Recursive Time Series Data Augmentation
Authors: Amine Mohamed Aboussalah
Minjae Kwon
Raj G Patel
Cheng Chi
Chi-Guhn Lee
Venue: ICLR 2023 poster (2023)
Keywords: Time Series
Data augmentation
Representation Learning
Deep Learning
Reinforcement Learning

Abstract: Time series observations can be seen as realizations of an underlying dynamical system governed by rules that we typically do not know. For time series learning tasks we create our model using available data. Training on available realizations, where data is limited, often induces severe over-fitting thereby preventing generalization. To address this issue, we introduce a general recursive framework for time series augmentation, which we call the Recursive Interpolation Method (RIM). New augmented time series are generated using a recursive interpolation function from the original time series for use in training. We perform theoretical analysis to characterize the proposed RIM and to guarantee its performance under certain conditions. We apply RIM to diverse synthetic and real-world time series cases to achieve strong performance over non-augmented data on a variety of learning tasks. Our method is also computationally more efficient and leads to better performance when compared to state of the art time series data augmentation.


Published as a conference paper at ICLR 2023
RECURSIVE TIME SERIES DATA AUGMENTATION
Amine Mohamed Aboussalah' MMinaee Kwon', Raj G Patel", Cheng Chi,, Chi-Guhn Lee NNww York University, *University of Virginia, UUniversity of Toronto ama10288@nyu.
hbt nbttsuevirginia.cc
(rajg.patel, cheng. chi @mail wtw.onto..ccoooo cgleeêmie
utorento.ca utt
ABSTRACT
Time series observations can be seen as realizations of an underlying dynamical system governed by rules that we typically do know. For time series learning tasks we create our model using available data. Training available realizations, where data is limited, often induces severe over-fitting thereby preventing gen- eralization. To address this issue, we introduce general recursive framework for time series augmentation, which we call the Recursive Interpolation Method (RIM). New augmented time series are generated using a recursive interpolation function from the original time series for use in training. We perform theoreti- cal analysis characterize the proposed RIM and guarantee its performance under certain conditions. We apply RIM to diverse synthetic and real-world time series cases to achieve strong performance over non-augmented data on variety of learning tasks. Our method is also computationally more efficient and leads better performance when compared state of the time series data augmenta- tion.
INTRODUCTION
The recent success of machine learning (ML) algorithms depends on the availability of a large amount of data and prodigious computing power, which practice are not always available. real world applications, it is often impossible indefinitely sample and ideally, model to make good decisions with limited number of samples. To overcome these issues, we can exploit additional information such as the structure or invariance in the data that help the ML. algorithms efficiently learn and focus on the most important features for solving the task. In ML, the exploitation of structure in the data has been handled using four different yet complementary approaches: I) Architecture design, 2) Transfer learning, 3) Data representation, and 4) Data aug- mentation. Our focus in this work is on data augmentation approaches in the context of time series learning.
would like the MI.
Time series representations do not expose the full information of the underlying dynamical system Prado (1998) in a way that MI. can easily recognize. For instance, in financial time series data, there are patterns at various scales that can be learned to improve performance. At a more fundamental level, time series one-dimensional projections of hypersurface of data called the phase space of a dynamical system. This projection results in loss of information regarding the dynamics of the system. However, we can still make inferences about the dynamical system that projects a time series realization. Our approach is use these inferences generate additional time series data from the original realization to build richer representations and improve time series pattern identifi- cation resulting in more optimal parameters and reduced variance. We show that our methodology applicable variety of ML algorithms.
Time series learning problems depend on the observed historical data used for training. We often use a set of time series data to train the ML model. Each element in the set can be viewed as sample derived from the underlying stochastic dynamical system. However, each historical time series data sample is only particular realization of the underlying stochastic dynamical system in the real

---

Published as a conference paper at ICLR 2023
world that trying learn. Our work focuses problems where available realizations limited but is not limited to these problems. In fact, our method can be applied to any time series learning task such as stock price prediction where we often have single realization or problems with numerous realizations such speech recognition where many audio clips are available for training. Let us consider the stock price prediction problem. The task is to predict or classify the trend of future price. Ideally we want our model to perform well by capturing the stochastic dy- namics of stock markets. However, we only train the model using a single time series realization or limited historical realizations. As result, we do not truly capture the characteristic behaviour of the underlying dynamical system. Using the original training data and hence one or a few realizations of the underlying dynamical system usually induces over-fitting. This is not ideal as we want our model perform well in the stochastic system instead of just a specific realization of that system.
Contributions. The contributions of our work are as follows:
We present a time series augmentation technique based on recursive interpolation.
We provide theoretical analysis of learning improvement for the proposed time series augmentation method:
We show that our recursive augmentation allows to control by how much the aug- mented time series trajectory deviates from the original time series trajectory (Theo- rem 3.1) and that there is a natural trade-off that is induced when our augmentation deviates considerably from the original time series (Theorem 3.2).
We demonstrate that our learning bound depends on the dimension and properties of the time series, as well as the neural network structure (Theorems 3.3 and 3.4).
We believe that this work is the first offer a theoretical ML framework for time series data augmentation with guarantees for variance reduction in the learned model (Theorem 3.5).
We empirically demonstrate learning improvements using synthetic data well as real world time series datasets.
Outline of the paper. Section 2 presents the literature review. Section 3 defines the notations, the problem setting, and provides the main theoretical results. Section 4 describes the experimental results, and Section 5 concludes with summary and a discussion of future work.
2 RELATED WORK
Augmentation for Computer Vision. In the computer vision context, there multiple ways to augment image data like cropping, rotation, translation, flipping, noise injection and so on. Among them, the mixup technique proposed in Zhang et al. (2018) is similar to our approach. They train a neural network on convex combinations of pairs of images and their labels. However, just applying static technique to dynamic time series data is not appropriate. Chen et al. (2020) showed that data augmentation has similar effect to an averaging operation over the orbits of certain group of transformation that keep the data distribution invariant.
Augmentation for Time Series. There is an exhaustive list of transformations applied to time series that are usually used as data augmentation Wen et al. (2020a). Fawaz et al. (2018) described trans- formations the time domain such as time warping and time permutation. There are methods that belong to the magnitude domain such as magnitude warping, Gaussian noise injection, quantization, scaling, and rotation Wen & Keyes (2019). There exists other transformations on time series in the frequency and time-freyuency domains that are based on Discrete Fourier Transform (DFT). In this context, they apply transformations in the amplitude and phase spectra of the time series and apply the reverse DFT to generate new time series signal Gao al. (2020). Besides the transforma- tions in different domains, there are also more advanced methods, including decomposition based methods such as the Seasonal and Trend decomposition using Loess (STL) method and its variants Cleveland al. (1990); Wen et al. (2020b), statistical generative models Kang et al. (2020), and ceaming-based methods. The learning- based methods can be further divided into embedding space DeVries & Taylor (2017), and deep generative models (DGMs) Esteban et al. (2017); Yoon et al. (2019). These approaches are problem dependent and do not offer theoretical guaranteed learning improvement. In addition, the beasd methods require large amounts of training data.
2

---

Published as a conference paper at ICLR 2023
Augmentation for Reinforcement Learning (RL). Laskin et al. (2020) presented the RL with Augmented Data (RAD) module which can augment most RL algorithms that use image data. They have demonstrated that augmentations such as random translate, random convolutions, crop, patch cutout, amplitude scale, and color jitter can enable RL algorithms to outperform complex advanced methods on standard benchmarks. Kostrikov et al. (2021) presented a data augmentation method that can be applied to conventional model-free reinforcement learning (RL) algorithms, enabling learning directly from pixels without the need for pre-training or auxiliary losses. The inclusion of this augmentation method improves performance substantially, enabling Soft Actor-Critie agent to reach advanced functioning capability on the DeepMind control suite, outperforming model-based methods and contrastive learning. Laskin et al. (2020) and Kostrikov et al. (2021) show the benefit of RL augmentation using convolutional neural networks (CNNs) for static data but do not handle dynamic data such as time series.
Ideally, we would like to have access to more data that is representative the underlying dynamics of the system or the regime under which we operate. However, we can not randomly add more data as there is a probability that the added data might not be representative of our stochastic dynamical system. To ensure that we are able to add meaningful data without disturbing the properties of the original data, we introduce a new approach called Recursive Interpolation Method. Our paper proposes recursive interpolation method for time series a tool for generating data augmentations.
3 THEORETICAL FRAMEWORK FOR RECURSIVE INTERPOLATION METHOD
3.1 RECURSIVE INTERPOLATION METHOD (RIM)
In our setting, we consider each time series sample as one realization from the underlying dynamical system. The realization consists of features along the time axis. Let + 1 be the dimension of the time series sample and {0, L......... } be the label set for each sample. Then, each sample belongs to gd+1 x , kj. Let S = {so, s1...., 8N3 be the collection of the time series samples. Let D distribution with support [0, 1) and X be drawn from independently denoted by X ~ D for time i E [1 dj. Let us denote = (AI,..., Aa) to be the vector of interpolations. For the sake of notational simplicity, in the rest of the paper, we denote X X and X D means that each component A, of À is sampled independently from D.
For given vector À E [0, 1)d and time series sample E S, we generate an augmented time series sample 8 as follows. Let consider original time series sample s (Iu, Ti,... Td. y) where the time series features (zo, Ti...., xz) E IR++1 and {0, 1,.... k} being the label of the corresponding time series sample. For each À = (A1...., da) E 10, (Ao is considered to be a dummy value), we define an augmented sample 8 (2o, L1,A1 Ld 3) such that
IA,,, = (1 - Aa)a + A,T and 20,20 TD
(1)
X,
X., 1.5
HIM (1, 2 (.8)
X,
X
Eq.1
X. 2.00
Xap 1.92
Figure 1: Illustration of RIM.
The newly generated augmented sample 8x has the same label as the original sample s. Our re- cursive methodology allows us to generate new time series realization that preserves the trajectory of the original data within some bound (Theorem 3.1).
3

