# regularizationfree_diffeomorphic_temporal_alignment_nets__408457a8.pdf Regularization-free Diffeomorphic Temporal Alignment Nets Ron Shapira Weber 1 Oren Freifeld 1 In time-series analysis, nonlinear temporal misalignment is a major problem that forestalls even simple averaging. An effective learning-based solution for this problem is the Diffeomorphic Temporal Alignment Net (DTAN) (Shapira Weber et al., 2019), that, by relying on a diffeomorphic temporal transformer net and the amortization of the joint-alignment task, eliminates drawbacks of traditional alignment methods. Unfortunately, existing DTAN formulations crucially depend on a regularization term whose optimal hyperparameters are dataset-specific and usually searched via a large number of experiments. Here we propose a regularization-free DTAN that obviates the need to perform such an expensive, and often impractical, search. Concretely, we propose a new well-behaved loss that we call the Inverse Consistency Averaging Error (ICAE), as well as a related new triplet loss. Extensive experiments on 128 UCR datasets show that the proposed method outperforms contemporary methods despite not using a regularization. Moreover, ICAE also gives rise to the first DTAN that supports variablelength signals. Our code is available at https: //github.com/BGU-CS-VIL/RF-DTAN. 1. Introduction Nonlinear temporal misalignment between different signals is a major obstacle to time-series statistical analysis. For example, physicians may be interested in the average Electrocardiogram (ECG) signal from a few minutes of recording, but the temporal misalignment across the patient s different heartbeats implies that naively averaging the data will distort the true underlying signal. A popular attempt to solve the problem relies on pairwise 1Ben-Gurion University. Correspondence to: Ron Shapira Weber , Oren Freifeld . Proceedings of the 40 th International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s). T θ2 T θ3 T θ4 T θ2 T θ3 T θ4 (a) Centroids computed using forward warps (b) The ICAE loss computed using backward warps Figure 1. The Inverse Consistency Averaging Error loss in a twoclass example. (a) The signals u1, u2, and u3 are in class c; u4 and u5 are in class c . Within each class, the centroid (µc or µc ) is obtained by averaging the warped signals ((ui T θi)i {1,2,3} or (ui T θi)i {4,5}) using the forward warps. (b) The loss is computed using the backward warps; i.e., we measure dissimilarity between each ui and its class centroid, where the latter is first warped backward ( unwarped ) using T θi (the inverse of T θi). alignments. Let ui = (ui(t))n t=1 and uj = (uj(t))m t=1 be two real-valued discrete-time signals of lengths n and m, respectively. The optimal pairwise alignment of uj towards ui, under some dissimilarity measure D, is defined by T = arg min T T D(ui, uj T) (1) where denotes function composition and T is a family of warps (or warping functions); namely, every T T is a function T : Ω R where Ω R is an interval containing {1, . . . , m}. For instance, Dynamic Time Warping (DTW) provides the optimal discrete warping path between the time indices of ui and uj via dynamic programming, where D is (usually) a Euclidean distance (Sakoe, 1971). More generally, while ui and uj are defined over discrete domains (i.e., {1, . . . , n} and {1, . . . , m}), the notation uj T in Equation 1 implicitly assumes that the value of uj(t ) at every t R is determined, using interpolation techniques, from (possibly a subset of) the m given values, (uj(t))m t=1. In this paper we focus on continuously-defined warps that are order-preserving diffeomorphisms. A diffeomorphism (namely, a differentiable invertible function whose inverse is differentiable), is a natural choice for representing time warping (Mumford & Desolneux, 2010). Since spaces of Regularization-free Diffeomorphic Temporal Alignment Nets diffeomorphisms are large, and in order to discourage unfavorable solutions, typically some regularization term, denoted by T 7 R(T; λ) and parameterized by so-called hyperparameters (HP), λ, is added to the objective function; e.g., R might penalize lack of smoothness (in the machinelearning sense, not calculus) or large deviations from the identity map. Hence, Equation 1 is commonly replaced with T = arg min T T D(ui, uj T) + R(T; λ) (2) where T is a space of 1D diffeomoprhisms from Ωinto R. In the case of an ensemble of N signals, (ui)N i=1 where N > 2, the pairwise approach usually does not generalize well, is prone to drift errors, and might introduce inconsistent solutions. This motivates approaches for joint alignment (JA), also known as global alignment or multiple-sequence alignment. The JA problem is often formulated as (T i )N i=1, µ = arg min (Ti)N i=1 T ,u i=1 D(u, ui Ti) + R(Ti; λ) where T , R( ; λ), and D are as before, Ti is the latent warp associated with ui, and µ is a latent signal, conceptually thought of as the average signal (or centroid) of the ensemble. This optimization task may also be amortized via the training of a deep net (e.g., (Shapira Weber et al., 2019; Huang et al., 2021; Martinez et al., 2022)). We emphasize that the success of JA methods, including deep-learning (DL) ones, depends crucially on the choice of R( ; λ) and, more importantly, the choice of its HP, λ. In this work, we propose a regularization-free DL approach based on a new loss, called the Inverse Consistency Averaging Error (ICAE), for time-series JA and averaging. This well-behaved loss, denoted by LICAE, alleviates the need for warp regularization and can be used within any JA method, as long as the warps are invertible. Concretely, the ICAE loss encourages both the warps and the latent µ to be consistent with the original signals by warping µ backward (also known as unwarping) towards each of those signals and then penalizing the difference between each of them and its signal-dependent version of the unwarped µ. That is, letting θi parameterize the ith warp (so Ti = T θi), and using the fact that T θi = (T θi) 1, we apply T θi to the current estimate of µ and penalize the difference between µ T θi and ui. See Figure 1 for a conceptual illustration. Importantly, the proposed approach frees us from the need to use a regularization over (T θi)N i=1. Another positive aspect of LICAE is that it lends itself immediately to support variable-length signals, a capability lacking by existing implementations of leading DL methods for JA and averaging. We demonstrate the validity of the approach on 128 datasets (Dau et al., 2019), and show that when other methods are realistically restricted in their HP search, our method outperforms them by a large margin. To summarize, our contributions are: 1) Introducing the ICAE loss for the JA and averaging task, thereby obviating the need for using a regularization over the predicted warps. 2) A triplet-loss variant of the proposed loss for better interclass separation. 3) An explicit formulation for JA and averaging of variable-length data. 4) Setting new state-ofthe-art (SOTA) results on 128 datasets from the UCR time series classification archive (Dau et al., 2019). 2. Related Work Dynamic Time Warping (DTW) is a popular distance measure (or discrepancy) between a time-series pair (Sakoe, 1971; Sakoe & Chiba, 1978). Given two signals of lengths n and m, DTW computes the best discrete alignment path in the n m pairwise distance matrix. While its complexity is O(nm), enforcing certain constraints on DTW results in a linear complexity. However, generalizing DTW from the pairwise case to the JA of multiple signals is prohibitively expensive since the complexity of finding the optimal discrete alignment between N signals of length n is O(n N). To overcome this limitation, several JA methods, working under the DTW geometry, were proposed. The DTWBarycenter Averaging (DBA) (Petitjean et al., 2011; 2014) employs expectation-maximization (EM) to refine a signal that minimizes the sum of DTW distances from the data; i.e., it alternates between finding µ (while fixing (Ti)N i=1), µ = arg min u i=1 D(u, ui Ti) , (4) and finding discretely-defined (Ti)N i=1 (while fixing µ), (T i )N i=1 = arg min (Ti)N i=1 T i=1 D(µ, ui Ti) . (5) Soft DTW (Cuturi & Blondel, 2017), a soft-minimum variant of DTW, extends DBA. Instead of using EM, Soft DBA computes µ via gradient-based optimization. Soft DTW has one HP, γ, that controls the smoothness of the alignment (γ = 0 leads to the original DTW score). Soft DTW-divergence (Blondel et al., 2021) modifies Soft DTW to a proper positive-definite divergence. Both of these optimization-based methods do not learn how to find the JA of new data; i.e., when new signals arrive, they must be run from scratch in order to achieve JA of the new ensemble. While it is possible to align the new data to the previously-found µ in a pairwise manner, this leads to inferior results (see 4). Additionally, the time/memory complexity of Soft DTW is O(mn). Soft DTW-div suffers from Regularization-free Diffeomorphic Temporal Alignment Nets an even worse complexity for a large n or m; e.g., results on Hand Outlines (the largest UCR dataset in terms of n N) were not reported by Blondel et al. (2021), and when we tried to run Soft DTW (using tslearn (Tavenard, 2017)) on it, it failed due to memory limitations. Other methods include the Global Alignment Kernel (GAK) (Cuturi, 2011) on which Soft DTW is based, DTW with Global Invariances which generalizes DTW/Soft DTW to both time and space (Vayer et al., 2020), and Neural Time Warping that relaxes the original problem to a continuous optimization using a neural net (albeit limited in the number of signals it can jointly align) (Kawano et al., 2020). Spaces of Diffeomorphisms are often used for modeling warping paths between sequences; e.g., Srivastava et al. (2010; 2011) proposed differomoprhisms based on the square-root velocity function (SRVF) representation. However, the employment of diffeomorphisms in DL used to be hindered by the associated expensive computations and/or approximation/discretization schemes. For example, this is why diffeomorphisms could not initially be used effectively within a Spatial Transformer Net (STN) (Jaderberg et al., 2015) since training the latter requires a large number of evaluations of both x 7 T θ(x) and x 7 θT θ(x) (where θ parameterizes the chosen diffeomorphism family), and these quantities are computed at multiple values of x. This has changed, however, with the