREN, STL requires prior knowledge of the frequency of the signal's seasonality.
Table I: Comparison using MSE on time space and MAE in frequency space (FFT) of implicit networks using different activation functions and of iSIREN on univariate and multivariate datasets.
Dataset
iIREN (Ours)
SIREN
P.E.
ReLU
Tanh MSE
FFT
MSE
FFT
MSE
MSE
FFT
MSE
Univariate
Crop
1.4e-3
5.6e-6
1.4e-3
1.6e-6
6.8e-4
7.3e-7
5.4e-1
2.1e-2
8.6e-1
6.0e-2
Energy
4.1e-3
5.3e-6
1.8e-2
1.2e-5
1.3e-1
7.7e-4
1.5e+0
4.9e-2
1.9++0
8.3e-2
FordA
1.7e-2
4.9c-6
1.9e2
6.2e-6
3.1e-1
2.1e-3
2.5e+0
1.3e-1
2.8c+0
1.4e-1
NonInv
3.6e-2
1.2c-5
4.0e-2
1.3e-5
L.fe-
1.3e4
1.0e40
2.2e-2
1.3e+0
4.6e-2
Phalanges
1.4c-3
2.1c-6
3.8c-3
1.8c-6
7.6c-3
1.2c-5
2.4c-1
3.8c-3
7.5c-1
8.4e-2
Stock
2.5e-3
5.1e-6
4.4e-3
1.4-66
4.3e-2
1.2e-4
6.2e-1
1.2e-2
8.9e-1
3.8e-2
Multivariote
Cricket
3.9c-1
4.1c-4
4,5e-1
4.2c-4
1.7e40
3.7e-3
3,5e40
1.7e-2
3,9e+0
3.1e-2
Moturlmagery
5.1e40
2.1e-3
7,2c40
6.2c-3
1.1e+1
2.4e-2
1.0c+1
2.6c-2
1.le+1
3.0e-2
PhonemeSpectra
2.9e-2
2.1e-6
4.2e-1
2.7e-4
1.8e+0
5.9e-3
3.0e+0
1.5e-2
3.4e+0
2.0e-2
0.
-18
14
TRIDO BEACK
SEACREAALITY
02
(a) Reconstruction
(b) Trend
(c) Seasonality
(d) STL: trend
STL:
Figure 4: Interpretable decomposition of iSIREN on two time series from the Stock dataset. (a) Ground truth and iSIREN reconstruction. (b) Trend Block output. (c) Seasonality Block output.
Columns (d) and (e) compare the output of iSIREN blocks with classic STL decomposition.
4.2 TIME SERIES GENERATION
We leverage our proposed hypernetwork architectures for time series generation via the interpolation of latent representations. We follow the experimental set up proposed by Alaa al. (22T) to evaluate our model multiple datasets in terms of two metrics: 1) Predictive score (MAE), which measures how well the synthetic samples can be used train a model and predict on real samples in forecasting task. 2) F1-score, measuring the quality of the generated data via precision and recall metries averaged over all time steps (Sajaadi et al. (2018).
we show qualitative evaluations of the generation performance for [HyperTimee and compare against multiple methods, including RealNVP Dinh et al. (2017) and current state-of -ftte-------rtt for time series generation (TimeGAN Yoon al. (2019b) and Fourier Flows Alaa et al. (2021) We see good agreement of iHyperTima for all datasets, while other methods fail to match the original data in some cases. Performance results using the Predictive summarized Table In addition to iHyperTime, we evaluate the following 3 other variations of the architecture:
In Figure 5]
and F1-score metrics
7

---

Under review as a conference paper at ICLR 2023
Crop
Nonlav
Phalanges
Energy
Stock
FordA
RealN
Tourier Flow
SIT (Ouns)
Figure 5: t-SNE visualization of real (blue) and synthetic (orange) data for all univariate datasets (in columns), using different time series generation methods (in rows).