---

Published as a conference paper at ICLR 2023
Notations. Let [s., 0) be loss function defined sample and model parameter 0 and similarly I(sx. 0) is the loss defined on the augmented sample using a model parameter 0. We denote that the distribution P parametrized by 0, on the sample space S. Given few realizations [EeJe]) from the distribution P, set:
0. argmin Eg-p (s, 0)
N
0 argmin +
Σ
(ss; 8)
auug argmin E Egup
(2)
N
Bang = argmin
Σ
ExDD (si2. 0]
N + 1
1=0
R₂ (1 θ) = Beja
sup 8E6
cll(s,
N
+1
1=0
Using the notations above, our recursive time series data augmentation minimizes the augmented loss under which we take the expectation over the augmented sample space. We call average augmented loss and denote it by auug (s, 0) = (x>.
The three most important aspects of our theoretical framework are the characterization of our recur- sive time series augmentation: the trade-off that is induced in the learning parameter space (Section 3.2), the learning bound showing the impact of the structural properties of time series and the neu- ral network on the learning parameters (Section 3.3), and better parameter learning with reduced variance when using augmented samples compared to the non augmented ones (Section 3.4).
3.2 LEARNING BOUND CONNECTING ORIGINAL TIME SERIES AND AUGMENTED TIME SERIES
We define the recursively interpolated time series and show that the augmented time series samples deviate from the original ones. We measured this deviation from the original time series using norm distance between the augmented time series and the original one. We show that this distance is bounded, where the bound depends on the characteristics of the time series features (Theorem 3.1).
0
if ( = 0
(1 if a = and D be a distribution with support [0, 1).
Let sign(t) (f)
if 0, 80b
if + b'
-1 ift < 0
Theorem 3.1. (Characterization of recursive augmentation) If A ~ D and g(An) = (1 - -1a((1 don) + (1 sngn then the following holds. Let n E N].
(1) Tn.A. E ())())) where A ~ D for j 2 and An is dummy value.
(2) Let ||+|| be norm, E(D], m' maxie[1:N]] { | -z 1l)) and m maxee ||]= Then
B [Zn,d - xn)]ll S min (3e)m, T-e m', Nem'). Expr.., [11( - xx)ill 2),
(3)
(4)
Recall that 8 = (zo, 21, Ld. v) and 88 = (I0 T1.Ap1... Ad y), we have
||s sx1l2
Σ
c 3 S
Σ - ..a.l ||s sxlli
Note that we used the la norm for simplicity but in general, any norm satisfies the inequality. Hence, measuring the distance each feature between the augmented time series sample and the original one gives information about how far they deviate from each other.
4

---

Published as a conference paper at ICLR 2023
Theorem 3.2. (Learning bound using characterization of recursive augmentation) Without any
loss of generality. let I(-.-) E [0, 1] be loss function with Lipschitz condition and S {so; $1,...,88' 8 the collection of the time series samples. Then with probability at least over the samples {8))1E(0:N] have
Ba-pl[(s, 0mug)] (((s, < 2R 0) +
2 N+1
2T Bea.ipeBpBBBaaaa a|]]
(5)
Moreover, we have
RN o 0 S (x)
max iE (E.......V
Bx-elss - -silll-
(6)
with Ex-D[[|sE. [|]] S 2mde, where e E(P], d d is the dimension of the time series sample, and m = max4ejo:N] (1|=:]]}.
Theorem 3.2 represents the standard argument for regret bounds using Rademacher complexity. The term in the left in (Eq.5) is what we call the generalization error or sometimes simply the risk, and term in right is called the regret excess risk. Theorem 3.2 tells us how good is our parameter estimation compared to the optimal parameters. Our bound is governed by three terms: 1) Rademacher complexity using the augmented data, 2) the sample size, 3) and the distance between our original time series sample and the augmented one. Now if the distance in (Eq.5) between time series samples before and after augmentation goes to zero then the learning bound is more tight. Another implication of the distance being small is that our recursive augmentation method decreases the Rademacher complexity as demonstrated by (Eq.6). Decreasing Rademacher complexity simply ends up producing a tighter bound on the parameter space.
On the other hand, if the distance in (Eq.5 and Eq.6) becomes large (i.c. the augmented time series samples deviate considerably from the original ones), then we can not guarantee with high probabil- ity that our method will always outperform the non augmented case. This trade-off frequently observed in learning theory.
3.3 LEARNING BOUND CONNECTING THE STRUCTURAL PROPERTIES OF TIME SERIES AND NEURAL NETWORKS
Theorem 3.3. Let fo be a neural network with a model parameter 0, ReLU activations, and sigmoid
function the activation function for the last layer denoted by fa(x) a(g((z) where ge (x) V + is the pre-activation signal for the last layer. Then the cross entropy loss function ((,,) has error bound as follows
-(ssx,))-(ss0 vd va (IAIE +
Σ 1=1
Bille
|/
(7)
where
0
G
8" 0'x1,AL 8x,9x A==
a d' 9X,8X,
0'
Amm6
A-d
A-6
A=
8x,8A lA-0
0trg.
of 0'
BAg
A=
A=
Bi
=
20,00
A=
A==
1
① E TRd denoted by column vector, and
||=|| is Frobenius norm.
Theorem 3.3 shows how the structural properties of the time series (A and B) and the neural network architecture (Vgu) can affect the learning process. The interpolation vector À will determine velocity and accelerations (B, [ |e(1:d) of the features for the augmented sample 8x-
We can choose the range of the loss function to be in any compact and connected subset of R under the usual topology.
5

---

Published as a conference paper at ICLR 2023
Theorem 3.4. (Learning bound with structural properties time series and neural network) Let /
be the cross entropy loss function such that ((:-)) E [0, 1]. Then with probability at least 8 over the samples {& (E(ONN] we have
(swp|l(s, aaug)] pl(s,,0) < 2R (haug 0) +
2lg(22/8) +2vd A B,)
|Vgell (8)
N+1
Moreover, have
Rr = (A
A+ Σ B,
81)1 VV
(9)
Theorem 3.4 reveals that there is trade off between the dimension of features of time series sample, the X, and the neural network architecture. A and (e [1:d]] are the bounds for the velocity and accelerations, respectively, which are computed using the interpolation vector. The last term of (Eq.8) constructed by the gradient of the pre-activation Go with respect to an input, the dimension of features of time series sample, and the bound for the velocity and accelerations induced by the interpolation vector. Theorem 3.4 tells us how good is our parameter estimation compared to the optimal parameters. Our bound is governed by three terms: 1) Rademacher complexity using the augmented data, 2) the sample size, and term that depends on the dimension of the features the time series sample d, the structural properties of the time series (A and B,), and the neural network architecture (Vgg). Because our RIM method uses a recursive interpolation between two consecutive features, the order of magnitude of and do not change drastically, and hence do not blow up the bound. Another implication of the A and B being small is that recursive augmentation method decreases the Rademacher complexity as demonstrated by (Eq.9). Decreasing Rademacher complexity simply ends up producing a tighter bound on the parameter space.
3.4 VARIANCE REDUCTION
Suppose that we observe set {so, $1,..., SNl of N + samples from the underlying sample space S. Using our RIM method, we can augment the observed sample s; with a distribution D, which results in the set of augmented samples {si,) A ~ D) for s;. Based on mild assumptions (please refer Appendix A.5) on the regularity of the loss function and the underlying sample space, we have the following results.
Theorem 3.5. (Asymptotic normality) Assume θ is open. Then 1 and dang admit the following Bahadur representation;
N V((s 0.) + (1)
VN + 1(0 10) +
(10)
VN 1(6ang - -.))
aug (s,, 0.) + (1))
0.
Therefore, both Ô and Ó dang are asymptotically normal
V 0.) + (V Eo) and VN + I(0ang 0.) - N Eaug)
(11)
where the covariance is given by
Eo V E.- V
(12)
Daug = En Ean XXT)
where VI(s,0.) Vlang (s, S.0.). As a consequence, the asymptotic relative efficiency 0 compared to is RE
(r(En)
> 1.
Er(San)) j
1Eq.11) deseribes the asymptotic behaviour of the learning parameters. (Eq.12) shows that our recursive time series augmentation reduces the variance of the learning parameters.

