# cnn_kernels_can_be_the_best_shapelets__e4879846.pdf Published as a conference paper at ICLR 2024 CNN KERNELS CAN BE THE BEST SHAPELETS Eric Qu1, Yansen Wang2, Xufang Luo2, Wenqiang He3, Kan Ren4, Dongsheng Li2 1University of California, Berkeley, 2Microsoft Research Asia, 3University of Science and Technology of China, 4Shanghai Tech University ericqu@berkeley.edu, {yansenwang,xufluo,dongsheng.li}@microsoft.com, wenqianghe@mail.ustc.edu.cn,renkan@shanghaitech.edu.cn 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 Shape Conv, 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 Shape Conv by adjusting its initialization and model performance is boosted with it. Experiments show that Shape Conv can achieve state-of-the-art performance on time-series benchmarks without sacrificing interpretability and controllability. 1 INTRODUCTION 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. 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 a shapelet and all conceivable sub-sequences of the raw input is ascertained, yielding features that signify a shapelet s imprint on a sequence. The allure of shapelets lies in their capacity to 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 to fit the sequence with a high dimensional non-convex function using deep neural networks such as RNN (Guo et al., 2019), CNN (Franceschi et al., 2019), Transformer (Wu et al., 2021; Qu et al., 2022; Cheng et al., 2023), etc. to model the time series. These deep-learning-based methods have attracted much more attention than shapelets, thanks to their great performance when the number of data is sufficient, but they are more likely to overfit when the signal-to-noise 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., 2023) 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 advantages and characteristics of both. Despite the apparent disparity between shapelets and deep models in 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 internship at MSRA. Correspondence to: Xufang Luo (xufluo@microsoft.com; Dongsheng Li (dongsheng.li@microsoft.com). Published as a conference paper at ICLR 2024 series to a specific convolutional layer (1-layer CNN) with a squared norm and pooling. This finding provides us the basis for combining shapelets with deep models. To implement the equivalence, we devise Shape Conv, a CNN layer wherein its kernel functions as shapelets, adeptly and interpretably addressing the time-series modeling challenge. We introduce several ingenious designs to make Shape Conv effective in practice. During the optimization process, due to the difference of candidate space, the subsequences derived via gradient-based techniques might diverge significantly from the sub-sequence prototypes, rendering them less suitable as interpretable shapelets. Hence, Shape Conv 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 to 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, Shape Conv, being a 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 approaches, our kernels provide controllability, facilitated by our strategic initialization that incorporates human expertise to yield more human-comprehensible results. Empirical evaluations underscore Shape Conv s prowess in both supervised classification and unsupervised clustering tasks and various datasets. It surpasses other learning-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 a marked reduction in time complexity when compared to earlier shapelet methodologies. We summarize our contribution as 4 folds: (1) 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 Shape Conv, an interpretable CNN layer with its kernel serving as shapelets. Several regularizations and initializations are accompanied to enforce similarity and diversity, making Shape Conv effective in practice. (3) By treating the CNN kernel as a shapelet, we claim another advantage of facilitating the incorporation of human prior knowledge by initialization. (4) Extensive experiments on real-world datasets validates Shape Conv s superior performance on interpretable time-series modeling tasks. 2 BACKGROUND Figure 1: An example of time-series classification task and learned shapelets by Shape Conv. Both shapelets are for class 1, and they capture two distinguish shapes of the class, respectively (indicated by green frames). Lines of class 1 exhibit more pronounced fluctuations at these two locations. Shapelets Shapelet (Ye & Keogh, 2009), originally defined as a maximally discriminative sub-sequence in time-series data (shown in Figure 1), is designed to capture inter-class features in terms of small subsequences rather than the full sequence. Shapeletbased methods will usually find a subset of shapelets S ˆS from data to maximize information gain by splitting data with the shapelets as nodes of a decision tree. The candidate shapelet set ˆS contains all possible sub-sequences of the original data. Afterwards, a shapelet transform (Lines et al., 2012) is introduced to decouple the shapelet discovering step and downstream classifier. This shapelet transform step will extract features for an input signal X based on the distances between shapelets and the input. Specifically, for each selected shapelet s S , we calculate the minimal distance between s and all sub-sequences of X, i.e., ds,X = min x ˆX dist(s,x), (1) Published as a conference paper at ICLR 2024 where ˆX is the set containing all sub-sequences with the same length as s, and dist(s,x) is the distance function (usually the squared Euclidean distance) measuring the similarity between a shapelet and a raw sub-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 tree-based models. We provide a more vivid example on shapelets and shapelet transform in Appendix B.2. Beyond traditional costly shapelet discovering methods, following works extend the candidate set to Rls where ls is the length of shapelet, and try to solve the problem with optimization-based methods. Convolutional Neural Networks CNN is one of the most commonly used network structures and its several variants have been used in time-series modeling for years (Ismail Fawaz et al., 2019). A CNN layer is essentially some sliding filters known as 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 Rnin lx with nin input channels and lx time steps. Applying a 1-D convolution over it can be formulated as: nin k=1 Wk Xk, (2) Where W Rnin nout ls denotes the weights of convolutional kernels. The symbol denotes the cross-correlation operator on the k-th input channel between each raw vector of Wk and the input Xk. Note that for most CNN kernels, weights Wk is learnable so this cross-correlation is equivalent to the convolution in terms of optimization. After that, a non-linear activation function and a pooling layer are often applied to extract the aggregated value among its neighbors N(j) for each channel i at location j, which can be written as: Ypool ij = pool j N(j) σ(Yij + bi). (3) Here, The bias term b is sometimes set to zero for simplicity. The symbol σ denotes the activation function such as Re LU, Tanh, and pool denotes a pooling function such as max or mean. 3 HOW CAN CNN KERNELS BE THE BEST SHAPELETS? In this section, we give a comprehensive answer to the question in the title. First, we provide a formal proof to show the equivalence between CNN and shapelets in Sec. 3.1. Then, we introduce Shape Conv, a novel convolutional layer well utilizing the equivalence in Sec. 3.2. Finally, we show how Shape Conv 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 is 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 X with nout kernels from a 1-D CNN layer added by squared L2 norm, followed by a maximum pooling. The difference between these two can be easily handled and omitted in practice. We summarize the finding in the following theorem: Theorem 3.1 Assume the input X Rlx is a 1-dimensional single-variate signal of length lx, and nout shapelets S = {s1,s2,...snout} with length ls are discovered. The feature extracted from X with si and squared Euclidean distance is dsi,X. Then we have dsi,X = 2 max j {1,2,...,lx ls+1}[Yij N(si,Xj j+ls 1)], (4) where Y = si X is the cross-correlation defined in Eq. 2 and N(si,X ) = ( si 2 2 + X 2 2)/2 is squared L2 norm term. Detailed proof can be found in the Appendix C. Eq. 4 bridges shapelets and the learnable CNN kernel. The left-hand side is the feature extracted by a specific shapelet, and the right-hand side contains the Published as a conference paper at ICLR 2024 maximum pooling over the convolution between the kernel and the input time series and a 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 σ is monotonically increasing, the order to apply maximum pooling and activation function can be swapped, i.e., max(σ( )) = σ(max( )). This fits for most cases in practice with Re LU, Tanh, Sigmoid and their variants as activation functions. Another is the bias term. Since b is often designed to be independent of the position, we can have maxj(Yij + bi) = maxj(Yij) + bi. Now we can rewrite Eq. 3 as: Ymax ij = max j N(j)σ(Yij + bi) = σ( max j N(j)(Yij ) + bi). (5) 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 a complete equivalence. 3.2 SHAPECONV: AN INTERPRETABLE CNN LAYER WITH ITS KERNELS SERVING AS SHAPELETS Motivated by the above established equivalence, we introduce Shape Conv, 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 a squared norm term N(si,X ) in Eq. 4 before the pooling function to the CNN layer. Consequently, Shape Conv 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 Shape Conv play the same role as shapelets. Although initially univariate, Shape Conv can directly adapt to multivariate data tasks by adjusting the number of input channels while maintaining its other designs, enabling the model to learn multiple kernels/shapelets for each variate. Shaping Kernels To serve as a good shapelet, Shape Conv 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 Rls 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.e., shapelet si, and sub-sequences in the input, and the shape regularizer is defined as the minimal distances, i.e., Rshape = 1 nout nout i=1 min x ˆXi dist(si,x), (6) where ˆXi is the set containing all sub-sequences with same length as si. 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.3. Therefore, we further introduce a diversity regularizer following Zhang et al. (2018) to relieve this issue. We first use the pairwise ℓ2 distance between different kernels (or shapelets) to construct a distance matrix Ds, where Ds(i,j) = exp( ℓ2(si,sj)), and the diversity regularizer is defined as F-norm of Ds, i.e., Rdiv = Ds F . (7) Published as a conference paper at ICLR 2024 Class 1 Class 2 Sliding Window Random Sample Clustering Shapelets Class 1 Class 2 Shapelets Samples Figure 2: Illustration of initialization methods for Shape Conv kernels. We let nout = 4, ncls = 2 for both figures. Left part shows initialization in supervised learning, k = 2 here. Right part shows initialization in unsupervised learning, ncut = 2 and k = 2 here. To summarize, the above basic designs make Shape Conv a differentiable CNN layer with its kernels serving as shapelets. Shape Conv can keep the advantages of both CNN and shapelets. On the one hand, Shape Conv is a fully differentiable CNN layer, so it can be optimized effectively with lower time complexity than traditional methods. On the other hand, kernels of Shape Conv have good interpretability since they are similar to discriminative sub-sequences in original data. 3.3 SUPERVISED LEARNING In this section, we show the method to apply Shape Conv in a supervised learning task. Here, we focus on the time-series classification task, which is a typical application for shapelets. Initialization The target of initialization is to provide convolutional kernels with a good starting point capturing class-specific sub-sequences in the data. As illustrated in the left part of Figure 2, 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 nout output channels, corresponding to nout shapelets, with each shapelet of length ls. The input length is l X, and there are ncla classes. First, we assign k = nout/ncla shapelets to each class. Within each class, we divide the time series into k equal parts of length lk = l X/k (along the time axis). For each part, we use a sliding window to find all possible candidates of length ls. 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 a specific region of the time-series data in the dataset. As a result, the model can more effectively capture local information within the initialization region. The initialization is also paralleled on GPU, making it very time efficient. Classifier and Loss Function The output of the Shape Conv layer is the shapelet transformed distances, representing features extracted by learned shapelets, and these features are further utilized by the downstream classifier. Here, we append a multi-layer perceptron (MLP) after the Shape Conv layer, to map the shapelet transformed distance to the class labels. This method allows for end-to-end training of the entire model, optimizing both the Shape Conv layer and the classifier simultaneously. The loss function is designed as follows, L = Lcls + λshape Rshape + λdiv Rdiv. (8) Here, Lcls is the task-specific classification loss, such as cross-entropy. The term Rshape is the shape regularizer defined in Eq. 6, and the term Rdiv is the diversity regularizer defined in Eq. 7. The hyperparameter λshape and λdiv controls the balance between each terms. Note that Shape Conv is compatible with other classifiers. We discuss this in Appendix D.1 and include this variant in our experiment in Sec. 4.1. Published as a conference paper at ICLR 2024 3.4 UNSUPERVISED LEARNING We now apply Shape Conv to the unsupervised learning task, where Shape Conv needs to capture the most representative sub-sequences in data and perform K-means on the shapelet-transformed 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 a 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 to find initial shapelets, as illustrated in the right part of Figure 2. Suppose we have nout shapelets in the Shape Conv layer, each with a length of ls. The input length is l X. First, all input time series are divided into ncut equal parts along the time axis, each with a length of lcut = l X/ncut. Each part is assigned with k = nout/ncut shapelets. We then sample a large number of subsequences (e.g., 10,000) of length ls from each part and perform KMeans clustering with k centers on them. Finally, the cluster centers are utilized as the initialization of the shapelets. In this approach, the class information is implicitly introduced during 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. Loss Function As for the task-specific loss, we employ Davies-Bouldin Index (DBI) (Davies & Bouldin, 1979) to optimize Shape Conv 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, L = LDBI + λshape Rshape + λdiv Rdiv. (9) Here, LDBI is DBI loss. The term Rshape is the shape regularizer defined in Eq. 6, and the term Rdiv is the diversity regularizer defined in Eq. 7. The hyperparameter λshape and λdiv controls the balance between each terms. Incorporating Human Knowledge The characteristic of Shape Conv makes it easy to incorporate human knowledge, which means that human experts can tell the model what some key subsequences look like, and the model can 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. 4 EXPERIMENTS AND ANALYSIS 4.1 SUPERVISED TIME-SERIES CLASSIFICATION Settings We evaluate our Shape Conv model on time-series classification tasks using the UCR univariate time-series dataset (Dau et al., 2019) and UAE multivariate times series dataset (Bagnall et al., 2018). Hyperparameters are tuned via grid search based on the validation set performance, and they are reported in Appendix G.2. Compared Methods We compare Shape Conv with three kinds of baselines: (1) shapelet-based methods (IGSVM (Hills et al., 2014), FLAG (Hou et al., 2016), LTS (Grabocka et al., 2014), ADSN (Ma et al., 2020b)), (2) common deep learning methods (MLP, CNN, Res Net (Wang et al., 2017)), and (3) state-of-the-art time-series classification models (DTW (Chen et al., 2013), TNC (Tonekaboni et al., 2021), TST (Zerveas et al., 2021), TS-TCC (Eldele et al., 2021), T-Loss (Franceschi et al., 2019), TS2Vec (Yue et al., 2022)). The results for the baseline methods are taken directly from the Published as a conference paper at ICLR 2024 1 3 5 7 9 11 2.4 Shape C. 4.2 ADSN 4.6 CNN 4.7 Shape C. w/o div 5.8 Shape C. w/ SVM LTS 6.1 Res Net 6.2 Shape C. w/o init 7.2 (a) Supervised time-series classification 2 4 6 8 10 12 2.08 Shape C. w/ Human 2.93 Shape C. 3.73 Shape C. w/o DBI 4.71 Auto S. 5.04 STCN 5.37 USSL DTCR 6.08 Shape C. w/o Init 7.87 k-Shape 9.38 U-Shape L 9.48 KMeans 10.62 (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 30 UEA datasets Method Avg. Acc.1 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 0.806 4.52 38.0 0.658 3.90 15.1 TS2Vec 0.836 2.86 0.9 0.704 3.12 0.6 ROCKET 0.842 2.59 0.89 Shape Conv 0.851 2.39 0.5 0.750 2.07 0.8 original paper. In addition, we also design some variants of Shape Conv 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 Shape Conv and LTS on the Herring dataset. Shapelet Length 50 100 150 200 250 300 350 400 Accuracy Shape Conv 61.3 67.2 67.2 68.8 75.0 70.3 71.9 71.9 LTS 64.1 60.9 59.4 59.4 59.4 59.4 59.4 59.4 Time Shape Conv 9.20s 9.09s 9.15s 9.30s 9.27s 9.19s 9.02s 9.09s LTS 3.37h 3.20h 3.23h 3.17h 2.69h 2.25h 1.87h 1.21h Results The performance of Shape Conv, its variants, and shapelet-based baselines, is evaluated on the 25 UCR datasets and presented in Figure 3 (a). For this experiment, we 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-art time-series classification models on 125 UCR and 29 UEA datasets are shown in Table 1. The full results are presented in the Appendix G.3 In general, Shape Conv consistently outperforms all other baselines and variants, ranking first on average. These results demonstrate that Shape Conv not only provides interpretability but also excels in performance, making it a competitive choice for time-series classification compared to state-of-the-art methods. Ablation studies tell that the effectiveness of our initialization method and diversity loss contributes to improved performance compared to the variants. Lastly, the choice of downstream classifier, either SVM or MLP, does not significantly impact the performance of the Shape Conv model, indicating its flexibility and robustness in different classification settings. Analysis In this section, we investigate two main research questions (RQs): (1) why does Shape Conv outperform other shapelet-based methods? (2) how are Shape Conv s interpretability results compared to 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 1This 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, but we sincerely call for metrics with more practical values here. Published as a conference paper at ICLR 2024 0 100 200 300 400 500 0.0 1.0 Ours Shapelet 1 Ours Shapelet 2 (a) Shape Conv 0 100 200 300 400 500 0.0 1.0 LTS Shapelet 1 LTS Shapelet 2 0 100 200 300 400 500 0.0 LTS Long Shapelet 1 LTS Long Shapelet 2 (c) LTS (shapelet length = 200) Figure 4: Learned shapelets by different methods on the Herring dataset. plotted in Figure 4. In the Herring dataset, Shape Conv (test accuracy 75.0) significantly outperforms the LTS method (test accuracy 64.1). In response to the first RQ, we observe that Shape Conv s shapelets (Figure 4 (a)) cover the most discriminative regions of the time series (the turning points), while LTS s shapelets (Figure 4 (b)) do not. This indicates that Shape Conv s learned shapelets are better at distinguishing classes, leading to improved performance. 0 20 40 60 80 100 120 140 0.0 1.0 LTS Shapelet 1 LTS Shapelet 2 Figure 5: Shapelets learned by LTS in the Gun Point dataset. We find that the learned shapelet by Shape Conv is much longer than that by LTS. The result of forcing the shapelet learned by LTS to be longer (Figure 4 (c)) reveals that LTS fails to learn a high-quality long shapelet. We also provide an ablation on the shapelet length in Table 2. It shows Shape Conv s accuracy increases with shapelet length up to a certain point, while LTS s accuracy does not benefit from the increased length. This is likely due to optimization issues in the LTS method, which cannot handle long-length shapelets. Shape Conv, on the other hand, can be efficiently computed in parallel, leading to better optimization results and significantly faster training time (about 1000 faster). In the Gun Point dataset, both Shape Conv and LTS methods achieve saturated accuracy (100). In response to the second RQ, we utilize this dataset to compare the interpretability of the learned shapelets by Shape Conv (Figure 1) and LTS (Figure 5). It is evident that the shapelet learned by Shape Conv captures the distinguishing features of the class effectively. Here, the shapelet 1 (blue) captures the gesture of reaching for the gun and drawing it out of the holster. The shapelet 2 (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 1 in blue. Based on this observation, we conclude that Shape Conv is capable of learning more interpretable shapelets compared to LTS. 4.2 UNSUPERVISED TIME-SERIES CLUSTERING Settings We evaluate our Shape Conv model on time-series clustering task using 36 UCR univariate time-series datasets (Dau et al., 2019). We first learn shapelets using a Shape Conv layer, then apply KMeans on the shapelet-transformed distance. We use the Normalized Mutual Information (NMI) metric to evaluate the models. Hyperparameters are tuned via grid search based on validation set performance, and they are reported in Appendix G.2. Compared Methods We compare Shape Conv 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), Auto Shape (Li et al., 2022)), and (3) state-of-the-art time-series clustering models (k-Shape (Paparrizos & Gravano, 2015), DTC (Madiraju et al., 2018), 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 Shape Conv for ablation studies, including training with random initialization (w/o Init), training without DBI Loss (w/o DBI), and using human knowledge to initialize shapelets (w/ Human). Published as a conference paper at ICLR 2024 0 20 40 60 80 0.0 Random Init 1 Random Init 2 0 20 40 60 80 0.0 Sample Init 1 Sample Init 2 0 20 40 60 80 0.0 Human Init 1 Human Init 2 0 20 40 60 80 Random Shapelet 1 Random Shapelet 2 0 20 40 60 80 0.0 Sample Shapelet 1 Sample Shapelet 2 0 20 40 60 80 0.0 Human Shapelet 1 Human Shapelet 2 Figure 6: Visualization of cases from ECG200 dataset. First row: illustration of the shapelet initializations; second row: illustration of the learned shapelets. Left: random initialization; middle: cut and sample initialization (Sec. 3.4); right: human knowledge initialization (Sec. 3.4) Results The results of all models on 36 UCR datasets are shown in Figure 3 (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 Shape Conv outperforms all other baselines and variants, achieving the highest average rank among the compared methods. Shape Conv s superior performance, particularly against its randomly initialized variant, underscores the importance of proper initialization 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, Shape Conv stands out as a potent, interpretable tool for unsupervised time-series clustering, outperforming existing methods while adeptly learning interpretable shapelets and assimilating human knowledge. Analysis of the Initialization We now provide a case study to analyze the effect of initialization for Shape Conv in unsupervised learning tasks. We select the ECG200 dataset from UCR (Dau et al., 2019) for this analysis and results are plotted in Figure 6. First, we 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, we solely minimize the shapelet-transformed distance, which tends to optimize within the local region. Therefore, determining the initialization of the shapelets is 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, we choose the two regions with the most significant differences between the classes (the shaded regions in Figure 6) and use the average of those regions as initialization. Consequently, the shapelets are converged in the these regions, effectively capturing the differences between the classes. 5 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 Shape Conv, an interpretable convolutional kernel with its kernels serving as shapelets accompanied by shaping regularizations, and we apply Shape Conv to both supervised and unsupervised tasks. Shape Conv is designed to maintain the advantages of both CNNs and shapelets, providing excellent performance without sacrificing interpretability and controllability. Our experiments on various benchmark datasets showed that Shape Conv outperforms other shapelet-based methods and state-of-the-art time-series classification and clustering models. Moreover, the incorporation of human knowledge can further enhance the performance of Shape Conv, highlighting its potential in real-world applications where expert knowledge is available. Published as a conference paper at ICLR 2024 N Alvarez, CT Lombroso, C Medina, and B Cantlon. Paroxysmal spike and wave activity in drowsiness in young children: its relationship to febrile convulsions. 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Multiview unsupervised shapelet learning for multivariate time series clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(4):4981 4996, 2022. Qin Zhang, Jia Wu, Peng Zhang, Guodong Long, and Chengqi Zhang. Salient subsequence learning for time series clustering. IEEE transactions on pattern analysis and machine intelligence, 41(9): 2193 2207, 2018. Published as a conference paper at ICLR 2024 A RELATED WORK Learning Shapelets How to find the best shapelets from data has long been an intriguing problem since (Ye & Keogh, 2009) firstly proposed it. Traditional practice is to search the raw datasets with some speed-up strategies, like paralleling computing (Chang et 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-based methods. (Grabocka et al., 2014) firstly proposed to learn shapelets with gradient descent. (Shah et al., 2016; Lods et al., 2017) extended this idea to learn more discriminative shapelets based on DTW measure. To encourage the interpretability of learned shapelets, (Ma et al., 2020b; Wang et al., 2019) designed adversarial strategies to guide model training. Besides, in (Li et al., 2022; Zhang & 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; Middlehurst et al., 2023) like HIVE-COTE (Lines et al., 2018), MUSE (Sch afer & Leser, 2017) 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 et al., 2019), newly devised CNN models have obtained more attention in this area. (Fortuin et al., 2018) 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 et al., 2021) designed a wavelet convolution layer to help CNNs discover filters with certain physical meaning, while (Tang et al., 2021b; Xiao et 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 Shape Conv an interpretable model. B DETAILED EXPLANATIONS ON SHAPELETS AND SHAPELET TRANSFORM B.1 INTERPRETABILITY AND EXPAINABILITY The interpretability of shapelet comes from its human-comprehensible nature. However, interpretability is sometimes confounded with the concept of explainability , introduced in many post-hoc explainable methods (Tonekaboni et al., 2020; Leung et al., 2021). 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-interpretable methods such as shapelets do not have these issues as the model itself provides explanations during the learning process. Furthermore, the proposed Shape Conv 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 Gun Point, aiming at classifying whether a person is pointing with a finger or a gun (Figure B.1). The sequences in the dataset record the normalized x-axis of the hand position, i.e., 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 1 are the small flat stage soon after the beginning and symmetrically Published as a conference paper at ICLR 2024 in the end. These two shapes corresponds to the motion of a hand pulling a gun out of the holster and put it back, and would not exist when a person is pointing with his finger. Therefore, when we 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 subfigure, we found the two classes are linear separable as expected. Data points at 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 can be discriminative while providing interpretability at the same time. In order to further verify our interpretations, we 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 in the data of the Point class. This observation matches our hypothesis made upon the learned shapelets. Figure B.1: Illustration of the Gun Point Dataset (Ratanamahatana & Keogh, 2005) 2 Aligned Shapelet with Sample Distance Between Sample and Shapelet Shapelet-Transformed Data Learned Shapelet Figure B.2: Illustration of shapelets and the shapelet transform step C PROOF OF THEOREM 3.1 Theorem 3.1 Assume the input X Rlx is a 1-dimensional single-variate signal of length lx, and nout shapelets S = {s1,s2,...snout} with length ls are discovered. The feature extracted from X with si and squared Euclidean distance is dsi,X. Then we have dsi,X = 2 max j {1,2,...,lx ls+1}[Yij N(si,Xj j+ls 1)], (10) where Y = si X is the cross-correlation defined in Eq. 2 and N(si,X ) = ( si 2 2 + X 2 2)/2 is squared L2 norm term. 2Image Source: http://www.timeseriesclassification.com/description.php? Dataset=Gun Point Published as a conference paper at ICLR 2024 Figure B.3: Illustration of the hand movements in Gun Point dataset Proof. According to Eq.1, shapelet transform step extract features using minimal distance and can be expanded as: dsi,X = min x ˆX dist(si,x) = min j {1,2, ,lx ls+1} si Xj j+ls 1 2 2 = min j {1,2, ,lx ls+1}( si 2 2 + Xj j+ls 1 2 2 2 ls k=1 sik Xj+k) = min j {1,2, ,lx ls+1}[ si 2 2 + Xj j+ls 1 2 2 2(si X)k], = 2 max j {1,2, ,lx ls+1}[Yij N(si,Xj j+ls 1)] with Y and N(si,Xj j+ls 1) defined as in the theorem. D DETAILS ON MODEL DESIGNS D.1 COMPATIBLE WITH OTHER CLASSIFIERS Shape Conv can 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 an end-to-end fashion, so we decompose the shapelet learning step and classification. Specifically, no additional module is appended to the Shape Conv layer, and the output of the layer is directly optimized using the above loss function, but with Lcls term in Eq. 8 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 k i=1 max j=1...k,j/=i ri + rj Here, ri is the diameter of cluster i, which is defined as the average distance between each element in cluster i and the center of cluster i. The distance between the center of cluster i and cluster j is di,j. However, this formulation is not tractable for optimization due to the max operator. Thus, following Li et al. (2022), the max operator is replaced and approximated by the following calculation, k j=1 mij eαmij k j=1 eαmij , (12) Published as a conference paper at ICLR 2024 0 100 200 300 400 500 0.0 1.0 Ours Shapelet 1 Ours Shapelet 2 0 100 200 300 400 500 0.0 1.0 w/o div Shapelet 1 w/o div Shapelet 2 Figure D.1: Learned shapelets by different methods on the Herring dataset. (a) Shape Conv; (b) Shape Conv without diversity loss. where mij = ri+rj dij . By numerical verification, α = 100 is enough for Eq. 12 to approximate the true maximum. D.3 ILLUSTRATION OF DIVERSITY LOSS Figure D.1 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 Shape Conv 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, Shape Conv turns the extraction of shapelets from the original data into a stackable layer can be combined with more sophisticated deep models and optimized in an end-to-end manner, leaving the possibility to keep the interpretability and effectiveness to the maximum extent. To verify the effectiveness of Shape Conv embedded in a deep model, we apply Shape Conv 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 electroencephalograph (EEG) data (Obeid & Picone, 2016). 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 to 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 Shape Conv with the most commonly used deep neural network models GRU (Cho et al., 2014) and TCN (Bai et al., 2018). The empirical results are illustrated in Table E.1. Our model has significantly outperformed the compared baselines, which showed the superiority of the proposed Shape Conv 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 Bai et al. (2018) 0.817(0.004) 0.383(0.010) Shape Conv + GRU 0.837(0.007) 0.414(0.008) We further investigate whether our Shape Conv 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 wellstudied and recognized in previous medical research (Marcuse et al., 2015). To this end, we visualized Published as a conference paper at ICLR 2024 EEG Examples from Textbook Learned Shapelet Polymorphic Delta Waves Paroxysmal Rhythm Waves (a) Comparison between two types of textbook waveform and the two learned shapelets, polymorphic delta waves related to lesions (Marshall et al., 1988) and paroxysmal rhythm waves related to sleep stage (Alvarez et al., 1983). (b) Shapelet transform by the two shapelets. Some data points from Class 1 are mixed with Class 2 need to be separated by other shapelets. Figure E.1: Demonstration of the learned shapelets for EEG data. 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 is shown in Figure E.1a. By aligning them with the mostly similar part of the original data (Figure E.2), we found these shapelets can match a specific type of seizure status and provides solid classification criteria (Figure E.1b). This showcases the possibility of how our interpretable method can benefit medical practitioners in 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 as similarly strong indicators, may not be included in existing studies. We believe these shapelets can provide inspirations and boost further research in related area. F MORE VISUALIZATIONS OF SHAPECONV F.1 MORE VISUALIZATIONS ON LEARNT SHAPELETS In this section, we provide more visualizations of the learnt shapelets of Shape Conv in different UCR datasets. The results clearly shows that Shape Conv could learn the determining regions of the time series. F.2 VISUALIZATIONS WITH SHAP VALUE In this section, we further substantiate our claim regarding the interpretability of our model using the SHAP (SHapley Additive ex Planations) Value (Lundberg & Lee, 2017). The analysis employs the Gun Point dataset from the UCR archive. We examine two variations of our model: the original Published as a conference paper at ICLR 2024 0 50 100 150 200 250 300 0.0 Class 2 Class 1 Shapelet 1 Shapelet 2 0 10 20 30 40 50 60 70 0.0 Class 1 Class 2 Shapelet 1 Shapelet 2 (a) Dodger Loop Weekend (b) Sony AIBORobot Surface1 0 5 10 15 20 0.0 Class 2 Class 1 Shapelet 1 Shapelet 2 0 20 40 60 80 100 120 0.0 Class 1 Class 2 Class 3 Shapelet 1 Shapelet 2 Shapelet 3 (c) Italy Power Demand (d) BME Figure F.1: More visualizations of the shapelets learnt by Shape Conv in UCR datasets. Shape Conv and a modified version where the term λshape in the loss function (Equation 8) is set to zero. Setting λshape to zero eliminates the L2 norm term, effectively transforming the layer into a 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 F.2, with the mean SHAP value of each class depicted. The blue and orange lines represent the Gun Class (Class 1) 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 of drawing the gun out of the holster and putting it back, underscoring the model s reliance on 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, Shape Conv 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. Published as a conference paper at ICLR 2024 Class 1 Class 2 Shapelet 1 Shapelet 2 0.1 0.0 0.1 SHAP Value - Class 1 0 50 100 150 0.1 0.0 0.1 SHAP Value - Class 2 Class 1 Class 2 Kernel 1 Kernel 2 0.25 0.00 0.25 SHAP Value - Class 1 0 50 100 150 0.25 0.00 0.25 SHAP Value - Class 2 (a) Shape Conv (b) λshape = 0 Figure F.2: Illustration of the trained average SHAP value for different class across all the Gun Point testing dataset. Blue: Gun Class. Orange: No Gun Class. Left: our proposed Shape Conv method. Right: Shape Conv with λshape = 0. G DETAILS ON EXPERIMENTS AND ANALYSIS G.1 ENVIRONMENT All experiments are performed on the Py Torch framework using a 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 are tuned via grid search based on validation set performance. The number of shapelets is chosen from {1,2,3,4,5} times the number of classes, and the shapelet length is evaluated over {0.1,0.2, ,0.8} times the time series length. The parameter λshape is chosen from {0.01,0.1,1,10} and the parameter λdiv is evaluated over {0.01,0.1,1,10}. Learning rate is chosen from {0.001,0.005,0.01,0.05,0.1}. 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 {1,2,3,4,5} 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 λshape is chosen from {0.01,0.1,1,10} and the parameter λdiv is evaluated over {0.01,0.1,1,10}. Learning rate is chosen from {0.001,0.005,0.01,0.05,0.1}. G.3 RESULTS OF SUPERVISED LEARNING TASKS In this section, we present the full results of supervised time-series classification tasks. We compared Shape Conv with (1) shapelet-based methods, common deep learning methods, and ablations (described in Sec. 4.1) across 25 UCR Datasets (Table G.1) (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-art times series classification methods (described in 4.1) across 30 UEA datasets (Table G.4). Published as a conference paper at ICLR 2024 Table G.1: Shape Conv Compared with Shapelet-Based 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. Dataset MLP CNN Res Net IGSVM FLAG LTS ADSN Shape C. Shape C. Shape C. Shape C. w/o init w/ SVM w/o div Adiac 0.752 0.857 0.826 0.235 0.752 0.519 0.798 0.691 0.007 0.813 0.047 0.852 0.014 0.867 0.020 Beef 0.833 0.750 0.767 0.900 0.833 0.767 0.933 0.849 0.005 0.898 0.031 0.921 0.001 0.936 0.008 Chlorine. 0.872 0.843 0.828 0.571 0.760 0.730 0.880 0.825 0.026 0.904 0.025 0.907 0.008 0.924 0.014 Coffee 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.000 1.000 0.000 1.000 0.000 1.000 0.000 Diatom. 0.964 0.930 0.931 0.931 0.964 0.942 0.987 0.992 0.008 0.991 0.001 0.992 0.005 0.994 0.009 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 0.013 DPMiddle 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 0.705 0.696 0.705 0.752 0.736 0.729 0.010 0.715 0.037 0.738 0.005 0.753 0.015 ECGFive Days 0.970 0.985 0.955 0.990 0.920 1.000 1.000 1.000 0.000 1.000 0.000 1.000 0.000 1.000 0.000 Face Four 0.830 0.932 0.932 0.977 0.909 0.943 0.977 0.941 0.023 0.916 0.001 0.942 0.015 0.961 0.007 Gun Point 0.933 1.000 0.993 1.000 0.967 0.996 0.987 0.994 0.016 0.995 0.008 0.993 0.003 0.997 0.016 Herring 0.641 0.681 0.641 0.641 0.641 0.641 0.703 0.703 0.014 0.713 0.005 0.702 0.006 0.724 0.028 Italy Power. 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 0.003 0.723 0.001 0.748 0.001 0.781 0.004 Medicallmages 0.729 0.792 0.772 0.552 0.714 0.713 0.720 0.682 0.012 0.694 0.000 0.752 0.009 0.774 0.003 Mote Strain 0.869 0.950 0.895 0.887 0.888 0.900 0.906 0.884 0.013 0.886 0.016 0.898 0.002 0.913 0.000 MPLittle 0.703 0.758 0.726 0.707 0.693 0.743 0.758 0.701 0.012 0.733 0.002 0.741 0.008 0.749 0.032 MPMiddle 0.750 0.800 0.775 0.769 0.750 0.775 0.791 0.736 0.020 0.759 0.011 0.779 0.013 0.807 0.020 PPLittle 0.710 0.753 0.761 0.721 0.671 0.710 0.715 0.661 0.013 0.694 0.029 0.676 0.000 0.732 0.001 PPMiddle 0.707 0.784 0.753 0.759 0.738 0.749 0.786 0.717 0.023 0.726 0.010 0.764 0.002 0.791 0.000 PPThumb 0.726 0.745 0.708 0.755 0.674 0.705 0.695 0.685 0.018 0.712 0.010 0.728 0.010 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 0.853 0.962 0.872 0.846 0.875 0.945 0.963 0.942 0.026 0.968 0.006 0.974 0.007 0.980 0.015 Synthetic C. 0.950 0.990 1.000 0.873 0.997 0.973 1.000 1.000 0.000 1.000 0.000 0.997 0.029 1.000 0.000 Trace 0.820 1.000 1.000 0.980 0.990 1.000 1.000 1.000 0.000 1.000 0.000 1.000 0.000 1.000 0.000 Two Lead ECG 0.853 1.000 1.000 1.000 0.990 1.000 0.986 1.000 0.000 1.000 0.000 1.000 0.000 1.000 0.000 Avg. Acc 80.5 86.4 84.8 79.4 82.6 83.2 86.6 83.8 85.4 86.4 87.8 Avg. Rank 8.6 4.7 6.1 7.8 8.4 6.2 4.2 7.2 5.8 4.6 2.4 Published as a conference paper at ICLR 2024 Table G.2: Shape Conv 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. Dataset RNTK NTK RBF POLY Gaussian Identity GRU OS-CNN Shape Conv RNN RNN Adiac 0.766 0.719 0.734 0.778 0.514 0.169 0.606 0.835 0.882 0.009 Arrowhead 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 0.941 0.001 Car 0.833 0.788 0.800 0.800 0.233 0.583 0.267 0.933 0.992 0.008 Chlorine Concentration 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 0.000 Computers 0.592 0.552 0.588 0.564 0.532 0.552 0.588 0.707 0.656 0.009 Cricket X 0.605 0.595 0.621 0.626 0.085 0.636 0.264 0.855 0.914 0.009 Cricket Y 0.639 0.590 0.605 0.597 0.159 0.592 0.362 0.867 0.729 0.020 Cricket Z 0.603 0.592 0.621 0.592 0.085 0.579 0.413 0.863 0.764 0.003 Distal Phalanx Outline C. 0.775 0.775 0.754 0.739 0.699 0.696 0.750 0.766 0.804 0.030 Distal Phalanx TW 0.662 0.698 0.669 0.674 0.676 0.647 0.691 0.664 0.781 0.008 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 0.012 Faceall 0.741 0.833 0.833 0.824 0.537 0.705 0.707 0.845 0.853 0.007 Faces UCR 0.817 0.802 0.803 0.830 0.532 0.753 0.795 0.967 0.957 0.004 Fifty Words 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 0.037 Freezer Regular Train 0.974 0.944 0.965 0.968 0.761 0.075 0.866 0.997 0.997 0.036 Gun Point 0.980 0.953 0.953 0.940 0.820 0.747 0.807 0.999 1.000 0.000 Gun Point Age Span 0.965 0.946 0.959 0.940 0.478 0.478 0.956 0.992 1.000 0.000 Gun Point Male VSFemale 0.991 0.997 0.994 0.997 0.687 0.525 0.997 0.999 1.000 0.000 Gun Point Old VSYoung 0.987 0.975 0.987 0.946 0.540 0.524 0.984 1.000 1.000 0.000 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 Insect EPGRegular 0.996 0.992 0.996 0.968 1.000 1.000 0.984 0.951 1.000 0.000 Lightning2 0.787 0.738 0.705 0.689 0.459 0.705 0.672 0.807 0.819 0.008 Lightning7 0.616 0.603 0.630 0.603 0.233 0.699 0.767 0.793 0.808 0.023 Meat 0.933 0.933 0.933 0.933 0.006 0.550 0.333 0.947 0.950 0.016 Medical Images 0.745 0.733 0.753 0.746 0.482 0.649 0.691 0.769 0.709 0.028 Middle Phalanx Outline C. 0.571 0.571 0.487 0.643 0.763 0.570 0.746 0.814 0.856 0.001 Middle Phalanx TW 0.578 0.610 0.597 0.604 0.584 0.584 0.591 0.519 0.642 0.002 Olive Oil 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 Power Cons 0.972 0.972 0.967 0.917 0.961 0.950 0.994 0.990 0.911 0.004 Proximal Phalanx Outline C. 0.890 0.880 0.873 0.869 0.828 0.746 0.869 0.908 0.913 0.006 Refrigeration Devices 0.469 0.371 0.365 0.411 0.360 0.509 0.467 0.503 0.613 0.002 Screen Type 0.416 0.432 0.435 0.384 0.400 0.411 0.363 0.526 0.493 0.010 Semg Hand Subject Ch2 0.842 0.853 0.861 0.867 0.200 0.367 0.891 0.718 0.981 0.025 Small Kitchen Appliances 0.675 0.384 0.403 0.379 0.602 0.760 0.715 0.721 0.803 0.005 Smooth Subspace 0.960 0.873 0.920 0.867 0.940 0.953 0.927 0.989 1.000 0.000 Star Light Curves 0.959 0.962 0.946 0.944 0.821 0.868 0.962 0.975 0.987 0.014 Strawberry 0.984 0.976 0.970 0.968 0.943 0.754 0.916 0.982 0.919 0.000 Swedish Leaf 0.906 0.910 0.914 0.907 0.592 0.459 0.910 0.971 0.961 0.015 Synthetic Control 0.987 0.967 0.980 0.977 0.927 0.977 0.990 0.999 1.000 0.000 Trace 0.960 0.810 0.760 0.760 0.700 0.710 1.000 1.000 1.000 0.000 Two Patterns 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 0.000 UWave Gesture Library X 0.796 0.787 0.785 0.658 0.560 0.753 0.736 0.822 0.832 0.006 UWave Gesture Library Y 0.716 0.706 0.704 0.703 0.445 0.652 0.654 0.757 0.754 0.003 UWave Gesture Library Z 0.740 0.739 0.729 0.719 0.433 0.678 0.703 0.764 0.805 0.003 Word Synonyms 0.580 0.585 0.611 0.621 0.177 0.458 0.538 0.742 0.784 0.022 Worms 0.571 0.507 0.558 0.507 0.351 0.494 0.416 0.765 0.803 0.021 Worms Two Class 0.623 0.623 0.610 0.597 0.519 0.468 0.571 0.657 0.815 0.028 Yoga 0.849 0.846 0.846 0.849 0.464 0.767 0.618 0.911 0.772 0.006 Average Accuracy 80.1 77.7 78.2 77.7 56.0 63.1 69.5 84.8 87.0 Average Rank 4.2 5.0 4.9 5.4 7.8 7.2 6.1 2.7 1.8 Published as a conference paper at ICLR 2024 Table G.3: Shape Conv Compared with State-Of-The-Art Time-Series Classification Methods: Evaluating Testing Accuracy for Supervised Time-Series Classification Tasks Across 128 UCR Datasets. Mean accuracy std over 3 independent experiments with different random seeds is reported. DTW TNC TST TS-TCC T-Loss TS2Vec ROCKET Shape Conv Adiac 0.604 0.726 0.550 0.767 0.675 0.775 0.783 0.867 0.000 Arrow Head 0.703 0.703 0.771 0.737 0.766 0.857 0.814 0.903 0.005 Beef 0.633 0.733 0.500 0.600 0.667 0.767 0.833 0.936 0.014 Beetle Fly 0.700 0.850 1.000 0.800 0.800 0.900 0.900 1.000 0.000 Bird Chicken 0.750 0.750 0.650 0.650 0.850 0.800 0.900 1.000 0.000 Car 0.733 0.683 0.550 0.583 0.833 0.883 0.847 0.974 0.000 CBF 0.997 0.983 0.898 0.998 0.983 1.000 1.000 1.000 0.000 Chlorine Concentration 0.648 0.760 0.562 0.753 0.749 0.832 0.815 0.924 0.003 Cin CECGTorso 0.651 0.669 0.508 0.671 0.713 0.827 0.836 0.778 0.007 Coffee 1.000 1.000 0.821 1.000 1.000 1.000 1.000 1.000 0.000 Computers 0.700 0.684 0.696 0.704 0.664 0.660 0.761 0.647 0.000 Cricket X 0.754 0.623 0.385 0.731 0.713 0.805 0.819 0.895 0.009 Cricket Y 0.744 0.597 0.467 0.718 0.728 0.769 0.852 0.726 0.016 Cricket Z 0.754 0.682 0.403 0.713 0.708 0.792 0.856 0.773 0.017 Diatom Size Reduction 0.967 0.993 0.961 0.977 0.984 0.987 0.970 0.994 0.005 Distal Phalanx Outline Correct 0.717 0.754 0.728 0.754 0.775 0.775 0.770 0.753 0.026 Distal Phalanx Outline Age Group 0.770 0.741 0.741 0.755 0.727 0.727 0.759 0.784 0.017 Distal Phalanx TW 0.590 0.669 0.568 0.676 0.676 0.698 0.719 0.763 0.022 Earthquakes 0.719 0.748 0.748 0.748 0.748 0.748 0.748 0.731 0.010 ECG200 0.770 0.830 0.830 0.880 0.940 0.920 0.906 0.992 0.001 ECG5000 0.924 0.937 0.928 0.941 0.933 0.935 0.947 0.953 0.007 ECGFive Days 0.768 0.999 0.763 0.878 1.000 1.000 1.000 1.000 0.000 Electric Devices 0.602 0.700 0.676 0.686 0.707 0.721 0.729 0.743 0.013 Face All 0.808 0.766 0.504 0.813 0.786 0.805 0.947 0.827 0.037 Face Four 0.830 0.659 0.511 0.773 0.920 0.932 0.977 0.961 0.019 Faces UCR 0.905 0.789 0.543 0.863 0.884 0.930 0.961 0.930 0.000 Fifty Words 0.690 0.653 0.525 0.653 0.732 0.774 0.830 0.699 0.009 Fish 0.823 0.817 0.720 0.817 0.891 0.937 0.979 0.917 0.006 Ford A 0.555 0.902 0.568 0.930 0.928 0.948 0.944 0.954 0.020 Ford B 0.620 0.733 0.507 0.815 0.793 0.807 0.805 0.835 0.022 Gun Point 0.907 0.967 0.827 0.993 0.980 0.987 1.000 0.997 0.002 Ham 0.467 0.752 0.524 0.743 0.724 0.724 0.726 0.733 0.032 Hand Outlines 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 0.003 Herring 0.531 0.594 0.594 0.594 0.594 0.641 0.692 0.724 0.008 Inline Skate 0.384 0.378 0.287 0.347 0.371 0.415 0.457 0.432 0.023 Insect Wingbeat Sound 0.355 0.549 0.266 0.415 0.597 0.630 0.657 0.613 0.008 Italy Power Demand 0.950 0.928 0.845 0.955 0.954 0.961 0.970 0.974 0.017 Large Kitchen Appliances 0.795 0.776 0.595 0.848 0.789 0.875 0.901 0.917 0.003 Lightning2 0.869 0.869 0.705 0.836 0.869 0.869 0.759 0.819 0.000 Lightning7 0.726 0.767 0.411 0.685 0.795 0.863 0.823 0.781 0.007 Mallat 0.934 0.871 0.713 0.922 0.951 0.915 0.956 0.932 0.010 Meat 0.933 0.917 0.900 0.883 0.950 0.967 0.948 0.943 0.017 Medical Images 0.737 0.754 0.632 0.747 0.750 0.793 0.799 0.774 0.003 Middle Phalanx Outline Correct 0.698 0.818 0.753 0.818 0.825 0.838 0.838 0.827 0.004 Middle Phalanx Outline Age Group 0.500 0.643 0.617 0.630 0.656 0.636 0.590 0.669 0.011 Middle Phalanx TW 0.506 0.571 0.506 0.610 0.591 0.591 0.560 0.637 0.007 Mote Strain 0.835 0.825 0.768 0.843 0.851 0.863 0.915 0.919 0.029 Non Invasive Fetal ECGThorax1 0.790 0.898 0.471 0.898 0.878 0.930 0.913 0.913 0.012 Non Invasive Fetal ECGThorax2 0.865 0.912 0.832 0.913 0.919 0.940 0.929 0.942 0.013 Olive Oil 0.833 0.833 0.800 0.800 0.867 0.900 0.917 0.827 0.015 OSULeaf 0.591 0.723 0.545 0.723 0.760 0.876 0.941 0.905 0.004 Phalanges Outlines Correct 0.728 0.787 0.773 0.804 0.784 0.823 0.834 0.813 0.017 Phoneme 0.228 0.180 0.139 0.242 0.276 0.312 0.280 0.204 0.004 Plane 1.000 1.000 0.933 1.000 0.990 1.000 1.000 1.000 0.000 Proximal Phalanx Outline Correct 0.784 0.866 0.770 0.873 0.859 0.900 0.899 0.913 0.019 Proximal Phalanx Outline Age Group 0.805 0.854 0.854 0.839 0.844 0.844 0.856 0.869 0.004 Proximal Phalanx TW 0.761 0.810 0.780 0.800 0.771 0.824 0.817 0.831 0.033 Refrigeration Devices 0.464 0.565 0.483 0.563 0.515 0.589 0.537 0.594 0.026 Screen Type 0.397 0.509 0.419 0.419 0.416 0.411 0.485 0.423 0.007 Shapelet Sim 0.650 0.589 0.489 0.683 0.672 1.000 1.000 1.000 0.000 Shapes All 0.768 0.788 0.733 0.773 0.848 0.905 0.907 0.853 0.001 Small Kitchen Appliances 0.643 0.725 0.592 0.691 0.677 0.733 0.818 0.741 0.005 Sony AIBORobot Surface1 0.725 0.804 0.724 0.899 0.902 0.903 0.922 0.962 0.005 Sony AIBORobot Surface2 0.831 0.834 0.745 0.907 0.889 0.890 0.913 0.914 0.018 Star Light Curves 0.907 0.968 0.949 0.967 0.964 0.971 0.981 0.987 0.004 Strawberry 0.941 0.951 0.916 0.965 0.954 0.965 0.981 0.903 0.008 Swedish Leaf 0.792 0.880 0.738 0.923 0.914 0.942 0.964 0.952 0.012 Symbols 0.950 0.885 0.786 0.916 0.963 0.976 0.974 0.980 0.041 Synthetic Control 0.993 1.000 0.490 0.990 0.987 0.997 1.000 1.000 0.000 Toe Segmentation1 0.772 0.864 0.807 0.930 0.939 0.947 0.968 0.957 0.003 Toe Segmentation2 0.838 0.831 0.615 0.877 0.900 0.915 0.924 0.931 0.020 Trace 1.000 1.000 1.000 1.000 0.990 1.000 1.000 1.000 0.000 Published as a conference paper at ICLR 2024 Two Lead ECG 0.905 0.993 0.871 0.976 0.999 0.987 0.999 1.000 0.000 Two Patterns 1.000 1.000 0.466 0.999 0.999 1.000 1.000 1.000 0.000 UWave Gesture Library X 0.728 0.781 0.569 0.733 0.785 0.810 0.815 0.805 0.034 UWave Gesture Library Y 0.634 0.697 0.348 0.641 0.710 0.729 0.744 0.738 0.013 UWave Gesture Library Z 0.658 0.721 0.655 0.690 0.757 0.770 0.732 0.792 0.018 UWave Gesture Library All 0.892 0.903 0.475 0.692 0.896 0.934 0.925 0.941 0.028 Wafer 0.980 0.994 0.991 0.994 0.992 0.998 0.998 0.973 0.006 Wine 0.574 0.759 0.500 0.778 0.815 0.889 0.813 0.894 0.018 Word Synonyms 0.649 0.630 0.422 0.531 0.691 0.704 0.753 0.765 0.025 Worms 0.584 0.623 0.455 0.753 0.727 0.701 0.740 0.783 0.002 Worms Two Class 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 0.887 0.910 0.742 0.047 ACSF1 0.640 0.730 0.760 0.730 0.900 0.910 0.886 0.902 0.020 All Gesture Wiimote X 0.716 0.703 0.259 0.697 0.763 0.777 0.790 0.831 0.042 All Gesture Wiimote Y 0.729 0.699 0.423 0.741 0.726 0.793 0.773 0.826 0.017 All Gesture Wiimote Z 0.643 0.646 0.447 0.689 0.723 0.770 0.766 0.848 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 0.968 0.983 0.954 0.016 Crop 0.665 0.738 0.710 0.742 0.722 0.756 0.751 0.703 0.005 EOGHorizontal Signal 0.503 0.442 0.373 0.401 0.605 0.544 0.539 0.609 0.018 EOGVertical Signal 0.448 0.392 0.298 0.376 0.434 0.503 0.441 0.521 0.013 Ethanol Level 0.276 0.424 0.260 0.486 0.382 0.484 0.583 0.704 0.018 Freezer Regular Train 0.899 0.991 0.922 0.989 0.956 0.986 0.998 0.993 0.003 Freezer Small Train 0.753 0.982 0.920 0.979 0.933 0.894 0.950 0.972 0.012 Fungi 0.839 0.527 0.366 0.753 1.000 0.962 1.000 0.954 0.003 Gesture Mid Air D1 0.569 0.431 0.208 0.369 0.608 0.631 0.617 0.541 0.026 Gesture Mid Air D2 0.608 0.362 0.138 0.254 0.546 0.515 0.561 0.585 0.029 Gesture Mid Air D3 0.323 0.292 0.154 0.177 0.285 0.346 0.315 0.405 0.008 Gesture Pebble Z1 0.791 0.378 0.500 0.395 0.919 0.930 0.906 0.871 0.002 Gesture Pebble Z2 0.671 0.316 0.380 0.430 0.899 0.873 0.830 0.874 0.030 Gun Point Age Span 0.918 0.984 0.991 0.994 0.994 0.994 0.997 1.000 0.000 Gun Point Male Versus Female 0.997 0.994 1.000 0.997 0.997 1.000 0.998 1.000 0.000 Gun Point Old Versus Young 0.838 1.000 1.000 1.000 1.000 1.000 0.991 1.000 0.000 House Twenty 0.924 0.782 0.815 0.790 0.933 0.941 0.964 0.953 0.018 Insect EPGRegular Train 0.872 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.000 Insect EPGSmall Train 0.735 1.000 1.000 1.000 1.000 1.000 0.979 1.000 0.000 Melbourne Pedestrian 0.791 0.942 0.741 0.949 0.944 0.959 0.904 0.926 0.016 Mixed Shapes Regular Train 0.842 0.911 0.879 0.855 0.905 0.922 0.921 0.965 0.021 Mixed Shapes Small Train 0.780 0.813 0.828 0.735 0.860 0.881 0.918 0.927 0.011 Pickup Gesture Wiimote Z 0.660 0.620 0.240 0.600 0.740 0.820 0.830 0.871 0.026 Pig Airway Pressure 0.106 0.413 0.120 0.380 0.510 0.683 0.095 0.594 0.005 Pig Art Pressure 0.245 0.808 0.774 0.524 0.928 0.966 0.954 0.872 0.022 Pig CVP 0.154 0.649 0.596 0.615 0.788 0.870 0.934 0.831 0.006 PLAID 0.840 0.495 0.419 0.445 0.555 0.561 0.903 0.904 0.008 Power Cons 0.878 0.933 0.911 0.961 0.900 0.972 0.940 0.901 0.000 Rock 0.600 0.580 0.680 0.600 0.580 0.700 0.900 0.700 0.016 Semg Hand Gender Ch2 0.802 0.882 0.725 0.837 0.890 0.963 0.927 0.972 0.004 Semg Hand Movement Ch2 0.584 0.593 0.420 0.613 0.789 0.893 0.645 0.924 0.005 Semg Hand Subject Ch2 0.727 0.771 0.484 0.753 0.853 0.951 0.881 0.981 0.002 Shake Gesture Wiimote Z 0.860 0.820 0.760 0.860 0.920 0.940 0.898 0.834 0.001 Smooth Subspace 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 1.000 0.000 Dodger Loop Day 0.500 0.000 0.200 0.000 0.000 0.562 0.573 0.628 0.003 Dodger Loop Game 0.877 0.000 0.696 0.000 0.000 0.841 0.873 0.906 0.006 Dodger Loop Weekend 0.949 0.000 0.732 0.000 0.000 0.964 0.975 0.971 0.022 Average Accuarcy 0.728 0.743 0.639 0.740 0.787 0.836 0.842 0.851 Average Rank 6.102 5.285 7.055 5.191 4.523 2.855 2.590 2.398 Published as a conference paper at ICLR 2024 Table G.4: Shape Conv Compared with State-Of-The-Art Time-Series Classification Methods: Evaluating Testing Accuracy for Supervised Multivariate Time-Series Classification Tasks Across 30 UEA Datasets. Mean accuracy std over 3 independent experiments with different random seeds is reported. DTW TNC TST TS-TCC T-Loss TS2Vec Shape Conv Articulary Word Recognition 0.987 0.973 0.977 0.953 0.943 0.987 0.994 0.001 Atrial Fibrillation 0.200 0.133 0.067 0.267 0.133 0.200 0.521 0.015 Basic Motions 0.975 0.975 0.975 1.000 1.000 0.975 0.997 0.016 Character Trajectories 0.989 0.967 0.975 0.985 0.993 0.995 0.981 0.018 Cricket 1.000 0.958 1.000 0.917 0.972 0.972 0.998 0.008 Duck Duck Geese 0.600 0.460 0.620 0.380 0.650 0.680 0.648 0.006 Eigen Worms 0.618 0.840 0.748 0.779 0.840 0.847 0.802 0.008 Epilepsy 0.964 0.957 0.949 0.957 0.971 0.964 0.972 0.009 ERing 0.133 0.852 0.874 0.904 0.133 0.874 0.774 0.003 Ethanol Concentration 0.323 0.297 0.262 0.285 0.205 0.308 0.253 0.001 Face Detection 0.529 0.536 0.534 0.544 0.513 0.501 0.635 0.025 Finger Movements 0.530 0.470 0.560 0.460 0.580 0.480 0.587 0.029 Hand Movement Direction 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 0.006 Heartbeat 0.717 0.746 0.746 0.751 0.741 0.683 0.784 0.011 Japanese 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 0.002 LSST 0.551 0.595 0.408 0.474 0.509 0.537 0.608 0.023 Motor Imagery 0.500 0.500 0.500 0.610 0.580 0.510 0.674 0.002 NATOPS 0.883 0.911 0.850 0.822 0.917 0.928 0.937 0.004 PEMS-SF 0.711 0.699 0.740 0.734 0.676 0.682 0.801 0.005 Pen Digits 0.977 0.979 0.560 0.974 0.981 0.989 0.968 0.018 Phoneme Spectra 0.151 0.207 0.085 0.252 0.222 0.233 0.192 0.002 Racket Sports 0.803 0.776 0.809 0.816 0.855 0.855 0.863 0.012 Self Regulation SCP1 0.775 0.799 0.754 0.823 0.843 0.812 0.858 0.003 Self Regulation SCP2 0.539 0.550 0.550 0.533 0.539 0.578 0.624 0.054 Spoken Arabic Digits 0.963 0.934 0.923 0.970 0.905 0.988 0.979 0.020 Stand Walk Jump 0.200 0.400 0.267 0.333 0.333 0.467 0.587 0.013 UWave Gesture Library 0.903 0.759 0.575 0.753 0.875 0.906 0.936 0.002 Insect Wingbeat - 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 G.6. 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. Dataset KMeans k-Shape U-Shape L DTC USSL DTCR STCN Auto S. Shape C. Shape C. Shape C. Shape C. w/o Init w/o DBI w/ Human Arrow 0.4816 0.5240 0.3522 0.5000 0.6322 0.5513 0.5240 0.5624 0.5134 0.6123 0.6064 0.010 0.6445 0.013 Beef 0.2925 0.3338 0.3413 0.2751 0.3338 0.5473 0.5432 0.3799 0.3077 0.3854 0.4039 0.003 0.6188 0.017 Beetle Fly 0.0073 0.3456 0.5105 0.3456 0.5310 0.7610 1.0000 0.5310 0.4897 1.0000 1.0000 0.000 1.0000 0.000 Bird Chicken 0.0371 0.3456 0.2783 0.0073 0.6190 0.5310 1.0000 0.6352 0.5824 1.0000 1.0000 0.000 1.0000 0.000 Car 0.2540 0.3771 0.3655 0.1892 0.4650 0.5021 0.5701 0.4970 0.3672 0.4690 0.4770 0.006 0.5013 0.013 Chlorine. 0.0129 0.0000 0.0135 0.0013 0.0133 0.0195 0.0760 0.0133 0.0024 0.0368 0.0527 0.007 0.0641 0.014 Coffee 0.5246 1.0000 1.0000 0.5523 1.0000 0.6277 1.0000 1.0000 1.0000 1.0000 1.0000 0.000 1.0000 0.000 Diatom. 0.9300 1.0000 0.4849 0.6863 1.0000 0.9418 1.0000 1.0000 1.0000 1.0000 1.0000 0.000 1.0000 0.000 Dist.age G 0.1880 0.2911 0.2577 0.3406 0.3846 0.4553 0.5037 0.4400 0.4237 0.4464 0.4786 0.004 0.5291 0.007 Dist.correct 0.0278 0.0527 0.0063 0.0115 0.1026 0.1180 0.2327 0.1333 0.0699 0.0885 0.1074 0.009 0.1836 0.012 ECG200 0.1403 0.3682 0.1323 0.0918 0.3776 0.3691 0.4316 0.3928 0.3002 0.5413 0.5552 0.009 0.6240 0.014 ECGFive Days 0.0002 0.0002 0.1498 0.0022 0.6502 0.8056 0.3582 0.7835 0.6355 0.8150 0.8246 0.010 0.7669 0.027 Gun Point 0.0126 0.3653 0.3653 0.0194 0.4878 0.4200 0.5537 0.4027 0.3803 0.4248 0.4476 0.003 0.5652 0.025 Ham 0.0093 0.0517 0.0619 0.1016 0.3411 0.0989 0.2382 0.3211 0.1764 0.3859 0.3911 0.001 0.4467 0.005 Herring 0.0013 0.0027 0.1324 0.0143 0.1718 0.2248 0.2002 0.2019 0.1423 0.2293 0.2630 0.012 0.2317 0.014 Lighting2 0.0038 0.2670 0.0144 0.1435 0.3727 0.2289 0.3479 0.3530 0.3040 0.3756 0.4282 0.008 0.4723 0.002 Meat 0.2510 0.2254 0.2716 0.2250 0.9085 0.9653 0.9393 0.9437 0.2901 0.9423 0.9481 0.007 1.0000 0.000 Mid.age G 0.0219 0.0722 0.1491 0.1390 0.2780 0.4661 0.5109 0.3940 0.1830 0.4214 0.4576 0.003 0.6654 0.013 Mid.correct 0.0024 0.0349 0.0253 0.0079 0.2503 0.1150 0.0921 0.2873 0.1942 0.2109 0.2096 0.026 0.3499 0.002 Mid.TW 0.4134 0.5229 0.4065 0.1156 0.9202 0.5503 0.6169 0.9450 0.7836 0.9161 0.9241 0.004 0.9310 0.006 Mote Strain 0.0551 0.2215 0.0082 0.0094 0.5310 0.4094 0.4063 0.4257 0.2976 0.5406 0.5738 0.001 0.5982 0.014 OSULeaf 0.0208 0.0126 0.0203 0.2201 0.3353 0.2599 0.3544 0.4432 0.2952 0.5131 0.5159 0.017 0.4846 0.018 Plane 0.8598 0.9642 1.0000 0.8678 1.0000 0.9296 0.9615 0.9982 1.0000 1.0000 1.0000 0.000 1.0000 0.000 Prox.age G 0.0635 0.0110 0.0332 0.4153 0.6813 0.5581 0.6317 0.6930 0.5164 0.6057 0.6453 0.013 0.7292 0.013 Prox.TW 0.0082 0.1577 0.0107 0.6199 1.0000 0.6539 0.7330 0.8947 0.5948 0.8368 0.8284 0.006 0.8117 0.005 Sony. 0.6112 0.7107 0.5803 0.2559 0.5597 0.6634 0.6112 0.6096 0.5423 0.6256 0.6215 0.047 0.6107 0.005 Sony.II 0.5444 0.0110 0.5903 0.4257 0.6858 0.6121 0.5647 0.7020 0.4985 0.6765 0.6767 0.015 0.7343 0.034 Swedish Leaf 0.0168 0.1041 0.3456 0.6187 0.9186 0.6663 0.6106 0.9340 0.5834 0.8592 0.8418 0.006 0.8412 0.010 Symbols 0.7780 0.6366 0.8691 0.7995 0.8821 0.8989 0.8940 0.9147 0.7778 0.9313 0.9250 0.030 0.9118 0.002 Toe Seg.1 0.0022 0.3073 0.3073 0.0188 0.3351 0.3115 0.3671 0.4610 0.2830 0.4700 0.4863 0.001 0.5851 0.007 Toe Seg.2 0.0863 0.0863 0.1519 0.0096 0.4308 0.3249 0.5498 0.4664 0.1293 0.4959 0.5178 0.002 0.6636 0.007 Two Patterns 0.4696 0.3949 0.2979 0.0119 0.4911 0.4713 0.4110 0.5150 0.3083 0.5030 0.5177 0.004 0.6154 0.019 Two Lead ECG 0.0000 0.0000 0.0529 0.0036 0.5471 0.4614 0.6911 0.5654 0.4220 0.6006 0.6289 0.033 0.7045 0.001 Wafer 0.0010 0.0010 0.0010 0.0008 0.0492 0.0228 0.2089 0.0520 -0.0063 0.0741 0.0802 0.008 0.0477 0.021 Wine 0.0031 0.0119 0.0171 0.0000 0.7511 0.2580 0.5927 0.6045 0.2840 0.6090 0.6328 0.016 0.6710 0.003 Words S. 0.5435 0.4154 0.3933 0.3498 0.4984 0.5448 0.3947 0.5112 0.3861 0.5884 0.5952 0.003 0.6654 0.001 Avg. Acc. 0.2132 0.2841 0.2777 0.2332 0.5427 0.4818 0.5478 0.5558 0.4183 0.5897 0.6017 0.6464 Avg. Rank 9.5556 8.4028 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 3 independent experiments with different random seeds is reported. Dataset MC2PCA SWIMDFC TCK m-k AVG+ED m-k DBA m-k Shape m-k SC De TSEC NESE MUSLA Shape Conv Articulary Word R. 0.934 0.523 0.873 0.834 0.741 0.344 0.843 0.792 0.849 0.838 0.867 0.010 Atrial Fibrilation 0.514 0.532 0.191 0.515 0.317 0.116 0.387 0.293 0.346 0.538 0.579 0.015 Basic Motions 0.674 0.510 0.776 0.543 0.639 0.341 0.554 0.800 0.525 1.000 1.000 0.000 Epilepsy 0.173 0.190 0.533 0.409 0.471 0.163 0.381 0.345 0.760 0.601 0.681 0.003 Ering 0.336 0.422 0.399 0.400 0.406 0.268 0.348 0.392 0.378 0.722 0.736 0.004 Hand Movement D. 0.067 0.151 0.103 0.168 0.265 0.079 0.151 0.112 0.030 0.398 0.362 0.011 Libras 0.577 0.500 0.620 0.622 0.622 0.447 0.724 0.602 0.542 0.724 0.738 0.004 NATOPS 0.698 0.472 0.679 0.643 0.643 0.339 0.600 0.043 0.314 0.855 0.878 0.007 PEMS-SF 0.011 0.441 0.066 0.491 0.402 0.447 0.474 0.424 0.586 0.614 0.630 0.004 Pen Digits 0.713 0.652 0.693 0.738 0.605 0.634 0.738 0.563 0.645 0.826 0.784 0.004 Stand Walk Jump 0.349 0.483 0.536 0.559 0.466 0.116 0.461 0.555 0.399 0.609 0.586 0.028 UWave Gesture L. 0.570 0.482 0.710 0.713 0.582 0.419 0.758 0.557 0.559 0.728 0.742 0.014 Average Acc. 0.4680 0.4466 0.5150 0.5530 0.5132 0.3095 0.5348 0.4566 0.4943 0.7044 0.7154 Average Rank 7.0833 7.3333 5.8333 4.8333 6.4167 10.0833 5.5833 7.6667 7.2500 2.2917 1.6250