HyperTime (no FFT):
this architecture is directly based of the original hypernetwork archi- tectures from SIREN (Sitzmann et al.).
HyperTime (w/FFT): same as previous, but including the FFT loss in the training.
HyperTime (iSIREN); same as previous, but using iSIREN as the predicted hyponetwork, thus generating time-series representations with built-in interpretable TS decomposition.
Table 2: Top: Performance scores for data generation using baselines and iHyperTime. Bottom: HyperTime ablations (in blue: values that improve over the baselines).
Crop
NonIny
Phalan.
Energy
Stock
FordA
RealNVP
MAE
0.170
0.038 0.986
0.073 0.976
0.036 0.964
0.019
0.115
Fl Score
0.981
0.977
0.999
TimeGAN
MAE
0.048
0.108
0.056
0.173
FI Score
0.831
0.960
0.479
0.938
Fourier Flows
MAE
0.040
0.018
0.056
0.030
0.010
0.024
FI Score
0.991
0.990
0.992
0.936
0,990
0.998
iHT (Ours)
MAE
0.039
0.004
0.024
0.056
0.011
0.009
FI Score
0.999
0.997
0.997
0.997
0.995
0.996
Ablations
(no FFT)
MAE
0.040
0.005
0.023
0.058
0.012
0.170
FI Score
0.999
0.996
0.996
0.998
0.995
0.084
HT (w/ FFT)
MAE
0.040
0.005
0.023
0.057
0.013
0.007
FI Score
0.999
0.997
0.999
0.997
0.994
0.998
HT (iSiren)
MAE
0.039
0.004
0.024
0.057
0.013
0.008
FI Score
0.999
0.997
0.999
0.997
0.995
0.997
All hypernetwork architectures consistently generate high-quality synthetic time series, outperforming the baselines across most datasets and metrics. This indicates that the data generated by the HT and iHT architectures presents a high predictive utility for model training (Predictive score) and large overlap with the distribution of the real data (F1 score).
An interesting failure case is presented with the generation of Ford& samples by the HT (no FFT) model (in red). Here both evaluation metries indicate low performance. As shown in the supplemen- tal material, the unusual diversity of spectral distributions present in the Forda dataset suggests that HT unable to learn effective reduced representation due the complexity of the data. However,
8

---

Under review as a conference paper at ICLR 2023
the issue is solved by the introduction of the spectral loss CFFT, observed the metrics for the HT (w/FFT) model. Moreover, the inclusion this loss does not result in loss of performance over other datasets.
Finally, in Figure we illustrate the interpolation of interpretable embeddings from iHT. This model produces independent encodings Zr and Zş for the trend and seasonality signals, enabling additional control over the synthesis process by interpolating them separately. Figure shows an interpolation of Zs from source to target, leaving Zr unchanged. By the end of the process, the interpolated time series has acquired the seasonality of the target signal, but maintains its original trend. Conversely, in Figure 6p we show an interpolation of the trend Zr while fixing the seasonality Zs. In both cases, we observe a smooth transition of the signal corresponding to the interpolated embedding, indicating a good generalization of the iHT architecture.
Origin
A-05
A-075
A-1.0
Torpel
T
(a) Interpolation in seasonality, leaving trend lixed.
Origin
X-0.25
2-05
A-0
A-1.0
Torpet
0.00
1000
330
(b) Interpolation in trend, leaving seasonality fixed.
Figure 6: (a) Interpolation of seasonality, with fixed trend. (b) Interpolation in trend, with fixed seasonality. red: original TS (Ist column) and target seasonality/trend (last column).
5 CONCLUSIONS
In this work, we explored the use of INRs for the encoding and analysis of both univariate and multivariate time series data. We proposed novel INR architecture (iSIREN) that enables learning an interpretable (seasonality-trend) continuous representation of time series data. We showed that our model outperforms other INRs in terms of the accurate reconstruction of the signal and its spectral distribution, with particular high performance for multivariate time series. Following this, we presented HyperTime, hypernetwork architecture that leverages iSIRENs to learn a prior over time series dataset. We demonstrated that by interpolating between embeddings learned by HT, we can generate new synthetic time series. To guide the training of HT, we introduced a Fourier-based loss that enforces the accurate reconstruction of the spectral distribution of the ground truth signals.