---

Published as a conference paper at ICLR 2023
4 EXPERIMENTS
4.1 DESIGN
We show our results time series classification tasks with different time series augmentation meth- ods. We compare the improvements of downstream task performance due to different data augmen- tation methods by enlarging the training set when we train the classifiers. To benchmark our RIM approach, compare results with that achieved by TimeGAN approach Yoon al. (2019). We specifically select TimeGAN as they are known to preserve the temporal dynamics thereby maintaining the correlation between variables across time. Furthermore. with the flexibility of the unsupervised GAN framework and the control offered by the supervised training in autoregressive models, comparing our results against those by TimeGANs can provide us rigorous benchmark against a tested state-of-the-art approach.
We use a relatively small training set with large testing set so that it is more challenging for classifiers generalize and data augmentations are favorable. Note that methods on two classes separately as data augmentations should only preserve properties of that particular class. We use different data augmentation methods on these two classes of time series in our training set as follows: RIM methods can be directly applied time series within each class to generate new time series for that class to enlarge the original training set such that the generated series are close to original series in that class according to Theorem 3.1. For the TimeGAN baseline, we train two TimeGANs separately using time series from each class. Once these two TimeGANs trained, they used generate time series for each class enlarge the original training set. We consider four tasks: the first two use synthetic datasets generated by solving I-dimensional ODEs, and the last two use real-world datasets. We compare testing accuracy using the original data, augmented data with RIM, and augmented data with TimeGANs.
data augmentation
4.2 RESULTS
For task 1, consider solutions to ODEs containing exponential functions, and two classes in our binary classification correspond to the two ODEs with different parameters. For task 2, we consider solutions to ODEs with trigonometric functions. In this setting, ODEs can be thought of as generators that generate time series on which we are performing classification, and ODEs with different parameters invoke different dynamical behaviors their solutions (time series). For each class, we generate multiple solutions using corresponding ODE with different initial values. To make the learning tasks harder, we add random noise to the solution generated by these ODEs.
Test Accuracy for Augmented Data Non-Augmented Data
Class
Class
No Ne-Aug
0.9
RIM
6
Valuo
0.7
4
0.5
0.4
75
100
125
150
175
200
20
70
Epachs
Figure 2: Time series from two classes on the left plot, Test Accuracy for the exponential synthetic ODE system using Convolutional Neural Network with kemel size=3, filter=32, batch sizc= 16, using BatchNorm and Adam optimizer. The test accuracy plot indicates the resulting mean standard deviation from 10 runs.
Task 1: Synthetic data Exponential ODEs solutions
Two ODEs (time series generators) containing exponential functions we use:
class dy
-0.5y² 2 e class 2 dy -0.3y² + 1.5e-

---

Published as a conference paper at ICLR 2023
Task 2: Synthetic data- Trigonometric ODEs solutions
Two ODEs (time series generators) we use:
class
dt 0.6 + 0.5 sin (y)) class 2 dt + cos
17.5
Test Accurary for Augmentedd Data vs Augmentte Data
Cless Class
0.9
RIM
12.5
GAN
0.8
7.5
Test
0.6
0.5
2.5
0
25
50
75
100
125
150
175
200
0 1.0
20
30
40
50
60
70
Time
Epochs
Figure 3: Time series from two classes and Test Accuracy for the trigonometric synthetic ODE system using Convolutional Neural Network with kernel size=3, filter=32, batch size=16, using BatchNorm and Adam optimizer. The test accuracy plot indicates the resulting mean + standard deviation from 10 runs.
Task 3: Real dataset Indoor User Movement from the Radio Signal Strength (RSS) Data This binary classification task from Bacciu et (2014) associated with predicting the pattern of user movements in real world office environments from time series generated by a Wireless Sensor Network (WSN). The input data contains RSS measured between the nodes of WSN, comprising of 5 sensors:
the environment and for the user. Data has been collected during movement of the users and labelled to indicate whether the user's trajectory will lead to change in the room or not. In experiments, we use subset of the data to form small training set to challenge our algorithm. We achieve better and more robust test accuracy than the TimeGAN and the non aug- mented case when using augmented data reflected in Figure 4. Since À is the only parameter used in our RIM augmentation technique, our ablation study paid very precise attention to the choice of the parameter. Under this, we tested different À distributions. Given that we were interested in convex combinations between 24 and -ı-1,A; we had to restrict between and Two ways perform this would be: (1) uniformly distribute the weights while sampling A; (2) concentrating on specific part of À distribution. To address (1), we use (4(0. 1) which is the main test-bed for all the experiments in the current main text. Whereas to address the (2), we perform studies using beta distribution by varying its shape parameters to focus on specific parts of densities. We tested Beta(2, 2), (0.a 0.5) and Beta(2, 5), for which the resulting plots can be found in Appendix C. For all these cases, we observed improvements from using RIM compared to con-augmented training, both in terms of a higher final testing accuracy and with fewer training iterations thereby solidifying the effectiveness of RIM.
Test Accuracy for Augmented Data us Non-Augmented Data
1.00
No-Aug
Class
RIM
0.50
0.65
0.25
0.60
-0.25
Test
0.55
0.50
-1.00
0.40
20
60
Time
Epochs
Figure Time series from two classes and Test Accuracy the Indoor Movement Classification using Convolutional Neural Network with kernel size=3, filter=32, batch size=16, using BatchNorm and Adam optimizer. The test accuracy plot indicates the resulting mean d: standard deviation from 10 runs.
8

---

Published as a conference paper at ICLR 2023
Task 4: Real dataset- Ford Engine Condition We use subset of the FordA dataset from 2008 WCCI Ford classification challenge Abou-Nast & Feldkamp (2007). This dataset contains time series corresponding measurements of engine noise captured by motor sensor. The goal is detect the presence of specific issue with the engine by classifying each time series into issue/no issue classes. We sample 100 time series from FordA to form a small training set to challenge our algorithm and 100 time series for testing. As shown in Figure 5, RIM outperforms the TimeGAN and the non augmented case on the test accuracy.
Test Accuracy for Augmented Data vs Non- Augmented Data
Class
3
Class
0.75
No-Aug
RIM
0.70
Engine Nobe
Test Accuracy
0
-1
0.55
0.45
100
200 300
400
500
20
40
60
80
100
Time
Epochs
Figure 5: Time series from two classes and Test Accuracy for the Ford Engine Classification with Convolutional Neural Network with kernel size=3, filter=32, batch size=16, 16, using BatchNorm and Adam optimizer. The test accuracy plot indicates the resulting mean + standard deviation from 10 runs.
From Figure to Figure 5, we can see that RIM achieves superior performance over the non aug- mented case. Furthermore, RIM also able to achieve better comparable performance than TimeGAN on these tasks without going through the extensive training process associated with GANs, For our experiments, we train the TimeGAN for 2500 epochs (3 hours on Xeon Processors CPU) for synthetic datasets and 5000 epochs (6 hours on Xeon Processors CPU) for real datasets. Visual comparisons of the time series generated by RIM and TimeGAN with the original time series are shown in Appendix B. As expected from Theorem 3.5, RIM has smaller variance and better convergence compared to the non augmented case across all experiments.
4.3 EXTENSION TO OTHER LEARNING TASKS
Section 4.2 demonstrates results for time series classification. In this section, we show that RIM can also he used in other learning tasks including continuous time series forecasting and RL. In continuous time series forecasting, we generally have one historical realization and we leverage that to form training composed of (x.y) pairs where is the previous time steps' data and is the future step target by decomposing the time series into smaller components. RIM can then be used to generate more time series from the unique realization so that we can enlarge our training by adding more (x.y) pairs from the original realizations. We also used RIM to augment state trajectories in RL tasks (please refer to pseudo code in Appendix E.2). Preliminary experiments for continuous time series forecasting can be found in the Appendix D and RL tasks can be found in the Appendix E.
5 CONCLUSION
We developed a Recursive Interpolation Method (RIM) for time series as a data augmentation tech- nique to learn models accurately with limited data. The RIM is simple, yet effective, supported by theoretical analysis guaranteeing faster convergence. Theoretically, we proved that the RIM guar- antees better parameter convergence with reduced variance. Empirically, our methodology outper- forms the current state-of-the- approaches for different real world problem domains and synthetic datasets by obtaining higher accuracy with reduced variance. Because our approach operates on the input time series data, is invariant to the choice of the ML algorithm. The methodology described in this paper can be used to enhance ML solutions to wide variety of time series learning problems.