emergence of new methods (Skafte Detlefsen et al., 2018; Balakrishnan et al., 2018). In particular, Skafte Detlefsen et al. (2018) built on the CPAB diffeomorphisms (see below) to propose the first diffeomorphic STNs. CPAB Diffeomorphisms (Freifeld et al., 2015; 2017). The name CPAB, short for CPA-Based, stems from the fact that these parametric diffeomorphisms are based on the integration of Continuous Piecewise-Affine (CPA) velocity fields. Of note, in 1D, the CPAB warp, x 7 T θ(x), has a closed form (Freifeld et al., 2015). While the CPAB warps were proposed by Freifeld et al. (2015) with no relation to DL, it turns out that their expressiveness and efficiency make them an invaluable tool in DL (Hauberg et al., 2016; Skafte Detlefsen et al., 2018; Skafte Detlefsen & Hauberg, 2019; Shapira Weber et al., 2019; Kaufman et al., 2021; Shacht et al., 2021; Schw obel et al., 2022; Martinez et al., 2022; Neifar et al., 2022) and thus this work uses them too. However, our method is not limited to this choice of T . A Temporal Transformer Net (TTN) is the 1D variant of the STN, where the latter is a DL module which, given a transformation family, predicts and applies a transformation to its input for a downstream task. Lohit et al. (2019) use TTNs with discretized diffeomorphisms for learning rateinvariant discriminative warps. The SRVF framework was integrated into TTNs to either predict DTW-based warping functions (Nunez & Joshi, 2020), learn a generative model Table 1. Comparing JA/averaging methods. Learning gives the ability to generalize JA to new data. VL indicates whether the method supports variable-length signals. METHOD REG.-FREE OPTIMIZATION LEARNING VL EUCLIDEAN N/A DBA EM SOFTDTW L-BFGS DTAN W/ WCSS DL TRAINING DTAN W/ LICAE DL TRAINING over the distribution of SRVF warps (Nunez et al., 2021), and time-series JA (Chen & Srivastava, 2021). However, computations in these nonparametric warps do not scale well with the signal length. Shapira Weber et al. (2019) propose the Diffeomorphic Temporal Alignment Net (DTAN), a diffeomorphic TTN that, using the parametric and highly-expressive CPAB warps, is an effective learning-based solution for JA and averaging. Shapira Weber et al. (2019) based their DTAN implementation on libcpab (Detlefsen, 2018). Recently, Martinez et al. (2022) released another CPAB library, Diffeomorphic Fast Warping (DIFW), which, while being similar to libcpab (and is, in fact, based on it), is even faster, largely due to the smart discovery of a closed-form gradient (Martinez et al., 2022) for CPAB warps. Together with some other changes and an extensive HP tuning on the test data, this let them propose a DTAN implementation with SOTA results in terms of Nearest Centroid Classification (NCC) accuracy, a standard metric for time-series averaging. Henceforth will refer to the DTAN implementations from Shapira Weber et al. (2019) and Martinez et al. (2022) as DTANlibcpab and DTANDIFW, respectively. Lastly, Res Net-TW (Huang et al., 2021) also predicts CPAB warps albeit via the Large Deformation Diffeomorphic Metric Mapping framework (Beg et al., 2005). Warp Regularization. As is typical with diffeomorphisms, CPAB warps too are usually regularized. In particular, the three works above (Shapira Weber et al., 2019; Huang et al., 2021; Martinez et al., 2022), who all use the withinclass-sum-of-squares (WCSS) loss, also use the following regularization from Freifeld et al. (2015), R(T θi; λ) = θ i Σ 1 CPAθi. The matrix ΣCPA is the covariance of a zeromean Gaussian smoothness prior over CPA velocity fields and has two HPs: λΣ, which controls the overall variance, and λsmooth, which controls the smoothness of the fields. Additionally, all these three methods predict a varying number of warps (denoted by Nwarps), such that their composition yields the final warp. We conclude the section with Table 1 that summarizes differences between several JA/averaging methods and ours. Regularization-free Diffeomorphic Temporal Alignment Nets 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Soft DTW( =0.1) 0 20 40 60 80 100 120 Soft DTW( =1) 0 20 40 60 80 100 120 Soft DTW( =10) 0 20 40 60 80 100 120 DTAN-No Regularization 0 20 40 60 80 100 120 DTAN-Weak Reg. 0 20 40 60 80 100 120 DTAN-Strong Reg. 0 20 40 60 80 100 120 0 20 40 60 80 100 120 DTAN-ICAE-triplet Figure 2. The effect of the regularization HP. The figures shows 10 samples (gray) from the ECGFive Days dataset with their estimated average (blue), and compares Euclidean averaging, DBA, Soft DTW, and several DTAN methods. DBA requires no HP but falls to poor local minima. Soft DTW s barycenter is severely affected by the choice of its smoothing HP, γ: γ = 0.1 results in a visible pinching effect while γ = 10 smoothens out peaks/valleys. DBA and Soft DTW are computed per class and do not learn how to generalize to new data, unlike DTAN which is learning-based and requires a single model for all classes. The regularization often used with DTAN has 2 HPs, (λΣ, λsmooth), where a weak regularization (λΣ, λsmooth : .5, .01) is insufficient and a strong regularization (λΣ, λsmooth : .001, .1), is too restrictive. Our LICAE and LICAE triplet are regularization-free, yet provide barycenters that represent the data well. We propose a regularization-free approach for time-series JA and averaging using DTAN. Our method leverages the fact that T is a diffeomorphism family and thus its elements are invertible. Our motivation stems in part from the fact that leading JA methods depend on warp regularization to avoid unrealistic deformation and/or trivial solutions (see Figure 2). Its optimal HPs, however, are dataset-specific. As time-series data varies considerably across different application domains (ECG compared with audio recording, for instance), determining a proper value of λ is difficult. For example, Martinez et al. (2022) ran 8064 different experiments (96 different configurations per each of 84 datasets) when evaluating on the UCR archive (Chen et al., 2015) (with such an approach, the 128 datasets of the updated UCR archive (Dau et al., 2019) will require 12288 experiments). Our approach eliminates this issue. The remainder of this section is constructed as follows. In 3.1, in a presentation that follows Shapira Weber et al. (2019), we briefly explain CPAB warps, and refer to Freifeld et al. (2015; 2017) for more details. in 3.2 we touch upon the TTN mechanics and our choice of architecture for it. In 3.3 and 3.4 we explain our proposed losses, ICAE, and a new tripletloss variant of it, respectively. 3.5 details how we handle variable-length data. Finally, in 3.6 we discuss limitations. 3.1. CPAB Diffeomorphisms Let Ωbe a partition of the signal s time domain into subintervals. Let V be the linear space of CPA velocity fields 0 20 40 60 80 100 x 0 20 40 60 80 100 x 0 20 40 60 80 100 x Figure 3. Examples of CPAB warps for three different partitions of Ω. Top: CPA velocity fields. Bottom: The resulting CPAB warps. w.r.t. Ω, let d = dim(V), and let vθ : Ω R, a velocity field parameterized by θ Rd, denote the generic element of V, where θ stands for the coefficient w.r.t. some basis of V. The corresponding space of CPAB warps, obtained via integration of elements of V, is T n T θ : x 7 ϕθ(x; 1) s.t. ϕθ(x; t) solves ϕθ(x; t) = x + Z t 0 vθ(ϕθ(x; τ)) dτ where vθ V o . These order-preserving warps are (C1) diffeomorphisms (Freifeld et al., 2015; 2017). See Figure 3 for typical CPAB warps. The fineness of Ωdetermines a trade-off between the expressiveness of T on the one hand and the computational complexity and dimensionality on the Regularization-free Diffeomorphic Temporal Alignment Nets other hand. CPA velocity fields support fast and accurate integration methods. Particularly useful in the context of DL is the fact that CPAB warps lend themselves to fast and accurate computation of the so-called CPAB gradient, x 7 θT θ(x). In fact, Martinez et al. (2022), showed that this gradient even has a closed form. Other types of efficient diffeomorphisms (e.g., (Zhang & Fletcher, 2018; Arsigny et al., 2006; Durrleman et al., 2013; Allassonniere et al., 2015)) may also be used in DTAN, provided that there is also an efficient way to evaluate x 7 θT θ(x). 3.2. Temporal Transformer Networks A TTN, which predicts the warping parameters θ and applies T θ to the input signals, consists of three modules. The first is the so-called localization net. This is a neural net, denoted by floc( ), which takes as an input a batch of sequences (ui)N i=1 and predicts the corresponding warping parameters, (θi)N i=1. The second is a grid generator which creates a grid G Ωof evenly-spaced points which are then warped by T θi. Lastly, a grid sampler computes the warped signal vi = ui T θi by interpolating its values, using ui, at (T θi) 1(G). See Jaderberg et al. (2015) for details. In this work, we set floc to be Inception Time (Ismail Fawaz et al., 2020) instead of a Temporal Convolutional Net (TCN) used in (Shapira Weber et al., 2019; Martinez et al., 2022). Originally designed for time-series classification, Inception Time was inspired by the Inception-v4 architecture and consists of several Inception modules leveraging the bottleneck design popular in image classification. Notably, we incorporate the Global Average Pooling (GAP) operator before the penultimate layer of floc, which allows the model to remain fixed in its number of trainable parameters w.r.t. the input size (e.g., we use the same architecture for all UCR datasets). It is also one of the reasons why we can process variable-length input at ease (see 3.5). 3.3. The Inverse Consistency Averaging Error Christensen & Johnson (2001) introduced the Inverse Consistency Error (ICE) as a regularizer for the task of pairwise image alignment. Given two images, I1 and I2, with domains Ω1 and Ω2 respectively, the latent spatial maps f1 : Ω2 Ω1 and f2 : Ω1 Ω2 should be consistent; i.e., f2 = f 1 1 and f1 = f 1 2 . The ICE, defined as Z Ω2 f2(f1(x)) x 2dx + Z Ω1 f1(f2(x)) x 2dx , penalizes deviations from that consistency. We propose a new form of inverse consistency that renders it useful for the JA task as well. Unlike the original ICE, which is pairwise and acts as a regularization term added to the main loss, our proposed LICAE measures the consistency between the estimated average sequence and each of its respective group members. Moreover, rather than being a regularizer term added to another loss, our LICAE is the entire loss by itself. That is, our generalization (of the ICE) stands on its own as a dedicated loss function and results in consistent JA. Importantly, and as we will show, it removes the need to use any form of regularization, and this, in turn, removes (trivially) the need to tune regularization HPs. Following the formulation in (Shapira Weber et al., 2019), let us first recall the previously-used JA loss function in the singleand multi-class cases. For a single class, the loss was the variance of the aligned signals: where ℓ2 is the ℓ2 norm, (floc(ui))N i=1 = (θi)N i=1 are the warp parameters predicted by floc, and i=1 ui T θi (9) is the post-alignment average signal. In the multi-class case, the loss was the sum of the within-class variances, often called the within-class sum of squares (WCSS): where K is the number of classes, yi is class label of ui, Nk is the number of signals in class k, and µk = 1 NK X i:yi=k ui T θi (11) is the post-alignment average of class k (this is a semisupervised problem in the following sense: training is done with known (yi)N i=1 and unknown (θi)N i=1). It is clear why, with these losses, the warp-regularization term, R(T θi; λ), is needed. First, the data term, Ldata, does not encourage warp consistency. Secondly, it is possible to reduce the variance (even to zero!) by severely distorting the signals, and this issue only worsens due to interpolation artifacts. However, optimal regularization is dataset-specific. For example, penalizing deformations that are too large might not be ideal in many cases. Likewise, with a temporal smoothness prior, it is hard to determine the right amount of smoothness. Figure 2 illustrates the critical role of regularization on the barycenter computation using DBA, Soft DTW, and DTAN. Improper values of γ (for Soft DTW) or λΣ, λsmooth (for DTAN) usually result in unrealistic warps or overly restrict the warps (e.g., a strong prior for DTAN). Regularization-free Diffeomorphic Temporal Alignment Nets Algorithm 1 The JA training with an ICAE loss Input: Nepochs, floc Data: (ui, yi)N i=1 Output: floc( ), trained for joint alignment 1 for each epoch and each batch j {1, . . . , Nbatches} do 3 (ui, yi)Nj i=1 batchj 4 (θi)Nj i=1 (floc(ui))Nj i=1 5 for k {1, . . . , K} do 6 µk = 1 Nk P i:yi=k(ui T θi) 7 LICAE = 1 NK P i:yi=k µk T θi ui 2 ℓ2 8 Lbatch += LICAE 9 Perform an optimization step to minimize Lbatch Instead, we propose a new loss that is minimized when the average sequence is both a minimizer of the variance and consistent with its class. Concretely, we propose the Inverse Consistency Averaging Error loss (ICAE), defined as: LICAE measures how well the average signal, µk, fits each signal ui in its class using the inverse warp T θi. It does so by first aligning all of the signals in class k using the predicted warps, then computing their average µk, and finally warping µk back toward each ui using T θi, thereby ensuring consistency between them. A key insight is that Equation 12 strongly discourages trivial solutions or unrealistic warps as this would result in a poor estimate of µk, which in turn would yield a high discrepancy between it and the original signals. In other words, the loss favors realistic deformations without the need to add a regularization term. The full training procedure is described in Algorithm 1. 3.4. Inverse Consistent Centroids Triplet Loss While LICAE implies consistency, it is agnostic about the separation between different classes. That said, while metrics such as DTW are completely data-driven, our learningbased can be utilized to learn task-driven representations. As such, we introduce the centroid triplet loss into our framework to encourage inter-class separation. Traditionally, e.g. in classification tasks, a triplet loss is defined over a triplet (ua i , up i , un i ) of an anchor, a positive, and a negative examples, respectively. As our task is intra-class JA and computing class averages (also known as centroids), adopting a centroid-based triplet loss is more adequate here (Doras & Peeters, 2020). We define the Inverse Consistent Centroids Triplet Loss over the triplet (ua i , µp i , µn i ) as LICAE triplet(ua i , µp, µn) max(0, ua i µp T θi 2 ℓ2 ua i µn T θi 2 ℓ2 + α) (13) 0 10 20 30 40 50 60 2 Misaligned signals 0 10 20 30 40 50 60 Misaligned average signal 0 10 20 30 40 50 60 2 Aligned signals 0 10 20 30 40 50 60 2 Aligned average signal Figure 4. JA of variable-length data (Dataset: Shake Gesture Wiimote Z) using the proposed LICAE. Shaded area is std. dev. where µp, µn are the positive and a negative class centroids, respectively, and α is the margin between them (α = 1 in all our experiments and is dataset-independent). As both µp and µn are compared via an inverse warp, LICAE triplet does not break the consistency between samples and their mean. The LICAE triplet is used in tandem with LICAE. 3.5. Variable-Length Joint Alignment Our proposed LICAE also allows for the JA and averaging of variable-length sequences without having to use a specialized loss function or tweak the boundary conditions on T θ (as mentioned in (Shapira Weber et al., 2019; Martinez et al., 2022) as a hypothetical possibility). Instead, our formulation (as well as our code) handles both fixed and variable-length data. It does so in the following manner. First, the post-alignment average signal is produced by dividing, at each time step, the sum of the relevant values by the number of non-missing values. That is, for each time step t along the duration of the mean signal µ, we compute: µ[t] = 1 Nvalid i:(ui T θi)[t] =null (ui T θi)[t] (14) where Nvalid is the number of signals whose domain includes a point mapped to t. Then, when µ is warped backward, Equation 12 is computed with no modifications. See, e.g., Figure 4. From an implementation standpoint, we note that any null value in either the input and/or loss would break the computational graph. To avoid for-loops and compute back-propagation in batches, it is computationally effective to first pad all samples with zeros (w.r.t. the longest signal) and create an indicator mask for missing values. The mask is also warped by T θ in Equation 14. 3.6. Limitations Limitations w.r.t. DTW: DTW-based methods are optimization-based, and thus, when the sample size is very small and/or the signal length is very short, running those methods on such a small training data, might be faster than Regularization-free Diffeomorphic Temporal Alignment Nets Table 2. Nearest Centroid Classification Accuracy. METHOD OBJECTIVE NCCmedian NCCbest #CONFIGS #DATASETS #EXPERIMENTS PART 1: ALLOWING HP SEARCH (PREVIOUSLY-REPORTED RESULTS) EUCLIDEAN N/A - 0.611 1 84 84 DBA DTW - 0.657 1 84 84 SOFTDTW SOFTDTW - 0.703 9 84 756 SOFTDTW SOFTDTW-DIV - 0.708 9 84 756 DTANlibcpab WCSS + REG - 0.705 12 84 1008 RESNET-TW WCSS + REG - 0.711 20 84 1680 DTANDIFW WCSS + REG - 0.749 96 84 8064 PART 2: SINGLE HP CONFIGURATION IN ALL DATASETS (SAME UCR DATASETS AS REPORTED BY OTHER WORKS ABOVE) DTANDIFW WCSS + REG 0.604 0.607 1 84 84 DTANDIFW LICAE (OURS) 0.665 0.694 1 84 84 DTANDIFW LICAE triplet (OURS) 0.707 0.739 1 84 84 PART 3: SINGLE HP CONFIGURATION IN ALL DATASETS (INCLUDING ADDITIONAL NEWER FIXED-LENGTH UCR DATASETS) DTANDIFW WCSS 0.609 0.65 1 117 117 DTANDIFW WCSS + REG 0.603 0.605 1 117 117 DTANDIFW LICAE (OURS) 0.656 0.686 1 117 117 DTANDIFW LICAE triplet (OURS) 0.709 0.741 1 117 117 PART 4: SINGLE HP CONFIGURATION IN ALL DATASETS (FULL UPDATED UCR ARCHIVE, INCLUDING VARIABLE-LENGTH DATASETS) DTANDIFW LICAE (OURS) 0.623 0.653 1 128 128 DTANDIFW LICAE triplet (OURS) 0.67 0.701 1 128 128 our training time (see Appendix A). We emphasize, however, that if the training data is large (in either dimension) our method is, in fact, usually faster. Additionally, like most learning-based methods, a small train set might result in over-fitting, which will damage performance on test data. Optimization-based methods may not suffer from this issue. Finally, The Soft DTW (Cuturi & Blondel, 2017) smoothness HP, γ, may provide more robustness to amplitude jitter than our method. However, it must be tuned (and this can be expensive or even infeasible), and in practice, the results show that our method still outperforms such methods. Limitations w.r.t. WCSS loss: During training, our complexity is slightly larger: the proposed LICAE requires two warps per sample (i.e., forward and inverse warps), and LICAE triplet requires 3, while the WCSS requires only the forward warp. Thus, the training times can be slightly longer. However, the difference is small, since the warps are computed very efficiently (using the DIFW package (Martinez et al., 2022)) and most of the computation time during training is spent on other parts of the network which are identical regardless which of the losses (WCSS or ICAE) is used. In any case, inference time is identical in both cases since then only a single forward warp is used. 4. Experiments and Results To evaluate our approach and compare with others, we used the UCR time-series classification archive benchmark. The most updated version (Dau et al., 2019) of the UCR archive has 128 datasets with inter-dataset variability in the number of samples, signal length, application domain, and the number of classes. Eleven of those datasets also present intra-dataset variability of the signal length; such datasets are referred to as variable-length (VL) datasets. In all of the experiments, we used the train/test splits provided by the archive. To quantify performances we used, as is customary, the NCC accuracy. This performance index is viewed as an evaluation metric for measuring how well each centroid describes its class members (and thus, implicitly, also measures the JA quality). The NCC framework has 2 steps: 1) compute the centroid, µk, for each class of the train set; 2) label each test sample by the class of its closest centroid. As we explain below, Table 2, which summarizes the NCC results, is divided into several parts. The full results, together with many illustrative figures and computation-time evaluation, appear in our Supplemental Material (Sup Mat). Technical details. In all of our DTAN experiments, training was done via the Adam optimizer (Kingma & Ba, 2014) for 1500 epochs, batch size of 64, Np (the number of subintervals in the partition of Ω) was 16, and the scaling-andsquaring parameter (used by DIFW) was 8. These values were previously reported to yield the highest number of Wins in (Martinez et al., 2022). As Shapira Weber et al. (2019) used a recurrent variation of DTAN (RDTAN) while Martinez et al. (2022) stacked TCNs, we fixed the number of recurrences to 4 (we did not find it necessary to stack Inception Time models). The Py Torch TSAI implementation of the Inception Time was taken from (Oguiza, 2022). In the timing experiments ( 4.3), for DTW, DBA, and Soft DTW we used the tslearn package (Tavenard, 2017). Regularization-free Diffeomorphic Temporal Alignment Nets 4.1. Nearest Centroid Classification Part 1: 84 datasets allowing an extensive HP search (previously-reported results). An older version (Chen et al., 2015) of the UCR archive had only 85 datasets (a subset of the 128 mentioned above). Several previous works reported results on only 84 datasets out of those 85, possibly due to the size of the largest dataset. Part 1 of Table 2 contains the results, on those 84 datasets, obtained by several key methods, as reported by their authors, as well as those obtained by a simple Euclidean averaging (i.e., a no-alignment baseline). The methods are DBA, Soft DTW, DTANlibcpab, Res Net-TW, and DTANDIFW. The regularization-free DBA requires no HP configurations. The Soft DTW methods have one HP for controlling the smoothness. Their results, reported in (Blondel et al., 2021), were obtained by those authors using cross-validation. The other works (Shapira Weber et al., 2019; Huang et al., 2021; Martinez et al., 2022) reported only their best results across different configurations. Shapira Weber et al. (2019) evaluated DTANlibcpab using 12 different configurations per dataset (4 configurations for (λΣ, λsmooth) and 3 different numbers of recurrences). In (Huang et al., 2021), Res Net-TW used the same regularization configurations as in Shapira Weber et al. (2019), but also tested varying numbers of Res Net blocks (4 to 8) per dataset. Martinez et al. (2022) evaluated DTANDIFW using 96 different configurations (various options of λΣ, λsmooth, Np, #stacked TCNs, boundary conditions, and the scaling-and-squaring parameter) per dataset. We note that: 1) tuning Np and the boundary conditions is another form of tweaking the regularization; 2) as stated in supplemental material of (Martinez et al., 2022), their reported results were chosen among those 96 configurations, per dataset, based on the best performance on the test set. Part 2: Using a single HP configuration in all 84 datasets. Part 1 of Table 2 suggests that increasing the number of tried HP configurations translates to better performance due to the large variability across the UCR datasets. However, the compact summary in Part 1 of Table 2 also hides an ugly truth: there is no one-size-fits-all configuration. For example, DTANDIFW produced the best performance but this is largely due to the fact they performed an expensive search over a large number of HP configurations. In fact, inspecting the full results of either DTANlibcpab, Res Net TW, or DTANDIFW, reveals that the optimal choice of HP varies across the datasets and affects results drastically. To demonstrate this crucial point, we ran a new set of experiments. We picked the HP configuration that according to Martinez et al. (2022) achieved the highest number of wins among their 96 configurations. Next, using that configuration we ran, on those 84 datasets, exactly the same DTAN but with 3 different losses: 1) WCSS plus the smoothness regularization (λΣ and λsmooth, 0.001 and 0.1, respectively); 2) our proposed LICAE; 3) our proposed LICAE triplet. In the last 2 cases, which are regularization-free, the values of λΣ and λsmooth from that configuration were ignored. In all 3 cases, we used DTANDIFW with the same Inception Time backbone (Oguiza, 2022) (in all 3 cases this gave better results than using a TCN). To account for random initializations and the stochastic nature of DL training, in each of the 3 cases we performed 5 runs on each dataset and report both the median and best results; see part 2 in Table 2. The results illustrate the merits of the proposed method: a single HP configuration for the regularization, even the one stated as the best, does not properly fit the entirety of the UCR datasets. In contrast, dropping the regularization term and using our LICAE increases performance by a large margin, which is only further increased when utilizing LICAE triplet, which increases separability between class centroids (a feat current DTW-based methods are incapable of) and achieves SOTA results. Part 3 & 4: Using a single HP configuration in all of the 128 datasets. To produce the results in part 3 of the table, we again repeated the procedure from part 2, except that 1) we added another case where the loss is only WCSS with no regularization, and 2) the results, on 117 datasets, also take into account additional fixed-length datasets that were added in the newer UCR archive. The results in, and conclusions from, Part 3 are consistent with Part 2. WCSS did slightly better than WCSS+Reg, probably since even though it distorts the signals, it makes it a bit easier (than in the WCSS+Reg case) to differentiate between classes. In any case, our losses outperform both of these methods. Part 4 extends the results of Part 3 by adding, for the DTANs with our proposed losses, the 11 VL datasets (for a total of 128). 4.2. Ablation Study An ablation study w.r.t. the backbones and losses is presented in Table 3. Note that Smooth Subspace dataset (length=15) was omitted for the TCN experiments since it was too short for the Max Pooling operations. Table 3. Ablation study Backbone Objective NCC #Datasets TCN LICAE 0.611 127 TCN LICAE triplet 0.632 127 Inception Time LICAE 0.623 127 Inception Time LICAE triplet 0.67 127 Inception Time WCSS-triplet 0.642 117 Inception Time WCSS-triplet + Reg. 0.603 117 Inception Time LICAE 0.656 117 Inception Time LICAE triplet 0.709 117 Regularization-free Diffeomorphic Temporal Alignment Nets Two Lead ECG ECGFive Days Star Light Curves Hand Outlines Method DBA Soft DTW = 0.01 Soft DTW = 0.1 Soft DTW = 1 (a) Time to compute train set barycenters Two Lead ECG ECGFive Days Star Light Curves Hand Outlines Method DBA Soft DTW = 0.01 Soft DTW = 0.1 Soft DTW = 1 (b) Inference - computing barycenters for 30 new samples Two Lead ECG ECGFive Days Star Light Curves Hand Outlines Method DTW Soft DTW = 0.01 Soft DTW = 0.1 Soft DTW = 1 (c) Distance to barycenter using the corresponding metric Figure 5. Timing comparison (the y-axis is log-scaled). See Table 4 in Appendix A for full details. 4.3. Computation-time Comparison A key advantage of learning-based approaches is fast inference on new data. We performed several timing experiments between DBA, Soft DTW (whose HP, γ {0.01, 0.1, 1}, must be searched in each dataset), and DTAN, trained with the proposed LICAE. We used a machine with 12 CPUcores, 32Gb RAM, and an RTX 3090 graphic card. We chose a subset of the UCR archive, spanning different lengths and sample sizes, and compared the time it took to compute the centroids on the entire train set. Then, since DBA and Soft DTW are optimization-based we provide timing for two approaches: (1) barycenter computation time of a new batch (N = 30, average of 5 runs) and (2) computing DTW/Soft DTW between the batch and its barycenter (which, after warping, can be averaged again). For DTAN, this is just the inference time. Figure 5 presents the result (while Appendix A presents the full datasets details). On training data, for smaller datasets (in terms of n, N), Soft DTW/DBA is faster than DTAN, but this trend is reversed for the larger ones. Soft DTW and DBA runs out of memory on the largest dataset (Hand Outlines). During inference, using DTAN is orders of magnitude faster (x10 x104) than recomputing barycenters, and, on the larger datasets, is x10 faster than computing DTW/Soft DTW . 4.4. Multivariate Data Joint alignment of multivariate time-series data requires special attention due to the usually-complicated inter-channel relationships. When the channels are highly-correlated, a single warp (i.e., a single θ) may suffice. Otherwise, warping each channel independently is preferable. The proposed loss, LICAE, supports both options. While complete analysis of multivariate data is outside the scope of this paper, as a proof of concept we trained DTAN with LICAE (using a single warp for all channels) on the Spoken Arabic Digits dataset (Bagnall et al., 2018) which contains 13 channels and 10 classes. The NCC accuracy for the baseline and the proposed ICAE are 0.08 and 0.402 respectively, demonstrating the potential efficacy of the approach such data. 5. Conclusion We have proposed the Inverse Consistency Averaging Error, LICAE, a novel loss function for regularization-free time-series joint alignment and averaging via diffeomorphic temporal transformer nets. The approach utilizes the invertibility of diffeomorphic warps and yields an effective JA while alleviating the need for extensive HP search. We also proposed the LICAE triplet which allows for a better inter-class separation using a warp-consistent variant of the triplet centroid loss. Additionally, we introduced a formulation of the joint alignment of variable-length time-series data via the proposed framework. Extensive experiments on 128 datasets demonstrate the validity of our approach, resulting in SOTA performance while requiring no warp regularization. Finally, our approach may also be used in conjunction with another regularization-free method for joint alignment which was suggested in (Erez et al., 2022) for spatial warps that relied on a memory-based formulation or with transformation-invariant clustering (Monnier et al., 2020) Acknowledgments This work was supported by the Lynn and William Frankel Center at BGU CS, by the Israeli Council for Higher Education via the BGU Data Science Research Center, and by Israel Science Foundation Personal Grant #360/21. R.S.W was also funded in part by the BGU Kreitman School Negev Scholarship. Regularization-free Diffeomorphic Temporal Alignment Nets Allassonniere, S., Durrleman, S., and Kuhn, E. 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IJCV, 2018. 5 Supplemental Material The Supplemental Material is organized as follows: Appendix A: A computation-time study between time-series averaging methods. Appendix B: t-SNE projections. Appendix C: An illustration of the different warps obtained by DTW on the one hand, and DTAN trained with the proposed LICAE on the other hand. Appendix D: A training procedure illustration between the WCSS and our LICAE. Appendix E: An illustration of unwarping the class mean to the original samples. Appendix F: Joint-alignment results on various datasets. Appendix G: A visual comparison of time-series averaging methods. Appendix H: UCR archive details. Appendix I: Full NCC results for all of the UCR archive datasets. Regularization-free Diffeomorphic Temporal Alignment Nets A. Computation Time Table 4. Timing comparison for several datasets of the UCR archive. (Top) During the fitting/training step, Soft DTW/DBA are computed per class while DTANICAE uses one model for all classes. (Middle) During inference, 30 new samples are averaged. Soft/DBA needs to be called again as it is optimization-based, while DTANICAE requires a single forward pass. (Bottom) Finally, each new sample is compared to its train-set barycenter using the corresponding metric. N/A = Out Of Memory (on a machine with 12 CPU cores and 32Gb RAM) . Dataset Nsamples Nclass Length DBA Soft DTWγ=0.01 Soft DTWγ=0.1 Soft DTWγ=1 ICAE Training time - full train set (sec) Two Lead ECG 23 2 82 0.39 0.64 0.31 0.09 164.78 ECGFive Days 23 2 136 0.90 0.65 0.64 0.31 157.39 Yoga 300 2 426 52.4104 265.493 283.566 50.3923 565.65 Star Light Curves 300 3 1024 1140.79 3399.90 964.21 441.33 2657.20 Hand Outlines 1000 2 2709 N/A N/A N/A N/A 6483.50 Inference time, averaged over 5 runs (sec) Two Lead ECG 30 1 82 0.25 0.41 1.19 0.36 0.35 0.08 0.13 0.0 0.03 0.04 ECGFive Days 30 1 136 0.3 0.03 3.39 1.32 2.22 0.36 0.73 0.25 0.02 0.013 Yoga 30 1 426 4.21 0.9 31.46 5.92 27.58 4.33 4.73 0.38 0.02 0.01 Star Light Curves 30 1 1024 19.08 3.06 80.52 12.71 61.5 22.07 15.2 0.15 0.02 0.01 Hand Outlines 30 1 2709 70.69 28.39 209.2 58.93 155.53 18.37 68.54 0.3 0.04 0.02 Distance to barycenter using the corresponding metric, averaged over 5 runs (sec) Two Lead ECG 30 1 82 0.04 0.05 0.02 0.0 0.03 0.0 0.03 0.0 0.034 0.001 ECGFive Days 30 1 136 0.05 0.0 0.04 0.0 0.04 0.0 0.05 0.0 0.024 0.0 Yoga 30 1 426 0.09 0.0 0.25 0.0 0.28 0.0 0.3 0.0 0.024 0.0 Star Light Curves 30 1 1024 0.11 0.01 1.39 0.01 1.61 0.0 1.76 0.0 0.023 0.0 Hand Outlines 30 1 2709 0.4 0.01 10.03 0.01 11.5 0.03 12.54 0.07 0.045 0.002 B. t-SNE Projection 60 40 20 0 20 40 (a) Original data 40 20 0 20 40 60 80 60 40 20 0 20 40 (c) LICAE triplet Figure 1. Comparison of t-SNE projections (Van der Maaten & Hinton, 2008) of the original and aligned test data (i.e., not embedding) of the 14-class Faces UCR dataset with their respective class centroids. Our proposed LICAE decreases the within-class variance, while LICAE triplet increases the inter-class variance further. Regularization-free Diffeomorphic Temporal Alignment Nets C. DTW vs. DTANICAE warping (a) DTW (b) DTANLICAE Figure 2. Warping paths computed by Dynamic Time Warping (DTW) and predicted by DTAN using the proposed LICAE, between a test sample (blue) and the class average (red, computed by DTAN). DTW is prone to overfit the signal s noise, whereas our method manages to capture the underlying structure of the time series and provide robust alignment. D. Training Procedure Illustration 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 (a) Input data 0 100 200 300 400 500 (b) Epoch 10 0 100 200 300 400 500 (c) Epoch 20 0 100 200 300 400 500 (d) Epoch 50 0 100 200 300 400 500 (e) Epoch 100 Figure 3. Training procedure on the Beetle Fly dataset. The first column depicts the input data (for better visualization, the top panel shows 3 random signals while the bottom 10 signals and their average are in blue). (Top) The Within-Class Sum of Squares (WCSS) loss reduces variance by applying an unrealistic deformation to the data, resulting in visible pinching effect (i.e., bad local minima). (Bottom) The proposed LICAE, while requiring no regularization, avoids such an undersired solution by maintaining consistency between the average sequence and its class members. Regularization-free Diffeomorphic Temporal Alignment Nets E. Inverse Warping Examples 0 20 40 60 80 (a) Class average, µk, (blue) with 4 samples, ui s, (grey). 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 2.0 0 20 40 60 80 (b) Uwarping µk to each sample (i.e., µk T θi) Figure 4. Unwarping the class average to the original data for the ECG200 dataset. 0 20 40 60 80 100 120 (a) Class average, µk, (blue) with 4 samples, ui s, (grey). 0 20 40 60 80 100 120 2 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 2 (b) Uwarping µk to each sample (i.e., µk T θi) Figure 5. Unwarping the class average to the original data for the CBF dataset. 0 20 40 60 80 100 120 (a) Class average, µk, (blue) with 4 samples, ui s, (grey). 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 (b) Uwarping µk to each sample (i.e., µk T θi) Figure 6. Unwarping the class average to the original data for the ECGFive Days dataset. Regularization-free Diffeomorphic Temporal Alignment Nets F. Joint Alignment Results Here we provide additional results for the joint alignment and averaging of various datasets of the UCR time series classification archive (Dau et al., 2019) using our proposed LICAE. The results are provided for both the train and test sets. F.1. Train data 0 20 40 60 80 100 120 6 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 6 Aligned signals 0 20 40 60 80 100 120 6 Aligned average signal (a) Class 0 0 20 40 60 80 100 120 4 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 4 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (b) Class 1 Figure 7. Joint alignment and averaging of the ECGFive Days dataset. Shaded area corresponds to σ. 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 2 Aligned signals 0 20 40 60 80 100 120 2 Aligned average signal (a) Class 0 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (b) Class 1 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 2 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (c) Class 2 Figure 8. Joint alignment and averaging of the CBF dataset. Shaded area corresponds to σ. 0 20 40 60 80 Misaligned signals 0 20 40 60 80 2 Misaligned average signal 0 20 40 60 80 Aligned signals 0 20 40 60 80 Aligned average signal (a) Class 0 0 20 40 60 80 4 Misaligned signals 0 20 40 60 80 Misaligned average signal 0 20 40 60 80 Aligned signals 0 20 40 60 80 Aligned average signal (b) Class 1 Figure 9. Joint alignment and averaging of the ECG200 dataset. Shaded area corresponds to σ. Regularization-free Diffeomorphic Temporal Alignment Nets 0 200 400 600 800 1000 2 Misaligned signals 0 200 400 600 800 1000 Misaligned average signal 0 200 400 600 800 1000 2 Aligned signals 0 200 400 600 800 1000 Aligned average signal (a) Class 0 0 200 400 600 800 1000 Misaligned signals 0 200 400 600 800 1000 1 4 Misaligned average signal 0 200 400 600 800 1000 Aligned signals 0 200 400 600 800 1000 1 4 Aligned average signal (b) Class 1 0 200 400 600 800 1000 2 Misaligned signals 0 200 400 600 800 1000 2 Misaligned average signal 0 200 400 600 800 1000 2 Aligned signals 0 200 400 600 800 1000 2 Aligned average signal (c) Class 2 Figure 10. Joint alignment and averaging of the Star Light Curves dataset. Shaded area corresponds to σ. 0 10 20 30 40 50 60 Misaligned signals 0 10 20 30 40 50 60 Misaligned average signal 0 10 20 30 40 50 60 2 Aligned signals 0 10 20 30 40 50 60 Aligned average signal (a) Class 0 0 10 20 30 40 50 60 2 Misaligned signals 0 10 20 30 40 50 60 Misaligned average signal 0 10 20 30 40 50 60 2 2 Aligned signals 0 10 20 30 40 50 60 Aligned average signal (b) Class 1 0 10 20 30 40 50 60 Misaligned signals 0 10 20 30 40 50 60 2 Misaligned average signal 0 10 20 30 40 50 60 Aligned signals 0 10 20 30 40 50 60 2 Aligned average signal (c) Class 2 0 10 20 30 40 50 60 Misaligned signals 0 10 20 30 40 50 60 Misaligned average signal 0 10 20 30 40 50 60 Aligned signals 0 10 20 30 40 50 60 Aligned average signal (d) Class 3 0 10 20 30 40 50 60 2 Misaligned signals 0 10 20 30 40 50 60 Misaligned average signal 0 10 20 30 40 50 60 2 Aligned signals 0 10 20 30 40 50 60 Aligned average signal (e) Class 4 0 10 20 30 40 50 60 Misaligned signals 0 10 20 30 40 50 60 Misaligned average signal 0 10 20 30 40 50 60 Aligned signals 0 10 20 30 40 50 60 Aligned average signal (f) Class 5 Figure 11. Joint alignment and averaging of the Synthetic Control dataset. Shaded area corresponds to σ. Regularization-free Diffeomorphic Temporal Alignment Nets 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (a) Class 0 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (b) Class 1 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (c) Class 2 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (d) Class 3 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 2 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (e) Class 4 0 20 40 60 80 100 120 4 Misaligned signals 0 20 40 60 80 100 120 4 Misaligned average signal 0 20 40 60 80 100 120 4 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (f) Class 5 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (g) Class 6 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (h) Class 7 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 2 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (i) Class 8 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 3 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (j) Class 9 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (k) Class 10 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 3 Aligned average signal (l) Class 11 0 20 40 60 80 100 120 4 4 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 4 Aligned signals 0 20 40 60 80 100 120 4 Aligned average signal (m) Class 12 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (n) Class 13 Figure 12. Joint alignment and averaging of the Faces UCR dataset. Shaded area corresponds to σ. Regularization-free Diffeomorphic Temporal Alignment Nets F.2. Test data 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 6 Aligned average signal (a) Class 0 0 20 40 60 80 100 120 4 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (b) Class 1 Figure 13. Joint alignment and averaging of the ECGFive Days dataset. Shaded area corresponds to σ. 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (a) Class 0 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 2 Misaligned average signal 0 20 40 60 80 100 120 2 3 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (b) Class 1 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 2 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (c) Class 2 Figure 14. Joint alignment and averaging of the CBF dataset. Shaded area corresponds to σ. 0 20 40 60 80 4 Misaligned signals 0 20 40 60 80 Misaligned average signal 0 20 40 60 80 4 Aligned signals 0 20 40 60 80 Aligned average signal (a) Class 0 0 20 40 60 80 4 Misaligned signals 0 20 40 60 80 Misaligned average signal 0 20 40 60 80 Aligned signals 0 20 40 60 80 Aligned average signal (b) Class 1 Figure 15. Joint alignment and averaging of the ECG200 dataset. Shaded area corresponds to σ. 0 200 400 600 800 1000 2 Misaligned signals 0 200 400 600 800 1000 Misaligned average signal 0 200 400 600 800 1000 2 Aligned signals 0 200 400 600 800 1000 Aligned average signal (a) Class 0 0 200 400 600 800 1000 Misaligned signals 0 200 400 600 800 1000 1 4 Misaligned average signal 0 200 400 600 800 1000 Aligned signals 0 200 400 600 800 1000 1 4 Aligned average signal (b) Class 1 0 200 400 600 800 1000 Misaligned signals 0 200 400 600 800 1000 Misaligned average signal 0 200 400 600 800 1000 Aligned signals 0 200 400 600 800 1000 Aligned average signal (c) Class 2 Figure 16. Joint alignment and averaging of the Star Light Curves dataset. Shaded area corresponds to σ. Regularization-free Diffeomorphic Temporal Alignment Nets 0 10 20 30 40 50 60 2 Misaligned signals 0 10 20 30 40 50 60 Misaligned average signal 0 10 20 30 40 50 60 2 Aligned signals 0 10 20 30 40 50 60 Aligned average signal (a) Class 0 0 10 20 30 40 50 60 Misaligned signals 0 10 20 30 40 50 60 Misaligned average signal 0 10 20 30 40 50 60 2 2 Aligned signals 0 10 20 30 40 50 60 Aligned average signal (b) Class 1 0 10 20 30 40 50 60 Misaligned signals 0 10 20 30 40 50 60 Misaligned average signal 0 10 20 30 40 50 60 Aligned signals 0 10 20 30 40 50 60 Aligned average signal (c) Class 2 0 10 20 30 40 50 60 Misaligned signals 0 10 20 30 40 50 60 Misaligned average signal 0 10 20 30 40 50 60 Aligned signals 0 10 20 30 40 50 60 Aligned average signal (d) Class 3 0 10 20 30 40 50 60 2 Misaligned signals 0 10 20 30 40 50 60 Misaligned average signal 0 10 20 30 40 50 60 2 2 Aligned signals 0 10 20 30 40 50 60 Aligned average signal (e) Class 4 0 10 20 30 40 50 60 2 Misaligned signals 0 10 20 30 40 50 60 Misaligned average signal 0 10 20 30 40 50 60 Aligned signals 0 10 20 30 40 50 60 Aligned average signal (f) Class 5 Figure 17. Joint alignment and averaging of the Synthetic Control dataset. Shaded area corresponds to σ. Regularization-free Diffeomorphic Temporal Alignment Nets 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (a) Class 0 0 20 40 60 80 100 120 4 Misaligned signals 0 20 40 60 80 100 120 4 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (b) Class 1 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 3 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 3 Aligned average signal (c) Class 2 0 20 40 60 80 100 120 4 Misaligned signals 0 20 40 60 80 100 120 3 Misaligned average signal 0 20 40 60 80 100 120 4 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (d) Class 3 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (e) Class 4 0 20 40 60 80 100 120 4 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 4 Aligned signals 0 20 40 60 80 100 120 3 Aligned average signal (f) Class 5 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (g) Class 6 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (h) Class 7 0 20 40 60 80 100 120 4 Misaligned signals 0 20 40 60 80 100 120 2 Misaligned average signal 0 20 40 60 80 100 120 4 4 Aligned signals 0 20 40 60 80 100 120 2 Aligned average signal (i) Class 8 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (j) Class 9 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (k) Class 10 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (l) Class 11 0 20 40 60 80 100 120 4 6 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 4 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (m) Class 12 0 20 40 60 80 100 120 Misaligned signals 0 20 40 60 80 100 120 Misaligned average signal 0 20 40 60 80 100 120 Aligned signals 0 20 40 60 80 100 120 Aligned average signal (n) Class 13 Figure 18. Joint alignment and averaging of the Faces UCR dataset. Shaded area corresponds to σ. Regularization-free Diffeomorphic Temporal Alignment Nets G. Barycenters Comparison (a) CBF - Class 1 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Soft DTW( =0.1) 0 20 40 60 80 100 120 Soft DTW( =1) 0 20 40 60 80 100 120 Soft DTW( =10) 0 20 40 60 80 100 120 DTAN-No Regularization 0 20 40 60 80 100 120 DTAN-Weak Prior 0 20 40 60 80 100 120 DTAN-Strong Prior 0 20 40 60 80 100 120 0 20 40 60 80 100 120 DTAN ICAE triplet (b) Class 2 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Soft DTW( =0.1) 0 20 40 60 80 100 120 Soft DTW( =1) 0 20 40 60 80 100 120 Soft DTW( =10) 0 20 40 60 80 100 120 DTAN-No Regularization 0 20 40 60 80 100 120 DTAN-Weak Prior 0 20 40 60 80 100 120 DTAN-Strong Prior 0 20 40 60 80 100 120 0 20 40 60 80 100 120 DTAN ICAE triplet (c) Class 3 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Soft DTW( =0.1) 0 20 40 60 80 100 120 Soft DTW( =1) 0 20 40 60 80 100 120 Soft DTW( =10) 0 20 40 60 80 100 120 DTAN-No Regularization 0 20 40 60 80 100 120 DTAN-Weak Prior 0 20 40 60 80 100 120 DTAN-Strong Prior 0 20 40 60 80 100 120 0 20 40 60 80 100 120 DTAN ICAE triplet (d) The effect of regularization hyperparameters (HP) on barycenter computation. 10 samples of the CBF dataset and their mean (blue). Regularization-free Diffeomorphic Temporal Alignment Nets (a) Synthetic Control - Class 1 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Soft DTW( =0.1) 0 10 20 30 40 50 60 Soft DTW( =1) 0 10 20 30 40 50 60 Soft DTW( =10) 0 10 20 30 40 50 60 DTAN-No Regularization 0 10 20 30 40 50 60 DTAN-Weak Prior 0 10 20 30 40 50 60 DTAN-Strong Prior 0 10 20 30 40 50 60 0 10 20 30 40 50 60 DTAN-ICAE-triplet (b) Class 2 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Soft DTW( =0.1) 0 10 20 30 40 50 60 Soft DTW( =1) 0 10 20 30 40 50 60 Soft DTW( =10) 0 10 20 30 40 50 60 DTAN-No Regularization 0 10 20 30 40 50 60 DTAN-Weak Prior 0 10 20 30 40 50 60 DTAN-Strong Prior 0 10 20 30 40 50 60 0 10 20 30 40 50 60 DTAN-ICAE-triplet (c) Class 3 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Soft DTW( =0.1) 0 10 20 30 40 50 60 Soft DTW( =1) 0 10 20 30 40 50 60 Soft DTW( =10) 0 10 20 30 40 50 60 DTAN-No Regularization 0 10 20 30 40 50 60 DTAN-Weak Prior 0 10 20 30 40 50 60 DTAN-Strong Prior 0 10 20 30 40 50 60 0 10 20 30 40 50 60 DTAN-ICAE-triplet Regularization-free Diffeomorphic Temporal Alignment Nets (d) Synthetic Control - Class 4 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Soft DTW( =0.1) 0 10 20 30 40 50 60 Soft DTW( =1) 0 10 20 30 40 50 60 Soft DTW( =10) 0 10 20 30 40 50 60 DTAN-No Regularization 0 10 20 30 40 50 60 DTAN-Weak Prior 0 10 20 30 40 50 60 DTAN-Strong Prior 0 10 20 30 40 50 60 0 10 20 30 40 50 60 DTAN-ICAE-triplet (e) Class 5 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Soft DTW( =0.1) 0 10 20 30 40 50 60 Soft DTW( =1) 0 10 20 30 40 50 60 Soft DTW( =10) 0 10 20 30 40 50 60 DTAN-No Regularization 0 10 20 30 40 50 60 DTAN-Weak Prior 0 10 20 30 40 50 60 DTAN-Strong Prior 0 10 20 30 40 50 60 0 10 20 30 40 50 60 DTAN-ICAE-triplet (f) Class 6 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Soft DTW( =0.1) 0 10 20 30 40 50 60 Soft DTW( =1) 0 10 20 30 40 50 60 Soft DTW( =10) 0 10 20 30 40 50 60 DTAN-No Regularization 0 10 20 30 40 50 60 DTAN-Weak Prior 0 10 20 30 40 50 60 DTAN-Strong Prior 0 10 20 30 40 50 60 0 10 20 30 40 50 60 DTAN-ICAE-triplet (g) The effect of regularization hyperparameters (HP) on barycenter computation. 10 samples of the Synthetic Control dataset and their mean (blue). Regularization-free Diffeomorphic Temporal Alignment Nets (a) ECG200 - Class 1 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 Soft DTW( =0.1) 0 20 40 60 80 Soft DTW( =1) 0 20 40 60 80 Soft DTW( =10) 0 20 40 60 80 DTAN-No Regularization 0 20 40 60 80 DTAN-Weak Prior 0 20 40 60 80 DTAN-Strong Prior 0 20 40 60 80 0 20 40 60 80 DTAN-ICAE-triplet (b) Class 2 0 20 40 60 80 4 Euclidean 0 20 40 60 80 0 20 40 60 80 4 Soft DTW( =0.1) 0 20 40 60 80 4 Soft DTW( =1) 0 20 40 60 80 4 Soft DTW( =10) 0 20 40 60 80 4 DTAN-No Regularization 0 20 40 60 80 4 DTAN-Weak Prior 0 20 40 60 80 4 DTAN-Strong Prior 0 20 40 60 80 4 DTAN-ICAE 0 20 40 60 80 4 DTAN-ICAE-triplet (c) ECGFive Days - Class 2 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Soft DTW( =0.1) 0 20 40 60 80 100 120 Soft DTW( =1) 0 20 40 60 80 100 120 Soft DTW( =10) 0 20 40 60 80 100 120 DTAN-No Regularization 0 20 40 60 80 100 120 DTAN-Weak Reg. 0 20 40 60 80 100 120 DTAN-Strong Reg. 0 20 40 60 80 100 120 0 20 40 60 80 100 120 DTAN-ICAE-triplet Regularization-free Diffeomorphic Temporal Alignment Nets H. UCR time series classification archive details ID Type Name Train Test Class Length 1 Image Adiac 390 391 37 176 2 Image Arrow Head 36 175 3 251 3 Spectro Beef 30 30 5 470 4 Image Beetle Fly 20 20 2 512 5 Image Bird Chicken 20 20 2 512 6 Sensor Car 60 60 4 577 7 Simulated CBF 30 900 3 128 8 Sensor Chlorine Concentration 467 3840 3 166 9 Sensor Cin CECGTorso 40 1380 4 1639 10 Spectro Coffee 28 28 2 286 11 Device Computers 250 250 2 720 12 Motion Cricket X 390 390 12 300 13 Motion Cricket Y 390 390 12 300 14 Motion Cricket Z 390 390 12 300 15 Image Diatom Size Reduction 16 306 4 345 16 Image Distal Phalanx Outline Age Group 400 139 3 80 17 Image Distal Phalanx Outline Correct 600 276 2 80 18 Image Distal Phalanx TW 400 139 6 80 19 Sensor Earthquakes 322 139 2 512 20 ECG ECG200 100 100 2 96 21 ECG ECG5000 500 4500 5 140 22 ECG ECGFive Days 23 861 2 136 23 Device Electric Devices 8926 7711 7 96 24 Image Face All 560 1690 14 131 25 Image Face Four 24 88 4 350 26 Image Faces UCR 200 2050 14 131 27 Image Fifty Words 450 455 50 270 28 Image Fish 175 175 7 463 29 Sensor Ford A 3601 1320 2 500 30 Sensor Ford B 3636 810 2 500 31 Motion Gun Point 50 150 2 150 32 Spectro Ham 109 105 2 431 33 Image Hand Outlines 1000 370 2 2709 34 Motion Haptics 155 308 5 1092 35 Image Herring 64 64 2 512 36 Motion Inline Skate 100 550 7 1882 37 Sensor Insect Wingbeat Sound 220 1980 11 256 38 Sensor Italy Power Demand 67 1029 2 24 39 Device Large Kitchen Appliances 375 375 3 720 40 Sensor Lightning2 60 61 2 637 41 Sensor Lightning7 70 73 7 319 42 Simulated Mallat 55 2345 8 1024 43 Spectro Meat 60 60 3 448 44 Image Medical Images 381 760 10 99 45 Image Middle Phalanx Outline Age Group 400 154 3 80 46 Image Middle Phalanx Outline Correct 600 291 2 80 47 Image Middle Phalanx TW 399 154 6 80 48 Sensor Mote Strain 20 1252 2 84 49 ECG Non Invasive Fetal ECGThorax1 1800 1965 42 750 50 ECG Non Invasive Fetal ECGThorax2 1800 1965 42 750 51 Spectro Olive Oil 30 30 4 570 52 Image OSULeaf 200 242 6 427 53 Image Phalanges Outlines Correct 1800 858 2 80 54 Sensor Phoneme 214 1896 39 1024 55 Sensor Plane 105 105 7 144 56 Image Proximal Phalanx Outline Age Group 400 205 3 80 57 Image Proximal Phalanx Outline Correct 600 291 2 80 58 Image Proximal Phalanx TW 400 205 6 80 59 Device Refrigeration Devices 375 375 3 720 60 Device Screen Type 375 375 3 720 61 Simulated Shapelet Sim 20 180 2 500 Regularization-free Diffeomorphic Temporal Alignment Nets ID Type Name Train Test Class Length 62 Image Shapes All 600 600 60 512 63 Device Small Kitchen Appliances 375 375 3 720 64 Sensor Sony AIBORobot Surface1 20 601 2 70 65 Sensor Sony AIBORobot Surface2 27 953 2 65 66 Sensor Star Light Curves 1000 8236 3 1024 67 Spectro Strawberry 613 370 2 235 68 Image Swedish Leaf 500 625 15 128 69 Image Symbols 25 995 6 398 70 Simulated Synthetic Control 300 300 6 60 71 Motion Toe Segmentation1 40 228 2 277 72 Motion Toe Segmentation2 36 130 2 343 73 Sensor Trace 100 100 4 275 74 ECG Two Lead ECG 23 1139 2 82 75 Simulated Two Patterns 1000 4000 4 128 76 Motion UWave Gesture Library All 896 3582 8 945 77 Motion UWave Gesture Library X 896 3582 8 315 78 Motion UWave Gesture Library Y 896 3582 8 315 79 Motion UWave Gesture Library Z 896 3582 8 315 80 Sensor Wafer 1000 6164 2 152 81 Spectro Wine 57 54 2 234 82 Image Word Synonyms 267 638 25 270 83 Motion Worms 181 77 5 900 84 Motion Worms Two Class 181 77 2 900 85 Image Yoga 300 3000 2 426 86 Device ACSF1 100 100 10 1460 87 Sensor All Gesture Wiimote X 300 700 10 Vary 88 Sensor All Gesture Wiimote Y 300 700 10 Vary 89 Sensor All Gesture Wiimote Z 300 700 10 Vary 90 Simulated BME 30 150 3 128 91 Traffic Chinatown 20 343 2 24 92 Image Crop 7200 16800 24 46 93 Sensor Dodger Loop Day 78 80 7 288 94 Sensor Dodger Loop Game 20 138 2 288 95 Sensor Dodger Loop Weekend 20 138 2 288 96 EOG EOGHorizontal Signal 362 362 12 1250 97 EOG EOGVertical Signal 362 362 12 1250 98 Spectro Ethanol Level 504 500 4 1751 99 Sensor Freezer Regular Train 150 2850 2 301 100 Sensor Freezer Small Train 28 2850 2 301 101 HRM Fungi 18 186 18 201 102 Trajectory Gesture Mid Air D1 208 130 26 Vary 103 Trajectory Gesture Mid Air D2 208 130 26 Vary 104 Trajectory Gesture Mid Air D3 208 130 26 Vary 105 Sensor Gesture Pebble Z1 132 172 6 Vary 106 Sensor Gesture Pebble Z2 146 158 6 Vary 107 Motion Gun Point Age Span 135 316 2 150 108 Motion Gun Point Male Versus Female 135 316 2 150 109 Motion Gun Point Old Versus Young 136 315 2 150 110 Device House Twenty 40 119 2 2000 111 EPG Insect EPGRegular Train 62 249 3 601 112 EPG Insect EPGSmall Train 17 249 3 601 113 Traffic Melbourne Pedestrian 1194 2439 10 24 114 Image Mixed Shapes Regular Train 500 2425 5 1024 115 Image Mixed Shapes Small Train 100 2425 5 1024 116 Sensor Pickup Gesture Wiimote Z 50 50 10 Vary 117 Hemodynamics Pig Airway Pressure 104 208 52 2000 118 Hemodynamics Pig Art Pressure 104 208 52 2000 119 Hemodynamics Pig CVP 104 208 52 2000 120 Device PLAID 537 537 11 Vary 121 Power Power Cons 180 180 2 144 122 Spectrum Rock 20 50 4 2844 123 Spectrum Semg Hand Gender Ch2 300 600 2 1500 124 Spectrum Semg Hand Movement Ch2 450 450 6 1500 Regularization-free Diffeomorphic Temporal Alignment Nets ID Type Name Train Test Class Length 125 Spectrum Semg Hand Subject Ch2 450 450 5 1500 126 Sensor Shake Gesture Wiimote Z 50 50 10 Vary 127 Simulated Smooth Subspace 150 150 3 15 128 Simulated UMD 36 144 3 150 Regularization-free Diffeomorphic Temporal Alignment Nets I. UCR Nearest Centroid Classification (NCC) Results I.1. Comparison with results reported in the literature (84 datasets (Chen et al., 2015)) Table 6: NCC results for 84 datasets of the UCR archive. Comparison between our LICAE and LICAE triplet (titled Ltriplet due to space limitations; median and best results across 5 runs) and various joint alignment and barycenter computation methods in terms of NCC accuracy. Euclidean (Euc.), DBA, Soft DTW (SDTW), and Soft DTW Divergence (SDTW-div) results are taken from (Blondel et al., 2021), Res Net-TW from (Huang et al., 2021), DTANlibcpab from (Shapira Weber et al., 2019) and DTANDIFW from (Martinez et al., 2022). Dataset Euc. DTW SDTW SDTW div DTAN Res Net DTAN Median Best libcpab TW DIFW LICAE Ltriplet LICAE Ltriplet adiac 0.550 0.471 0.675 0.685 0.696 0.698 0.719 0.696 0.752 0.703 0.775 arrowhead 0.611 0.509 0.514 0.577 0.749 0.754 0.726 0.737 0.783 0.754 0.846 beef 0.533 0.433 0.467 0.367 0.633 0.633 0.700 0.567 0.733 0.600 0.733 beetlefly 0.850 0.800 0.700 0.700 0.800 0.800 0.950 0.700 0.600 0.850 0.650 birdchicken 0.550 0.600 0.650 0.600 0.800 0.950 0.950 0.600 0.800 0.750 0.900 car 0.617 0.617 0.700 0.733 0.817 1.000 0.989 0.783 0.833 0.833 0.883 cbf 0.763 0.969 0.971 0.971 0.914 0.850 0.982 0.961 0.847 0.993 0.857 chlorineconcentration 0.333 0.325 0.352 0.322 0.333 0.352 0.397 0.324 0.779 0.325 0.812 cincecgtorso 0.385 0.403 0.719 0.704 0.616 0.543 0.741 0.445 0.521 0.514 0.550 coffee 0.964 0.964 0.964 0.964 1.000 0.964 1.000 0.964 1.000 0.964 1.000 computers 0.416 0.632 0.516 0.568 0.592 0.676 0.616 0.468 0.448 0.480 0.520 cricketx 0.239 0.577 0.569 0.567 0.423 0.341 0.428 0.474 0.482 0.526 0.518 crickety 0.349 0.526 0.556 0.549 0.541 0.415 0.513 0.562 0.600 0.572 0.641 cricketz 0.305 0.600 0.610 0.600 0.421 0.333 0.451 0.518 0.474 0.544 0.556 diatomsizereduction 0.958 0.951 0.967 0.964 0.971 0.974 0.984 0.971 0.974 0.987 0.977 distalphalanxoutlineagegroup 0.818 0.840 0.845 0.848 0.848 0.863 0.748 0.719 0.712 0.727 0.727 distalphalanxoutlinecorrect 0.472 0.482 0.480 0.473 0.472 0.505 0.775 0.493 0.775 0.518 0.793 distalphalanxtw 0.748 0.757 0.745 0.745 0.780 0.797 0.683 0.626 0.619 0.647 0.633 earthquakes 0.755 0.581 0.823 0.652 0.773 0.973 0.820 0.698 0.683 0.719 0.698 ecg200 0.750 0.750 0.720 0.730 0.790 0.795 0.914 0.790 0.900 0.830 0.920 ecg5000 0.860 0.845 0.867 0.860 0.891 0.800 0.999 0.854 0.907 0.855 0.912 ecgfivedays 0.690 0.653 0.806 0.834 0.978 0.932 0.993 0.859 0.791 0.922 0.947 electricdevices 0.483 0.536 0.571 0.616 0.535 0.519 0.574 0.521 0.427 0.549 0.508 faceall 0.492 0.807 0.816 0.886 0.805 0.841 0.856 0.738 0.744 0.782 0.825 facefour 0.841 0.830 0.864 0.898 0.830 0.855 0.920 0.773 0.830 0.841 0.864 facesucr 0.539 0.792 0.890 0.911 0.857 0.857 0.801 0.808 0.808 0.896 0.886 fiftywords 0.516 0.598 0.763 0.780 0.653 0.516 0.631 0.609 0.587 0.611 0.622 fish 0.560 0.657 0.811 0.840 0.903 0.903 0.914 0.829 0.891 0.891 0.909 forda 0.496 0.556 0.556 0.524 0.605 0.568 0.652 0.574 0.669 0.604 0.855 fordb 0.500 0.607 0.476 0.559 0.580 0.566 0.546 0.499 0.515 0.531 0.623 gunpoint 0.753 0.680 0.820 0.813 0.880 0.807 0.847 0.913 0.967 0.933 0.973 ham 0.762 0.733 0.714 0.752 0.790 0.762 0.810 0.790 0.752 0.800 0.790 handoutlines 0.818 0.792 0.824 nan 0.850 0.835 0.908 0.773 0.938 0.800 0.949 haptics 0.393 0.357 0.461 0.461 0.458 0.464 0.487 0.419 0.377 0.435 0.403 herring 0.547 0.609 0.641 0.641 0.703 0.766 0.781 0.625 0.609 0.672 0.656 inlineskate 0.193 0.227 0.234 0.264 0.260 0.244 0.287 0.205 0.233 0.242 0.271 insectwingbeatsound 0.601 0.298 0.582 0.586 0.587 0.571 0.607 0.533 0.517 0.554 0.536 italypowerdemand 0.918 0.742 0.881 0.905 0.962 0.965 0.967 0.939 0.955 0.950 0.964 largekitchenappliances 0.440 0.715 0.720 0.736 0.483 0.501 0.517 0.392 0.408 0.421 0.435 lightning2 0.688 0.623 0.672 0.721 0.721 0.754 0.738 0.557 0.672 0.623 0.689 lightning7 0.589 0.726 0.781 0.836 0.712 0.685 0.726 0.562 0.562 0.562 0.589 mallat 0.967 0.949 0.957 0.948 0.969 0.967 0.974 0.957 0.957 0.965 0.959 meat 0.933 0.933 0.850 0.850 0.933 0.933 0.933 0.933 0.883 0.933 0.917 medicalimages 0.385 0.442 0.404 0.409 0.468 0.474 0.483 0.479 0.563 0.521 0.613 middlephalanxoutlineagegroup 0.733 0.725 0.728 0.728 0.738 0.752 0.636 0.604 0.578 0.610 0.604 middlephalanxoutlinecorrect 0.552 0.485 0.522 0.528 0.543 0.532 0.698 0.656 0.801 0.670 0.835 middlephalanxtw 0.592 0.566 0.582 0.582 0.596 0.634 0.539 0.487 0.552 0.506 0.552 motestrain 0.861 0.824 0.904 0.902 0.904 0.913 0.875 0.843 0.855 0.857 0.890 noninvasivefetalecgthorax1 0.770 0.701 0.816 0.823 0.853 0.839 0.874 0.844 0.926 0.855 0.934 noninvasivefetalecgthorax2 0.802 0.763 0.872 0.877 0.905 0.839 0.917 0.889 0.949 0.891 0.950 oliveoil 0.867 0.767 0.833 0.867 0.867 0.867 0.900 0.833 0.700 0.867 0.800 Regularization-free Diffeomorphic Temporal Alignment Nets osuleaf 0.360 0.459 0.521 0.512 0.463 0.459 0.933 0.409 0.426 0.426 0.512 phalangesoutlinescorrect 0.626 0.636 0.637 0.645 0.642 0.663 0.676 0.652 0.837 0.656 0.845 phoneme 0.079 0.177 0.201 0.206 0.102 0.117 0.101 0.088 0.083 0.093 0.090 plane 0.962 0.990 0.990 0.990 1.000 1.000 1.000 0.981 0.981 1.000 1.000 proximalphalanxoutlineagegroup 0.820 0.829 0.844 0.844 0.854 0.873 0.873 0.849 0.844 0.854 0.844 proximalphalanxoutlinecorrect 0.646 0.650 0.650 0.650 0.643 0.687 0.725 0.643 0.911 0.643 0.928 proximalphalanxtw 0.708 0.735 0.812 0.815 0.818 0.823 0.790 0.756 0.766 0.766 0.780 refrigerationdevices 0.355 0.579 0.581 0.552 0.467 0.483 0.485 0.331 0.339 0.339 0.365 screentype 0.443 0.381 0.373 0.400 0.445 0.469 0.461 0.443 0.408 0.472 0.459 shapeletsim 0.500 0.617 0.733 0.728 0.539 0.589 0.572 0.461 0.483 0.539 0.533 shapesall 0.513 0.622 0.655 0.687 0.628 0.682 0.643 0.565 0.577 0.578 0.587 smallkitchenappliances 0.419 0.645 0.680 0.688 0.621 0.560 0.592 0.429 0.400 0.435 0.419 sonyaiborobotsurface1 0.812 0.829 0.827 0.829 0.894 0.860 0.892 0.699 0.725 0.734 0.742 sonyaiborobotsurface2 0.793 0.766 0.798 0.765 0.811 0.830 0.875 0.790 0.826 0.817 0.831 strawberry 0.669 0.612 0.656 0.688 0.843 0.786 0.892 0.654 0.976 0.676 0.981 swedishleaf 0.702 0.704 0.794 0.811 0.806 0.837 0.858 0.798 0.827 0.843 0.862 symbols 0.864 0.958 0.951 0.956 0.857 0.907 0.912 0.865 0.860 0.885 0.882 syntheticcontrol 0.917 0.983 0.980 0.987 0.950 0.950 0.980 0.970 0.983 0.990 0.993 toesegmentation1 0.575 0.627 0.733 0.711 0.640 0.654 0.794 0.583 0.583 0.618 0.610 toesegmentation2 0.546 0.869 0.862 0.854 0.754 0.746 0.785 0.569 0.592 0.669 0.700 trace 0.580 0.980 0.980 0.970 0.780 0.800 0.980 0.780 1.000 0.960 1.000 twoleadecg 0.555 0.762 0.780 0.831 0.956 0.955 0.989 0.908 0.985 0.942 0.994 twopatterns 0.465 0.984 0.987 0.982 0.556 0.701 0.716 0.988 0.999 1.000 1.000 uwavegesturelibraryall 0.850 0.835 0.893 0.909 0.921 0.912 0.944 0.895 0.887 0.903 0.913 uwavegesturelibraryx 0.631 0.700 0.680 0.697 0.681 0.722 0.710 0.685 0.680 0.694 0.697 uwavegesturelibraryy 0.548 0.532 0.613 0.621 0.612 0.617 0.641 0.630 0.628 0.643 0.642 uwavegesturelibraryz 0.537 0.606 0.633 0.645 0.642 0.646 0.652 0.627 0.617 0.634 0.631 wafer 0.654 0.319 0.688 0.689 0.989 0.983 0.986 0.976 0.993 0.978 0.997 wine 0.556 0.537 0.574 0.556 0.574 0.593 0.833 0.556 0.778 0.556 0.815 wordsynonyms 0.271 0.343 0.522 0.517 0.475 0.502 0.475 0.433 0.414 0.458 0.425 worms 0.215 0.403 0.436 0.448 0.260 0.343 0.338 0.351 0.338 0.351 0.351 wormstwoclass 0.541 0.630 0.680 0.707 0.619 0.619 0.649 0.494 0.558 0.532 0.571 yoga 0.497 0.600 0.571 0.617 0.632 0.697 0.681 0.620 0.825 0.628 0.838 Regularization-free Diffeomorphic Temporal Alignment Nets I.2. Comparison between objective functions (128 datasets (Dau et al., 2019)) Table 7: NCC results for 128 datasets of the UCR archive (including ones containing variable length time series). Comparison between our LICAE and LICAE triplet (titled Ltriplet due to space limitations) and the standard Within-Class Sum of Squares (WCSS) w/o regularization prior. Otherwise, all other parameters, including floc, are identical. Prior values are set to λΣ = 0.001 and λsmooth = 0.1. Median and best results across 5 runs. N/A in the results refers to datasets containing signals of variable length . Dataset Median Best WCSS WCSS+reg LICAE Ltriplet WCSS WCSS+reg LICAE Ltriplet acsf1 0.410 0.640 0.500 0.810 0.560 0.700 0.580 0.830 adiac 0.660 0.550 0.696 0.752 0.668 0.550 0.703 0.775 allgesturewiimotex N/A N/A 0.250 0.197 N/A N/A 0.307 0.209 allgesturewiimotey N/A N/A 0.390 0.193 N/A N/A 0.501 0.261 allgesturewiimotez N/A N/A 0.096 0.126 N/A N/A 0.129 0.140 arrowhead 0.640 0.611 0.737 0.783 0.691 0.611 0.754 0.846 beef 0.500 0.533 0.567 0.733 0.567 0.533 0.600 0.733 beetlefly 0.700 0.850 0.700 0.600 0.800 0.850 0.850 0.650 birdchicken 0.750 0.550 0.600 0.800 0.950 0.550 0.750 0.900 bme 0.853 0.647 0.907 0.933 0.880 0.647 0.967 0.980 car 0.683 0.617 0.783 0.833 0.800 0.617 0.833 0.883 cbf 0.782 0.762 0.961 0.847 0.990 0.762 0.993 0.857 chinatown 0.959 0.959 0.980 0.980 0.965 0.959 0.983 0.983 chlorineconcentration 0.318 0.331 0.324 0.779 0.323 0.333 0.325 0.812 cincecgtorso 0.324 0.407 0.445 0.521 0.372 0.408 0.514 0.550 coffee 1.000 0.964 0.964 1.000 1.000 0.964 0.964 1.000 computers 0.560 0.412 0.468 0.448 0.640 0.416 0.480 0.520 cricketx 0.156 0.241 0.474 0.482 0.190 0.241 0.526 0.518 crickety 0.174 0.349 0.562 0.600 0.251 0.354 0.572 0.641 cricketz 0.195 0.303 0.518 0.474 0.233 0.305 0.544 0.556 crop 0.492 0.472 0.549 0.657 0.526 0.472 0.556 0.658 diatomsizereduction 0.954 0.958 0.971 0.974 0.958 0.958 0.987 0.977 distalphalanxoutlineagegroup 0.719 0.698 0.719 0.712 0.727 0.698 0.727 0.727 distalphalanxoutlinecorrect 0.645 0.688 0.493 0.775 0.663 0.688 0.518 0.793 distalphalanxtw 0.612 0.576 0.626 0.619 0.633 0.583 0.647 0.633 dodgerloopday 0.450 0.463 0.475 0.438 0.450 0.463 0.487 0.487 dodgerloopgame 0.812 0.812 0.812 0.826 0.812 0.812 0.841 0.841 dodgerloopweekend 0.986 0.986 0.986 0.986 0.986 0.986 0.986 0.986 earthquakes 0.719 0.669 0.698 0.683 0.763 0.683 0.719 0.698 ecg200 0.880 0.750 0.790 0.900 0.910 0.750 0.830 0.920 ecg5000 0.597 0.860 0.854 0.907 0.614 0.861 0.855 0.912 ecgfivedays 0.886 0.747 0.859 0.791 0.908 0.785 0.922 0.947 electricdevices 0.342 0.487 0.521 0.427 0.371 0.487 0.549 0.508 eoghorizontalsignal 0.213 0.359 0.425 0.434 0.235 0.359 0.428 0.478 eogverticalsignal 0.243 0.279 0.304 0.309 0.246 0.279 0.337 0.365 ethanollevel 0.262 0.284 0.314 0.834 0.328 0.284 0.316 0.842 faceall 0.799 0.495 0.738 0.744 0.876 0.496 0.782 0.825 facefour 0.761 0.830 0.773 0.830 0.784 0.841 0.841 0.864 facesucr 0.737 0.548 0.808 0.808 0.856 0.549 0.896 0.886 fiftywords 0.035 0.516 0.609 0.587 0.116 0.516 0.611 0.622 fish 0.680 0.560 0.829 0.891 0.697 0.566 0.891 0.909 forda 0.515 0.501 0.574 0.669 0.527 0.504 0.604 0.855 fordb 0.486 0.502 0.499 0.515 0.516 0.504 0.531 0.623 freezerregulartrain 0.793 0.769 0.768 0.993 0.942 0.769 0.776 0.995 freezersmalltrain 0.769 0.763 0.791 0.806 0.782 0.763 0.815 0.881 fungi 0.823 0.823 0.823 0.823 0.828 0.823 0.828 0.828 gesturemidaird1 N/A N/A 0.569 0.608 N/A N/A 0.600 0.631 gesturemidaird2 N/A N/A 0.562 0.531 N/A N/A 0.585 0.554 gesturemidaird3 N/A N/A 0.354 0.354 N/A N/A 0.385 0.400 gesturepebblez1 N/A N/A 0.192 0.192 N/A N/A 0.203 0.203 gesturepebblez2 N/A N/A 0.234 0.228 N/A N/A 0.297 0.285 gunpoint 0.933 0.753 0.913 0.967 0.967 0.753 0.933 0.973 gunpointagespan 0.892 0.854 0.642 0.981 0.978 0.854 0.668 0.987 Regularization-free Diffeomorphic Temporal Alignment Nets Table 7: NCC results for 128 datasets of the UCR archive (including ones containing variable length time series). Comparison between our LICAE and LICAE triplet (titled Ltriplet due to space limitations) and the standard Within-Class Sum of Squares (WCSS) w/o regularization prior. Otherwise, all other parameters, including floc, are identical. Prior values are set to λΣ = 0.001 and λsmooth = 0.1. Median and best results across 5 runs. N/A in the results refers to datasets containing signals of variable length . Dataset Median Best WCSS WCSS+reg LICAE Ltriplet WCSS WCSS+reg LICAE Ltriplet gunpointmaleversusfemale 0.956 0.690 0.965 1.000 0.959 0.690 0.968 1.000 gunpointoldversusyoung 0.511 0.775 0.670 0.981 0.565 0.775 0.705 0.987 ham 0.667 0.762 0.790 0.752 0.752 0.762 0.800 0.790 handoutlines 0.778 0.819 0.773 0.938 0.819 0.819 0.800 0.949 haptics 0.364 0.399 0.419 0.377 0.396 0.399 0.435 0.403 herring 0.578 0.547 0.625 0.609 0.672 0.547 0.672 0.656 housetwenty 0.706 0.756 0.706 0.689 0.723 0.765 0.765 0.714 inlineskate 0.209 0.195 0.205 0.233 0.224 0.196 0.242 0.271 insectepgregulartrain 0.622 0.482 0.586 0.719 0.635 0.490 0.671 0.727 insectepgsmalltrain 0.618 0.586 0.683 0.651 0.663 0.586 0.747 0.695 insectwingbeatsound 0.318 0.604 0.533 0.517 0.364 0.605 0.554 0.536 italypowerdemand 0.934 0.920 0.939 0.955 0.948 0.920 0.950 0.964 largekitchenappliances 0.456 0.443 0.392 0.408 0.488 0.443 0.421 0.435 lightning2 0.689 0.672 0.557 0.672 0.705 0.689 0.623 0.689 lightning7 0.671 0.575 0.562 0.562 0.726 0.603 0.562 0.589 mallat 0.922 0.967 0.957 0.957 0.950 0.967 0.965 0.959 meat 0.917 0.933 0.933 0.883 0.950 0.933 0.933 0.917 medicalimages 0.261 0.386 0.479 0.563 0.284 0.387 0.521 0.613 melbournepedestrian 0.789 0.609 0.733 0.839 0.795 0.609 0.743 0.845 middlephalanxoutlineagegroup 0.591 0.571 0.604 0.578 0.597 0.571 0.610 0.604 middlephalanxoutlinecorrect 0.608 0.478 0.656 0.801 0.612 0.478 0.670 0.835 middlephalanxtw 0.448 0.442 0.487 0.552 0.500 0.442 0.506 0.552 mixedshapesregulartrain 0.791 0.731 0.839 0.845 0.805 0.731 0.843 0.851 mixedshapessmalltrain 0.729 0.729 0.779 0.802 0.773 0.729 0.800 0.812 motestrain 0.844 0.861 0.843 0.855 0.850 0.862 0.857 0.890 noninvasivefetalecgthorax1 0.737 0.770 0.844 0.926 0.749 0.770 0.855 0.934 noninvasivefetalecgthorax2 0.827 0.803 0.889 0.949 0.835 0.803 0.891 0.950 oliveoil 0.833 0.867 0.833 0.700 0.867 0.867 0.867 0.800 osuleaf 0.360 0.364 0.409 0.426 0.459 0.364 0.426 0.512 phalangesoutlinescorrect 0.613 0.626 0.652 0.837 0.628 0.626 0.656 0.845 phoneme 0.080 0.080 0.088 0.083 0.094 0.082 0.093 0.090 pickupgesturewiimotez N/A N/A 0.080 0.120 N/A N/A 0.100 0.120 pigairwaypressure 0.019 0.005 0.005 0.024 0.029 0.010 0.038 0.038 pigartpressure 0.154 0.096 0.197 0.159 0.173 0.096 0.231 0.212 pigcvp 0.053 0.038 0.053 0.048 0.072 0.038 0.053 0.048 plaid N/A N/A 0.069 0.019 N/A N/A 0.076 0.032 plane 1.000 0.962 0.981 0.981 1.000 0.962 1.000 1.000 powercons 0.783 0.861 0.889 0.928 0.867 0.861 0.911 0.944 proximalphalanxoutlineagegroup 0.839 0.820 0.849 0.844 0.844 0.820 0.854 0.844 proximalphalanxoutlinecorrect 0.643 0.646 0.643 0.911 0.643 0.646 0.643 0.928 proximalphalanxtw 0.741 0.698 0.756 0.766 0.756 0.698 0.766 0.780 refrigerationdevices 0.352 0.355 0.331 0.339 0.384 0.365 0.339 0.365 rock 0.660 0.620 0.540 0.780 0.680 0.620 0.620 0.860 screentype 0.395 0.443 0.443 0.408 0.397 0.445 0.472 0.459 semghandgenderch2 0.655 0.688 0.692 0.833 0.683 0.688 0.697 0.893 semghandmovementch2 0.380 0.393 0.393 0.369 0.407 0.411 0.400 0.467 semghandsubjectch2 0.556 0.560 0.560 0.638 0.624 0.567 0.580 0.664 shakegesturewiimotez N/A N/A 0.120 0.160 N/A N/A 0.160 0.200 shapeletsim 0.511 0.494 0.461 0.483 0.561 0.517 0.539 0.533 shapesall 0.457 0.513 0.565 0.577 0.470 0.513 0.578 0.587 smallkitchenappliances 0.467 0.437 0.429 0.400 0.547 0.456 0.435 0.419 smoothsubspace 0.713 0.707 0.713 0.700 0.873 0.707 0.747 0.807 sonyaiborobotsurface1 0.754 0.815 0.699 0.725 0.772 0.822 0.734 0.742 sonyaiborobotsurface2 0.801 0.792 0.790 0.826 0.812 0.793 0.817 0.831 starlightcurves 0.830 0.762 0.845 0.897 0.853 0.762 0.875 0.938 Regularization-free Diffeomorphic Temporal Alignment Nets Table 7: NCC results for 128 datasets of the UCR archive (including ones containing variable length time series). Comparison between our LICAE and LICAE triplet (titled Ltriplet due to space limitations) and the standard Within-Class Sum of Squares (WCSS) w/o regularization prior. Otherwise, all other parameters, including floc, are identical. Prior values are set to λΣ = 0.001 and λsmooth = 0.1. Median and best results across 5 runs. N/A in the results refers to datasets containing signals of variable length . Dataset Median Best WCSS WCSS+reg LICAE Ltriplet WCSS WCSS+reg LICAE Ltriplet strawberry 0.651 0.584 0.654 0.976 0.686 0.584 0.676 0.981 swedishleaf 0.770 0.704 0.798 0.827 0.786 0.706 0.843 0.862 symbols 0.836 0.865 0.865 0.860 0.848 0.865 0.885 0.882 syntheticcontrol 0.943 0.920 0.970 0.983 0.987 0.920 0.990 0.993 toesegmentation1 0.583 0.575 0.583 0.583 0.605 0.579 0.618 0.610 toesegmentation2 0.608 0.554 0.569 0.592 0.746 0.554 0.669 0.700 trace 0.760 0.580 0.780 1.000 0.930 0.580 0.960 1.000 twoleadecg 0.917 0.556 0.908 0.985 0.930 0.556 0.942 0.994 twopatterns 0.256 0.464 0.988 0.999 0.260 0.465 1.000 1.000 umd 0.778 0.542 0.806 0.910 0.861 0.542 0.979 0.958 uwavegesturelibraryall 0.723 0.850 0.895 0.887 0.762 0.850 0.903 0.913 uwavegesturelibraryx 0.595 0.631 0.685 0.680 0.599 0.631 0.694 0.697 uwavegesturelibraryy 0.536 0.549 0.630 0.628 0.574 0.549 0.643 0.642 uwavegesturelibraryz 0.475 0.538 0.627 0.617 0.545 0.538 0.634 0.631 wafer 0.769 0.655 0.976 0.993 0.801 0.655 0.978 0.997 wine 0.574 0.556 0.556 0.778 0.630 0.556 0.556 0.815 wordsynonyms 0.155 0.271 0.433 0.414 0.183 0.271 0.458 0.425 worms 0.273 0.208 0.351 0.338 0.351 0.208 0.351 0.351 wormstwoclass 0.468 0.532 0.494 0.558 0.558 0.532 0.532 0.571 yoga 0.664 0.497 0.620 0.825 0.683 0.497 0.628 0.838