Finally we introduced HHyperTime, a hypernetwork model that incorporates interpretability into its latent representation of time series. We compared both HT and iHT against state-of-the-arr methods for time series generation, showing improved results across most datasets and generative evaluation showed that the disentangled seasonal-trend embeddings learned by iHT
metrics. Furthermore, can be leveraged to introduce constraints into the generation, with potential applications for the incorporation of expert knowledge in the synthesis of time series.

---

Under review as a conference paper at ICLR 2023
REFERENCES
Ahmed Alaa, Alex James Chan, and Mihaela van der Schaar. Generative time-series modeling with fourier flows. In International Conference on Learning Representations, 2021.
Ali Alqahtani, Mohammed Ali, Xianghua Xie, and Mark W. Jones. Deep time-series clustering: A
review. Electronics, 10(23), 2021. ISSN 2079-9292. doi: 10.3390/electrronics 10233001. URL
https://vin www. wttp:/wwwwwwpp...ppmpppppp.. com/ 207 00111999999221022200000
Ivan Anokhin, Kirill V. Demochkin, Taras Khakhulin, Gleb Sterkin, Victor S. Lempitsky, and Denis Korzhenkov. Image generators with anddtionally-indeeennee pixel synthesis. 2021 IEEE/CVF Conference on Computer Vision and Puttern Recognition (CVPR). pp. 14273-14282, 2021.
Samuel A. Assefa, Danial Dervovic, Mahmoud Mahfouz, Robert Tillman, Prashant Reddy, and Manuela Veloso. Generating synthetic data in finance: Opportunities, challenges and pitfalls. In Proceedings of the First ACM International Conference on Al in Finance, ICAIF '20, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450375849, doi: 10.1145738345553422554 URL
http://oii.rrr.mmcccceeeeeeeee idm. ocle. org/10
3422554
A. Bagnall, J. Lines, A. Bostrom, J. Large, and E. Keogh, The great time series classification bake off: review and experimental evaluation recent algorithmic advances. Data Mining and Knowledge Discovery,
2017.
Kasun Bandara, Rob J. Hyndman, and C. Bergmeir. Mstl: A seasonal-trend decomposition algorithm for time series with multiple seasonal patterns. International Journal of Operational Research, 2022.
Dimitrios Bellos, Mark Basham, Tony P. Pridmore, and Andrew P. French. A convolutional neural net- work for fast upsampling of undersampled tomograms in x-ray ct time-series using a representative highly sampled tomogram. Journal of Synchrotron Radiation, 26:839 853, 2019.
Nuri Benbarka, Timon Höfer, Hamd ul Moqeet Riaz, and Andreas Zell. Seeing implicit neural representations as fourier series. 2022 JEFF/CVF Winter Conference on Applications of Computer Vision (WACV), pp. 2283-2292 2022.
Wei Cao. Dong Wang, Jian Li, Hao Zhou, Lei Li, and Yitan Li. Brits: Bidirectional recurrent imputation for time series. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and Garnett (eds.), Advances in Neural Information Processing Systems, volume 31. Curran As- sociates, Inc., 2018, URL https: http::://ppoceeddnng/ meereeeddddinnnnennessee....eeee neurips htcplpss.ccc/eee httpappppr//////eppe 734 htt.eeccceee2.ccc.ooo22223555111
Eric Chan, Marco Monteiro, Petr Kellnhofer, Jiajun Wu, and Gordon Wetzstein. pi-gan: Periodic
implicit generative adversarial networks for 3d-aware image synthesis. 202 IFFF/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5795-5805, 2021.