---

Published as a conference paper at ICLR 2023
ACKNOWLEDGEMENT
Amine is grateful to Hafida lnes Bouzaouia and Douglas Tweed for their help and constructive comments regarding this work. This research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) [grant number 41 1266449), Canada's federal funding agency for university based research.
REFERENCES
Mahmoud Abou-Nast and Lee Feldkamp. Ford classification challenge. Zip Archive http://www..ii messerissclassfication www////mm////ieeeeeeeee conrpppecrppioon.ppp 2007.
Davide Bacciu, Paolo Barsocchi, Stefano Chessa, Claudio Gallicchio, and Alessio Micheli. An perimental characterization of reservoir computing in ambient assisted living applications. Neural 2014.
Computing and Applications, 24(6):1451
Shuxiao Chen, Edgar Dobriban, and Jane H. Lee. A group-theoretic framework for data augmen- tation. Advances Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, pp. -. 2020.
Robert Cleveland, William S Cleveland, Jean McRae, and Irma Terpenning. Stl: seasonal- trend decomposition. Journal of Official Statistics, 6(1):3-73, 1990.
Saverio De Vito, Ettore Massera, Marco Piga, Luca Martinotto, and Girolamo Di Francia. On field calibration of an electronic nose for benzene estimation in urban pollution monitoring scenario. Sensors and Actuators B: Chemical, 125(2):50--757, 2008.
Terrance Vries and Graham W Taylor. Dataset augmentation in feature space. ICLR, Toulon, pp- 2017.
Cristóbal Esteban, Stephanie L Hyland, and Gunnar Rätsch. Real-valued (medical) time series 2017.
generation with recurrent conditional gans. arXiv preprint
Hassan Ismail Fawaz, Germain Forestier, Jonathan Weber, Lhassane Idoumghar, and Pierre-Alain Muller. Data augmentation using synthetic data for time series classification with deep residual networks. ECML/PKDD Workshop AALTD, 2018.
Center for Machine Learning and Intelligent Systems UCI Machine Learning Repository. Air quality pp.
data set. In http:hhhwwweec.......ddde//aaeae
Jingkun Gao, Xiaomin Song, Qingsong Wen, Pichao Wang, Liang Sun, and Huan Xu. Robusttad: Robust time series anomaly detection via decomposition and convolutional neural networks. In MileTS'20: 6th KDD Workshop on Mining and Learning from Time Series, pp. 1-6, 2020.
Frank Hutter, Holger H Hoos, and Kevin Leyton-Brown. Sequential model-based optimization for general algorithm configuration. In International conference on learning and intelligent optimiza- fion, pp. 507-523. Springer, 2011.
Yanfei Kang, Rob Hyndman, and Feng Li. Gratis: Generating time series with diverse and con- trollable characteristics. Statistical Analysis and Data Mining: The ASA Data Science Journal, 13(4):354-376, 2020.
Ilya Kostrikov, Denis Yarats, and Rob Fergus. Image augmentation is all you need: Regularizing deep reinforcement learning from pixels. In International Conference on Learning Representa- tions, 2021.
Michael Laskin, Kimin Lee, Adam Stooke, Lerrel Pinto, Picter Abbeel, and Aravind Srinivas. Re- inforcement learning with augmented data, Advances Neural Information Processing Systems, 2020.
Raquel Prado. Latent structure in non-stationary time series. PhD thesis, Duke University, 1998.
10

---

Published as a conference paper at ICLR 2023
Aad W Van der Vaart. Asymptotic statistics. Cambridge university press, 1998.
Qingsong Wen, Liang Sun, Fan Yang, Xiaomin Song, Jingkun Gao, Xue Wang, and Huan Xu. Time series data augmentation for deep learning: A survey. arXiv preprint 11X14:2002.124788, 2020a.
Qingsong Wen, Zhe Zhang, Yan Li, and Liang Sun. Fast robuststl: Efficient and robust seasonal- trend decomposition for time series with complex patterns. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 2203-2213, 2020b.
Tailai Wen and Roy Keyes. Time series anomaly detection using convolutional neural networks and transfer learning. IJCAI Workshop on A14IoT, 2019.
Jinsung Yoon, Daniel Jarrett, and Mihaela van der Schaar. Time-series generative adversarial net- works. In Proceedings of the 33rd International Conference on Neural Information Processing Systems, pp. 5508-5518, 2019.
Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In International Conference on Learning Representations, 2018.

---

Published as a conference paper at ICLR 2023
Appendix
The code for all the experiments can be found at the following link.
A PROOFS
A.1 PROOF OF THEOREM 3.1
J ift =0 0
if =
Let sign(1) =
ift > 0, Sab
and a distribution with support [0, 1).
0 if a + b"
if < 0
Proof of Theorem 3.1 (1) Note that An a dummy value for mathematical convenience and IT= A; 1 if n < k + 1. We prove this by induction on n. The case n shows that FLA = (>1)a + Xig(Aa)z )ag (1 A1 )1)=1 + Aizo since g\o) 1 and g(\1) A1. We now assume that the inequality holds for n 1 and prove it for n. By construction of Zn.An
Fn,An = (1 - )m) + Ant
(1 An)a + An Σ
II kin--
de
km0
n
X
g(An)an
+
II X1 o(de)ox \i=++1
[in++1
A=
Σ k=
II A
g(Ak)ao
Limk+1
Thus (1) holds by mathematical induction.
(2) Consider the equation ||,n,,,.
&nll-
|l. Then by (1), the first bound can be found by
xx]]ll
EE
II
Agg(AA) - xn]ll
k=0
i=
= EE
{))(Ax)e - AnænJl
k=0 i=#+1
k (1 e) + e e--1x0 ex
(1 - |x + e-~~1|=001 + e||xn|l
n-2
c nーk + en + e)m
km0
Σ a- e)m S (ca-1- -c" - =22mm 3em
where E and m ma { [1x1
Now we prove the second bound. Since
|| FnI
|| - ((1 An 2n + -.
Am Anl|1: In' In-I + In-1 In 1,2
(13)
5 Nn (1 x--1ll + | In In
12

---

Published as a conference paper at ICLR 2023
By recursively applying (Eq. 13), obtain
In Znll - E >)) xk-1ll-
(14)
k=1 i=
By Jensen's inequality and (Eq. 14), we obtain the second bound
[[[[(,,, rn)]] x B - In
E
X xR-1ll
(15)
i=
e" m- m' S min( T-e" m Σ
mmm nem')
c m',
where e E'D] and m' = [ f1 1i-1 Now we show that
E Ex...., An [1 In)lll S m.
AA (In,
xn)1ll 11I
II Ar)(Aa) Antn
k=0 b=
)))(Ae)llExxI1 + |M
1=++1
E
Agg(AR) + Am.m
k=0 i=1+1
... An + A2d3.... .. An(1 - A1) + A.d4 -n A2) + .....n(1 An-1) + An )m
2
(16)
where = [) [|. The first inequality holds by Minkowski inequality.
A.2 PROOF OF THEOREM 3.2
Neural Networks with ReLU Activations. Using ReLU activation functions, neural networks are constructed by piecewise linear functions of an input. Such a neural network go can be formulated by Vgz + b where is input, Vgy is the gradient of g0 along
Lemma A.1. (Structure of partial derivative) Let (ro,... xa)) be one sample drawn from time series and À E 1j.. Then
<fi<j
OzgAA. ad
(2i-1A-1 - xi)
ifi=j
(17)
(TT
(T if>>j
02c 0.
Proof. if <.j, then Ti,A; does not depend on Ay. Hence
If = j. then
Fi,A (1 Ac)xs A =x; + de 11 - -1)).
Hence
(Z -I;) Iz)..
If j. we prove this by induction on i. The case = j+ + shows that
Tjt1AA+1 = > + Ayt1 #j+1 + )5+1(23,A, - Rÿ+1).
Since 0z+1 is not depedent on X and B = (K5- : Iz), we have 00541.2541 ax, Az+1 (15-1 - 2z). We now assume that the inequality holds for and prove for i 1. Since T141.1411 = (1 dit1)aa11 A1t1FEA = Is+1 + A4ta(ii, I:t1) and
db;
II (
k=j+1
13

---

Published as a conference paper at ICLR 2023
021+1.44
1+1
Ast1(
II Xx) - - - II
Ax) - -zz).
0hj
R=++1
k=
As consequence of mathematical induction, the conclusion holds.
We neural networks with ReLU activations and sigmoid function the activation function for the last layer. So, a neural network [a( = a (go (ar)) where go(x)) Vgh x x- b is the pre-activation
signal for the last layer. Recall that we denote by X _ À and À D means that each component X; of À is sampled independently from D.
Proof of Theorem 3.2. Let = (ro, : .d..a. y) and 8 be one sample. We denote the features by ΣΛ = (Iu, 21.A Id,As) and the label by y. Define the matrices
0
821.A1
Ax..Ad 8A1
0A1
x=
8A1
A=ū
x=ū
0zg.
A
0
A=ō
8A2
A=
(18)
Bard.
Thd
1=
0
0
Ø'zd,11
TheaA A==
x==
aA,0A, A=
8 22.22
B;
8X,8A2
x=0
8A,8A2
A=ū
(19)
Paasa dA,0Aa
-==ū
where E Rd denoted by column vector and is vector whose i-th component is 1 and 0 otherwise. Let ((x,, 0) log(fe(zx)) + +11 y) log(1 - fo(xx)). Denote (A) = 1(5x, 0). Using
Taylor expansion of the loss / around X, we have
lo(A) = (o(0 + Σ (1)
A Σ (lo(A)
Aedy + +0(1A131))
(20)
ar A=Ō
2 ij
axda, 1==
Note that
(A)
2 [lg(ff(ZZ)
Olog (1 - fo(xx))
+ (11 (1 1)
0
Ori.A
0folax)
0fo(ax)
- (1 - 1) fo(ox)
fo(ox)
(21)
Ofo(IN) y(1 fo(xx)) + (y 1)fo(xx)
(1 - fo(xx))(o(xx))
0fo(IA)
y v-fo(a>)
(1- fe(=x))fo(xx)
and
af((zx)
(o(g)(α>))
Oxii
FriM
(gx( (100(2)))11
a(Vgg xN) a(go(=x))(1 (xg((x))))
(22)
gV FX Vib dxi.
Oxs (1)))((2))((1
Vig (ooo - (gg(22))))
14

