CNN Kernels Can Be the Best Shapelets
Authors: Eric Qu
Yansen Wang
Xufang Luo
Wenqiang He
Kan Ren
Dongsheng Li
Venue: ICLR 2024 poster (2024)
Keywords: Shapelet
Covolutional Neural Network
Time-series
Primary area: representation learning for computer vision, audio, language, and other modalities

Abstract: Shapelets and CNN are two typical approaches to model time series. Shapelets aim at finding a set of sub-sequences that extract feature-based interpretable shapes, but may suffer from accuracy and efficiency issues. CNN performs well by encoding sequences with a series of hidden representations, but lacks interpretability. In this paper, we demonstrate that shapelets are essentially equivalent to a specific type of CNN kernel with a squared norm and pooling. Based on this finding, we propose ShapeConv, an interpretable CNN layer with its kernel serving as shapelets to conduct time-series modeling tasks in both supervised and unsupervised settings. By incorporating shaping regularization, we enforce the similarity for maximum interpretability. We also find human knowledge can be easily injected to ShapeConv by adjusting its initialization and model performance is boosted with it. Experiments show that ShapeConv can achieve state-of-the-art performance on time-series benchmarks without sacrificing interpretability and controllability.

Published as a conference paper at ICLR 2024
CNN KERNELS CAN BE THE BEST SHAPELETS
Eric Qu', Yansen Wang", Xufang Luo", Wenqiang He", Kan Ren', Dongsheng L.i² 'University of California, Berkeley, Miirosooft Research Asia, University of Science and Technology of China, "ShanghaiTech University ericqu@berkeler wtw.beereeeeee.......eeee edu,
(yansenwang, xufluo, dongsheng. . 11 ww com, wenqi venqianghe@maii 1. ustc.uttuee edu. en, renkan@@shanghaiitech. www.eeitiieeeee...eeee.eeeeeeeee
ABSTRACT
Shapelets and CNN are two typical approaches to model time series. Shapelets aim at finding set of seq-seeueeces that extract feature-based interpretable shapes, but may suffer from accuracy and efficiency issues. CNN performs well by encoding sequences with a series of hidden representations, but lacks interpretability. In specific
this paper, demonstrate that shapelets are essentially equivalent type of CNN kernel with a squared norm and pooling. Based on this finding, we propose ShapeConv, an interpretable CNN layer with its kernel serving as shapelets to conduct time-series modeling tasks in both supervised and unsuper- vised settings. By incorporating shaping regularization, we enforce the similarity for maximum interpretability. We also find human knowledge can be easily injected to ShapeConv by adjusting its initialization and model performance is boosted with it. Experiments show that ShapeConv can achieve cate-of-the-ard performance on time-serics benchmarks without sacrificing interpretability and controllability.
INTRODUCTION
In the realm of machine learning, interpretable time-series modeling stands as a pivotal endeavor, striving encode sequences and forecast in manner that resonates with human comprehension. Among an array of early methods to distill interpretable features from sequences, shapelets (Ye & Keogh 2009) have garnered significant attention, finding applications in diverse downstream tasks. These shapelet are discriminative sub-sequences culled from the primary time series and the minimal distance between shapelet and all conceivable sub-seue subb-sqqences of the raw input ascertained, yielding features that sigmify shapelet's imprint on sequence. The allure of shapelets lies in their capacity discern local discriminative patterns inherent in the data. However, conventional shapelets grapple with inefficiencies, attributed to their exhaustive search demands and elevated time complexity.
As the new era of deep learning comes, more and more works seek fit the sequence with a high dimensional con-co function using deep neural networks such as RNN (Guo et al. (019), CNN (Franceschi et al. 2019), Transformer (Wu al., 2021 Qu et al. 2022 Cheng et al.) 2023), etc. to model the time series. These deep-learning bheepleeaniin--buseed based methods have attracted much more attention than shapelets, thanks to their great performance when the number of data is sufficient, but they more likely to overfit when the signal-to- ratio is relatively low and the data are scarce. Also, the representations (often called hidden representations) are almost impossible to interpret and control due to the black-box nature of neural networks. While subsequent research endeavors (Ma et al. 2020b Li et al. 2022; He et al., (202)) are proposed aiming at fusing the interpretability of shapelets and the promising performance of deep methods, they often fail with striking a harmonious equilibrium between performance and interpretability.
In this paper, we aim to seamlessly inject the interpretability of shapelets into the convolutional layer while retaining the ad advantages and characteristics of both. Despite the apparent disparity between shapelets and deep models time-series modeling. for the forward process, we first theoretically prove that extracting features with shapelets can be equivalently conducted by passing the input time
This work was done during Eric Qu and Wenqiang He's intersshi at MSRA. Correspondence to: Xufang Luo (xufluo microsoft.com Dongsheng Lii (dongsheng.hl li mierosoft.com..

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Published as a conference paper at ICLR 2024
series specific convolutional layer (I-layer CNN) with squared norm and pooling. This finding provides us the basis for combining shapelets with deep models.
To implement the equivalence, we devise ShapeConr, CNN layer wherein its kernel functions as shapelets, adeptly and interpretably addressing the time-serics modeling challenge. We introduce several ingenious designs to make ShapeCony effective in practice. During the optimization process, due to the difference of candidate space, the subsequences derived via gradient-basee techniques might diverge significantly from the sub-sequence prototypes, rendering them less suitable as interpretable shapelets. Hence, ShapeConv incorporates an additional shaping regularization to enforce similarity, Besides, another regularization term is utilized to relieve the issue that the model tends to fall into the local optimal point where kernels are similar but not catching diverse and discriminative features. As for the initialization, we also design separate judicious strategies make model weights close to different discriminative sub-sequences in data in different tasks, capturing class-specific and cluster-specific information for supervised classification and unsupervised clustering, repsectively.
In contrast to traditional shapelet techniques, ShapeConv, being deep model, facilitates end-to-end optimization. This paves the way for parallel computing for acceleration, effortless stacking with deep modules for improved performance. When compared to learning-based maaningg approaches, our kernels provide controllability. facilitated by our strategic initialization that incorporates human expertise to yield more human-cooie comprrheccmme results. Empirical evaluations underscore ShapeConv's 's prowess in both supervised classification and unsupervised clustering tasks and various datasets. It surpasses other learning-basee based shapelet techniques and contemporary deep models tailored for time-series classification and clustering, all the while preserving interpretability. Furthermore, the infusion of human knowledge amplifies the model's performance, and there's marked reduction in time complexity when compared to earlier shapelet methodologies.
We summarize our contribution 4 folds: (I) We have formally and theoretically proven the equivalence of a specific CNN layer, when combined with square norm and pooling. to the shapelet. (2) Based on the discovered equivalence, we introduce ShapeConv, an interpretable CNN layer with its kernel serving shapelets. Several regularizations and initializations are accompanied to enforce similarity and diversity, making ShapeConv effective in practice.
(3) By treating the CNN kernel as shapelet, we claim another advantage of facilitating the incorporation of human prior knowledge by initialization. (4) Extensive experiments on real-world datasets validates ShapeConv's 's superior performance on interpretable time-series modeling tasks.
2 BACKGROUND
Shapelets Shapelet (Ye Keogh 2009), originally defined as a maximally discriminative sub-sequence in time-series data (shown in Figure D. is designed to capture inter-class features in terms of small sub- sequences rather than the full sequence. Shapelet- based methods will usually find subset of shapelets S* c S from data maximize information gain by splitting data with the shapelets as nodes of a deci- sion tree. The candidate shapelet set contains all possible sub-sequences of the original data.
Clas
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Afterwards, a shapelet transform (Lines et al. 2012) introduced to decouple the shapelet discovering step and downstream classifier. This shapelet trans-
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Figure 1: An example of time-series clas- sification task and learned shapelets by ShapeConv. Both shapelets are for class 1, and they capture two distinguish shapes of the class, respectively (indicated by green frames). Lines of class exhibit more pro- nounced fluctuations at these two locations.
form step will extract features for an input signal X based on the distances between shapelets and the in- put. Specifically, for each selected shapelet e S* calculate the minimal distance between and all sub-sequee sequencueees of X, i.c.,
dexx = min ds,
(1)
x
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where X is the set containing all sub-sequences with the same length s, and (s, is the distance function (usually the squared Euclidean distance) measuring the similarity between a shapelet and raw sub-sequee sequence. Using all shapelets in S the feature vector with length |S*|| can be obtained for each X, and these features are used for building different kinds of classifiers besides base models. We provide more vivid example shapelets and shapelet transform in Appendix [8.2
Beyond traditional costly shapelet discovering methods, following works extend the candidate set to R where the length of shapelet, and try solve the problem with aptimizattionbasses methods.
Convolutional Neural Networks CNN is one of the most commonly used network structures and its several variants have been used time-series modeling for years (Ismail Fawaz et al., 2019). A CNN layer is essentially some sliding filters known convolutional kernels followed by the activation function and pooling layer. Here, we consider a simple case to illustrate the CNN layer, where the input is 2-dimensional signal X e Rniia with Bin input channels and la time steps. Applying a 1-D convolution over it can be formulated as:
Y Y= Y W k=1
Where We (oox denotes the weights of convolutional kernels. The symbol denotes the cross-correlation operator on the k-th input channel between each raw vector of W& and the input Xg. Note that for most CNN kernels, weights W₁ is learnable so this corsscorrel equivalent to the convolution in terms of optimization. After that, a non-linear activation function and pooling layer are often applied extract the aggregated value among its neighbors Nj) for each channel at location j. which can be written as:
Ypool pool ooY b.). J(eN(i)
Here, The bias term b is sometimes set to zero for simplicity. The symbol a denotes the activation function such as ReLU, Tanh, and pool denotes pooling function such as max mean.
How CAN CNN KERNELS BE THE BEST SHAPELETS?
In this section, we give comprehensive answer to the question in the title. First, we provide
formal proof to show the equivalence between CNN and shapelets in Sec. 3.1 Then, we introduce
ShapeConv, novel convolutional layer well utilizing the equivalence in Sec. 3.2 Finally, we show
how ShapeCony can be used for supervised learning and unsupervised learning in Sec. 3.3 and Sec.
3.4 respectively.
3.1
EQUIVALENCE BETWEEN CNN KERNELS AND SHAPELETS
The core idea is that when the calculation of squared Euclidean distance in the shapelet transform step expanded, one of the terms is exactly the same as the forward passing in a CNN layer. Therefore, using these shapelets to extract features with squared Euclidean distance for X can be equivalently done by convolving with nout kernels from a I-D CNN layer added by squared L2 norm, followed by maximum pooling. The difference between these two can be casily handled and omitted in practice. We summarize the finding in the following theorem:
Theorem 3.1 Assume the input X E R is a I-dimensional single-variate signal of length Ly and -(11 $2, Spowt } with length l, are discovered. The feature extracted from X
nou shapelets S* = (81. with S₁ and squared Euclidean distance is X. Then we have
d =22
max Je 1.+1
[Yg -N(s X X1g-1,1)11
(4)
where Y Si * X is the cross-correlation defined in Eq. and N(ss X') (1s:1]3 |X'12)/2 is squared Lo norm term.
Detailed proof can be found in the Appendix C Eq. bridges shapelets and the learnable CNN kernel. The left-hand side is the feature extracted by specific shapelet, and the right-hand side contains the
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maximum pooling over the convolution between the kernel and the input time series and squared norm term. The difference between these two is a constant factor -2 which can be absorbed in the learnable parameters.
There are two more gaps between the practical use of CNN in Eq. 3 and Theorem 3.1 One is the non-linear activation function. When the activation function is monotonically increasing, the order to apply maximum pooling and activation function can be swapped, i.e., max(o(-)) = (max(() (-)) This fits for most cases in practice with ReLU, Tanh, Sigmoid and their variants activation functions. Another is the bias term. Since b is often designed to be independent of the position, we can have max, (Y; i5 + b;) max, (5) by. Now we can rewrite Eq. as:
Yo max oYYi + b,) a( (JE
max (Y (x,))
(5)
J'eN(J)
This suggests that we can optionally add the bias term and the non-linear activation after the features are extracted with the shapelets to obtain complete equivalence.
3.2 SHAPECONV: AN INTERPRETABLE CNN LAYER WITH ITS KERNELS SERVING AS SHAPELETS
Motivated by the above established equivalence, introduce ShapeConv, a novel interpretable CNN layer for time-series data, with its kernels serving as shapelets. Specifically, in addition to the cross-correlation operator and max pooling in the original CNN layer, we add squared norm term (s; X') in Eq. 4 before the pooling function to the CNN layer. Consequently, ShapeConv's forward pass mirrors the shapelet transform, calculating the minimum distance between a shapelet and all possible raw input sub-sequences, and convolutional kernels in ShapeCony play the same role as shapelets. Although initially univariate, ShapeConv can directly adapt multivariate data tasks by adjusting the number of input channels while maintaining its other designs, enabling the model to learn multiple kemels/shapelets for each variate.
Shaping Kernels To serve as good shapelet, ShapeConv's s kernel should be the maximally discriminative thus can provide solid criteria for downstream classification (discriminability). and as human-comprehensible as possible by the minimize the distance from the sub-sequence of the data (interpretability).
The discriminability is achieved by optimizing kernel weights via task-specific loss which will be discussed in Sec. 3.3 and Sec. (3.4] respectively. As for the interpretability, while extending the candidate set from S* to I allows the learning-based methods to achieve best classification results, shapelets which look way too different from the input data conversely downgrade the overall interpretability. Therefore, to shape kernels like original data, we first strategically design the initialization method. Depending on whether class labels are available, we suggest different methods for initial kernel-data proximity, elaborated in the following subsections. Besides, we introduce a shape regularizer to keep the kernel similar to data during training. Specifically, we calculate distances between the kernel weight. i.c., shapelet st and sub-sequences in the input, and the shape regularizer is defined as the minimal distances, i.c..
Rohape
min dist(si, x
(6)
Nout
where X, is the set containing all sub-sequences with same length as sj. We opt for squared Euclidean distance here to allow for the reuse of the distance calculated in the shapelet transform step. This approach ensures kernel weights are initialized with original data shapes and remain close during training, yielding interpretable kernels with discriminative shapes.
Increasing Diversity Since shapelets in our model are learnable weights with large flexibility, they tend to fall into local optimality where all kernels are similar but not catching diverse and discriminative features. We provide experimental results on this phenomenon in Appendix [D. Therefore, we further introduce a diversity regularizer following Zhang et al. (2018) to relieve this issue. We first use the pairwise lg distance between different kernels (or shapelets) to construct a distance matrix D., where Di(i.)) (xp(-(2(s).ss)) and the diversity regularizer is defined as F-nor of D., i.e.,
Ratio
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Sliding Window
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Figure 2:
Illustration of initialization methods for ShapeConv kernels. We let noat flele for both figures. Left part shows initialization in supervised learning, k here. Right part shows initialization in unsupervised learning, Reut and here.
To summarize, the above basic designs make ShapeConv differentiable CNN layer with its kernels serving as shapelets. ShapeConv can keep the advantages of both CNN and shapelets. On the one hand, ShapeConv a fully differentiable CNN layer, so it can optimized effectively with lower time complexity than traditional methods. On the other hand, kernels of ShapeConv have good interpretability since they are similar to discriminative sub-sequences in original data.
3.3 SUPERVISED LEARNING
In this section, show the method apply ShapeConv in supervised learning task. Here, focus on the time-series classification task, which is typical application for shapelets.
Initialization The target of initialization to provide convolutional kernels with good starting point capturing class-specific sub-sequences in the data. As illustrated in the left part of Figure Z the main idea is that for every class of data, we split them along the time axis, and calculate the mean of sub-sequences in each split part. Suppose we have nous output channels, corresponding to Nout shapelets, with each shapelet of length 1,. The input length is 1x, and there are Neta classes. First, we assign k = Mout/n,do shapelets to each class. Within each class, we divide the time series into k equal parts of length 4 /x/k (along the time axis). For each part, we use a sliding window to find all possible candidates of length 4,. The final initialization of the kernel is obtained by calculating the mean of all candidates. This method not only incorporates class information into the kernel but also associates each kernel with specific region of the time-series data in the dataset. As result, the model can more effectively capture local information within the initialization region. The initialization also paralleled GPU, making very time efficient.
Classifier and Loss Function The output of the ShapeConv layer the shapelet transformed distances, representing features extracted by learned shapelets, and these features further utilized by the downstream classifier. Here, we append a multi-layer perceptron (MLP) after the ShapeCony layer, map the shapelet transformed distance to the class labels. This method allows for end-to-end training of the entire model, optimizing both the ShapeConv layer and the classifier simultaneously. The loss function designed as follows.
C = Lets + Xahape Ryhape + Radiv
(8)
Here, Lote is the task-specific classification loss, such cross-entropy. The term Rahape is the shape regularizer defined Eq. 6, and the term Raiv is the diversity regularizer defined Eq. 7 The hyperparameter Àshape and Adiv controls the balance between each terms. Note that ShapeConv is compatible with other classifiers. We discuss this in Appendix [.I and include this variant in our experiment in Sec. 4.1
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3.4 UNSUPERVISED LEARNING
We now apply ShapeConv to the unsupervised learning task, where ShapeConv needs to capture the most representative sub-sequences in data and perform K-means on the shapelet-rransformed distance to cluster unlabelled time series.
Initialization Initialization in the unsupervised learning setting is more challenging, since no class information is given now. Therefore, we design pre-clustering method to find out some discriminative sub-sequences. The main idea is dividing the time-series data into portions along the time axis and performing separate clustering within each portion find initial shapelets, illustrated in the right part of Figure 2 Suppose we have nout shapelets in the ShapeCony layer, each with length of 1,. The input length is 1x- First, all input time series are divided into nout equal parts along the time axis, each with a length of lous ///te Each part is assigned with k= nuu nott shapelets. We then sample a large number of subsequences (e.g., 10,000) of length l, from each part and perform KMeans clustering with & centers on them. Finally, the cluster centers are utilized as the initialization of the shapelets. the clustering process, and the division into cuts allows the shapelets to focus on different regions of the time series. This enables the model to better capture the local patterns of different classes.
this approach, the class information is implicitly introduced during
Loss Function As for the task-specific loss, employ Davies -Bouldn Index (DBI) (Davies & Bouldin 1979) to optimize ShapeConv for better clustering results. Overall, DBI loss aims to minimize the intra-cluster distances while maximizing the inter-cluster distances, ensuring that the extracted shapelet transformed distances yield well-separated clusters (Li et al.. 2022). The detailed formulation of DBI loss can be found in Appendix D.2
The overall loss function in unsupervised
learning is designed as,
C = LDBI SchapeRsshape + R
Here, DBB DBI loss. The term Rshape is the shape regularizer defined in Eq. 6 and the term RRii is the diversity regularizer defined in Eq. 7 The hyperparameter Xshape and Xdiv controls the balance between each terms.
Incorporating Human Knowledge The characteristic of ShapeConv makes it easy to incorporate human knowledge, which means that human experts can "tell" the model what some key sub- sequences look like, and the model use these knowledge for improving its performance. On the other hand, in unsupervised learning tasks, the model will first learn to minimize the shapelet transform distance, which may lead it to converge to local minima. If the shapelet is initialized in a non-discriminative region, the model's performance may be negatively affected. Therefore, we propose using human knowledge for shapelet initialization.
Specifically, we first visualize the dataset and ask the human labeler to identify the most discriminative regions. Once these regions are labeled, we calculate the mean of each region and use it as the initialization for the shapelets. Then, the shapelet will tend to converge in the targeted region. As shown in experiments in Section [4.2 this approach makes the model learn high-quality shapelets.
EXPERIMENTS AND ANALYSIS
4.1 SUPERVISED TIME-SERIES CLASSIFICATION
Settings We evaluate ShapeConv model time-series classification tasks using the UCR univariate time-series dataset (Dau et al. (019) and UAE multivariate times series dataset (Bagnall al. 2018). Hyperparameters are tuned via grid search based the validation set performance, and they are reported in Appendix G.2
Compared Methods We compare ShapeConv with three kinds of baselines: (1) shapelet-based
methods (IGSVM (Hills al. 2014), FLAG (Hou et al. 2016). LTS (Grabocka et al. 2014). ADSN Ma et al. 2020b)), ), (2) common deep learning methods (MLP, CNN, ResNet (Wang et al. 201 7)), and (3) state-of-the of-the--rtt time-series classification models (DTW (Chen et al. 2013). TNC (Tonekaboni
et al. 202T), TST (Zerveas et al.) (02I). TS-TCC (Eldlee et al., 2021). T-Loss (Franceschi et al.
2019
TS2 (Yue et al. 2022 )) The results for the baseline methods are taken directly from the

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MLP
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AA At-Shopet: WAM
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(a) Supervised time-series classification
(b) Unsupervised time-series clustering
Figure 3: Critical difference diagram for supervised and unsupervised learning tasks. Pairwise statistical comparison is based on accuracy on 25 datasets from the UCR Archive for supervised classification, and NMI on 36 datasets from the UCR Archive for unsupervised clustering. Table 1: Testing accuracy of supervised time-series classification tasks on UCR and UEA datasets. Training Time includes initialization.
125 UCR datasets
UEA datasets
Method
Acc
Avg. Rank
Training Time (hours)
Avg. Acc.
Avg. Rank
Training Time (hours)
DTW
0.727
6.10
0,650
4.72
TNC
0.761
5.29
228.4
0.670
4.58
91.2
TST
0.641
7,06
17.1
0.617
5.28
28.6
TS-TCC
0.757
5.19
1.1
0.668
4.33
3.6
T-Loss
4.52
38.0
15.1
TS2Vec
0.836
2.86
0,9
0,704
3.12
0.6
ROCKET 0.842
2.59
0.89
ShapeCony
0.851
2.39
0.750
0.8
original paper. In addition, also design variants of ShapeConv for ablation studies, including training without diversity loss (w/o div), training with random initialization (w/o init), and using an SVM classifier (w/ SVM).
Table 2: Testing accuracy and training time of ShapeConv and LTS on the Herring dataset.
Shapelct Length
50
100
150
200
250
300
350
400
ShapeConv
61.3
67.2
67.2
68.8
75.0
70.3
71.9
71.9
Accuracy
LTS
64.1
60.9
59.4
59.4
59.4
59.4
59.4
59.4
ShapeConv
9.20s
9.09s
9.15s
9.30s
9.27s
9.19s
9.02s
9.09s
Time
LTS
3.37h
3.20h
3.23h
3.17h
2.69h
2.25h
1.87h
1.21h
Results The performance ShapeConv, its variants, and shapelet-based baselines, evaluated on the 25 UCR datasets and presented in Figure 3 (a). For this experiment, compared in the 25 subset of the 128 UCR dataset, because the baseline method only reported on this subset. Additionally, the summary of results for state-of-the-an time-series classification models 125 UCR and 29 UEA datasets are shown Table T The full results are presented the Appendix G.3 In general, ShapeConv consistently outperforms all other baselines and variants, ranking first average. These results demonstrate that ShapeConv not only provides interpretability but also excels in performance, making it competitive choice for time-series classification compared state-of-the-art methods. Ablation studies tell that the effectiveness of our initialization method and diversity loss contributes improved performance compared the variants. Lastly, the choice of downstream classifier, either SVM or MIP, does not significantly impact the performance of the ShapeConv model, indicating its flexibility and robustness different classification settings.
Analysis In this section, investigate main research questions (RQs): (1) why does ShapeCony outperform other shapelet-based methods? (2) how are ShapeConv's interpretability results compared other shapelet- based methods when they yield similar results?
In response to the first RQ, we examine the Herring datasets from UCR (Dau et al. 2019). Learned shapelets with minimum distance from the original data by the model with best validation accuracy are
'This average accuracy metric is meaningless to some extent, due to datasets of different sizes, class skews, number of classes, default rates, etc. We list these results here just for the comparison with previous works, hut sincerely call for metrics with more practical values here.
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Figure 4: Learned shapelets by different methods the Herring dataset.
plotted in Figure In the Herring dataset, ShapeCony (test accuracy 75.0) significantly outperforms the LTS method (test accuracy 64.1). In response to the first RQ, we observe that ShapeConv's shapelets (Figure g (a)) cover the most discriminative regions of the time series (the turning points), while LTS's shapelets (Figure({(b)) do not. This indicates that ShapeConv's learned shapelets are better at distinguishing classes, leading to improved performance.
We find that the learned shapelet by ShapeConv is much longer than that by LTS. The result of forcing the shapelet learned by LTS to be longer (Figure (c)) reveals that
1.0
LTS Shapelet
LTS fails learn high-quality long shapelet. We also provide an ablation on the shapelet length in Table 2 shows ShapeConv's accuracy increases with shapelet length up to a certain point, while LTS's accuracy does benefit from the increased length. This is likely due to optimization issues in the LTS method, which cannot handle long-length shapelets. ShapeConv, on the other hand, can be efficiently computed in parallel, leading to better optimization results and significantly faster training time (about 1000x faster).
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Figure 5: Shapelets learned by LTS the GunPoint dataset.
In the GunPoint dataset, both ShapeConv and LTS meth- ods achieve saturated accuracy (100). In response the second RQ, we utilize this dataset to compare the interpretability of the learned shapelets by ShapeConv (Figure [) and LTS (Figure 5). It is evident that the shapelet learned by ShapeConv captures the distinguishing features of the class effectively. Here, the shapelet (blue) captures the gesture of reaching for the gun and drawing it out of the holster. The shapelet (red) captures the gesture of putting the gun back to the holster. In contrast, the shapelets learned by LTS do not align well with either of the classes, especially for shapelet in blue. Based on this observation, we conclude that ShapeCony is capable of learning more interpretable shapelets compared to LTS.
4.2 UNSUPERVISED TIME-SERIES CLUSTERING
Settings evaluate our ShapeConv model on time-series clustering task using UCR univariate time-series datasets (Dau et al. 2 We first learn shapelets using a ShapeConv layer, then apply KMeans on the shapelet- -rrnseroret distance. We use the Normalized Mutual Information (NMI) metric to evaluate the models. Hyperparameters are tuned via grid search based validation set performance, and they are reported Appendix G.2
Compared Methods We compare ShapeConv with three kinds of baselines: (1) pure clustering
methods (KMeans (Hartigan & Wong 1979) applied to the entire time series), (2) shapelet- based
methods (U-Shapelet (Zakaria et al., 2012), AutoShape (Li et al. 2022 ). and (3) state-of of-the--rt time-series clustering models (k-Shape (Paparrizos & Gravano 2015). DTC Madiraju et al. 201 8), USSL (Zhang et al. 2018), DTCR (Ma et al. 2019). STCN (Ma et al. 2020a)). The results for the baseline methods are taken directly from the original paper. We also design variants of ShapeCony for ablation studies, including training with random initialization (w/ Init), training without DBI Loss (w/o DBI), and using human knowledge to initialize shapelets (w/ Human).
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Jimen Stapelei
0.0
Bastoo Shapoler
0.0
Stepole:
Human Stapele:
20
40
20
40
60
80
200
40
Figure 6: Visualization of cases from ECG200 dataset. First row: illustration of the shapelet initializations; second row: illustration the learned shapelets. Left: random initialization; middle: cut and sample initialization (Sec. 3.4) right: human knowledge initialization (Sec. 3.4
The results of all models on 36 UCR datasets are shown Figure (b), and details are in Appendix G.4 We compared on the 36 subset of the UCR datasets because the baseline methods only reported on this subset. In general, our ShapeConv outperforms all other baselines and variants, achieving the highest average rank among the compared methods. ShapeConv's superior performance, particularly against its randomly initialized variant, underscores the importance of proper initializa- tion for effective clustering. Its best performance with human knowledge initialization highlights the model's ability to incorporate human knowledge to guide the learning process and improve clustering results. Overall, ShapeConv stands out potent, interpretable tool for unsupervised time-series clustering, outperforning existing methods while adeptly learning interpretable shapelets and assimilating human knowledge.
Results
Analysis of the Initialization We now provide case study to analyze the effect of initialization for ShapeConv unsupervised learning tasks. We select the ECG200 dataset from UCR (Dau et al. 2019) for this analysis and results plotted in Figure 6 First, observe that in the time-series clustering task, the learned shapelets are close to their initializations. This is because, during the first step of learning shapelets, solely minimize the transfeetran distance, which tends to optimize within the local region. Therefore, determining the initialization of the shapelets critical for unsupervised learning.
In both the random and sample initialization, one of the shapelets matches the right part of the time series, where the two classes are indistinguishable. In contrast, when using human initialization, choose the two regions with the most significant differences between the classes (the shaded regions Figure 6) and use the average of those regions initialization. Consequently, the shapelets are converged in the these regions, effectively capturing the differences between the classes.
CONCLUSION
In this paper, we bring together CNNs and shapelets in time-series modeling by finding the equivalance between them. Upon the findings, we further proposed ShapeConv, an interpretable convolutional kernel with its kernels serving as shapelets accompanied by shaping regularizations, and we apply ShapeConv to both supervised and unsupervised tasks. ShapeConv is designed to maintain the advan- tages of both CNNs and shapelets, providing excellent performance without sacrificing interpretability and controllability. Our experiments on various benchmark datasets showed that ShapeConv out- performs other shapelet-based methods and state-of-the-an time-series classification and clustering models. Moreover, the incorporation of human knowledge can further enhance the performance of ShapeConv, highlighting its potential in real-world applications where expert knowledge is available.

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APPENDIX
A RELATED WORK
Learning Shapelets How to find the best shapelets from data has long been intriguing problem since (Ye & Keo 2009) firstly proposed it. Traditional practice is to search the raw datasets with some speed-up strategies, like paralleling computing (Chang al., 2012), SAX transformation (Rakthanmanon Keogh 2013) and procedure simplification through newly designed measurements
Lines et al.] 2012 Guillaume et al.) 2022 Zakaria et al.] 2012). However, despite ingenious techniques, the performance of these methods are quite limited in large-scale real-world scenarios due to their inefficiency and inflexibility. Cutting-edge research mostly focus on learning shapelets via optimization--an bassed methods. (Grabocka et al. 2014) firstly proposed to learn shapelets with
gradient descent. (Shah et al., 2016 Lods et al. (017) extended this idea to learn more discriminative
shapelets based on DTW measure. To encourage the interpretability of learned shapelets, (Ma et al. 20206 Wang et al.] 2019) designed adversarial strategies guide model training. Besides, in (Li et al. 2
& Sun 2022). autoencoder and neighbour graph structure were also leveraged to capture high-quality shapelets in an unsupervised manner. Nevertheless, none of these methods have rigorously shown and made full use of the equivalence between the CNN layer and shapelet to achieve both good interpretability and efficiency like ours.
Interpretable Time Series Modeling Despite noticeable progress made by feature engineering
methods (Ruiz et al. 2021| Bagnall et al., 2017, Middlchurst et al. 2023) like HIVE-COTE (Lines) al. 2018). MUSE (Shääfer & Leser 2017) 7) and ROCKET (Dempster et al. 2020), recent years have also witnessed the rapid advancement of deep learning methods in interpretable time series modeling.
Apart from RNN models (Choi et al. 2016; Guo al. 2019), newly devised CNN models have obtained more attention in this area. (Fortuin et al. (018)) developed a CNN-based SOM- VAE method to learn the topologically interpretable discrete representations of time series in a probabilistic fashion. (Luo et al. 2022) employed convolutional kernels to approximate the partial differential equations on data distribution so as to explain the nonlinear dynamics of their sequential patterns. (Li al. 202I) designed a wavelet convolution layer to help CNNs discover filters with certain physical mcaning, while [Tag et al., 2021b Xiao al., 2022) studied the best kernel size for time series modelling. Innovatively, we view convolution kernels from the shapelet perspective and endow shapelet- based interpretability to incomprehensible model parameters, making ShapeConv an interpretable model.
B DETAILED EXPLANATIONS ON SHAPELETS AND SHAPELET TRANSFORM
B.I INTERPRETABILII AND EXPAINABILITY
The interpretability of shapelet comes from its human-comprehensible nature. However, "inter- pretability" is sometimes confounded with the concept of "explainability" introduced in many post-hoc explainable methods (Tonekaboni et al. 2020 Leung et al., (021). These explainable methods focus on explaining an already learned, non-interpretable model. They may not reflect the actual behavior of the original model and may disagree with each other (Krishna et al. 2022). self-interpreetale methods such as shapelets do not have these issues as the model itself provides explanations during the learning process. Furthermore, the proposed ShapeCony in our paper whose kernel serves as shapelet inherit the same interpretability of the traditional shapelet, utilizing the prototypical shape information of the data to perform classification.
B.2 VISUALIZING SHAPELETS AND SHAPELET TRANSFORM BY SHAPECONV
To illustrates shapelets and shapelet transform more vividly, we take a real-dataset example named GunPoint, aiming at classifying whether a person is pointing with a finger a gun (Figure[B.I). The sequences in the dataset record the normalized x-axis of the hand position, i.c., how far away the hand is from the main body through time.
Figure B.2 demonstrates the result. The leftmost two subfigures show the learned shapelet. When we try to align the shapelets with typical samples one with "gun" (Class 1) in blue and one with "finger" (Class 2) in orange in the middle subfigure, we can immediately find that the most distinguishable shapes underlying data from Class are the small flat stage soon after the beginning and symmetrically
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in the end. These two shapes corresponds to the motion of hand pulling gum out of the holster and put it back, and would not exist when a person is pointing with his finger.
Therefore, when perform the shapelet transform by calculating the distance between data points and the learned shapelet and draw them in the 2-D feature plane as is shown in the right subtigure, we found the two classes are linear separable expected. Data points the left-bottom of the plane from the "gun" class contain the shapelets indicating the gun-related actions, while data from the "finger" class do not. This illustrate how the model be discriminative while providing interpretability the same time.
In order further verify interpretations, obtained the frames of videos during the data collection process from the collector and visualized the trajectory of hand movements for the first half of the sequence, as is shown in Figure B.3 For data of the Gun class, we observe clear stops when pulling the gun out of the holster as is highlighted in green circles, and such movements do not exist the data of the Point class. This observation matches our hypothesis made upon the learned shapelets.
Figure B.1: Illustration of the GunPoint Dataset [atanaaaahaaaaa & Keogh 2005
Learned Shapelet
Aligned Shapelet with Sample
Shapelet- formed Data
1.0
1.00
Class Class
Shapslet
0.5
0.5
Shapelet
0.0
Distance Between Sample and Shapeler
0.50
1.0
Shapelo
0.5
0.00
25
0
-25
150
75
150
0.25
0.50
0.75
Shapelet 1
Figure B.2: Illustration of shapelets and the shapelet transform step
C PROOF OF THEOREM 3.1
Theorem 3.1
Assume the input X e IR!= is I-dimensional single-variate signal of length L,, and Nout shapelets S* (S1.82.- S2, ...Sn with length la are discovered. The feature extracted from X with s; and squared Euclidean distance is ds,x. Then have
dg
max 1e(1,2....4a-.41
[Y -N(s X x1)1,,1)],
(10)
where Y Si X is the cross-correlation defined in Eq. and N(s X') - (1s
X
/2 is
squared L2 norm term.
Image Source:
http://www.timeseet seseerieeccssss assiaassiii fi cat wtt.:iocccmmmerriii.... pt php?
bataset Guuneint
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Gun
Point
Figure B.3: Illustration of the hand movements in GunPoint dataset
Proof. According [ shapelet transform step extract features using minimal distance and can be expanded as:
dx,x= min dist
min je(1.2.-.4.., 11,+1
||s:
X Xjj+t,-11
xeX
min
(ss: + X Ktttt.-1 -2
Sik x+tk)
21 +1)
k=1
min
|| |X [xjj+ز,-1
2(s) X)R].
= -22
max
Y (si X X
with and W 11))-,-11 defined as in the theorem.
D DETAILS ON MODEL DESIGNS
D.1 COMPATIBLE WITH OTHER CLASSIFIESS
ShapeConv also be used together with traditional classifiers, such as support vector machines (SVMs), decision trees, or random forests. In this case, the whole model cannot be optimized via end-to-end fashion, so decompose the shapelet learning step and classification. Specifically, no additional module is appended to the ShapeConv layer, and the output of the layer is directly optimized using the above loss function, but with Lats term in Eq. removed. Then, the learned features are fed into the chosen classifier for training and prediction. We also include this variant in our experiment in Sec. 4.1
D.2 DAVIES-BOULDIN INDEX (DBI) LOss
When the number of cluster is set to k; DBI can be denoted as
1 IDBI Σ max
T3 r
(11)
k ke..
di
Here, r; is the diameter of cluster i, which is defined as the average distance between each element in cluster and the center of cluster The distance between the center of cluster i and cluster j di, However, this formulation is not tractable for optimization due to the max operator. Thus, following Li al. 2022). the max operator replaced and approximated by the following calculation.
$
E Σjmmi mij aam
LDBI
Σ 1=1
(12)
Σj eamis
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1.0
1.0
Our Shapelet
wio div Shapelet
0.8
Ous Shopelet 2
0.8
wio div Shapelor 2
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
100
200
300
400
500
100
200
300
400
500
(a)
(b)
Figure D.1:
Learned shapelets by different methods the Herring dataset. (a) ShapeConv; (b) ShapeConv without diversity loss.
where may
TETts d.y By numerical verification, 100 enough for Eq. 12 to approximate the true
maximum.
D.3 ILLUSTRATION OF DIVERSITY LOSS
Figure D.I depicts the trained shapelet both with and without diversity loss. The illustration reveals that without the application of diversity regularization, the shapelets tend to converge to the same local minimum. Employing diversity loss can mitigate this issue.
E SHAPECONV AS FEATURE EXTRACTOR WITH DEEP MODELS
Another advantage brought by our ShapeConv to find shapelets with a special kind of convolutional layer is the flexibility. While the traditional shapelet works suffer from handling large time-series data of long sequence efficiently, ShapeConv turns the extraction shapelets from the original data into a stackable layer can be combined with more sophisticated deep models and optimized in end-to-end manner, leaving the possibility to keep the interpretability and effectiveness to the maximum extent.
To verify the effectiveness of ShapeConv embedded in deep model, we apply ShapeConv as the first layer to extract features which are further processed by GRU(Cho et al., 2014). a widely used time-series model for long-term modeling. We then conducted the experiment of the proposed method on the seizure detection task based on electroencephalograpp (EEG) data (Obeid & Picone, (016)) The dataset contains 97,859 samples (83,647 for training and 14,212 for testing), and each sample contains 20-channel 30-second EEG signal sampled at 200Hz. The goal of the prediction models is predict the probability of seizure event within the given EEG signal piece, following (Tang et al.
2021a
Li et al. 2023)
We compare our model ShapeConv with the most commonly used deep neural network models GRU (Cho et al. 2014) and TCN (Bai et al. 20188.. The empirical results are illustrated in Table [E.] Our model has significantly outperformed the compared baselines, which showed the superionty of the proposed ShapeConv paradigm even embedded in another neural architectures.
Table E.1: Performance comparison on seizure detection task.
Model
AUROC
AUPRC
GRU Cho et al. (2014)
0.814(0.009)
0.386(0.018)
TCN Bar et al. (018)
0.817(0.004)
0.383(0.010)
ShapeConv + GRU
0.837(0.007)
0.414(0.008)
We further investigate whether our ShapeConv can preserve its interpretability when stacking with deep models. As clinical practice, the morphology of waveform in EEG describes its overall shape, and is important for both interpreting a tracing and communicating findings, which has been well- studied and recognized in previous medical research (Marcuse al. 2015). To this end, we visualized
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EEG Examples from Teexthook
Learned Shapelet
Shapelat Wansformed
Clas 002
Polymorphic Delta Waves
0.5
Waveform
Shapelet
M
Paroxysmal Rhythm Waves
-00
0.0
Waveform
Shepelet 2
(a) Comparison hetween two types of texthook waveform and the two learned shapelets, polymorphic delta related lesions
(h) Shapelet transform by the two shapelets. Some data points from Class are mixed with Class need to be sep- arated by other shapelets.
MMarshall al. 1988) and paroxysmal rhythm waves related to sleep
stage (Alvarez et al. 1983).
Figure E.1: Demonstration of the learned shapelets for EEG data.
1.0
Class
Class
Shapelet
0.8
2hapelet
0.5
0.25
0.4-
0.00
0.225
-04-
0.75
80 -06-
40
20
40 60
80
(a)
(b)
Figure E.2: Alignment between the learned shapelets and original data.
a few obtained shapelets out of 2,688 shapelets (128 shapelets per variate, 21 variates in total) and excitedly found that some of them accord with some textbook waveform, as shown in Figure E.la By aligning them with the mostly similar part of the original data (Figure E.2). we found these shapelets can match specific type of seizure status and provides solid classification criteria (Figure[..1) This showcases the possibility of how our interpretable method can benefit medical practitioners practice by not only offering an accurate judgement, but also pointing out the area of interests with respect to their expert knowledge.
It's also noteworthy that since the amount of summarized waveform in textbook is limited, some shapelets, while serving similarly strong indicators, may not be included in existing studies. We believe these shapelets can provide inspirations and boost further research related area.
F MORE VISUALIZZATIONS OF SHAPECONV
F.1 MORE VISCALIZATIONS ON LEARNT SHAPELETS
In this section, provide more visualizations of the learnt shapelets of ShapeConv in different UCR datasets. The results clearly shows that ShapeConv could leam the determining regions of the time series.
F.2 VISUALIZATIONS WITH SHAP VALUE
In this section. further substantiate claim regarding the interpretability of our model using the SHAP (SHapley Additive exPlanations) Value (Lundberg & Lee|2011 The analysis employs the GunPoint dataset from the UCR archive. We examine two variations of model: the original
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Class
Class 1
Sbapelet
Shapolet
Class
Class
Shapelet
Slopsicl
1.0
1.0
0.8
0.8
0.6-
0.6
0.4
0.4
0.2
0.2
0.0
0.0
50
100
150
200
250
300
10
20
30
40
50
60
70
(a) Dodgerlooop Weekend
(b) Sony SonyAllOORoosSurfacce
Class
Class 3
Shapelet
Class
Class
Shopelet 1
Shopelet
Class 2
Shapdet
Shapelet 3
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0
5
10
15
20
40
60
80
100
120
(c) ItalyPowerDemand
(d) BME
Figure F.l: More visualizations of the shapelets learnt by ShapeConv in UCR datasets.
ShapeConv and modified version where the term Ashope in the loss function (Equation ( set to zero. Setting Ashape to zero eliminates the L2 norm term, effectively transforming the layer into standard CNN. Consequently, this variant lacks the interpretability feature.
Both models underwent training under identical hyperparameters. Post-training, we computed the SHAP Values for each model across the entire test dataset using the expected gradients approach. These values are illustrated in Figure F22 with the mean SHAP value of each class depicted. The blue and orange lines represent the Gun Class (Class and No Gun Class (Class 2), respectively.
The left side of Figure F.2 reveals that the model is particularly sensitive to the left and right turning points. These points symbolize the gesture drawing the gun out of the holster and putting it back, underscoring the model's is reliance these regions for decision-making. This observation aligns with our hypothesis about the model's interpretative capabilities. However, on the right, we first notice that the kernel does not match with the input sequence, indicating the lack of interpretability. Additionally, the model appears to base its decisions predominantly on the left region. This disparity highlights the limitations of the variant without the interpretability term.
In conclusion, our findings are twofold: firstly, the integration of Shape Loss successfully enhances interpretability. Secondly, ShapeCony not only encompasses all significant regions identified by the SHAP Value of the baseline CNN, but also surpasses conventional explainability methods like SHAP by capturing the shape of sensitive regions, rather than merely indicating their locations.
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Published as a conference paper at ICLR 2024
Class
Cliss
Shapelet
Shapelet 2
Class
Class 2
Kernel 1
Kemel
1.0
0.5
0.5
0.0
0.0
0.1
0.25
SHAP
0.0
0.00
-0.1
-0.25
0.25
SHAA Valee Clas
0.00
0.0
00.
-0,25
SHAP Valie Clas
50
100
150
50
100
150
(a) ShapeConv
(b) Ashape 0
Figure F.2: Illustration the trained average SHAP value for different class across all the GunPoint testing dataset. Blue: Gun Class. Orange: No Gun Class. Left: our proposed ShapeConv method. Right: ShapeConv with Ashope
0.
G DETAILS ON EXPERIMENTS AND ANALYSIS
G.1 ENVIRONMENT
All experiments are performed on the PyTorch framework using 24-cores AMD Epyc 7V13 2.5GHz CPU, 220GB RAM, and an NVIDIA A100 80GB PCIe GPU. The server is provided by the Azure cloud computing platform.
G.2 HYPERPARAMETERS
Supervised Learning The training set is divided into training and validation sets at an 8:2 ratio. Hyperparameters tuned via grid search based validation performance. The number of shapelets is chosen from {1, 3,4,5) times the number of classes, and the shapelet length is evaluated over (0.1, o.,, 0.8} times the time series length. The parameter Ashape chosen from (0.01, 0.1, 1, 10) and the parameter Adi is evaluated over (0.01, 0.1, 1, 10). Learning rate is chosen from (0.001, 0.005, 0.01, 0.05, 0.13.
Unsupervised Learning The training set is divided into training and validation sets at an 8:2 ratio. Hyperparameters are tuned via grid search based on validation set performance. The number of shapelets is chosen from 21,2,3,4, 4,5) times the number of classes, and the shapelet length is evaluated over (0.1, 0.15, 0.2, 0.25, :.., 0.8) times the time series length. The parameter Ashape is chosen from (0.01, 0.1, 10) and the parameter Adi is evaluated over (0.01, 0.1, 10). Learning rate is chosen from {0.001, 0.005, 0.01, 0.05, 0.13.
G.3 RESULTS OF SUPERVISED LEARNING TASKS
In this section, we present the full results supervised time-series classification tasks. We compared ShapeConv with (1) shapelet thapelet-based methods, common deep learning methods, and ablations (de- scribed Sec. 1 across 25 UCR Datasets (Table G.T (2) RNN-based methods (Tang et al.) 2021b) across 56 UCR Datasets (Table G.2 (3) state-of-the-art times series classification methods (described in Sec. 4.1 across 125 UCR datasets (Table G.3) (4) state-of-the-at times series classification
methods (described in $4.1p across 30 UEA datasets (Table G.4).
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Table G.I: ShapeConv Compared with Shapelet-Basee Methods, Common Deep Learning Methods, and Ablations: Evaluating Testing Accuracy for Supervised Time-Series Classification Tasks Across 25 UCR Datasets. Mean accuracy std over 3 independent experiments with different random seeds is reported.
Datasct
MLP
CNN
RosNet
IGSVM
FLAG
LTS
ADSN
ShupeC.
ShupeC. w/ SVM
ShupeC. wlo div
ShapoC.
w/o iit
0.691
0.867
Beef
0883
0.750
0.767
0.900
0.835
0.767
0.933
0.849+0.000 005
0.898=0003
0.92100.001
0.936-0.008
Chlorine.
0.872
0.843
0.828
0.571
0.760
0,730
0.880
0.82520.026
0.904=0.025
0.907=0.008
0.924-0.014
Coffee
1.000
1.000
1,000
1000000000
Diatom.
0.964
0.930
0.931
0.931
0.964
0.942
0.987
0.99220.008
0.991=0.001
0.992-0.005
0.994_00009
DPLittle
0.701
0.703
0.701
0.666
0,683
0.734
0.727
0.698.0.012
0.706-0.003
0.703-0.000
0.713-0.00 0,013
DPMiddie
0.721
0.736
0.723
0.695
0.713
0,741
0.784
0.778+0.018
0.782-0.003
0.789-0.001
0.807-0.039
DPThumb
0.705
0.701
0.205
0.696
0.705
0,752
0.736
0.729.0.011
0.015-0.037
0.738-0.005
0.753 0.015
BCGFiveDays
1.00000.0000
1.0000.0000
1.000-0.000
FaceFour
0.830
0.932
0,932
0.977
0.909
0.943
0.977
0.041=0.023
0.916-0.001
0.942-0.015
0.961-0.007
GunPoint
0,933
1.000
0.993
1,000
0.967
0.996
0,987
0.991
0.995=0.008
0. 0.993-0.003
0.097
Herring
0.713=0.005
0.724-0.028
ItalePower:
0.966
0.970
0.960
0.937
0.946
0.958
0.972
0.932: 0.027
0.953.0.005
0.961 0.001
0.974 0.007
Lightning7
0,644
0.863
0.836
0.630
0.767
0,790
0.808
0.758 003
0.723-0.001
0.748=0.001
0.701
Medicallmages
0.772
0.682+0.012
0.00000000
0.752-0.009
MoteStrain
0.869
0.950
0.895
0.887
0.888
0.900
0.906
0.83
0.886=0.016
0
0.913-0.000
MPLitle
0,703
0.758
0.726
0.707
0.693
0,743
0.758
0.701
0.733=0.002
0.741=0.008
0.749
MPMiddle
0.750
0.800
0,775
0.769
0.750
0.775
0.791
0.736+0.020
0.759-00111
0.779=0.013
0.807-00020
PPLinle
0.710
0.753
0.761
0.721
0.671
0.710
0.715
0.661
0.694-0.029
0.676=0.000
0.732-0.00
PPMiddle
0.707
0.784
0.753
0.759
0.738
0.7
0.726-0.010
0.764-0.002
0.791 0.000
PPThamb
0.726
0.745
0,708
0.755
0.674
0.705
0.695
0.685+0.018
0.712-00100
0.728=00010
0.731-0.011
Sony.
0.727
0.968
0.985
0.927
0.929
0.910
0.915
0,901+0.026
0.914-0.007
0.903-0.002
0.926-0.010
Symbols
0.968=0.006
0.974-0.007
0,980-0.015
SyntheticC.
0.950
0.990
1.000
0.873
0.997
0.973
1.000
1.000
1.0000.000
0.997-0.029
1.0000.000
Trace
0,820
1.000
1.000
0.980
0.990
1,000
1,000
1.00000.0000
1.000=0.000
1.000-0.000
1.00000000
TwoLeadECG
Avg. Ace
80.5
86.4
$4.8
79.4
82.6
83.2
86.6
83.8
85.4
$6.4
87.8
Rank
7.8
8.4
6.2
5.8
2.4
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Table G.2: ShapeConv Compared with RNN-Based Methods: Evaluating Testing Accuracy for Supervised Time-Series Classification Tasks Across 56 UCR Datasets. Mean accuracy + std over 3 independent experiments with different random seeds is reported.
Gaussian RNN
Identity RNN
Darasct
RNTK
NTK
RBF
POLY
GRU
OS-CNN
ShapeConv
Adiac
0.766
0.719
0.734
0.778
0.514
0.169
0.606
0.835
0.882
Arrowhcad
0.806
0.834
0.806
0.749
0.480
0.560
0.377
0.838
0.915=0.033
Beef
0.900
0.733
0.833
0.933
0.267
0.467
0.367
0.807
00.01
Car
0.833
0.788
0.800
0.800
0.233
0.583
0.267
0.933
0.992=0.008
ChlorineCommenntrationn
0,908
0.773
0.864
0.915
0.660
0.558
0.611
0.839
0.924 0.002
Coffee
1,000
1.000
0,929
0.929
1.000
0.429
0.571
1.000
1.000
Computers
0.592
0.552
0.588
0.564
0.532
0.552
0.588
0.707
0.656=0.009
CricketX
0.605
0,595
0.621
0.626
0.085
0.636
0,264
0.855
0.91410.009
CricketY
0.639
0.590
0.605
0.597
0.159
0.592
0.362
0.867
0.729
CricketZ
0,603
0.592
0.621
0.592
0.085
0,579
0.413
0.863
0.764
DistalPialaxOOutinnCC.
0.775
0.775
0.754
0.699
0.696
0.750
0.766
0.804=0.030
DistalPhalanxTWW
0.662
0.698
0.669
0.674
0.676
0.647
0.691
0.664
00.008
Earthquakes
0.748
0.748
0.748
0.748
0.655
0.770
0.770
0.670
0.784=0.031
ECG200
0,930
0.890
0.890
0.860
0.860
0.720
0.760
0.908
1.000 0.000
ECG5000
0,938
0.940
0.937
0,940
0.884
0.932
0.933
0.940
0.963
Faccall
0.741
0.833
0.833
0.824
0.537
0.705
0.707
0.845
0.853=0.007
FacesUCR
0.817
0.802
0.803
0.830
0.532
0.753
0.795
0.967
0.957 0000
FiftyWordh
0.686
0.686
0.697
0.688
0.343
0.602
0.653
0,816
0.699 0.024
Fish
0,903
0.840
0.857
0.880
0,280
0.383
0.240
0.987
0.920
FreezerRegular
0.974
0,944
0.965
0.761
0.075
0.866
0.997
0.997=0.036
GunPoint
0.980
0.953
0.953
0.940
0.820
0.747
0.807
0.999
1.000=0.000
GumPuintAgeSS AgeSpun
0.965
0.946
0.959
0.940
0.478
0.478
0.956
0.992
1.000
GunPointMale VSTemalee
0,991
0.997
0,994
0.997
0.687
0.525
0,997
0.999
1.000 0.0000
GunPointOLd Young
0,987
0.975
0.987
0.946
0,540
0,524
0.984
1.000
1.000
Ham
0.705
0.716
0.667
0.714
0.533
0.600
0.610
0.704
0.733=0.009
Herring
0.567
0.594
0.594
0.594
0.233
0.594
0.594
0.608
0.750=0.014
IGGeullpr
0.996
0,992
0.996
0.968
1.000
L,00
0,984
0.951
1.000
Lightning2
0.787
0.738
0.705
0.689
0.459
0.705
0.672
0.807
0.819
Lightning7
0.616
0.603
0.233
0.699
0.767
0.793
0.003
Meat
0.933
0.933
0.933
0.933
0.006
0.550
0.333
0.947
0.950=0.016
MedicalImages
0.745
0.733
0.753
0.746
0.482
0.649
0.691
0.769
0. 0.709
AiblleehallnnOuttinee
0,571
0.571
0.487
0.643
0.763
0.570
0.746
0.814
0.85610.001
MiddicPhalanxTTV
0,578
0.610
0.597
0.604
0.584
0,584
0.591
0.519
0.642
OliveOil
0.900
0.867
0.867
0.833
0.667
0.400
0,400
0.787
0.833=0.000
plane
0.981
0.962
0.971
0.971
0.962
0.848
0.962
1.000
1.000=0.000
PowerCons
0.972
0,972
0.967
0.917
0.961
0.950
0.994
0.990
0.911
ProximalPhalannOutiine
0.890
0.880
0,873
0.869
0.828
0.746
0.869
0.908
0.913
RefiigerationDevicess
0.469
0.365
0.360
0.509
0.467
0.503
0.613=0.002
ScreenType
0.416
0.432
0.435
0.384
0.400
0.411
0.363
0.526
0.493
Ianggannd
HandSubjectCh2
0.842
0.853
0.861
0.867
0.200
0.367
0.891
0.718
0,981 0
MaallKiichhAAppllana
0.675
0.384
0.403
0.379
0.602
0.760
0.715
0.721
0.803 005
SmoothSubspace
0,960
0,873
0,920
0.867
0.940
0.953
0.927
0.989
1.000
StarLightCCrrvss
0.959
0,962
0.946
0.821
0.868
0.962
0.975
0.087
Strawberry
0,984
0.976
0.970
0.968
0.943
0.754
0.916
0.982
0.90000.000
S Swedishl eaf SyntheticControl
0.906
0.910
0.914
0.907
0.592
0.459
0.910
0.971
00.015
0.987
0.967
0.980
0.977
0.927
0.977
0.990
0.999
1.000
0.960
0.810
0.700
0.710
1,000
1.000
0.000=0.000
TwoPatterns
0.943
0.905
0.913
0.939
0.997
0.999
1.000
1.000
1.000=0.000
UMD
0.917
0,924
0.972
0.910
0.444
0.715
1,000
0.993
1.000
UWaveGestuurLLibre Libraryy X
0.796
0.787
0.785
0.658
0.560
0.753
0.736
0.822
0.832
UWaveGGstureLLibaay
0.716
0.706
0.704
0.703
0.445
0.652
0.654
0.757
0.754
UWaveGasturLibbaary
0.740
0.739
0.729
0.433
0.678
0.703
0.764
0.003
WordSynonyms
0.580
0.585
0.611
0.621
0.177
0.458
0.538
0.742
0.784
Worms
0.571
0.507
0.558
0.507
0.351
0.494
0.416
0.765
0.003
WonnsTwoClass
0.623
0.623
0.610
0.597
0.519
0.468
0.571
0.657
0.815
Yoga
0.849
0.846
0.846
0.464
0.767
0.618
0.911
0.772+0.006
Average Accuracy
80.1
77.7
77.7
$6.0
63.1
84.8
Average Rank
4.2
5.0
4.9
5.4
7.8
7.2
6.1
2.7
1.8
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Published as a conference paper at ICLR 2024
Table G.3; ShapeConv Compared with Slate-Of-The-Af Time-Series Classification Methods: Evalu- ating Testing Accuracy for Supervised Time- SenieS Classification Tasks Across 128 UCR Datasets. Mean accuracy std over 3 independent experiments with different random seeds is reported.
DTW
TNC
TST
TS: TS-
T.sss
TS2Vec
ROCKET
ShapeCony
Adiac
0.604
0.726
0.550
0.767
0.675
0.775
0.7K3
0.867=0.000
ArrowHeadd
0.703
0.703
0.771
0.737
0.766
0.814
0.903=0.0005
Beef
0.633
0.733
0.500
0.600
0.667
0.767
0.833
0.034
BectleFly
0.700
0.850
1,000
0,800
0.800
0.900
0,900
1.000=0.000
BirdChicken
0.750
0.750
0,650
0.650
0.850
0.800
0,900
1.00000.000
Car
0.733
0.683
0.550
0.583
0.833
0.883
0.847
0.974=0.000
CBF
0.997
0,983
0.998
0,983
1,000
0.00000.0000
ChlorineConenttaaiinn
0.648
0.760
0.562
0.753
0.749
0.832
0.815
0.924=0.003
CinCEOGTonO
0.651
0.669
0.508
0.671
0.713
0.827
0.836
0.778
Coffee
1,000
1,000
0,821
1.000
1,000
1,000
1,000
1,000 000
Computers
0.700
0.684
0.696
0,704
0.664
0.660
0.761
0.647
CricketX
0.754
0.623
0.385
0.731
0.713
0.805
0.819
0.895 0.009
CricketY
0.744
0.597
0.467
0.718
0.728
0.769
0.852
0. 0.726 0.016
CticketZ
0.754
0.682
0.713
0.708
0.856
0.073
DistomiSizeReduction
0.967
0.993
0
0.977
0.984
0.987
0.970
1.994=0.000
DissiPPailanxOuttieeCoeeed
0.717
0.754
0.728
0.754
0.775
0.775
0.770
DissalPhllancOutiissgggooo
0,770
0.741
0,741
0.755
0.727
0.727
0.759
0.011
DistaPPhalanx
0,590
0.669
0.568
0.676
0.676
0.698
0.719
0.763 0.022
Earihquakes
0.719
0.748
0,748
0.748
0.748
0.748
0.748
0.0010
BCG200
0.770
0.830
0.830
0.880
0.940
0.920
0
0.001
BCG5000
0.924
0,937
0,928
0.941
0.933
0.947
0.007
ECGFiveDays
0.768
0.999
0,763
0.878
L,000
1,000
1.000
1,000 0.000
BlectrieDevices
0,602
0,700
0.676
0.686
0. 707
0.721
0.729
0.743 013
FaceAll
0.808
0.766
0,504
0.813
0.786
0,805
0.947
0.827
FaceFour
0,830
0.659
0.511
0.773
0.920
0.932
0.977
0.961
FacesUCR
0.905
0.789
0.543
0.863
0.884
0.930
0,961
0.930=
Fifty Words
0.690
0.653
0.525
0.653
0.732
0.774
0.830
0. 0.699 009
0.823
0.817
0,720
0.817
0.891
0.979
0.917
FordA
0.555
0.902
0.568
0.930
0.928
0.948
0.944
0.954 0 220
FordB
0.620
0.733
0.507
0.815
0.793
0.807
0.805
0.835 022
GunPoint
0:907
0.967
0,827
0.993
0,980
0,987
1.000
0.997
Ham
0.467
0.752
0.524
0,743
0.724
0.724
0.726
0.032
HandOutlines
0.881
0.930
0.735
0.724
0.922
0.930
0,942
0.947+0.016
Haptics
0.377
0.474
0.357
0.396
0,490
0.536
0,524
0,580 003
Herring
0
0.594
0.594
0.594
0.594
0.641
0.692
0.724=0.008
InlineSkate
0.384
0,378
0,287
0.347
0.371
0.415
0.457
0.432 023
InseetWinbbenSSound
0.355
0,549
0,266
0,415
0.597
0.630
0.657
0.613
Italy PowerDemand
0.950
0,928
0,845
0.955
0.954
0.961
0,970
0.974
I Lung::iicheeAAppliine
0.795
0.776
0.595
0.848
0.789
0.875
0,901
0.017
Lightning2
0.869
0.869
0.705
0.836
0.869
0.569
0.759
0.819 000
Lightning?
0.726
0.767
0.411
0.685
0.795
0.863
0.823
0.781
0.934
0.871
0.922
0.951
0.956
0.032
Mcat
0.933
0.917
0,900
0.883
0.950
0.967
0,948
0.043
Medicallmages
0.737
0.754
0.632
0.747
0.750
0.793
0.799
0.774
0.698
0.818
0.753
0.818
0.825
0.838
0.838
0.82740.004
0.500
0.643
0.617
0.630
0.656
0.636
0.590
0.669
MiddiePhalanxTTV
0.506
0.571
0.506
0.610
0.591
0.591
0.560
0.637 007
MoteStrain
0.835
0.825
0.768
0,843
0.851
0.863
0.915
0.919=0.029
0.790
0.898
0.898
0.878
0.913
0.913
0.865
0.912
0.832
0.913
0.919
0.940
0.929
0.942
OliveOil
0.833
0.833
0.800
0,800
0.867
0.900
0.917
0.827
OSULeaf
0.591
0.723
0.545
0.723
0.760
0.876
0.941
0.004
PhalangeSOutlinssComees
0.728
0.787
0.773
0.804
0. 787
0.823
0.834
0
Phoneme
0.228
0.180
0.139
0.242
0.276
0.312
0.280
0.204 004
Plane
1,000
1.000
0.933
1.000
0.990
1.000
1.000
1.000=0.000
0.784
0.866
0,770
0,873
0.859
0.900
0.899
0.019
ProximsaPPaalan Outtieiiiaaantttteeeeeeeoog
0.805
0.854
0.854
0.839
0.844
0.844
0.856
0.069
ProsinalPhallmsTTV TW
0.761
0.810
0.780
0,800
0.771
0.824
0.817
0.033
RefigerationDevices
0.464
0.565
0.483
0.563
0.515
0.589
0.537
0.594+0.026
ScreenType
0.397
0.509
0.419
0.419
0.416
0.411
0.485
0. 0.423 007
ShapeletSim
0.650
0.589
0.683
0.672
1.000
1.000
1.000=0.000 000
ShapesAll
0.768
0.78%
0.733
0.773
0.848
0.905
0.907
0. 0.051
SmullKichenAAppliacce
0.643
0.725
0.592
0
0.677
0.733
0.818
0.00S
Sony A.B
0.725
0.804
0.724
0.899
0.902
0.913
0.922
0.962=0.005
Sony Sony-ABORR AIBORabaSSSurfeee
0.831
0.834
0.745
0.907
0.889
0.890
0.913
0.014
Starl iughCueee
0.907
0.968
0.949
0.967
0.964
0.971
0.981
0.987 004
Strawberry
0.941
0.951
0.916
0.965
0.954
0.965
0.981
0.903 008
SwedishtLeaf
0.792
0.880
0.738
0.923
0.914
0.964
0.952#0.012
Symbols
0.950
0.885
0.786
0.916
0.963
0.976
0.974
0.980
SyntheticContral
0.993
1.0000
0.490
0.990
0.987
0.997
1.000
1.000
ToeSogmentationl
0.772
0.864
0,807
0.930
0.939
0.947
0.968
0.957
TheSegmentaiion2
0.838
0.831
0.615
0.877
0.900
0.915
0.924
Trace
1.000
1.000
1,000
1.000
0.990
1.000
1.000
1.000=0.000
23

---

Published as a conference paper at ICLR 2024
TwoLeadECG
0.976
0.900
0,990
6999
0.999
TwoPatterns
1,000
1.000
0,466
1.000
1,000
1,000
UWaveGestureLibrng
UWavoGesureLibrary
0.634
0.728
0.697
0.781
0,348
0,569
0.733
0,641
0.710
0.785
0.739
0.810
0.744
0.815
0.738=0.015
0.658
0.721
0,655
0,690
0.757
0.770
0.732
0.792
0.892
0.903
0,475
0.692
0.896
0.934
0.925
0.941=0.025
Wafer
0.980
0.994
0.991
0.994
0.992
0.998
0,998
0.973+
Wine
0.778
0.014
WondSynonyms
0.649
0.630
0.422
0.531
0.691
0.704
0.753
0.765 +
Wonns
0.584
0.623
0.455
0.753
0.727
0.701
0,740
0.783
WornsTWoCClas
0.623
0.727
0,584
0.753
0.792
0.805
0.797
0.815=0.011
Yoga
0.837
0.812
0.830
0.791
0.837
0.857
0,910
0.742
ACSFI
0.640
0.730
0.760
0.730
0.900
0.910
0.886
0.902 D
AllGeettreWimmteX
0.716
0.703
0.259
0.697
0. 763
0.777
0,790
0,831 042
W
0.017
AllGestureimmate
0.643
0.646
0.447
0.689
0.723
0.770
0.766
0 002
BME
0.900
0,973
0,760
0.933
0.993
0.993
1.000
1.000=0.000
Chinatown
0.957
0.977
0.936
0.983
0.951
0,9668
0,983
0.954
Crog
0.665
0.738
0.710
0,742
0.722
0.756
0.751
0.003
gnGHorizottaSiinal
0.503
0.442
0.373
0.401
0.605
0.544
0.539
0.609
EOGVarticaSignall
0.448
0,392
0.298
0.376
0.434
0.503
0.441
0.001
LlhanolLevel
0.704
PreezerRegularTrain
0.899
0.991
0.922
0.989
0.956
0.986
0.998
0.993 003
FreezerSmaal/raiin
0.753
0.982
0.920
0,979
0.933
0.894
0,950
0.972=0.011
Fungi
0.839
0.527
0,366
0.753
1,000
0.962
1.000
0.954
GestureMidAiiD1
0.569
0.431
0,208
0.369
0.608
0.631
0.617
O. 0.026
GestureMidAirD2
0,608
0.362
0,138
0.254
0.546
0.515
0,561
0 029
GesturMMidAirDS
0.323
0.292
0.154
0.177
0.285
0.346
0.315
0,405 0.008
GesturePebble211
0.919
0.
GesturcPoböle22
0.671
0.316
0.380
0.430
0.899
0.873
0.830
0.874
GuuiPointAgeSpan
0.918
0.984
0.991
0.994
0.994
0.914
0,997
1.000
Gan Versuseemlle
0.997
0,994
1,000
0.997
0.997
1,000
0,998
1.000=0.000
Venn Young
0.838
1,000
1,000
1.000
1,000
1.000
0.991
0.000
HouscTwenty
0.924
0,782
0.815
0.790
0.933
0.941
0.964
0.953+0.018
HasecEPGReguualllin TeaularTrrann
0.872
1,000
1,000
1.000
1.000
1.000
1.000
1.000
InseclEPGSmmlllTia
MelbouncPodestrian
0.791
0.942
0,741
0.949
0.944
0,959
0.904
0.92620.016
MixeddShapsRRegulee Train
0.842
0.911
0.879
0,855
0.905
0.922
0,921
0.005
MixedSShpesSSmill Small
0.750
0.813
0,828
0.735
0.860
0.881
0.918
0.027
Pic Pickup Gesture WiimoteZ
0.660
0.620
0,240
0.600
0.740
0.820
0.830
0.871 026
MggiiwayPressure
0.106
0.413
0.380
0.510
0.683
0.095
0.594 0.005
PigAriPressuree
0.245
0.808
0.774
0,524
0.928
0.966
0.954
0.872=0.022
PigCVP
PLAID
0.840
0.495
0.419
0.445
0.555
0.561
0,903
0.904
PowerCons
0.878
0.933
0.911
0.961
0.900
0.972
0,940
0.901
Rock
0.600
0.580
0.680
0.600
0.580
0.700
0,900
0.700+0.016
SemmHanndGemdsrhh22
0.802
0.882
0.725
0.837
0.890
0.963
0.927
0.972 004
SemggHandMovemmetthh22
0.584
0.924=0.00S
0.727
0.771
0.484
0.753
0.853
0.951
0.881
0.981
ShakeGestur Wimmttee
0.860
0,860
0.920
0.834
SmootSSubspace
0.827
0.913
0.827
0.953
0.960
0.993
0.979
1.000=0.000
UMD
0.993
0.993
0.910
0.986
0.993
1,000
0.992
L000=
DodgerL Doopaa
0.500
0.000
0,200
0,000
0.000
0.562
0.573
0.628 003
DodgerLLoppaaee
0.877
0.000
0.696
0.000
0.000
0.841
0.873
0.906 0.006
WodkettLoopWeeeen Weekend
0.975
0.971
Average Accuarcy
0.728
0.743
0.639
0,740
0.787
0.836
0.842
0.851
Averuge Rank
6.102
5.285
7.055
5.191
4.523
2.855
2.590
2.398
24

---

Published as a conference paper at ICLR 2024
Table G.4: ShapeConv Compared with State-Of-The-Art Time-Series Classification Methods: Eval- uating Testing Accuracy for Supervised Multivariate Time-Series Classification Tasks Across 30 UEA Datasets, Mean accuracy std over independent experiments with different random seeds is reported.
DTW
TNC
TST
TS-TCC
T-Loss
Vec
ShapeConv
Articulary WordRecognition
0.987
0.973
0.977
0.953
0.943
0.987
0.994=0.001
Atrilloibrillation
0.200
0.133
0.067
0.267
0.133
0.200
00,015
BasicMotions
0.975
0.975
0.975
1.000
1.000
0.975
0.991
Character Trajetorrss
0.989
0.967
0.975
0.985
0.993
0.995
0.98 +0.018
Cricket
1.000
0.958
1.000
0.917
0.972
0.972
0.998+0.008
DeekSue
0.600
0.460
0.620
0.380
0.650
0.680
0.64810.006
EigenWorms Worms
0.618
0.840
0.748
0.779
0.840
0.847
0.80210.008
Epilepsy
0.964
0.957
0,949
0.957
0.971
0.964
0.97210.009
ERing
0.133
0.852
0.874
0.904
0.133
0.874
0.774
ElhanooConnentrration
0.323
0.297
0.262
0.285
0.205
0.308
0.253
FaceDetection
0.529
0.536
0.534
0.544
0.513
0.501
0.635 10.025
FingerMovements
0.530
0,470
0.560
0.460
0,580
0.480
+..879
HandMovemennDirectton
0.231
0.324
0.243
0.243
0.351
0.338
0.413+0.020
Handwriting
0,286
0.249
0.225
0.498
0.451
0.515
0.527
Heartbeat
0.717
0.746
0.746
0.751
0.741
0.683
0.784=0.011
Japanese
Vowels Vowels
0.949
0.978
0.978
0.930
0.989
0.984
0.993+0.022
Libras
0.870
0.817
0.656
0.822
0.883
0.867
0.887
LSST
0.551
0.595
0.408
0.474
0.509
0.537
0.608
MotorImagery
0.500
0,500
0.500
0.610
0.580
0.510
0.674
NATOPS
0.883
0.911
0.850
0.822
0.917
0.928
0.937 0.0
PEMS-SF
0.711
0.699
0.740
0.734
0.676
0.682
0.801
PenDigits
0.977
0.979
0.560
0.974
0.981
0.989
0.96810.01
PhonemeSpectra
0.151
0.207
0.085
0.252
0.222
0.233
O. 0.09220.002
RacketSports
0.803
0.776
0.809
0.816
0.855
0.855
0..611
SeIlRegulationSCPP
0.775
0.799
0.754
0.823
0.843
0.812
0,858
SellRegullatonSSCP2
0.539
0.550
0.550
0.533
0.539
0.578
0.624
SpokenArrabiDDigitss
0.963
0.934
0.923
0.970
0.905
0.988
0.979
StandWalkJump
0.200
0.400
0.267
0.333
0.333
0.467
0.587+0.013
UWaveGestureiibraay
0.903
0.759
0.575
0.753
0.875
0.906
0.936=0.00Z
InsectWingbeat
0.469
0.105
0.264
0.156
0.466
0.509#0.026
Average Accuracy
0.650
0.670
0.617
0.668
0.658
0.704
0.743
Average Rank
4.717
4.583
5.283
4.333
3.900
3.117
2.067
G.4 RESULTS OF UNSUPERVISED LEARNING TASKS
In this section, we report the full results of unsupervised time-series clustering tasks across 36 UCR Datasets in Table G.5| We also report the result of unsupervised multivariate time-series clustering tasks compared with /Zhang & Sun (2022) across 12 UEA datasets in Table
25

---

Published as a conference paper at ICLR 2024
Table G.5: Unsupervised time-series clustering results (NMI on test data) across 36 UCR datasets. Mean accuracy + std over 3 independent experiments with different random seeds is reported.
Dataser
KMcans
1-Stape
U-Shuget.
DTC
USSI.
DTCR
STCN
AutoS.
ShapeC:
Shapel.. w/o
ShapeC.
Human
Arrow
0,4816
0.5240
0.5000
0.6322
0.5513
0.5240
0.5624
0.5134
0.6123
0.60
Reef
0.2925
0.3338
0.3413
0.2751
0.3338
0.5473
0.5433
0.3799
0.5077
0.3854
0.4039-0.000
0.618--0.017
BeetleFly
0.0073
0.3456
0.5105
0.4456
0.5310
0,7610
1,0000
0.5310
0.4397
10000
1.000=0.000
10000-000000
BirdChacken
0.0371
0.3456
0.2783
0.0073
0.6190
0.5300
1,0000
0.6882
0.5824
1,000
1.00000.0000
1.0000 0.000
Car
0.2540
0.3771
0.3655
1892
04650
0.9001
0,5261
0.4970
0.5672
0,4690
0.4770-0.006
0.5013=00.01
Chlorine
0.0129
0.000
0.0135
0.0133
0.0133
0.0195
0.0760
0,0133
0.0024
0.1358
0.0527200007
0.06-120.014
Coffee
0.5246
1,0000
1,0000
0.5523
1,0000
0.6277
1.0000
1,0000
1.0000
LIM00
L 0.000-0.000
L BOIND-C.00M
Diatom,
0.2300
1,0000
0.9418
1,0000
1,0000
1,000
1.000000:000
LIBND
0,80
0.3911
0,5037
04237
0.4464
0.4766-0.000
0.5291-0.00
Diatcooreet
0,0278
0.0527
0.9063
00115
0.1180
0,2327
0.133
0,0599
0.0885
0.1074200000
BCX0200
0.3403
0.3682
0.1323
00218
03276
033691
04516
0.3228
0,3002
0.5413
0.5552-0:009
0,6240
0.0002
0.0002
0.94
00022
06502
0,8056
0,352
0.7835
0.6355
0.8150
0.826600.010
0.7669-0.027
GunPoint
0.0126
0.3653
0,3633
0.019
0.4200
0,5537
0.4127
0.3405
0.4243
0.4676600000
0.5652:0.00
Hom
0.0093
00817
0.0619
01016
03411
0.0069
0.2582
0.3211
0.1764
0.3890
0.3011-0:001
0.4467- -C
Herring
0.0013
0.0027
0.1321
00145
0.1718
0.2248
0.2002
0.2019
0.1423
0.2291
0.2630=0.012
=2.01%=00014
Lighting?
0.0038
0.2670
0.0144
0.34
0.5527
0.2286
0,3479
0.3550
0.3040
0.3756
0.4282:0.000
0.4723-0
Mest
0.2510
0.2716
0.2250
0,9055
0.9653
0.9437
0.2901
0.9481
10000_
Mid ageri
0.219
0.0722
0,2780
0,4661
0,5109
0,3940
0.1530
0.4214
0.4576-
oireet
0,0253
0,002
0,200
0-1150
0,0921
0.2873
0,1912
0,2109
0.209600.00
0.3499-0.000
MidTW
0,134
0.5229
0,005
03120
0-5502
0.4480
0,756
02161
0.9241.0:004
0U3T00.C.0M
MoxeStrain
0.051
022115
0.0033
00094
05310
04004
0-257
0 0,2776
054006
6.738-00000
0.50N2-00014
OSULeaa
0.0288
0.0125
0.0200
022011
03353
0.259
0,3544
0.4632
0.9922
0,133
0.5159-0.017
0.
Plane
0,8598
0.9642
1.0000
03678
1.0000
0.9296
0.9115
0.9982
1,0000
1.0000
I. 0.000=0.000
L.0N0= 0.000
Prox. agets
0.0635
00110
0.033
04155
0.813
0.5581
116317
0.6930
0.5164
0.6857
0.6453500013
0,7 0.7242 C:013
Prox. TW
0.0022
011577
0.0107
06199
1,0000
0.6530
0.7330
0.8947
0.5948
0.836
0.8264-0.000
0.81 I-- C.05
Sony
06112
0.7107
0.5800
0.2559
0.5991
0.6634
0.6112
0.6096
0.5413
0.6256
0,610700.00 06215=0.047
S1ny.1
0.5444
0.0100
0.6121
0.5647
0.7022
0.4955
0.7343 C.055
SwedishLenf
0.1041
0.3456
0,1186
0.6663
0,6106
0.9340
0.8592
0.8415=0.00
024412:0.00
Symbols
0,7780
0.6366
0,8691
07955
18221
0.8999
0,8940
0.9147
0.7728
0,3313
0.91
T
0.0022
0.3073
0.3073
0.1888
03351
0.3115
0,367]
0,3610
0.2830
0,4700
0.4863.0.00 0.00
-
ToeSeg
0.0863
01866
0.1519
0009
01430
0.3249
0,5498
D.4664
0.1293
0.995
0.5178. 1.000000000
0.6636, CJIUT
TwoPulkern
0.4696
0.3949
0.2979
00119
0.4011
0,4713
0,4110
0.5150
0.3083
0,5030
0.007700.004
0.6154-0.019
TweLeaGECG
0.0000
0.0000
0,0129
0.0066
0.4471
0.4614
0.6911
0.5654
0.4220
0.6936
0.628%%00.033
0,7048
Wafer
0.0010
0.0010
0.0010
0.0088
010452
0.0128
0.2088
0.0120
-0:003
0.0741
0 0.00K
OLD477 C.021
Wine
0.0031
0.0199
0.0171
0.0000
0.5311
0.2580
0.5927
0.6045
0.3340
0.6090
06612500,116
0.6710-0.00
Wordes.
0.5435
0,3933
0.349
0.4984
0,3947
0.5112
0.3861
0,5884
0,5952.0.000
0.065
Avg Avc.
0,2777
2333
0.5427
0,5478
0.5558
0,4133
0.5897
0.6017
0.6464
Arg.
9.5556
S.4028
8,4306
9.6250
4,3611
5,0833
4,1250
3.7083
7.2639
3,0694
2.3750
NA
Table G.6: Unsupervised time-series clustering results (NMI on test data) across 12 UEA datasets. Mean accuracy + std over independent experiments with different random seeds is reported.
Daiaset
MC2PCA
SWIMDFC
TCK
In-LAVG+ED
m-LDBA
m-kShtaps
M-1SC
DeTSEC
NESE
MUSLA
ShapeCoom
Articulary WordR.
0.934
0.523
0.873
0.834
0.741
0.344
0.843
0.792
0.849
0.838
0.867.0.010
AtrialFibrilazion
0.514
0.191
0.313
0.116
0.387
0,538
0.57920.015
BasicMotions
0.674
0.510
0.776
0.543
0.639
0.341
0.554
0.800
0.525
1.000
1.000:0.000
Epilajay
0.175
0.190
0.533
(.479)
0.471
0.163
0.381
0.345
0.760
0.601
0.681
Ering
0.336
(1422
0.359
0.400
0.406
0.268
0.348
0.392
0.378
0.722
1.004
HandMooeemmnDD...
0.067
0.151
0.103
0.168
0.265
0.079
0.151
0,112
0.030
0.398
0.302+0.11
Libras
0.577
(1.00)
0.670
0,622
0,622
0.447
0.724
0.602
0.724
0.738 0003
NATOPS
0.698
0.472
0.679
0.643
0.643
0.339
0.600
0.043
0.314
0.855
0.878 00.00
PEMS-SP SP
0.011
0.44]
0.066
0.491
0.402
0.447
0.474
0.424
0.536
0.614
0.630
PenDigits
0.713
0.652
0.603
0.738
0.605
0.634
0.738
0.563
01645
01836
0.784-
StandWalkJJmp
0.349
01483
0.536
0.59)
0.466
0.116
0.461
0.555
0.399
(1609
UWWaeGeestuee..
0.570
0.482
0.710
0.582
0.419
0.758
0,557
0.728
0,74240.014
Average Ace.
0.4680
0.4466
0.5150
0.5530
0.5132
0.5348
0.4943
0,7044
0.7154
Average Kank
7.0833
7,3333
5,8333
4,833
6.4167
10.0833
5,5833
7.6667
7,2500
2.2917
162500
26


Reviews:
[{"review_id": "8S8eBMgH3q", "reviewer": "Reviewer_VPQr", "summary": "This article deals with the classification of time series. The authors describe the equivalence between a particular approach, shapelets, and convolutional layers. They provide several losses to enforce the diversity of learned shapelets and closeness to original data, as well as intuitive initialization methods. The proposed approach is compared to several algorithms in a thorough experimental study.", "questions": "In addition to addressing my comments about Theorem 3.1 and the comparison to the state-of-the-art, I have one question:\n- Convolutional layers are meant to be stacked. Unless I am mistaken, in the experiments, there is only one ShapeConv layer. Would the interpretability of ShapeConv remain if there are several layers?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 1, "strengths": "The article describes the methodology well, and, to the best of my knowledge, the proposed initializations and losses are novel in the context of shapelets. The experiment study is extensive (with one caveat, see below) and convincing.", "weaknesses": "- The main contribution is based on Theorem 3.1, which shows that the shapelet transform is somewhat equivalent to a convolution layer followed by a max pooling operation. However, this fact has been observed previously to provide accelerated shapelet transform: the authors of [1] show that computing the distance profile ($dist(\\mathbf{s}, \\mathbf{x})$ for a given sequence $\\mathbf{s}$ and all subsequences $\\mathbf{x}$ of $\\mathbf{X}$) is equivalent to a convolution. \n\n- An extensive review of time series classification algorithms exists on the same data sets, see [2] and more recently but unpublished [3]. None of the algorithms referenced in [2, 3] are compared to ShapeConv. The authors should at least compare themselves to the best-performing algorithms of the state-of-the-art.\n\n- (Minor comment.) It is considered bad practice to start sentences with mathematical symbols.\n\n[1] Yeh, C. C. M., Zhu, Y., Ulanova, L., Begum, N., Ding, Y., Dau, H. A., Zimmerman, Z., Silva, D. F., Mueen, A., & Keogh, E. (2016). Matrix Profile I: All Pairs Similarity Joins for Time Series: A Unifying View that Includes Motifs, Discords and Shapelets. Proceedings of the IEEE International Conference on Data Mining (ICDM), 1317–1322. https://doi.org/10.1007/s10618-017-0519-9\n\n[2] Bagnall, A., Lines, J., Bostrom, A., Large, J., & Keogh, E. (2017). The great time series classification bake off: a review and experimental evaluation of recent algorithmic advances. Data Mining and Knowledge Discovery, 31(3). https://doi.org/10.1007/s10618-016-0483-9\n\n[3] Middlehurst, M., Schäfer, P., & Bagnall, A. (2023). Bake off redux: a review and experimental evaluation of recent time series classification algorithms. ArXiv. http://arxiv.org/abs/2304.13029", "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": "", "extra_scores": {}}, {"review_id": "ptQrS5aYKg", "reviewer": "Reviewer_JCHN", "summary": "The paper bridges the divide between traditional shapelets and modern deep learning methods in time-series modeling. Shapelets, while interpretable, face efficiency issues; deep learning models offer performance but lack interpretability. The proposed ShapeConv melds these approaches, using a CNN layer with its kernel functioning as shapelets. This layer is both interpretable and efficient, achieving state-of-the-art results in experiments. The introduction of shaping regularization and human knowledge further enhances its performance and interpretability.", "questions": "This study opted for a combination of CNN and Shapelets to enhance interpretability while also boosting performance. For time series classification tasks, why not choose the stronger baseline models for research, such as RNN or Transformer?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper theoretically establishes an equivalence between traditional Shapelets and using a convolutional layer to derive similar features. It’s a fresh perspective in utilizing shapelets in combination with deep learning methods and structure.\n2. The comprehensive experiments empirically demonstrate the superior performance of ShapeConv, in both classification as well as clustering tasks.\n3. The paper is well-written and easy to understand.", "weaknesses": "1. An analysis of the computational complexity and resource requirements of ShapeConv could make the paper more comprehensive.\n2. Though the model's performance is promising, concerns may arise regarding the complexity of implementing ShapeConv compared to other traditional or deep learning models.", "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": "", "extra_scores": {}}, {"review_id": "wCkfgHfJM8", "reviewer": "Reviewer_nftB", "summary": "The paper combines Shapelets and CNNs", "questions": "NA", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Good empirical results.\nTests on many datasets (although, just download and test, no new datasets)", "weaknesses": "I appreciate the accuracy improvements, while small, are probably real.\n\nHowever, I have no confidence in any of the claims of interpretability and explainability.\nYou made no effort to obtain the original herring images or gun-point videos. You explanations here are \"just-so\" stories. [v].\n\nIf you wanted to make convincing claims here, you could obtain the herring images, modify them to add / remove effects, then see how this affects the shapelets. Or reenact the gun-point video, and  modify the protocol to add / remove effects, or...\n\nI do understand that most people in this space are too lazy to go beyond downloading the UCR datasets. But if that is all you do, it seems like you should temper your claims about interpretability and explainability.\n\n\n\n\n\n“In the realm of machine learning, interpretable time-series modeling stands as a pivotal endeavor, striving to encode sequences and forecast in a manner that resonates with human comprehension”\nThis (and the rest of the paper) read like flowery language [a].\n\n\nIn fig 1, can you move the legend away from the data?\n\n\n“is evaluated on the 25 UCR” Did you mean “125” or “25”?\n\n\n“Figure 5: Shaplets learned” typo (Shapelets)\n\n\n“It is evident that the shapelet learned by ShapeConv captures the distinguishing features of the class effectively”  Evident to whom? You should argue that the blue shapelets correctly represents the actors hand having to hover over the gun holster, then reach down to the gun, then draw the gun.\n\n\n“clustering task using 36 UCR univariate”\nWhy 36? Why this particular 36?\n\n\n\n“In response to the first RQ, we observe that ShapeConv’s shapelets (Figure 4 (a)) cover all turning points in the time series, where the two classes differ the most, while LTS’s shapelets (Figure 4 (b)) do not cover the targeted regions.”\nThis evaluation is tautological. If  “turning points” are the best places for shapelets, then we don’t need any search for shapelets at all.\n\n\n\n“In contrast, when using human initialization,..”\nHmm, it is a bit tricky to claim results based on human initialization. Which humans, how trained are they in the system, how are they briefed. In my mind, that is a separate “human in the loop” paper.\n\n\nHowever, despite ingenious, the performance (missing a word?)\nHowever, despite ingenious suggestions, the performance\n\n\ngun out of the gun pocket (holster) \n\n\n“while data from the “finger” class don’t.”\n“while data from the “finger” class do not.”  (avoid contractions in scientific writing) \n\n\nThis illustrate how\nThis illustrates how\n\nIn table E.5, why four significant digits? This is spurious  accuracy.\n\nIn table E.5 and elsewhere, you report the average accuracy.  This is meaningless for datasets of different sizes, class skews, number of classes, default rates etc. To be clear, it is not a flawed metric, it is just meaningless. \n\n\n[a] https://www.latimes.com/socal/daily-pilot/opinion/story/2022-05-03/a-word-please-flowery-writing-can-turn-off-readers\n\n[v] https://en.wikipedia.org/wiki/Just-so_story", "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": "", "extra_scores": {}}, {"review_id": "dwBj0B1jEv", "reviewer": "Reviewer_JpzU", "summary": "The paper proposes a modified convolutional layer for time series analysis inspired by the Shapelet distance that is widely used in the domain.\nThis new layer is then used at the core of neural networks for both supervised and unsupervised tasks.\nA regularization term for the task-specific losses is designed that enforces learned kernels to (i) look like actual subseries from the training set and (ii) form a diverse set.", "questions": "Apart from the questions/suggestions related to the evaluation of interpretability, I have a few remarks/questions that are listed below:\n\n* If you took your your ShapeConv model (with exact same initialization, regularization terms, etc.) and changed the ShapeConv layer with a convolutional one, what would you get in terms of performance? This experiment is required to fully assess if the norm terms a really helpful\n* In terms of evaluation:\n    * How are baseline model hyperparameters tuned (and which parameters are tuned)?\n    * How do you pick the datasets for the subsets (25 datasets for supervised learning and 36 datasets for unsupervised learning)?\n    * If the goal is to compare to state-of-the-art methods, other competitors should be included in the comparison (eg. ROCKET, COTE & variants, ...)\n\nBelow are some minor remarks/questions:\n* In Section 1, you write:\n    >  they are more likely to overfit when the signal-to-noise ratio is relatively large\n    * Don't you mean \"is relatively low\"?\n* Initialization\n    * Have you assessed how important it was to use supervised information at initialization?\n    * Have you tried simpler approaches (eg. kmeans++ on randomly selected subsequences of adequate length)?\n* If ShapeConv is faster than LTS, it is probably more an artifact of the implementation since the overall complexity of ShapeConv is probably higher than that of LTS (similar local representation extracted, but ShapeConv have additional loss terms that induce more computations)\n* Presentation\n    * Unsupervised learning: it is unclear from the presentation in Section 3.4 which clustering method is used on top of the features extracted from ShapeConv. This is detailed in Section 4.2, but should be explained in Section 3.4 imho", "limitations": "", "rating": 5, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The paper is very well written and the motivation for the method is clear.\nThe experimental validation is quite thorough and it is nice to showcase that the method can be used for both supervised and unsupervised learning.\nIn terms of the method, while the idea of having a Shapelet layer included in a neural network is not novel, both the initialization scheme and the regularization terms included in the loss lead to improvement on the performance of the resulting models.", "weaknesses": "In the abstract (it is also said in the introduction in other words), it is stated that:\n\n>  In this paper, we demonstrate that shapelets are essentially equivalent to a specific type of CNN kernel with a squared norm and pooling\n\nIn fact, this demonstration is not novel, it is for example stated in (Lods et al. 2017) (that is cited in the paper).\nHowever, it seems that here, the proof aims at more rigor, but Theorem 3.1 is not successful in this regard since it completely disregards the fact that the bias term in convolution is independent of the input, which is not the case in the $-\\mathcal{N}(s_i, X_{j:j+l_s-1})$ term. (Also, as a side note, in Theorem 3.1, Squared Euclidean distance is used, not Euclidean distance as stated.)\n\nMoreover, the review of the Related Work is very succinct and a more thorough presentation of competing interpretable Shapelet-based methods would have been a plus. \nSimilarly, a more detailed comparison of the interpretability of the ShapeConv model with those baselines is required to fully assess interpretability:\n* Only toy examples are presented (eg. Fig 4: 2 shapelets), what does it give when training with a large amount of shapelets?\n* Also, providing visualization for a large number of datasets instead of only GunPoint+Herring+ECG200 would be a real plus", "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": "", "extra_scores": {}}, {"review_id": "4Zff89qcAa", "reviewer": "Reviewer_Lmj7", "summary": "The authors introduce ShapeConv, an interpretable CNN layer whose kernels function as shapelets, designed for time series modeling in both supervised and unsupervised settings. They demonstrate that using the square norm in convolution, coupled with max pooling, is equivalent to computing the distance between a shapelet and a time series. Within this framework, a convolutional kernel essentially serves as a tunable shapelet. The authors also incorporate regularization to enforce similarity and diversity among shapelets, depending on whether the task is supervised (classification) or unsupervised (clustering). The methodology is validated through experiments on time series classification and clustering, using several competitor models and alternative implementations of ShapeConv for comparison. XAI is assessed via author-selected examples.", "questions": "1. **Lack of Comprehensive Review of Related Work**:\n   - The authors focus exclusively on optimization-based shapelet approaches. While space is limited, notable methods like Random Shapelet Forest and standard shapelet transform should not be omitted. Dictionary-based and interval-based approaches are also relevant and have achieved state-of-the-art performance in time series classification, yet they are not mentioned. Furthermore, the competitor models used in the experimental section are largely transformer-based or rely on embeddings, making for an unusual selection. I recommend that the authors thoroughly review relevant literature on time series classification, such as the paper by Ruiz et al. (2021) and models like ROCKET by Dempster et al. (2020).\n\n2. **Ambiguities in the Experimental Section**:\n    - Is the \"Initialization\" phase's cost included in the runtime?\n    - In Table 1, why do the methods differ with respect to the cd plots?\n    - Why is the evaluation limited to 25 UCR datasets, and what was the criteria for selection?\n    - Several state-of-the-art methods like Rocket, CIF, ShapeletTransform, and MUSE are absent from the comparison.\n\n3. **Limitations in XAI Evaluation**:\n    - While the author-selected examples support the paper's claims, they do not suffice to demonstrate the superiority of ShapeConv in terms of shapelet quality. Additionally, pairing ShapeConv with MLP or SVM models does not provide sufficient interpretability. I suggest testing the approach with tree-based or linear models, or employing explainers such as SHAP to determine the importance of shapelets, especially in supervised tasks.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper is generally well-structured and straightforward to follow.\n- It establishes an interesting link between convolutional operations and the shapelet transform.\n- The proposed methodology is versatile, applicable to both supervised and unsupervised tasks.", "weaknesses": "- The paper lacks a comprehensive review of related work, and the selection of competitor approaches for comparison is odd.\n- Parts of the experimental section are unclear and require further clarification. The XAI evaluation is restricted to examples selected by the authors.\n- There is no discussion or citation concerning code implementation.", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)