Yinbo Chen, Sifei Liu, and Xiaolong Wang. Learning continuous image representation with local implicit image function. In Proceedings of the IEEE/CVF Conference on Computer Vision and Puttern Recognition, pp.
2021.
Kukjin Choi, Jihun Yi, Changhwa Park, and Sungroh Yoon. Deep learning for anomaly detection in time-series data: Review, analysis, and guidelines. IEEE Access, 9: 9:120043-12006, 2021. doi: A1110//ACEES.20....11...
Robert B. Cleveland, William S. Cleveland, Jean E. McRae, and Irma Terpenning. Stl: A seasonal- trend decomposition procedure based loess (with discussion). Journal of Official Statistics, 6: 3-73, 1990.
Abhyuday Desai, Cynthia Freeman, Zuhui Wang, and Ian Beaver. Timevae: A variational auto- encoder for multivariate time series generation. arXiv preprint
2021.
Laurent Dinh, Jascha fohk-Dickstein, and Samy Bengio. Density estimation using real NVP. In International Conference on Learning Representations, 2017. TRLL/Attts::/ppeeeee
10

---

Under review as a conference paper at ICLR 2023
Chenguang Fang and Chen Wang. Time series data imputation: A survey on deep learning approaches. ArXiv, 0bbs/2011.11444 1347, 2020.
Elizabeth Fons, Paula Dawson, Xiao jun Zeng, John Keane, and Alexandros Iosifidis. Adaptive weighting scheme for automatic time-series data augmentation, 2021.
David Ha, Andrew M. Dai, and Quoc V. Le. Hypernetorrks. In 5th International Confer- ence on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference net forummidt
Track Proceedings. OpenReview.net, 2017. httph::tt//ppetttt
rkpACells
Kyle Hundman, Valentino Constantinou, Christopher Laporte, Ian Colwell, and Tom Soder- strom. Detecting spacecraft anomalies using Istms and nonparametric dynamic threshold- ing. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Dis- covery & Data Mining, KDD '18, pp. 387-395, New York, NY, USA. 2018. Association for Computing Machinery. ISBN 97814503555
doi: 00.1145/3219819.322998 idm. wol.. orw.co..r..111111.1 11e...115121111 ..0.9.3321199 3219845
URL
Hassan Ismail Fawaz, Benjamin I Lucas, Germain Forestier, Charlotte Pelletier, Daniel F. Schmidt, Jonathan Weber, Geoffrey I. Webb, Lhassane Idoumghar, Pierre-Alain Muller, and François Petitjean. Inceptiontime: Finding alexnet for time series classification. Data Mining and Knowledge Discovery, 2020.
Kyeong-Joong Jeong and Yong-Min Shin. Time-series anomaly detection with implicit neural repre- sentation. CoRR, abs/2201. 1950. 2022. URL httlt::////eet http///aavvvvv..a wttps//.rrggb/////ee//ee /200..22211112 11950
James Jordon, Jinsung Yoon, and Mihacla van der Schaar. Pate-gan: Generating synthetic data with differential privacy guarantees. In ICLR, 2019.
James Jordon, Daniel Jarrett, Evgeny Saveliev, Jinsung Yoon, Paul Elbers, Patrick Thoral, Ari Ercole, Cheng Zhang, Danielle Belgrave, and Mihaela van der Schaar. Hide-and-seek privacy challenge: Synthetic data generation vs. patient re-identification. In Hugo Jair Escalante and Katja Hofmann (eds.), Proceedings of the NeurIPS 2020 Competition and Demonstration Track, volume 133 of Proceedings of Machine Learning Research, pp. 206-215. PMLR, 06-12 Dec 2021. URL
httpss://proceeding mlr press v1 33/ www.ooorddndn...a html
Bryan Lim and Stefan Zohren. Time-series forecasting with deep learning: a survey. Phylosophical Transactions of the Royal Society A, 2021. doi: 10. 10.1098/stta.222000000
Gidi Littwin and Lior Wolf. Deep meta functionals for shape representation. In JEEE/CVF Interna- tional Conference on Computer Vision (ICCV), pp. 1824-1833, 10 2019. doi: 10. 10.1109//C,/:22119
00191.