---

Published as a conference paper at ICLR 2023
Since the i-th feature 21,7; of Z depends on (A1....,A.hh we have
(1)(2)
Oeid (1)
Tgg
Tr, dri
d Ori,d 0f((xx) -- fo(xx)
Qdg dma (1 - fo(x)))oo(x
(23)
Oridd Vgh Oh,
y y- fo(x)
(1- - fooox))
d
dri.d Q),
Σ
Vgh e:ly fo(xx))
Thus we have
d Σ
(l)(A)
>;= - () AV
(24)
bdg
1=1
1=
Hence have
d Alo(A)
A3 S
(25)
A=
Note that Iy - (xx))) S 1. Now consider the second partial derivative of the loss function (A).
(lo(2)
0 d
dri,d Vggeily e: fo(x))))
ax,
0%
i=
Prisd (y So(xx))- Thm
V
(26)
OoR. V (1oo(((11o(x))
ax,
We will put (Eq. 26) into (Eq. 20) to calculate the Taylor loss explicitly, For the first term, we have
PALN (u fo(x)))) 0x,0,,
Σ
9o eil
T
=
j=1
l=j
A=Ō
(27)
d
Xv - ( B.V
and for the second term, we have
OUL 0)
Jak.ds Voh en so(x)))
k=
drid
(28)
Σ
Vgla
yg e
T
fo(x) - (O, AV
j=1
=
- -
A
15

---

Published as a conference paper at ICLR 2023
By combining (Eq. 27 and 28), we have
d
Plo(1)
X:
A)" B:V
00,00, A=Ō
(29)
The last inequality holds due
f1(z)(1 - (O) (V
A 20
(30)
By (Eq. 20, 25 and 29), the conclusion holds.
A.3 PROOF OF THEOREM 3.2
We start with a basic inequality frequently used in the proofs.
Lemma A.2. (Supremum inequality) Let f and g be functions which have the same domain and range. Then
(0) suplg(0)] S sup) suppo(() 0
-
(31)
0
Proof.
sup)f(0) suplf(0) ( + s(0)
0
(up([f(0) ((0) + lo(0)))
(0) |(0) + suplg(0)
#
Thus the conclusion holds.
Notation. We assume that the distribution P is parametrized by 8, the sample space S. Let (si be the collection of samples from the distribution P. We set:
0. = argmin Ey-p
N
= argmin
(ss
N +
==
aaug argmin E,wp EANDI 0]]
N Exwd ((s.., 0)]
(32)
p aaug argmin
N
i=
N
R₂, (1 o +) EejnE
sup 0c0
Σ cl((s,, 8)
N
+1+1
laug (8 EALD ((sx, 0)]
Assumption 1. Assume that the loss function satifies Lipschitz condition with respect to the norm.
Proof of Theorem 3.2.
Expp (18 0ang)) Ranp U1 U2 Ug 244 115
16

---

Published as a conference paper at ICLR 2023
where
41 Es-p [1,,, Omnz)l - Ean (es_D|BBDlle (1s,, Onnz)]
= B [RDD (sx) Oaug)]]
1
B BAND (ANDl((S,AAi Osue)l
N
BAND BAwDD oaug)l
u3
N
+
(33)
i=0
N
|x 0-]1
N +1
2=0
BAND (88,,, 0.)] Esp [EA 0.)]
N
U5 E. T
0.)]] Ra-p (s)
We get
U1 Us < sup
Egep [ss Eapp BB
(34)
AE8
where we have
(Ba000_aPPPBB.... BB 0]] = Ea-pl [[, (B 0)]
E E BAND 0) 1(8). 0)]
(35)
< BAND =|].
Hence, from (Eq. 34 and 35)
+ us S PA - s|]..
(36)
By McDiarmid's 's inequality, in terms of the probability
2t2
N+1 Σ EALD(M(SLA,,)) 0.)] EspP BAND (x), 2 exx
(37)
ΣN
N
( )2
24 has the following bound with probability at least 5
log(1/0) 2(N + 1)
(38)
Moreover, Rademacher complexity holds for u2, so have
log(4/8)
< 2 (lang Q) AV
(39)
N +1
with probability at least 5. By (Eq. 38 and 39), get the following inequality
llog44/5) 4/8)
U2 + 24 u4 S 2R (laug o O) + +5
(40)
N +
with probability at least 5 where
RN(1o o θ) E (sup 6E
e B.EEDD((Ss., Σ
N
0=0
where is a Radamacher variable for all E [N].
Since Oang is optimal parameter for
D EAND ((si,, 0)], then
u3 $ 0.
(41)
17

---

Published as a conference paper at ICLR 2023
From (Eq. 47, 40 and 41), conclude that
log (4/6)
Egwp|i(s, 0aug)) Es (ls < 2R ol)
N +1 + Ba
-
(42)
Now will prove that
Rx o Rx + max
LuipBs-D(lls; 8:ll]
(43)
ie (O....))
Since
RNal (0)-Rr(1oO) EEEee
1
Σ edlang (si, 0)
sup
Σ edl(s,, 07
#g
N
N +
N
e,l(s,, 0) 0//1
< E
Eglaug (sés 0) N +
N
==
N
Ed elaug (st, 0) - -15t, 0)}
Esue dE
N
1
2=0
1
(suu o N
E ellaaug 0) 1(si, 01)1
+
N
I sup 9
(laug(S;, (8i, 0) -((s;;
+1
i=0
N
sup age N +
Σ
E EAND (81,2, 1(8i 0)]
1=0
S max iE N)
TLip
- -:ll]-
(44)
We will show BAND|llsE, - *|]] < 2dem. Let each sample s (or 8;) E RRd+1 x 1.... kj. Then s (To, F1,-.... ., Tel v). By the augmentation method, recursively applying convex combinations, have 8\ (No, 31,,;; ... (d..a. v). By Theorem 3.1 (Eq. 4), each E,,..., (1( "
Ill < 2A,m. Hence
Exwollsia - 8/] - Σ Exwolloe - x,
1=1
(45)
< 2E (()1 + d2 + + Xa)]m
<
2mde
where E.D), d+ is the dimension of the time series sample, and max [|::1]].
A.4 PROOF OF THEOREM 3.4
Let A = sup.es A|| and B, = sup.es B|] | Then for each 8 ~ P, we have:
|| 1 A and | S Br,
where and B, depend on s for e [d]. Please refer to (Eq. 18) and (Eq. 19).
Proof of Theorem 3.4. The proof almost same with the proof of Theorem 3.2. We only de- scribe the part should be replaced in order to prove Theorem 3.4. Similarly, We start with the
18

---

Published as a conference paper at ICLR 2023
decomposition of the equation follows
E Eg-p [(,,, aaug)] P ((s, u1 + u2 +
+ u1 us
where u1, u2, Ma. u4 and us defined in (Eq. 33). (Eq. 35) in the proof of Theorem 3.2 will be replaced with the following
Eg-p(((s, 0)] E [B (s 0)] Eg-p (1, 0) Ex (1sx,, 0]]
= Bgup BAND (ss, 0) 1s52. 0)]
(46)
S Vdl A+
Bi V gool-
Hence, from (Eq. 34) and (Eq. 46), we have
us S 22vd 2vd
A
Σ B: ||V
(47)
The last two lines of (Eq. 44) will be replaced with the following
N
d Σ 1=1
sup
Σ
EA-D(M(81,, 0) [(*; 0)]
S vd A+
<va(A+
B,
V
(48)
N
AE
Theorem 3.3 guarantees the inequality. Thus the conclusion holds.
A.5 PROOF OF THEOREM 3.5
Note that the sample space is contained in Rd+1 x {0,.... .o....,k k) and the param- eter space C IR" where {0..... ,k} the label set. Under regularity conditions, is well known that 0 is asymptotically normal with covariance given by the inverse Fisher information matrix. We will see that Aau is also asymptotically normal with the covariance. Suppose that we observe set (So, .1,...,88 of N + samples from the underlying sample space S. Using our RIM method, we can augment the observed sample s; with distribution D, which results in the of augmented samples {st.A X ~~ D) for s;. We then decompose [ X D} into disjoint union of some sets S; such that E S:.
Asymptotic Results.
Assumption 2. (Disjointness) the sample space S the disjoint countable union of all possible augmented sample spaces.
Consider the probability space is (PP where F is sigma algebra and P the corresponding probability measure. Let be a measurable function from S to IR for some E N. For each sample $ s= (zx E S, define ()s) = Elu S:] where E S. Let's assume that we observe + samples {so, ,1,...,88. from the underlying sample space. Then by Assumption the underlying sample space is decomposed into , S; i,e S = S This is well-defined by Assumption 2, and is a measurable function. Let μ4 be the expectation value of μ over 5,.
Lemma A.3. (Effects of the average function) With notation as above, the following holds.
1. The law of total expectation: Ee>P [M] Ea-p[T] [T]
2. The law of total covariance: Cov E (Cov(([])) + Cov
Proof. From the disjointedness of the underlying sample space, for each S, we have S, for
some i E N
EgpP
E
TT| Eg-p [R [4| [S:]] Ea-p Ea-r(u y 1T mi] Ea-p/y).
(49)
Thus the law of total expectation and the law of total covariance naturally follow.
Under mild assumptions for given loss function, we show that the average loss function lang inher- its the same properties from the augmented loss function, where lang(s, 0) EX-D (1x,, 0)).
19

---

Published as a conference paper at ICLR 2023
Assumption 3. (Regularity of the loss function) For the loss function ((=, ((,8), we assume that
1. For the minimizer 0, of the population risk and any 0, we have
sup
Eg-p (1s 0)] > Ds 0.)]
{ [0--8.12e (E@)
2. For every > O, there exists function 1' (P) such that for almost every and for every 01, 02 E (Nb,, <), we have
((s,0;) 1(s,02) - (s) 02l
3. Uniform weak law of large number holds
Σ Us, - Baap +0
sup e
+
4. For each θ, the map 8 + I(s, 0) measurable
5. The map 0 - [(8, 0) is differentiable at 0. for almost every
6. The map 0 + E.- ((,, 0)] admits second-order Taylor expansion at 0, with non- singular second derivatives matrix Vo.
Proposition A.4. For the pair (faug )a the following property holds. For every > 0, there exists function laug (P) such that for almost every s and for every 01. 02 (0.. c), we have
Maaux( (s,u 01) - laug (2) S Gaug (s)]10) - 021l
Proof. Since we have
Mang (8, 01) laug 02) = |EAD (o 01) ((5x, 02)1
< EAnD ((s,,
((xx, 01) - ((SA, 02)l
s EADD (2' (s)) 02 l]
(' 8211,
the conclusion holds and laug EX~D(('(ss)),
Proposition A.5. For the pair (Daug, lugg) the following property holds.
lug (si, 0) Banpllaug 8)] 0
sup 8E
Proof. We have
N
sup 00
laug (8i, 0) - Eeup Mang 0)]
N +
1=0
Σ
E ExDD (AnD((SE,x.00 Egup BBwp(EEND 0]]]
sup #E0
(1)
where the last equality holds because the underlying sample space S is the disjoint countable union of all possible augmented sample spaces.
Proposition A.6. For the pair (Maug. laug). the following property holds. For each θ the map lang (s, 0) is measurable
Proof. lang is measurable since is measurable and lang (s, EA-D (((ss,)).
20

---

Published as a conference paper at ICLR 2023
Proposition A.7. For the pair (Gaug: laug), the following property holds. For each 6 in O the map + (s, 0) is differentiable
Proof. we have
laug (s, 02 + 60
lang (s,
0.) 5t (saug(s,0,)
lim
|8||
EAND - 0 - (V(x,
lim
6-0
||||
ExnD 0++)) [11
lim
6-0
||ó||
BAND
lim
8-0
|[8||
Let us define
ExND (1
Fg(8)
|| G(s) = EX-D(l(((sx) + (.)(s)0.))] for all s E S
Since [Fa(s)) S G(s) for all by Lebesgue's dominated convergence theorem,
aug (8, 0. + 5) - lang (8, 0.) 5t Vlaug (s, 0.)
lim
6-20
|0||
EAND 1ss,
lim
|8||
18x, -1o0
Ex-D
lim 6-20
=
|6
where the last inequality holds because 1 is differentiable.
Proposition A.8. The map Eg-r(lang / (s, 0)) admits second-order Taylor expansion 0. with non-singular second derivatives matrix Vo.
Proof. By the total law of expectation of Lemma A.3, we get
(Langl 0)] = E (1, (ss 0)).
Hence the conclusion holds by the bullet from Assumption 3.
Combining all the Propositions [A.4 A.8] with some results shown in Van der Vaart (1998), we prove Theorem 3.5.
Proof of Theorem 3.5. The results for have already been proven in (Van der Vaart, 1998, Theorem 5.23). The Propositions [A.4 A.8] guarantee that (Van der Vaart, 1998, Theorem 5.23) can be applied to the pairs (0.,l) and (Qang, lang). Therefore Öang is asymptotically normal and satisfies (Eq. 10).
Let X 0.) Vlang (8, 0.). Recall that Cov Egsp (Cov((())))) Cov by
21

---

Published as a conference paper at ICLR 2023
Lemma A.3. Let's consider and μ to be Vi(s, and Vlaug(s, 0x), respectively. Then
Eg Cov(V((s,O.))
and
Eang Cov(VVaugge 0.)).
Hence have
Eo- Eaug CoV Cov Es- (TT)(
Egwp EEN (1 )
Eanp P [1/1 TD)(( " π
(50)
Eowp p [V((s, Vlaug (s, 0. (.))( 0s))) A
Egpp E, Eaup [XX*
Thus we get
Lang = Eo - Egup [XX]]
(51)
Since tr tr(Eepp(XXX 2 0, we have RE tr(Eu) (r(Faux)
B VISUALIZATIONS OF RIM TIMEGAN GENERATED TIME SERIES
This sections shows visualization time series generated by RIM and TimeGAN. We plot samples from original time series. RIM generated time series and TimeGAN generated time series from both classes for Synthetic exponential ODE classification in Figure 6.
pains
10
Original
Origina
RIM
GAN
GAN
Velue
0
25
50
75
100 Time
125
150
175
200
25
50
75
100
125
150
175
200
Time
Figure 6: Visualization of exponential ODE classification series generated by RIM and TimeGAN against original series each)
C ABLATION ANALYSIS
Teal Accuracy Augmented
Cata
TAET Acturey for Augivineed DVs velee Awg.ieoaeadddd
CAS
Figure 7: Test accuracy over epochs for synthetic data with Exponential ODE with À sampled from beta distributions different scales (Left) Beta(0.5, 0.5), (Middle) Beta 2), and (Right) Beta(2,
22

---

Published as a conference paper at ICLR 2023
Text Accurcc for Augmarcaa Data fon-kspmentes Deto
Tox. Accionss for dugir enued Duta NenvAggrarte Cala
0.76
C4
040
Figure 8: Test accuracy over epochs for Indoor dataset with sampled from beta distributions with different scales (Left) Bela(0.5, 0.5), (Middle) Beta(2, 2), and (Right) Beta(2, 5).
D TIME SERIES FORECASTING
In this section, we consider time series forecasting task where we use previous n time steps data to predict next times step data. We compare performances of regression model trained with original dataset and regression models trained with augmented dataset using RIM.
Predicting Stock Price Movement.
This regression task consists of predicting the next day SPY500 index Open price from historical SPY500 index using data from July 2008 to December 2012 training data and data from January 2013 to March 2014 as testing data. The input data contains the last days' historical Open, Close, High, Low, and Volume of the SPY500 index. After predicting the next day's Open price, take long position if the predicted next day Open is larger than today's Open, short otherwise. On comparing the results on the test data for the augmented case and the original case, we observe that the proportion of profitable trading signals is higher in the uugmentation-trainee model as observed also calculate the trading system's CAGR (Compound Annual Growth Rate) which we observe to be higher in the augmentation trained model. The test loss plot shows that the MSE for the augmentation trained model is consistently lower than the non-augmentation trained model.
in Figure 9. Using these trading signals,
Stuck
dstart
Aug
3.0002
0.00
20008
0.0000
20
Figure 9: Profitable trading signals (left), test set CAGR (middle), test set MSE (right) for the SPY500 Dataset using an LSTM model with 2 LSTM layers (200 neurons). 2 dense layers (100 neurons), Ir=le-4, batch sizc=16. The plots indicate resulting mean + standard deviation from 10 runs.
Predicting Air Quality.
The restricted air quality dataset contains 1200 instances of hourly averaged responses from array of 5 metal oxide chemical sensors. This is time series regression task where the target is the next time step's CO concentration. The input data contains the last six time steps' 10 features as used in De Vito et al. (2008) and for Machine Learning & Repository. Figure 10 shows that the test MSE of the augmentation trained model remains lower than the non-augmentation trained model during the training epochs validating our claims about the robustness of the approach. Accordingly. the proportion of correct predictions of CO up/down also remains higher for the augmented case.
23

---

Published as a conference paper at ICLR 2023
AP Quaily Supest
2.81
10-
6.00
Figure 10: Test accuracy (left) and Test MSE (right) for the Air Quality Dataset using an LSTM model with 2 LSTM layers (200 neurons), dense layers (100 neurons), Ir-le-4, and batch size=16. The plots indicate the resulting mean + standard deviation from 10 runs.
E
TIME SERIES RL
In this section, we consider portfolio management task using reinforcement learning. More specifi- cally, we compare performances agents (DPG) trained with original state trajectories (price evo- lution) and augmented state trajectories using RIM.
E.1 DATASET
closing price evolution
220
200001
2005
2011
7011-00
2011-10
2012--21
291206
2012-09
2912410 201301
Date
Figure
Evolution stock prices.
For RI, the data comes from the Quandl finance database that has daily data. Our RL models are trained over 600 trading days from 2010-08-09 2012-12-23 and tested over 200 trading days from 1012-12-26 to 2013-10- 11 on portfolio consisting of ten selected stocks: American Tower Corp. (AMT), American Express Company (AXP), Boeing Company (BA). Chevron Corporation (CVX), Johnson & Johnson (JNJ), Coca-Cola Co (KO), McDonald's Corp. (MCD), Microsoft Corporation (MSFT), AT&T Inc. (T) and Walmart Inc (WMT). To promote the diversification of the portfolio, these stocks were selected from different sectors of the S&P 500, so that they are uncorrelated as much as possible as shown in Fig. II.
24

---

Published as a conference paper at ICLR 2023
E.2 RL PSEUDOCODE
Algorithm 1: RIM: RL Training
input Observed time series data S = {so. 87) where = (24, for Rd and yi R for
C [0 T).
Initialize purameter for the policy network. Y epochs, and a distribution 2> with support [0, 1).
for e = 1 to Y do
Initialize A= (A1...., Ar) with D // Initialize interpolation coefficients vector
Augmented Path Simulator Generate augmented trajectory Sx= {eo. 81. 81, ST, where
St.A, (2,, Bi,A.) for to
(.18t,,e))
~ p(. Ae.
Si + S Sx Add transition to the replay buffer
UpdateCritic(S:)
UpdateeActor(Ss).// Data augmentation applied the samples actor training as well
end
end
E.3 POLICY DEPLOYMENT
In our RL experiments, the investment decisions to rebalance the portfolio are made daily and each input signal represents multidimensional tensor that aggregates historical open, low. high, close prices and volume. should be noted that our training and testing include the transaction costs (TC). We used the typical cost due to bid-ask spread and market impact that is 0.25%. We believe these reasonable transaction costs for the portfolio trades.
1.2
1.00
1.00
5.00
Figure 12: Training and testing results for DPG (above) and DDPG (below). The plots indicate the resulting mean = standard deviation from 20 runs with different seeds.
25

---

Published as a conference paper at ICLR 2023
F HYPERPARAMETERS
We note that the primary objective of the conducted experiments to show that RIM can improve model performance. Therefore, in all experiments, instead of finding optimal set of parameters for augmented and non-augmented trained model, we compare the performance of augmented trained model and non-augmented trained model with the same hyperparameter configuration. We demon- strated in Section that RIM indeed improves model performance with same hyperparameter con- figuration. However, these improvements robust to other hyperparameters? To answer this ques- tion, we conducted sensitivity analysis for two supervised learning tasks (Indoor user movement classification and Air Quality regression) and RL task (Portfolio Management).
F.1 HYPERPARAMETER SENSITIVITY FOR SUPERVISED TASKS
For Indoor movement classification task, conducted the same experiment in Section 4 with 9 dif- ferent hyperparameter configurations as shown in table and observed that for all the cases RIM out- performs Non-Augmented case (with smaller mean test loss and higher mean test accuracy) which solidifies our claim of enhancement observed in model performance when use RIM.
Table I: In the table, the Test Loss and Test Accuracy are mcan(standard deviation) over runs for 50 epochs for varying Filters and Kernel size for Indoor User Movement Classification Task
Filters
Kernel Size
Test Loss RIM
Test Loss NoAug
Test Acc RIM
Test Acc NoAug
3
1.09 (0.43)
2.72 (0.95)
0.71 (0.04)
0.63 (0.02)
16
4
0.90 (0.45)
1.70 (0.69)
0.76 (0.04)
0.68 (0.04)
16
5
0.86 (0.19)
1.00 (0.19)
0.72 (0.01)
0.60 (0.05)
32
3
1.37 (0.31)
3.07 (0.67)
0.72 (0.02)
0.62 (0.05)
32
4
1.75 (0.71)
2.70 (1.20)
0.74 (0.03)
0.68 (0.03)
32
5
1.70 (0.70)
2.57 (0.85)
0,66 (0.08)
0.64 (0.06)
64
3
2.99 (2.17)
3.31 (1.36)
0.68 (0.07)
0.68 (0.06)
64
4
2.60 (0.70)
4.80 (4.30)
0.73 (0.03)
0.68 (0.02)
64
5
3.30 (0.74)
4.90 (4.08)
0.66 (0.03)
0.60 (0.06)
For Air quality regression task, we again conducted the same experiment as in Section 4 with 8 different hyperparameter configurations as shown table 2. Here too, we find that RIM outperforms Non- oug-eneeete case (with smaller mean test MSE and higher mean test accuracy) which confirms claim of enhancement observed in model performance when we use RIM.
Table 2: In the table, the MSE and Accuracy the mean (ttandrd deviation) over 10 runs for 50 epochs for varying Epoch Initial, Batch Size, LSTM Layer and Layer for Air Quality Regression Task on Test Data
Epoch Init Batch Size LSTM Layer Dense Layer MSE RIM MSE NoAug Ace - RIM Acc NoAug
100
50
5.34 (0.11)
5.43 (0.25)
0.63 (0.03)
0.47 (0.08)
80
40
5.40 (0.08)
5.44 (0.04)
0.63 (0.05)
0.51 (0.02)
150
50
5.59 (0.01)
5.74 (0.35)
0.66 (0.03)
0.53 (0.10)
200
100
5.23 (0.10)
5.55 (0.19)
0.64 (0.02)
0.51 (0.06)
10
16
150
100
5.32 (0.05)
5.48 (0.11)
0.57 (0.02)
0.51 (0.35)
64
200
100
5.59 (0.07)
5.67 (0.23)
0.65 (0.02)
0.52 (0.03)
10
16
100
50
5,44 (0.24)
5.49 (0.38)
0.59 (0.05)
0.56 (0.07)
15
16
100
50
5.56 (0.08)
5.60 (0.16)
0.65 (0.01)
0.56 (0.02)
26

---

Published as a conference paper at ICLR 2023
F.2 HYPERPARAMETER SENSITIVITY FOR RL TASK
The hyperparameter space is represented by hypercube: the more values it contains the harder it to explore all the possible combinations. To efficiently find the optimal set of hperparameters, we explored the hyperparameter space using Bayesian optimization (BO) Hutter et al. (2011). Table shows the range of values for the hyperparameters used during the training and validation phase. The learning rate controls the speed at which neural network parameters are updated. The window used to allow the deep RL agents to utilize a range of historical data values to relax the Markov assumption. We allow the use of 2 days up 30 days of historical data. The number of filters and kernel strides are the hyperparameters for the convolution neural networks. It is important to carefully optimize these parameters in order to capture the best feature representations used by the policy networks. Finally, the training and testing sizes may also impact the RL performance. So, also consider them hyperparameters.
Table Hyperparameters used by our RL algorithms.
Parameters
Bounds
Type
Learning rate (Ir)
10 10-9-5.10
9-5.10
Discrete
Trading cost (tc)
2-30
Diserete
Number of filters (nf)
2-52
Discrete
Kerne Strides (ks)
2-10
Diserete
Window
2-30
Discrete
Training size (train)
20-500
Discrete
Testing size (test)
5-100
Discrete
15
Comulative cadze
Figure 13: Hyperparameter sensitivity DPG: the vertical axes
models we evaluated by its index.
The detailed hyperparameter configurations each index refers to are listed in Tables 4 and 5. The horizontal axis shows the cumulative total validation return. The blue line shows the validation performance for DPG without augmentation. The orange line shows validation performance for DPG using RIM. The worst to best models are ordered from bottom to top.
27

---

Published as a conference paper at ICLR 2023
Table 4: Sensitivity analysis for DPG configurations without augmentation
lr
tc
nf
ks
window
train
test
validation cumulative return
0.001
0.0225
[48, 20, 1710]
9
19
50
10
1,1563
1.0
0.0125
[50, 38, 1340]
3
6
500
100
1,1509
0.0005
0.0125
[14, 6, 1300]
4
16
50
50
1,1600
0.01
0.0125
[28, 28, 210]
4
13
100
10
1.1380
0.01
0.0175
[16, 2, 1080]
3
9
50
5
1.1597
0.001
0.0125
[18, 12, 1290]
8
16
50
50
1,1542
1.0
0.0175
[20, 20, 1090]
6
16
50
10
1,1507
0.0005
0.0075
[18, 2, 1270]
3
16
50
50
1.1614
1,00E-05
0.0225
[10, 10, 1310]
8
9
50
50
1.1591
0.1
0.0225
[10, 10, 770]
2
17
200
20
1.15470
5,00E-05
0.0075
[12, 16, 1300]
7
17
50
50
1.1558
0.05
0.0125
[16, 10, 1090]
3
50
1,1551
0.0001
0.0075
[24, 12, 1080]
5
15
50
5
1,1544
0.1
0.0175
[44, 2, 1500]
2
11
20
10
1.16082
0.005
0.0025
[2, 42, 270]
8
16
500
50
1,1374
5,00E-05
0.0175
[24, 10, 1090]
2
11
50
5
1.1537
1.0
0.0175
[32, 12, 1490]
2
20
20
1.1574
0.5
0.0075
[46, 6, 950]
4
20
20
1.1589
0.0001
0.0025
[30, 26, 1190]
7
500
100
1,1510
0.01
0.0025
[50, 16, 940]
7 15
20
5
1.1501
28

---

Published as a conference paper at ICLR 2023
Table 5: Sensitivity analysis for DPG configurations with augmentation
lr
tc
nf
ks
window
train
test
validation cumulative return
0.005
0.0175
[36, 36, 1960]
4
9
20
10
1.1513
0.0001
0.0025
[22, 14, 1270]
9
10
50
50
1.1582
0.1
0.0125
[40, 22, 880]
3
6
20
10
1.1503
1,00E-05
0.0075
[8, 16, 760]
2
6
200
20
1.1521
0.005
0.0175
[44, 6, 1500]
5
6
20
20
1.1593
1,00F-05
0.0025
[2, 2, 1310]
19
50
50
1,1619
0.001
0.0125
[36, 6, 960]
3
17
20
20
1.1560
0.0001
0.0075
[22, 6, 770]
9
13
200
20
1.1573
1,00E-05
0.0025
[22, 2, 1320]
2
19
50
50
1.1619
0.5
0.0225
[28, 2, 970]
7
12
20
5
1.1581
0.5
0.0125
[40, 14, 960]
9
18
10
1.1547
0.01
0.0175
[32, 16, 760]
5
200
20
1.1478
0.01
0.0025
[24, 10, 780]
7
16
200
5
1.1517
0.005
0.0025
[46, 12, 770]
9
19
200
20
1.1502
0.01
0.0225
[48, 44, 190]
4
10
100
50
1.1450
1.0
0.0175
[28, 8, 960]
6
13
20
5
1.1541
0.005
0.0125
[36, 2, 1500]
3
15
20
5
1.1607
0.005
0.0225
[18, 8, 1270]
5
9
50
1.1585
1.0
0.0125
[8, 1260]
7
8
50
50
1.1604
1.0
0.0025
[28, 2, 1250]
2
7
50
50
1.1607
29


Reviews:
[{"review_id": "rQpyBL7Gl3", "reviewer": "Reviewer_aEoM", "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 work propose a new time series augmentation technique, called Recursive Interpolation Method (RIM), to learn accurate models with limited data. They provide theoretical analysis with the guaranteeing faster convergence with reduced variance. Empirically, they show that the proposed method improves the performance on classification tasks over the SOTA on both real-world and synthetic datasets.", "summary_of_the_review": "This paper studies an interesting and important problems, but the proposed method lacks motivation ", "strength_and_weaknesses": "Pros:\n1. Data augmentation is critical for time series analysis. We have seen a lot of research for developing new data augmentation methods to improve the performance of downstream tasks, but few of them provided theoretical analysis. This work opens the door for analyzing the generalization capability of time series data augmentation. Besides, this work also presents a new data augmentation method based on recursive interpolation.\n\n2. Empirical studies verify the efficacy of the proposed data augmentation method on both synthetic and real-world datasets.\n\n\nCons:\n\nMajor:\n1. The proposed RIM in sec 3.1 lacks motivation. It would be better to provide more motivational examples showing how it can benefit the time series related tasks. Most of the existing data augmentation methods aim to enhance the temporal dependencies, but the proposed RIM in Eq 1. is temporal independent. I am unclear on how  RIM can help avoid overfitting and capture better temporal patterns. Since the proposed data augmentation method is temporal independent, it can be generalized to any tabular data. How does this method connect to time series data? Additionally, RIM in eq. 1 is a kind of mixup of two consecutive dimensions. How to justify its correctness, especially when there exists dependency over two non-consecutive dimensions?\n\n2. The theoretical analysis is interesting but needs more clarification. It is clear to me the effect of sample size and the distance between the original time series and the augmented one. Still, it is unclear how to control and decrease the Rademacher complexity for the augmented data. Eq.6 does not guide how the proposed data augmentation method can reduce it.\n\n3. The empirical studies are not promising. First, I am not sure why the authors do not compare with other data augmentation methods, but the time series generative model, TimeGAN. Besides, there are a lot of novel generative time series models, like the generative Fourier model,  the diffusion model beat TimeGAN with a margin. Finally, since the major contribution of this work is about the generalization capability, it would be more interesting to present the empirical analysis for each complexity term to show its vacuousness.\n\n4. The proposed method is limited to point-wise time series classification, and it can not generalize to the more common settings, like window-wise classification or conventional time series forecasting problems. This might reduce the impact of this work.\n\nMinor:\n1. what's the definition of 'dem'?\n2. It would be better to present assumptions 3 and 4 in the main content to make the whole manuscript self-contained.", "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": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "jucYijeWcAw", "reviewer": "Reviewer_Ftkr", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "I have reviewed this paper in another conference.\nThis paper introduces a technique called Recursive Interpolation Method (RIM) for time series augmentation. The augmented samples preserve the original inherent time-series dynamics. Theoretical analysis is given to guarantee the test performance. Experiments are conducted on multiple time series tasks such as classification, regression and RL.", "summary_of_the_review": "A good paper with theoretical insights and experimental support. But a flaw in the assumption of the model.", "strength_and_weaknesses": "Pros:\n\n(1) The motivation to perform data augmentation that preserves the original inherent dynamics makes sense.\n\n(2) The methodology is clearly described and theoretical analysis is given.\n\n(3) Experiments are extensively conducted and it is shown that the model with data augmentation clearly outperforms the model without augmentation.\n\nCons:\n\n(1) Same to my concerns in the previous review, the method in this paper assume that the data is stationary, which is not the case in real world. Though it is possible to use a large lambda close to 1 for non-stationary data, I still feel the assumption made in the method is a drawback.", "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": "Z7c1lbpWyuk", "reviewer": "Reviewer_LCdk", "summary": "", "questions": "", "limitations": "", "rating": 10, "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 a time-series augmentation method names Recursive Interpolation Method (RIM), designed to boost performance in learning tasks with limited data. The method is solidly based on a theoretical based, and the analytic results are constructed with numerical experiments that prove the goodness of the technique.", "summary_of_the_review": "The authors present a time-series data augmentation approach based on theoretical results. The work is solid and checked against experiments. This kind of work is in stark contrast to most of what can be seen in the field: ad-hoc pipelines with some experimental testing where it is almost impossible to assess the generality of the results. ", "strength_and_weaknesses": "Strengths:\n1. The pipeline is based on a principled approach.\n2. Stochastic type loss functions are used (cross-entropy) instead of deterministic ones.\n", "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": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "DyRaFWlyVb", "reviewer": "Reviewer_oRTw", "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": "Training set augmentation has been found to be an effective way of improving performance on downstream tasks such as classification accuracy, particularly for computer vision problems. Augmentation has also been considered for the time series domain, although much of the contributions in this space implicitly make assumptions on the structure of the time series, and can thus be less widely applied. It is also difficult to provide guarantees on the quality of generated samples using conventional methods. In this work, the authors propose a recursive approach for generating time series that follow the same dynamic process as a (possibly) limited number of observed time series, for which it is also possible to obtain guarantees on quality under certain conditions. The authors demonstrate how the approach compares favorably to augmentation methods leveraging GANs, while also being more computationally efficient.", "summary_of_the_review": "The contributions featured in this paper are interesting, and provide an intuitive way of overcoming sparsity in time series datasets. However, I believe there is still considerable room for improvement in the paper, and I encourage the authors to consider rewriting certain sections in order to include further illustrative example and ablation studies that will make the paper more complete and insightful to both readers and practitioners.  \n\n\n** Post-rebuttal comments **\n\nThank you for your replies, and the updates to the paper. I am raising my score to a '5', although I still think this paper would benefit from another round of reviewing. The ablation study included as part of the rebuttal is interesting, but I believe there could still be more clarity on aspects such as settings where the RIC approach doesn’t perform as well.", "strength_and_weaknesses": "- The recursive approach detailed in this work is intuitive, and I appreciated how it presents a fairly straightforward solution to a problem that manifests itself in any domain where there is risk of overfitting to limited representative time series.\n- The paper is mostly well-written, although there were several instances where I wished for more clarity or detail. I especially think that the authors should consider including more illustrative examples within the main paper itself such that the concepts become easier to follow. It is otherwise difficult to sometimes follow which time series structure the authors are referring to throughout the text, and it could also better illustrate the circumstances under which the guarantees put forward in the paper would not hold.\n- One of the main motivations for undertaking this work was that existing augmentation techniques are problem dependent, so I would have liked to see more concrete examples of this. I consider such insight and analysis to be a crucial contribution of the paper, and yet it currently feels very underdeveloped.\n- Another appealing aspect of this work is that the augmented time series can be leveraged in multiple different downstream tasks, which are highlighted in the introduction to the paper. However, besides a small concluding interlude in Section 4.3, experiments on next step prediction and reinforcement learning are left entirely to the Supplementary Material, with the focus in the main paper being exclusively on classification tasks. While there is a nice variety of synthetic and real-world classification problems here, featuring a different task entirely would have given a more complete picture.\n- The actual discussion of the results is also quite surface-level, and I would have liked to see more insights provided throughout as well as ablation studies that tweak different aspects of the augmentation procedure.", "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": {}}]

Author rebuttal:

Decision: Accept: poster