Yan Liu. Recurrent neural networks for multivariate time series with missing values. Scientific Reports, 8(1):6085, 2018. doi: 10. (0.10038/41198--18--221111 URL [tL/stttss///ol.o//// 1038/ 018 24271-9
Yonghong Luo, Xiangrui Cai, Ying ZHANG, Jun Xu, and Yuan xiaojie. Multivariate time series imputation with generative adversarial networks. In S. Bengio, H. Wallach, H. Larochelle, K. Grau- man, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. URL URL/httpsss/ppprceedimng newr.ss......................... cc/ paper/2018/ http:/eeeeeeeeeeeeeee00000. 22tp://000/////bffffooocccct1112
Qianli Ma, Jiawei Zheng, Sen Li, and Gary W Cottrel. Learning representations for time se- ries clustering. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché Buc, E. Fox, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 32, Curran As- sociates, Inc., 2019. URL httpt:ttt//.ccm htt.://ssss/prrcceeimim neurips...... wttpc//pppppp///////eoo
1359aa9 22
pdf
Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger. Occupancy networks: Learning 3d reconstruction in function space. In Proceedings IEEE Conf. Computer Vision and Pattern Recognition (CVPR), 2019.
11

---

Under review as a conference paper at ICLR 2023
Ben Mildenhall, Pratul P. Srinivasan, Matthew Tancik, Jonathan T. Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. In ECCV, 2020.
Cheolhwan Oh, Seungmin Han, and Jongpil Jeong. Time-series data augmentation based interpolation. Procedia Computer Science, 175:64-71, 2020. ISSN 1877-0509. doi: https://doi.org orgp//.....r//..oo.
URL https:/ /www www.sccennedirect... com/ The 17th International Conference
Mobile Systems and Pervasive Computing (MobiSPC), The 15th International Conference on
Future Networks and Communications (Fh 10th International Conference on Sustainable Energy Information Technology.
Boris N. Oreshkin, Dmitri Carpov, Nicolas Chapados, and Yoshua Bengio. N-beats: Neural basis ex- pansion analysis for interpretable time series forecasting. In International Conference Learning Representations, 2020. URL /atttss://pperrvviiie net/ wttp:eewwnee.e/tommeeedd-ddeeeee
Mathias Perslev, Michael Jensen, Sune Darkner, Poul Jø rgen Jennum, and Christian Igel. U-time: A fully convolutional network for time series segmentation applied to sleep stag- ing. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Gar- nett (eds.), Advances in Neural Information Processing Systems, volume 32. Curran Asso- ciates, Inc., 2019. URL http:: htactpp::/ppocceedinng htwp:eeweediinn....ieeeeeeeeeeee neurips httpppppppppppppe////eeeeee/ 57ba www.fbbc2dfeeeeeeeeeeeeee wwwaee99bbbb33e 1tt. Paa-pper.
Tamar Rott Shaham, Michael Gharbi, Richard Zhang. Eli Shechtman, and Tomer Michaeli. Spatially- adaptive pixelwise networks for fast image translation. In Computer Vision and Pattern Recognition (CVPR), 2021.
Mehdi S. M. Sajjadi, Olivier Bachem, Mario Lučić, Olivier Bousquet, and Sylvain Gelly. Assessing Generative Models via Precision and Recall. In Advances Neural Information Processing Systems (NeurIPS), 2018.
Vincent Sitzmann, Michael Zollhöfer, and Gordon Wetzstein. Scene representation networks: Con- tinuous 3d-structure-aware neural scene representations. Advances in Neural Information Processing Systems, 2019.
Vincent Sitzmann, Eric R. Chan, Richard Tucker, Noah Snavely, and Gordon Wetzstein. Metasdf: Meta-leaming signed distance functions. Proc. NeurIPS, 2020a.
Vincent Sitzmann, Julien N.P. Martel, Alexander W. Bergman, David B. Lindell, and Gordon Wetzstein. Implicit neural representations with periodic activation functions. In Proc. NeurIPS, 2020b.
Ivan Skorokhodov. Savva Ignatyev, and Mohamed Elhoseiny. Adversarial generation of continuous images. In Proceedings of the IEEF/CVF Conference on Computer Vision and Puttern Recognition (CVPR), pp. 10753-10764, June 2021.
Alejandro Sztrajman, Gilles Rainer, Tobias Ritschel, and Tim Weyrich. Neural brdf representation and importance sampling. Computer Graphics Forum, 40(6):332- 346, 2021. doi: https://doi.org/ 10.111 1/cgf. 14335. cgf 14335]
wwt.ee.wwwleee wiley com /ww//dd/////////////// 1111/
Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Kridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan T. Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. NeurIPS, 2020.
José F. Torres, Dalil Hadjout, Abderrazak Sebaa, Francisco Martínez-Alvand Álvarez, and Alicia Troncoso Lora. Deep learning for time series forecasting: A survey. Big data, 2021.
Svitlana Vyetrenko and Shaojie Xu. Risk-sensitive compact decision trees for autonomous execution in presence of simulated market response, 2019. URL /tt.psss/aaxxvvoog//abb////
02312
Qingsong Wen, Jingkun Gao, Xiaomin Song. Liang Sun, Huan Xu, and Shenghuo Zhu. Robuststl: A robust seasonal-tres -renn decomposition algorithm for long time series. Proceedings of the AAAI Conference Artificial Intelligence, 5:509-55416, Jul. 2019.
12

---

Under review as a conference paper at ICLR 2023
Magnus Wiese, Robert Knobloch, Ralf Korn, and Peter Kretschmer. Quant gans: deep generation of financial time series. Quantitative Finance, pp. 1-22, Apr 2020. ISSN 7696. doi: 10. htt dx. ob1.0.t9/1.1100/116533333 2020
01114666688.2220.7700000 1730426
Gerald Woo, Chenghao Liu, Doyen Sahoo, Akshat Kumar, and Steven C. H. Hoi. Deeptime:
Deep time-index meta-learning for non-stationary time-series forecasting. 2022. URL https:
//arxiv. 2twpv./..g//ab//222..0ee
Haowen Xu, Wenxiao Chen, Nengwen Zhao, Zeyan Li, Jiahao Bu, Zhihan Li, Ying Liu, Youjian Zhao, Dan Pei, Yang Feng, Jie Chen, Zhaogang Wang, and Honglin Qiao. Unsupervised anomaly
detection via variational auto-encoder for seasonal kpis in web applications. In Proceedings of the 2018 World Wide Web Conference, WWW 18, pp. 187-196, 2018. ISBN 9781450356398., doi:
10.114553178886633155966 URL 1145/3178876 71819889666119996
wawchesst..ccceeee........... ddm. www.ooll..oooo....
Jinsung Yoon, Daniel Jarrett, and Mihaela van der Schaar. Time-series generative adversarial networks. In NeurIPS, 2019a.
Jinsung Yoon, Daniel Jarrett, and Mihaela van der Schaar. Time-series generative adversarial networks. In NeurlPS, 2019b.
Li Zeng, Baifan Zhou, Mohammad AI-Rifai, and Evgeny Kharlamov. Segtime: Precise time series segmentation without sliding window, 2022. URL httpss///openeevveeeeeee//tuuuu CFFFMXXvVHQUTA
13


Reviews:
[{"review_id": "nfFI_p5VfM", "reviewer": "Reviewer_H1Gq", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "main_review": "", "summary_of_the_paper": "The authors present the iSIREN pipeline, designed to accurately reconstruct signals. To do it, they use latent representations of the data set. They also work with metrics in the FFT domain to enforce an even better reconstruction.", "summary_of_the_review": "As stated above, the work is incrementally novel. Even though very good results are obtained, it is difficult to assess whether they are general (not many experiments) or conclusive (marginal improvements).", "strength_and_weaknesses": "Strengths:\n1. I think that enforcing accuracy in the Fourier domain adds powerful constraints that should generate better results.\nWeaknesses:\n1. The f(t) = ftr(t) + fs(t) decomposition introduces a very strong assumption: What happens with non seasonal signals?\n2. Even though the results are good and generally better, the experiments are not conclusive in stating the superiority of the proposed pipeline. This is important because the method is not based on fundamental principles but on common sense ad-hoc decisions. Thus, it is the experiments that should state whether the pipeline is better or not.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "bzX8bjFOwG", "reviewer": "Reviewer_jVNg", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "main_review": "", "summary_of_the_paper": "In this work, the authors propose an implicit neural representations method for time-series data, namely iSIREN. The key different from the original SIREN is that iSIREN has modelled the trend and seasonality explicitly. Then authors show that the parameters of iSIREN can be produced by a hypernetwork (HyperTime or iHyperTime), such that the new generative model can be produced on-the-fly without any training by feeding the time-series data into hypernetwork. The proposed method is demonstrated for reconstruction and generation tasks cross many datasets and surpassing some other baselines. ", "summary_of_the_review": "Overall it is a nice paper with a good demonstration of implicit network for time-series data. I may upgrade the score if the seasonality modelling part is further elaborated.", "strength_and_weaknesses": "* Strength\n\n(i) The work is well written and the motivation of using implicit neural network is appreciated because of lack of the similar work on this direction.\n\n(ii) By separating the trend component from the seasonality, the model has better interpretability. Besides, it also offers a way for trend/seasonality decomposition.\n\n(iii) Hypernet is an interesting angle here, as it saves time from training a new model for incoming data.\n\n* Weakness\n\n(i) The work is largely built-on SIREN and the migrating to time-series data is not challenging, given the flexibility of implicit network itself.\n\n(ii) While the decomposition of tend and seasonality is interesting, it is hard to justify they are the components that are claimed to be, esp. for the case of seasonality, i.e., SIREN can model seasonality (more broadly, periodic pattern) well, but you may not claim that what SIREN has modelled is truly seasonality.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "L3COuOxQhr", "reviewer": "Reviewer_wjaa", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper proposes, based on SIREN, a new interpretable INR architecture for time series called iSIREN. Building up on that, the paper proposes to use the iSIREN set-up in the context of hypernetworks for time series generating. Both methods differ from its plain counter version (SIREN and SIREN-based hypernetwork) in that it splits the model into a classic time\ntime series additive decomposition, i.e. trend and seasonal component. The paper provides a variety of different experiments showing its superior performance compared to baseline models.", "summary_of_the_review": "The paper introduces iSIREN and a hypernetwork based on iSIREN, which consists of SIREN together with a time series additive decomposition. The novelty is very marginal and the results are not outstanding. Moreover, the results are not reproducible. Finally, major parts of the paper need to be rewritten. Hence I recommend to reject the manuscript in its current form.", "strength_and_weaknesses": "Using a series decomposition for SIREN and SIREN-based hypernetworks in novel up to my knowledge. However, using an additive time series decomposition into trend and seasonal component is one of the most standard approaches in time series analysis. The paper provides many and versatile experiments showing a strong performance of the provided models on many different tasks.\n\nHowever, there are open questions and weaknesses that need to be addressed:\n\n* Is the better performance of iSIREN due to the additive decomposition technique or simply because it uses slightly more parameters than standard SIREN?\n* What are the weaknesses of the provided models, e.g. slower training/inference time and so on?\n* Adding an FFT penalty term to the loss function seems to be a great idea. Is it novel or has it been used in this context already before?\n* No standard deviations are provided. It is not clear if the obtained results are robust.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "4j6KKCvalW", "reviewer": "Reviewer_Wscx", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper proposes to use coordinate-based networks for time-series data representation. This is done by proposing a new architecture which separates a time series into a trend (low-frequency) and seasonal (high-frequency) component by representing each independently with a set of weights and a SIREN, and then summing them. The method proposes to generalize over these coordinate-based representations of time series data using hypernetwork architectures: one which outputs standard SIREN representations of time series, and one which outputs the modified iSIREN architecture proposed with the decomposition of signal frequencies. The paper demonstrates that this new architecture results in improved time series fitting, and that the hypernetwork architecture places similar time series signals at similar places in the latent space.", "summary_of_the_review": "The paper proposes a method for representing time series data with a coordinate-based network architecture based on SIRENs, and generalizing across these representations with hypernetworks. Overall, I find the method to be described and evaluated well. However, the architecture proposed is heavily based on previous work, and is not compared to other state-of-the-art coordinate-based representation architectures which have emerged for other applications. The generalization aspect is interesting, but is not novel as it is just a drop-in of the method used in SIREN, and is not shown to be useful for a useful application in 1D signal processing, such as solving inverse problems.\n\nPOST REBUTTAL UPDATE: After reading the other reviews and responses, I am not inclined to change my score. I now understand that the purpose of iSIREN is not to improve directly on SIREN, but instead to provide an interpretable breakdown. However, the argument of using two separate SIRENs (as in [1]) requiring an additional hyperparameter is not convincing, as iSIREN also needs to choose the parametric form for one of the networks. While I believe the FFT loss is novel in this context (although it has been used before in imaging applications), I find that the paper still combines a lot of known methods into a new type of data. This is interesting and important for this application domain, but doesn't propose anything new or enable significantly more interpretable function fitting. For this reason, I view the paper as borderline for acceptance.", "strength_and_weaknesses": "In my opinion, the strengths of the paper are:\n- The paper tackles an important problem which is improving the quality of neural signal representations of 1D signals, and generalization across instances of these signals.\n- The paper is written very clearly, and the method is explained well. The figures are very detailed and describe the proposed architecture concisely.\n\nI think that the main weakness of the paper is novelty. Specifically:\n- The iSIREN architecture proposed is a SIREN, but with a factored out low-frequency component. This has already been proposed in [1] (where the output of a coordinate-based network is the sum of two different functions), although a slightly different function implementation is used here for the trend component instead of another low-frequency biased SIREN.\n- The generalization aspect, HyperTime, effectively just uses the SIREN + hypernetwork setup, with the same set-encoder used in the SIREN paper, for generalization. This generalization scheme has already been shown to work for coordinate-based representations of images, so it is not surprising it could also work for 1D signals, considering that the original SIREN paper also fits 1D signals in the form of audio.\n- It seems that the results don’t outperform the standard SIREN by much? Figure 3 shows that SIREN and iSIREN seem to converge similarly, and Table 1 shows that they are much more similar to each other than to any of the other methods. It is difficult to evaluate the relative magnitude of the difference in these numbers. \n- Additionally, I don’t find the baseline comparisons to be fair. There exist other methods for coordinate-based networks (ex: hybrid implict-explicit architectures, SAPE, etc…) which can be used for signal representation and generalization. Simply comparing versus other nonlinearities in the network architecture is not a fair comparison as the SIREN paper already has shown that they simply don’t work for high-frequency signal memorization, so they are not valid baselines to compare to an improvement on SIREN.\n- The generalization visualization in Figure 5 shows that the synthetic and real data distributions match, but the real data distribution doesn’t seem to be clustered. Thus I wonder if the prior learned by this hypernetwork would allow the useful aspects of generalization, such as solving inverse problems for example? In the original SIREN paper, this is used to perform inpainting of images - is something similar done here for time series?\n\n\n[1] Geometry-Consistent Neural Shape Representation with Implicit Displacement Fields. Yifan, Wang and Rahmann, Lukas and Sorkine-hornung, Olga. ICLR, 2021.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject