Original: T AKING ROCKET ON AN EFFICIENCY MISSION : A DISTRIBUTED SOLUTION FOR FAST AND ACCURATE MULTIVARIATE TIME SERIES CLASSIFICATION
Revised: T AKING ROCKET ON AN EFFICIENCY MISSION : A DISTRIBUTED SOLUTION FOR FAST AND ACCURATE MULTIVARIATE TIME SERIES CLASSIFICATION

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Nowadays, with the rising number of sensors in sectors such as healthcare and industry, the problem of multivariate time series classiﬁcation (MTSC) is getting increasingly relevant and is a prime target for machine and deep learning solutions.
Revised: Nowadays, with the rising number of sensors in sectors such as healthcare and industry, the problem of multivariate time series classiﬁcation (MTSC) is getting increasingly relevant and is a prime target for machine and deep learning solutions.

Original: Their expanding adoption in real-world environments is causing a shift in focus from the pursuit of ever higher prediction accuracy with complex models towards practical, deployable solutions that balance accuracy and parameters such as prediction speed.
Revised: Their expanding adoption in real-world environments is causing a shift in focus from the pursuit of ever higher prediction accuracy with complex models towards practical, deployable solutions that balance accuracy and parameters such as prediction speed.

Original: An MTSC solution that has attracted attention recently is
Revised: An MTSC solution that has attracted attention recently is

Original: ROCKET, based on random convolutional kernels, both because of its very fast training process and its state-of-the-art accuracy.
Revised: ROCKET, based on random convolutional kernels, both because of its very fast training process and its state-of-the-art accuracy.

Original: However, the large number of features it utilizes may be detrimental to inference time.
Revised: However, the large number of features it utilizes may be detrimental to inference time.

Original: Examining its theoretical background and limitations enables us to address potential drawbacks and present
Revised: Examining its theoretical background and limitations enables us to address potential drawbacks and present

Original: LightWaveS: a distributed solution for accurate MTSC, which is fast both during training and inference.
Revised: LightWaveS: a distributed solution for accurate MTSC, which is fast both during training and inference.

Original: Speciﬁcally, utilizing a wavelet scattering transformation of the time series and distributed feature selection, we manage to create a solution which employs just 2,5% of the ROCKET features, while achieving accuracy comparable to recent deep learning solutions.
Revised: Speciﬁcally, utilizing a wavelet scattering transformation of the time series and distributed feature selection, we manage to create a solution which employs just 2,5% of the ROCKET features, while achieving accuracy comparable to recent deep learning solutions.

Original: LightWaveS also scales well with more nodes and large numbers of channels.
Revised: LightWaveS also scales well with more nodes and large numbers of channels.

Original: In addition, it can give interpretability into the nature of an MTSC problem and allows for tuning based on expert opinion.
Revised: In addition, it can signiﬁcantly reduce the input size and also provide insight to an MTSC problem by keeping only the most useful channels.

Original: We present three versions of our algorithm and their results on training time, accuracy, inference speedup and scalability.
Revised: We present three versions of our algorithm and their results on training time, accuracy, inference speedup and scalability.

Original: We show that we achieve speedup ranging from 8x to 30x compared to ROCKET during inference on an edge device, on datasets with comparable accuracy.
Revised: We show that we achieve speedup ranging from 9x to 65x compared to ROCKET during inference on an edge device, on datasets with comparable accuracy.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Time series classiﬁcation is the task of characterizing a series of values observed in sequential moments in time as belonging to one of two or more categories, or classes.
Revised: Time series classiﬁcation is the task of characterizing a series of values observed in sequential moments in time as belonging to one of two or more categories, or classes.

Original: There has been extensive work on univariate time series classiﬁcation with machine and deep learning methods, as observed in surveys such as Bagnall et al.
Revised: There has been extensive work on univariate time series classiﬁcation with machine and deep learning methods, as observed in surveys such as Bagnall et al.

Original: In practice, as noted in Ruiz et al.
Revised: In practice, as noted in Ruiz et al.

Original: (2021), many problems are described by more than one channels of information, turning the task into multivariate time series classiﬁcation (MTSC).
Revised: (2021), many problems are described by more than one channels of information, turning the task into multivariate time series classiﬁcation (MTSC).

Original: Some recent contributing factors have been the development of smaller and cheaper sensors for the measurement of various quantities, the general advancement of
Revised: Some recent contributing factors have been the development of smaller and cheaper sensors for the measurement of various quantities, the general advancement of

Original: Internet of Things and the engagement with inherently more complex problems in sectors such as healthcare and industry, which beneﬁt from additional information channels.
Revised: Internet of Things and the engagement with inherently more complex problems in sectors such as healthcare and industry, which beneﬁt from additional information channels.

Original: In addition, the rise of edge computing and deployment of models in such environments makes it necessary to also consider the conditions and resources of the computational nodes on which those models will execute.
Revised: In addition, the rise of edge computing and deployment of models in such environments makes it necessary to also consider the conditions and resources of the computational nodes on which those models will execute.

Original: This change in perspective makes it important to include, apart from accuracy, variables such as prediction speed or throughput in the evaluation of a model’s suitability for a given task.
Revised: This change in perspective makes it important to include, apart from accuracy, variables such as prediction speed or throughput in the evaluation of a model’s suitability for a given task.

Original: In the recent evaluation of advances in MTSC (Ruiz et al., 2021), one solution that shows good performance is ROCKET (Dempster et al., 2019), both in terms of accuracy and training time.
Revised: In the recent evaluation of advances in MTSC (Ruiz et al., 2021), one solution that shows good performance is ROCKET (Dempster et al., 2019), both in terms of accuracy and training time.

Original: ROCKET utilizes random convolutions to transform the time series channels and then extracts two features per convolution that are used with a linear classiﬁer.
Revised: ROCKET utilizes random convolutions to transform the time series channels and then extracts two features per convolution that are used with a linear classiﬁer.

Original: The evolution of ROCKET,
Revised: The evolution of ROCKET,

Original: MINIROCKET (Dempster et al., 2020), is proposed as the new default variant of ROCKET by its authors and utilizes only one feature per kernel (percentage of positive values), thereby halving the features.
Revised: MINIROCKET (Dempster et al., 2020), is proposed as the new default variant of ROCKET by its authors and utilizes only one feature per kernel (percentage of positive values), thereby halving the features.

Original: It also utilizes other optimizations to speed up ROCKET in general, and uses a minimally random approach with a given set of kernels.
Revised: It also utilizes other optimizations to speed up ROCKET in general, and uses a minimally random approach with a given set of kernels.

Original: Another variant, MultiRocket (Tan et al., 2021), uses more features per convolution to achieve better accuracy at the expense of transformation time.
Revised: Another variant, MultiRocket (Tan et al., 2021), uses more features per convolution to achieve better accuracy at the expense of transformation time.

Original: Both ROCKET and MINIROCKET have been developed for univariate time series.
Revised: Both ROCKET and MINIROCKET have been developed for univariate time series.

Original: Although the authors of MINIROCKET characterize their provided repository code for multivariate time series as naive, it is the same algorithmic logic in the form of a ROCKET extension in the sktime (L¨oning et al., 2019) repository that achieves the impressive performance mentioned above.
Revised: Although the authors of MINIROCKET characterize their provided repository code for multivariate time series as naive, it is the same algorithmic logic in the form of a ROCKET extension in the sktime (L¨oning et al., 2019) repository that achieves the impressive performance mentioned above.

Original: In (MINI)ROCKET, the authors encourage research into more sophisticated approaches for multivariate time series.
Revised: In (MINI)ROCKET, the authors encourage research into more sophisticated approaches for multivariate time series.

Original: We can start by identifying potential points that can be improved.
Revised: We can start by identifying potential points that can be improved.

Original: One point with (MINI)ROCKET that we try to address in our work is that the default number of features, although beneﬁcial to accuracy, can be highly redundant, and extreme for some datasets, leading to unnecessarily high transformation and inference times during deployment.
Revised: One point with (MINI)ROCKET that we try to address in our work is that the default number of features, although beneﬁcial to accuracy, can be highly redundant, and extreme for some datasets, leading to unnecessarily high transformation and inference times during deployment.

Original: In addition, due to the ﬁxed process and number of generated features, there is a large discrepancy in the representation (features per channel) across datasets with different dimensions.
Revised: In addition, due to the ﬁxed process and number of generated features, there is a large discrepancy in the representation (features per channel) across datasets with different dimensions.

Original: Moreover, the indiscriminate inclusion of all channels in the feature generation is susceptible to uninformative series.
Revised: Moreover, the indiscriminate inclusion of all channels in the feature generation is susceptible to uninformative series.

Original: We present experiments to illustrate the above points in section 4.3.
Revised: We present experiments to illustrate the above points in section A.1.

Original: Finally, the stochastic element, especially the random channel combination, although beneﬁcial to accuracy, does not offer signiﬁcant interpretability.
Revised: Finally, the stochastic element, especially the random channel combination, although beneﬁcial to accuracy, does not offer signiﬁcant interpretability.

Original: Based on this analysis, we propose LightWaveS, a framework for fast, distributed transformation of multivariate time series based on convolutional kernels, wavelet scattering and feature selection, for lightweight and accurate classiﬁcation with linear classiﬁers.
Revised: Based on this analysis, we propose LightWaveS, a framework for fast, distributed transformation of multivariate time series based on convolutional kernels, wavelet scattering and feature selection, for lightweight and accurate classiﬁcation with linear classiﬁers.

Original: Our solution aims to keep the successful aspects of the ROCKET model family, such as the short transformation time, the arbitrary convolutional kernel approach and the few descriptive features per kernel.
Revised: Our solution aims to keep the successful aspects of the ROCKET model family, such as the short transformation time, the arbitrary convolutional kernel approach and the few descriptive features per kernel.

Original: On top of that, LightWaveS adds well-studied signal theory, multi-node 1 distribution and smart feature selection to address the drawbacks identiﬁed above.
Revised: On top of that, LightWaveS adds well-studied signal theory, multi-node 1 distribution and smart feature selection to address the drawbacks identiﬁed above.

Original: The trade-off that we propose compared to (MINI)ROCKET is clear: we take advantage of additional computational resources during training time to keep the short duration and also save time during inference.
Revised: The trade-off that we propose compared to (MINI)ROCKET is clear: we take advantage of additional computational resources during training time to keep the short duration and also save time during inference.

Original: Our contributions with LightWaveS are:
Revised: Our contributions with LightWaveS are:

Original: • We introduce the concept of wavelet scattering based on arbitrary kernels • We achieve accuracy comparable to state-of-the-art in the majority of the UEA datasets (Bagnall et al., 2018), using only a fraction of the number of features used by (MINI)ROCKET • We distribute the framework to achieve training time comparable to (MINI)ROCKET onnode, and shorter on multiple nodes • We achieve linear scaling of training time with the number of channels, tested in experiments with up to 900 channels • Depending on the dataset size, we achieve good speedup with additional nodes, tested in experiments with up to 8 nodes • We achieve inference speedups ranging from 8x to 30x compared to (MINI)ROCKET on an edge device in datasets with comparable accuracy
Revised: • We introduce the concept of wavelet scattering based on arbitrary kernels • We achieve accuracy comparable to state-of-the-art in the majority of the UEA datasets (Bagnall et al., 2018), using only a fraction of the number of features used by (MINI)ROCKET • We distribute the framework to achieve training time comparable to (MINI)ROCKET onnode, and shorter on multiple nodes • We achieve linear scaling of training time with the number of channels, tested in experiments with up to 900 channels • Depending on the dataset size, we achieve good speedup with additional nodes, tested in experiments with up to 8 nodes • We achieve inference speedups ranging from 9x to 65x compared to ROCKET on an edge device in datasets with comparable accuracy

Original: With this work, we take an efﬁciency-centric approach, focusing on the practicality and the inference speed of a model that may be deployed on an edge device, without necessarily trying to surpass the state-of-the-art in terms of accuracy.
Revised: With this work, we take an efﬁciency-centric approach, focusing on the practicality and the inference speed of a model that may be deployed on an edge device, without necessarily trying to surpass the state-of-the-art in terms of accuracy.

Original:  R ELATED W ORK 2. M ULTIVARIATE TIME SERIES CLASSIFICATION
Revised:  R ELATED W ORK 2. M ULTIVARIATE TIME SERIES CLASSIFICATION

Original: Due to the recent rise in popularity of the deep learning ﬁeld, there is a multitude of models that can be easily adapted to incorporate the additional dimension of MTSC (Ismail Fawaz et al., 2019).
Revised: Due to the recent rise in popularity of the deep learning ﬁeld, there is a multitude of models that can be easily adapted to incorporate the additional dimension of MTSC (Ismail Fawaz et al., 2019).

Original: A detailed evaluation of recent MTSC methods appears in Ruiz et al.
Revised: A detailed evaluation of recent MTSC methods appears in Ruiz et al.

Original: Convolutions in particular, either 1-D or with more dimensions, are a very popular module when dealing with time series, as seen in models such as TapNet (Zhang et al., 2020), InceptionTime
Revised: Convolutions in particular, either 1-D or with more dimensions, are a very popular module when dealing with time series, as seen in models such as TapNet (Zhang et al., 2020), InceptionTime

Original: (Fawaz et al., 2020) and OS-CNN (Tang et al., 2021) among others.
Revised: (Fawaz et al., 2020) and OS-CNN (Tang et al., 2021) among others.

Original: In these works, convolutions are combined in various architectures with other modules, such as fully connected networks, and achieve impressive classiﬁcation accuracy.
Revised: In these works, convolutions are combined in various architectures with other modules, such as fully connected networks, and achieve impressive classiﬁcation accuracy.

Original: The majority of those deep learning models takes a signiﬁcant amount of time and memory to train even on GPU nodes, ranging from hours to even days, depending on the dataset size (Ruiz et al., 2021).
Revised: The majority of those deep learning models takes a signiﬁcant amount of time and memory to train even on GPU nodes, ranging from hours to even days, depending on the dataset size (Ruiz et al., 2021).

Original: In contrast, our method takes less than 15 minutes to transform, train and test all 30 UEA datasets on a single CPU node with a linear classiﬁer.
Revised: In contrast, our method takes less than 20 minutes to transform, train and test all 30 UEA datasets on a single CPU node with a linear classiﬁer.

Original: If we distribute the solution on e.g. 8 CPU nodes, this time drops to under 4 minutes.
Revised: If we distribute the solution on e.g. 8 CPU nodes, this time drops to under 4 minutes.

Original: An additional challenge is the interpretability of deep learning models.
Revised: An additional challenge is the interpretability of deep learning models.

Original: The usage of wavelets by LightWaveS and their method of application gives an advantage to interpreting the result based on signal theory, allows to incorporate domain expert knowledge and  helps to extract useful conclusions about a given time series problem.
Revised: The usage of wavelets by LightWaveS and their method of application gives the potential to interpret the result based on signal theory and the input channel ﬁltering helps to extract useful conclusions about a given time series problem.

Original: 2. F EATURE EXTRACTION AND SELECTION
Revised: 2. F EATURE EXTRACTION AND SELECTION

Original: Deep learning models implicitly extract features from the input along their ﬁrst layers.
Revised: Deep learning models implicitly extract features from the input along their ﬁrst layers.

Original: In contrast, explicitly extracting features to use with classiﬁers has also been visited in multiple ways.
Revised: In contrast, explicitly extracting features to use with classiﬁers has also been visited in multiple ways.

Original: We have already described how the *ROCKET family extracts features from the convolutions of random kernels with the input, an idea that has also been explored before asnoted in Dempster et al.
Revised: We have already described how the *ROCKET family extracts features from the convolutions of random kernels with the input, an idea that has also been explored before, such as in Farahmand et al.

Original: Another example is tsfresh (Christ et al., 2018), which extracts a large number of predeﬁned statistical features from the time series and then through feature selection reduces them to the most useful.
Revised: Another example is tsfresh (Christ et al., 2018), which extracts a large number of predeﬁned statistical features from the time series and then through feature selection reduces them to the most useful.

Original: Similarly, catch22 (Lubba et al., 2019) is a solution that uses only 22 predeﬁned characteristics to transform time series, aiming for a very fast transformation.
Revised: Similarly, catch22 (Lubba et al., 2019) is a solution that uses only 22 predeﬁned characteristics to transform time series, aiming for a very fast transformation.

Original: A different approach is presented in
Revised: A different approach is presented in

Original: WEASEL+MUSE (Sch ¨ afer & Leser, 2018), which extracts features based on a bag-of-patterns approach and selects the most useful based on a χ 2 test.
Revised: WEASEL+MUSE (Sch¨afer & Leser, 2018), which extracts features based on a bag-of-patterns approach and selects the most useful based on a χ 2 test.

Original: LightWaveS, similarly to MINIROCKET, depends on an arbitrary set of convolutions and only 4 statistical features, extracted however from the coefﬁcients of wavelet scattering.
Revised: LightWaveS, similarly to MINIROCKET, depends on an arbitrary set of convolutions and only 4 statistical features, extracted however from the coefﬁcients of wavelet scattering.

Original: Due to the speed of the convolution operation and the simplicity of the features, we can achieve extremely fast training and inference times, while giving the model enough complexity to accommodate difﬁcult datasets, where predeﬁned statistical features on the raw time series values may not be descriptive enough.
Revised: Due to the speed of the convolution operation and the simplicity of the features, we can achieve extremely fast training and inference times, while giving the model enough complexity to accommodate difﬁcult datasets, where predeﬁned statistical features on the raw time series values may not be descriptive enough.

Original: 2. D ISTRIBUTED TRAINING
Revised: 2. D ISTRIBUTED TRAINING

Original: As the size of the deep learning models and the amount of input data increase, it is increasingly difﬁcult to perform training on a single node.
Revised: As the size of the deep learning models and the amount of input data increase, it is increasingly difﬁcult to perform training on a single node.

Original: For that reason, distribution of the training process across multiple nodes is becoming increasingly necessary (Mayer & Jacobsen, 2020).
Revised: For that reason, distribution of the training process across multiple nodes is becoming increasingly necessary (Mayer & Jacobsen, 2020).

Original: This distribution can be implemented with methods such as data parallelism, where each node applies the same operations on different parts of the input data, with frequent communication among the nodes to update the model.
Revised: This distribution can be implemented with methods such as data parallelism, where each node applies the same operations on different parts of the input data, with frequent communication among the nodes to update the model.

Original: Distribution can also be applied on solutions that do not depend on deep learning, such as tsfresh mentioned above, which supports operation on a cluster through Dask(Dask Development Team, 2016).
Revised: Distribution can also be applied on solutions that do not depend on deep learning, such as tsfresh mentioned above, which supports operation on a cluster through Dask (Dask Development Team, 2016).

Original: LightWaveS is distributed using MPI in a data-parallel way, but in contrast to the communication heavy training of DL models, there is efﬁcient and minimal communication between the worker nodes and the central coordinator.
Revised: LightWaveS is distributed using MPI in a data-parallel way, but in contrast to the communication heavy training of DL models, there is efﬁcient and minimal communication between the worker nodes and the central coordinator.

Revised: Speciﬁcally, the nodes only send a limited number of feature scores and descriptions once during the execution, making communication a trivial percentage of the whole process.

Original: 2. W AVELETS
Revised: 2. W AVELETS

Original: Wavelets are localized waveforms (as seen in Figure 1 on the right) and are a well-studied method in signal processing that have been used extensively in the analysis of time series (Mallat, 1999).
Revised: Wavelets are localized waveforms (as seen in Figure 1 on the right) and are a well-studied method in signal processing that has been used extensively in the analysis of time series (Mallat, 1999).

Original: There is vast literature with methods and applications of wavelets on all types of problems, ranging from healthcare to audio analysis, and well-studied and developed families of wavelets suitable for speciﬁc applications (Addison, 2017; Merry & Steinbuch, 2005).
Revised: There is vast literature with methods and applications of wavelets on all types of problems, ranging from healthcare to audio analysis, and well-studied and developed families of wavelets suitable for speciﬁc applications (Addison, 2017; Merry & Steinbuch, 2005).

Original: There are also numerous approaches that combine wavelets with machine and deep learning methods, either as implicit or explicit feature extractors (Li et al., 2021; Wang et al., 2018).
Revised: There are also numerous approaches that combine wavelets with machine and deep learning methods, either as implicit or explicit feature extractors (Li et al., 2021; Wang et al., 2018).

Original: A seminal work is Bruna & Mallat (2013), where the concept of a wavelet scattering network using a Morlet wavelet is introduced, in combination with linear and support vector machine classiﬁers for hand-written digit classiﬁcation and texture recognition.
Revised: A seminal work is Bruna & Mallat (2013), where the concept of a wavelet scattering network using a Morlet wavelet is introduced, in combination with linear and support vector machine classiﬁers for hand-written digit classiﬁcation and texture recognition.

Original: This method was constructed to be invariant to translations of the input and stable to small deformations.
Revised: This method was constructed to be invariant to translations of the input and stable to small deformations.

Original: It has also been recently used in combination with deep learning networks for speciﬁc applications (Soro & Lee, 2019; Jin & Duan, 2020).
Revised: It has also been recently used in combination with deep learning networks for speciﬁc applications (Soro & Lee, 2019; Jin & Duan, 2020).

Original: LightWaveS aims to combine the strong points of these works under a single generalized framework, with a focus on efﬁciency.
Revised: LightWaveS aims to combine the strong points of these works under a single generalized framework, with a focus on efﬁciency.

Original: We aim to bridge the gap between ROCKET and the wavelet theory, and we progress to the next logical step of wavelet scattering.
Revised: We aim to bridge the gap between ROCKET and the wavelet theory, and we progress to the next logical step of wavelet scattering.

Original: We keep this approach lightweight, both in depth and paths of the scattering, so we can apply it to time series channels on a massive scale in a very short time.
Revised: We keep this approach lightweight, both in depth and paths of the scattering, so we can apply it to time series channels on a massive scale in a very short time.

Original: The arbitrary base set of wavelets can  be extended based on expert opinion, backed by the solid theory behind wavelets and their applications, making LightWaveS a suitable platform for experimentation on solutions for MTSC problems.
Revised: The arbitrary base set of wavelets can potentially be extended based on expert opinion, backed by the solid theory behind wavelets and their applications, making LightWaveS a suitable platform for experimentation on solutions for MTSC problems.

Original: Finally, the hierarchical feature ﬁltering leads to the most relevant output features of the scattering coefﬁcients being selected.
Revised: Finally, the hierarchical feature ﬁltering leads to the most relevant output features of the scattering coefﬁcients being selected.

Original:  P ROPOSED FRAMEWORK 3. P RELIMINARIES 3.1. M INI R OCKET FUNDAMENTALS
Revised:  P ROPOSED FRAMEWORK 3. P RELIMINARIES 3.1. M INI R OCKET FUNDAMENTALS

Original: (MINI)ROCKET is primarily based on the convolution operation, in which a kernel k with size l ( k ∈ R l ), bias b and dilation factor d is used to calculate a sliding dot product with a 1-D input x ∈ R L of size L and produce an output x (cid:48) , where each element is calculated as (Dempster et al., 2019): 
Revised: (MINI)ROCKET is primarily based on the convolution operation, in which a kernel k with size l ( k ∈ R l ), bias β and dilation factor d is used to calculate a sliding dot product with a 1-D input x ∈ R L of size L and produce an output x (cid:48) , where each element is calculated as (Dempster et al., 2019): [Equation]

Original: [Equation]
Revised: [Equation]

Original: MINIROCKET generates a large number (10000 by default) of those kernels.
Revised: MINIROCKET generates a large number (10000 by default) of those kernels.

Original: The weights are selected from an empirically chosen subset of 84 kernels of length 9 with weights in { 2 , − 1 } .
Revised: The weights are selected from an empirically chosen subset of 84 kernels of length 9 with weights in { 2 , − 1 } .

Original: This has been chosen to limit the possible weight combinations but is not unique for the purpose, and other lengths, different values or weights drawn from ∼ N (0 , 1) are equally effective (Dempster et al., 2020).
Revised: This has been chosen to limit the possible weight combinations but is not unique for the purpose, and other lengths, different values or weights drawn from ∼ N (0 , 1) are equally effective (Dempster et al., 2020).

Original: The only thing that is important is that the kernel weights have sum 0.
Revised: The only thing that is important is that the kernel weights have sum 0.

Original: The bias is drawn from the convolution output, and the dilation is drawn from the set D
Revised: The bias is drawn from the convolution output, and the dilation is drawn from the set D

Original: = {(cid:98) 2 e (cid:99)} , where e ∼ U (0 , max ) , with max such that the length of the dilated kernel does not exceed that of the input.
Revised: = {(cid:98) 2 e (cid:99)} , where e ∼ U (0 , m ) , with m such that the length of the dilated kernel does not exceed that of the input.

Original: Finally, half of the kernels use padding, which appends zeros to either side of the input so that x (cid:48) and x have the same dimension.
Revised: Finally, half of the kernels use padding, which appends zeros to either side of the input so that x (cid:48) and x have the same dimension.

Original: The feature extracted from each of the 10000 outputs is the percentage of positive values, and is used for the ﬁnal classiﬁcation.
Revised: The feature extracted from each of the 10000 outputs is the percentage of positive values, and is used for the ﬁnal classiﬁcation.

Original: 3.1. W AVELET S CATTERING
Revised: 3.1. W AVELET S CATTERING

Original: Wavelet scattering is the process of applying wavelet transforms in a cascading manner (Bruna &
Revised: Wavelet scattering is the process of applying wavelet transforms in a cascading manner (Bruna &

Original: Mallat, 2013), combined with non-linearities and pooling.
Revised: Mallat, 2013), combined with non-linearities and pooling.

Original: The wavelet transform is a method used to approximate a signal using a set of wavelets which originate from a ”mother” wavelet Ψ( t ) , scaled by s and shifted by b (Young, 2012).
Revised: The wavelet transform is a method used to approximate a signal using a set of wavelets which originate from a ”mother” wavelet Ψ( t ) , scaled by s and shifted by b (Young, 2012).

Original: Each such wavelet can be described as:
Revised: Each such wavelet can be described as:

Original: [Equation]
Revised: [Equation]

Original: Intuitively, we can relate these wavelets to the convolution ﬁlters that we discussed above, with the dilation being the scale of the wavelets (how ”narrow” or ”wide” they are), and the shift parameter b corresponding to the starting point of the convolution on the input ( i + d ∗ j ).
Revised: Intuitively, we can relate these wavelets to the convolution ﬁlters that we discussed above, with the dilation being the scale of the wavelets (how ”narrow” or ”wide” they are), and the shift parameter b corresponding to the starting point of the convolution on the input ( i + d ∗ j ).

Revised: This connection between convolutional networks and the scattering architecture has also been thoroughly explored in Mallat (2016).

Original: We can see an example of a wavelet convolution in Figure 1, where the response is strong at the points where the sliding wavelet ”matches” the signal.
Revised: We can see an example of a wavelet convolution in Figure 1, where the response is strong at the points where the sliding wavelet ”matches” the signal.

Original: In addition, the wavelet set  from a mother wavelet is parallel to the way that MINIROCKET has a ﬁxed set of kernel weights, for which different paddings and dilations are randomly selected, generating the ”child” kernels.
Revised: In addition, the wavelet set created from a mother wavelet is parallel to the way that MINIROCKET has a ﬁxed set of kernel weights, for which different paddings and dilations are randomly selected, generating the child kernels.

Original: In the wavelet scattering transform, on each level λ , the previous result is convoluted with each wavelet ψ λ n (kernel) and a complex modulus operator is applied before propagating the result to the next level, such that:
Revised: In the wavelet scattering transform, on each level λ , the previous result is convoluted with each wavelet ψ λ n (kernel) and a complex modulus operator is applied before propagating the result to the next level, such that:

Original: [Equation]
Revised: [Equation]

Original: and the scattering coefﬁcients that result from each level are
Revised: and the scattering coefﬁcients that result from each level are

Original: [Equation]
Revised: [Equation]

Original: where φ is an averaging kernel.
Revised: where φ is an averaging kernel.

Original: Since on every level of the scattering transform there can be multiple candidate wavelets (kernels), there is a geometric progression of potential paths, as can be seen in the graphic representation of the process in Figure 1.
Revised: Since on every level of the scattering transform there can be multiple candidate wavelets (kernels), there is a geometric progression of potential paths, as can be seen in the graphic representation of the process in Figure 1.

Original: Throughout the above intuitive description, we connected the concepts of MINIROCKET with the wavelet theory.
Revised: Throughout the above intuitive description, we connected the concepts of MINIROCKET with the wavelet theory.

Original: In MINIROCKET the prototype kernels have zero mean, which corresponds to a desirable constraint of mother wavelets (Young, 2012).
Revised: In MINIROCKET the prototype kernels have zero mean, which corresponds to a desirable constraint of mother wavelets (Young, 2012).

Original: Under this new context, we can reconsider MINIROCKET as a classiﬁer based on convolutions with random child wavelets of an arbitrary set of mother wavelets.
Revised: Under this new context, we can reconsider MINIROCKET as a classiﬁer based on convolutions with random child wavelets of an arbitrary set of mother wavelets.

Original: In addition, although not being wavelet scattering, MINIROCKET can be intuitively mapped on the ﬁrst level, with its kernel set equivalent to the ﬁrst level set, { Ψ 0 , 1 ,s , ..., Ψ 0 ,n,s } , with s drawn from D as we said above.
Revised: In addition, although not being wavelet scattering, MINIROCKET can be intuitively mapped on the ﬁrst level, with its kernel set equivalent to the ﬁrst level set, { Ψ 0 , 1 ,s , ..., Ψ 0 ,n,s } , with s drawn from D as we said above.

Original: Since each kernel is slid across the input, we can think of b as taking all discrete values from 0 to l input − l kernel ∗ s for each candidate wavelet, and the convolution output (response) as being the combined response of all shifted wavelets, as portrayed in Figure 1.
Revised: Since each kernel is slid across the input, we can think of b as taking all discrete values from 0 to l input − l kernel ∗ s for each candidate wavelet, and the convolution output (response) as being the combined response of all shifted wavelets, as portrayed in Figure 1.

Original: However, MINIROCKET extracts the features before the application of any modulus operator, so it does not satisfy the rest of the wavelet scattering transform requirements.
Revised: However, MINIROCKET extracts the features before the application of any modulus operator, so it does not satisfy the rest of the wavelet scattering transform requirements.

Original: 3. A LGORITHM
Revised: 3. A LGORITHM

Original: Based on the above observations, we improve the approach and reach the crux of LightWaveS:
Revised: Based on the above observations, we improve the approach and reach the crux of LightWaveS:

Original: Lightweight Wavelet Scattering based on random wavelets.
Revised: Lightweight Wavelet Scattering based on random wavelets.

Original: The term lightweight refers both to the system optimizations of the framework that make it fast, such as the distribution, as well as the fact that not all paths ofthe wavelet scattering  are considered, for computational efﬁciency.
Revised: The term lightweight refers both to the system optimizations of the framework that make it fast, such as the distribution, as well as the fact that only a reduced set of wavelet scattering paths are computed.

Original: Speciﬁcally, we consider only the paths shown in bold in Figure 1.
Revised: Although it is established that not all paths need to be considered in a scattering network (Bruna & Mallat, 2013), since we are aiming for fast training and inference, we take this notion to its limit.

Original: We generate and utilize the same set of kernels  forboth levels, so the bold paths mean that the same kernel is applied twice consecutively.
Revised: We compute all coefﬁcients for the given kernels and dilations for the ﬁrst level, but we consider only one path per ﬁrst level output for the second level.

Original: In addition, we limit  the depth to two layers, as a good balance point between computational speed and informative coefﬁcients, as observed in Bruna & Mallat (2013).
Revised: In this way, we limit signiﬁcantly the memory and computation time required for the extracted features, while accepting the trade-off of losing some descriptive coefﬁcients.

Original: Samples N , Channels C , Rank R in MPI WORLD, S nodes, Feature target size F , Feature pool size FP
Revised: The choice of the kernels to be applied in the second layer is not immediately clear. We can consider heuristics such as selecting kernels whose ﬁrst level coefﬁcients gave features with high correlation to the classiﬁcation task.

Original: Result: Transformed X (cid:48) ∈ R NxF 1 X node ← GetDataSlice ( S, R, C ) ; 2 K node ← GenerateKernels ( S, R, C ) ; 3 X (cid:48) node ← WaveletScatteringFeatures ( X node
Revised: However, after experimenting with such heuristics, and even with up to four second layer paths, we found that the ﬁnal classiﬁcation results were not better than the simple approach of applying the same kernel and dilation, a concept intuitively and loosely similar to the process of a discrete wavelet transform (Sundararajan, 2016).

Original: K node ) ; 4 f topnode ← TopFeatureSelection ( X (cid:48) node y FP ) ; 5 if R is 0 then f topglobal ← GatherTopFeatures ( FP ) ; X top ← FeatureTransform ( X , f topglobal ) ; f final ← MRMRFeatureSelection ( X top , y , F X (cid:48) ← FeatureTransform ( X , f final ) ;
Revised: Thus, we end up computing only the paths shown in bold in Figure 1, which means that the same kernel is applied twice consecutively.

Original: 10 end
Revised: In addition, we limit the depth to two layers, as a good balance point between computational speed and informative coefﬁcients, as observed in Bruna & Mallat (2013).

Original: The main steps of LightWaveS appear in Algorithm 1. Initially, the number of channels is split among the nodes.
Revised: The main steps of LightWaveS are the following: Initially, the dataset is split among the nodes across the channel dimension.

Revised: For larger datasets, the nodes receive only a sample of the total training examples, in order to speed up computation.

Revised: This sample selection is the only source of randomness in the algorithm.

Original: Then the kernels are generated, which for our purpose are based on the same subset of 84 kernels that MINIROCKET uses.
Revised: Then, on each node, the kernels are generated, which for our purpose are based on the same subset of 84 kernels that MINIROCKET uses.

Original: We limit randomness even more, so each kernel is dilated with all dilations in the set { 2 0 , 2 1 , ..., 2 max } .
Revised: We limit randomness even more, so each kernel is dilated with all dilations in the set { 2 0 , 2 1 , ..., 2 5 } .

Original: We also completely remove bias, and the padding is common to all kernels so that the output dimension is equal to the input.
Revised: We also completely remove bias, and the padding is common to all kernels so that the output dimension is equal to the input.

Original: All those kernels are applied to each of the input channels according to the wavelet scattering process described above.
Revised: All those kernels are applied to each of the input channels according to the wavelet scattering process described above.

Original: The generated kernels have no complex part, so the non-linearity modulus operator is equal to the absolute value of the convolution output.
Revised: The generated kernels have no complex part, so the non-linearity modulus operator is equal to the absolute value of the convolution output.

Original: We also downsample the U [ λ 1] result by a factor of 2 before propagating it, while keeping the same scale for the kernel, so it operates in lower frequencies.
Revised: We also downsample the U [ λ 1 ] result by a factor of 2 before propagating it, while keeping the same scale for the kernel, so it operates in lower frequencies.

Original: This is according to the insight in Bruna & Mallat (2013), thatnot all paths are necessary and the most useful ones are frequency decreasing.
Revised: This is according to the insight in Bruna & Mallat (2013) that the most useful paths are frequency decreasing.

Original: As far as the pooling is concerned, since we want to limit the number of coefﬁcients that will be used as features, we use max and min pooling, instead of the averaging kernel.
Revised: As far as the pooling is concerned, since we want to limit the number of coefﬁcients that will be used as features, we use max and min pooling, instead of the averaging kernel.

Original: In this way, although we accept the loss of more information, we have fewer features and we also save two additional convolutions per path.
Revised: In this way, although we accept the loss of more information, we have fewer features and we also avoid two additional convolutions per path.

Original: The features that are extracted from each scattering path are 8: the ﬁrst four are the max and min value of U [ λ 1] and U [ λ 2] .
Revised: The features that are extracted from each scattering path are 8: the ﬁrst four are the max and min value of U [ λ 1 ] and U [ λ 2 ] .

Original: We selected those as the most straightforward non-linearities that can be quickly calculated during the convolution process.
Revised: We selected those as the most straightforward non-linearities that can be quickly calculated during the convolution process.

Original: The other four are the percentage of positive values (ppv) and normalized longer sequence of positive values (ls) in U [ λ 1] and U [ λ 2] before the application of the modulus operator.
Revised: The other four are the percentage of positive values (ppv) and normalized longer sequence of positive values (ls) in U [ λ 1 ] and U [ λ 2 ] before the application of the modulus operator.

Original: We keep these features based on MINIROCKET and MultiROCKET respectively, since they have proved to be useful during classiﬁcation.
Revised: We keep these features based on MINIROCKET and MultiROCKET respectively, since they have proven to be useful during classiﬁcation.

Original: After the feature extraction, the ﬁrst selection phase is performed  in a supervised way using ANOVA.
Revised: After the feature extraction, the ﬁrst selection phase is performed on each node in a supervised way using ANOVA.

Original: The F-values are calculated on different subsets of the data, and the average score is taken, making the method more robust (Li et al., 2017).

Original: The main node  gathers the top FP features and performs the ﬁnal feature selection, using the minimum-redundancy, maximum-relevance algorithm (Peng et al., 2005), incrementally selecting the feature with the highest score.
Revised: The main node then gathers the top scoring features and performs the ﬁnal feature selection, using the minimum-redundancy, maximum-relevance algorithm (Peng et al., 2005), incrementally selecting the feature with the highest score.

Original: This score is determined by its F-statistic (relevance to the class), divided by its average correlation to the previously selected features (redundancy).
Revised: This score is determined by its F-statistic (relevance to the class), divided by its average correlation to the previously selected features (redundancy).

Original: In our case, we use the Spearman correlation coefﬁcient as the correlation indicator.
Revised: In our case, we use the Pearson correlation coefﬁcient as the correlation indicator.

Original: In both phases the feature selection methods are ﬁlter based, which are fast, classiﬁerindependent and can be implemented to scale well with the number of features (Li et al., 2017).
Revised: In both phases the feature selection methods are ﬁlter based, which are fast, classiﬁer-independent and can be implemented to scale well with the number of features (Li et al., 2017).

Original: The design choices that are not guided or restricted by theory, such as the feature scoring function, are straightforward and focused on computational efﬁciency.
Revised: The design choices that are not guided or restricted by theory, such as the feature scoring function, are straightforward and focused on computational efﬁciency.

Original: We also performed empirical experimentation on a development subset, although without exhaustive parameter tuning of the method.
Revised: We also performed empirical experimentation on a development subset, although without exhaustive parameter tuning of the method.

Original:  E XPERIMENTS 4. D ATASETS
Revised:  E XPERIMENTS 4. D ATASETS

Original: We select as benchmark the UEA collection of multivariate datasets (Bagnall et al., 2018), excluding
Revised: We select as benchmark the UEA collection of multivariate datasets (Bagnall et al., 2018), excluding

Original: InsectWingbeat , since due to its large size it presented issues when training ROCKET.
Revised: InsectWingbeat , since due to its large size it presented issues when training ROCKET.

Original: The datasets are described in detail in Ruiz et al.
Revised: The datasets are described in detail in Ruiz et al.

Original: Following the example of Dempster et al.
Revised: Following the example of Dempster et al.

Original: (2019), we selected 15 of the 30 UEA datasets to work on when developing the method, in order to draw generic conclusions and avoid overﬁtting the whole collection.
Revised: (2019), we selected 15 of the 30 UEA datasets to work on when developing the method, in order to draw generic conclusions and avoid overﬁtting the whole collection.

Original: In addition, we prepare and use ﬁve machinery related datasets:
Revised: In addition, we prepare and use ﬁve machinery related datasets:

Original: • MAFAULDA , from Machinery Fault Database (MAF) is a dataset with 8 sensor measurements on a machine fault simulator, taken under normal conditions and ﬁve different fault types.
Revised: • MAFAULDA , from Machinery Fault Database (MAF) is a dataset with 8 sensor measurements on a machine fault simulator, taken under normal conditions and ﬁve different fault types.

Original: We downsample the measurements so that the input length is 1000 steps and we split the dataset in train and test with ratio 85-15 %.
Revised: We downsample the measurements so that the input length is 1000 steps and we split the dataset in train and test with ratio 85-15 %.

Original: • TURBOFAN (Saxena & Goebel, 2008), is an engine degradation simulation dataset collection, containing 4 datasets with operation simulations of engines that run until failure under different conditions, with measurements from 26 sensors.
Revised: • TURBOFAN (Saxena & Goebel, 2008), is an engine degradation simulation dataset collection, containing 4 datasets with operation simulations of engines that run until failure under different conditions, with measurements from 26 sensors.

Original: The goal is to predict the remaining useful life (RUL) of the engines.
Revised: The goal is to predict the remaining useful life (RUL) of the engines.

Original: In order to turn the problem into binary classiﬁcation, we prepare the dataset and the labels in a suitable way with the goal being classifying RUL as more or less than  operational cycles.
Revised: In order to turn the problem into binary classiﬁcation, we prepare the dataset and the labels in a suitable way with the goal being classifying RUL as more or less than 20 operational cycles.

Original: 4. E XPERIMENTAL SETUP
Revised: 4. E XPERIMENTAL SETUP

Original: All training experiments were run on nodes which have dual 8-core 2.4 GHz (Intel Haswell E52630-v3) CPUs and 64GB of RAM.
Revised: All training experiments were run on nodes which have dual 8-core 2.4 GHz (Intel Haswell E52630-v3) CPUs and 64GB of RAM.

Original: The inference experiments are executed on a Jetson Xavier board which has an 8-core ARM CPU.
Revised: The inference experiments are executed on a Jetson Xavier board which has an 8-core ARM CPU.

Original: We ran (MINI)ROCKET on the 29 UEA datasets and the additional datasets using the default number of features (20 and 10 thousand respectively).
Revised: We ran (MINI)ROCKET on the 29 UEA datasets and theadditional datasets using the default number of features (20 and 10 thousand respectively).

Original: As for
Revised: As for

Original: LightWaveS, we present three variants of the model, termed L1,L2 and L1L2.
Revised: LightWaveS, we present three variants of the model, termed L1,L2 and L1L2.

Original: These versions refer to keeping the features only from the scattering level 1, level 2 or both, although we consider
Revised: These versions refer to keeping the features only from the scattering level 1, level 2 or both, although we consider

Original: the
Revised: the

Original: L1L2 version as the default.
Revised: L1L2 version as the default.

Original: We select 500 features as the default variant, which has shown good balance between training time, inference speed and accuracy in the development set.
Revised: We select 500 features as the default variant, which has shown good balance between training time, inference speed and accuracy in the development set.

Original: We use a Ridge regression classiﬁer from the Scikit-learn package (Pedregosa et al., 2011) for all methods.
Revised: We use a Ridge regression classiﬁer from the Scikit-learn package (Pedregosa et al., 2011) for all methods.

Original: We use a critical difference diagram to present the results, a popular method of comparing classiﬁer performance across multiple datasets (Ruiz et al., 2021; Ismail Fawaz et al., 2019).
Revised: We use a critical difference diagram to present the results, a popular method of comparing classiﬁer performance across multiple datasets (Ruiz et al., 2021; Ismail Fawaz et al., 2019).

Original: This shows the ranking of the classiﬁers and groups the ones that do not show statistical difference, based on pairwise comparisons using a Wilcoxon signed-rank test (Wilcoxon, 1992) with Holm’s alpha correction (Holm, 1979; Garcia & Herrera, 2008).
Revised: This shows the ranking of the classiﬁers and groups the ones that do not show statistical difference, based on pairwise comparisons using a Wilcoxon signed-rank test (Wilcoxon, 1992) with Holm’s alpha correction (Holm, 1979; Garcia & Herrera, 2008).

Original: The grouped classiﬁers appear on the diagram connected by a thick horizontal line.
Revised: The grouped classiﬁers appear on the diagram connected by a thick horizontal line.

Original: 4. (M INI )R OCKET ANALYSIS

Original: Using the sktime repository version of the code, we can see in Figure 2a that even keeping the top half features of MINIROCKET using an unsophisticated chi-squared test does not result in a statistically signiﬁcant difference in the classiﬁer performance (rather, it slightly increases the rank).

Original: As for the channel inclusion, as an extreme case, we can add noise channels to the UEA datasets (10% of the original channel number) and we see that both solutions are affected negatively when trained on them, belonging in a statistically different group.

Original: [Figure]

Original:  A CCURACY RESULTS
Revised: 4. A CCURACY RESULTS

Original: We present the performance of LightWaveS in terms of accuracy in comparison with (MINI)ROCKET, as well as other recent solutions, namely (M)OSS-CNN (Tang et al., 2021),
Revised: We present the performance of LightWaveS in terms of accuracy in comparison with (MINI)ROCKET, as well as other recent solutions, namely (M)OSS-CNN (Tang et al., 2021),

Original: WEASEL+MUSE (Sch ¨ afer & Leser, 2018), TapNet (Zhang et al., 2020) and Catch22 (Lubba et al., 2019).
Revised: WEASEL+MUSE (Sch¨afer & Leser, 2018), TapNet (Zhang et al., 2020) and Catch22 (Lubba et al., 2019).

Original: Apart from LightWaveS and *ROCKET, the rest of the accuracy metrics are taken from the repositories of Tang et al.
Revised: Apart from LightWaveS and (MINI)ROCKET, the rest of the accuracy metrics are taken from the repositories of Tang et al.

Original: (2021) 2 and Dhariyal et al.
Revised: (2021) 2 and Dhariyal et al.

Original: Due to missing values in those metrics, we manage to compare the methods on 28 of the 30 UEA datasets.
Revised: Due to missing values in those metrics, we manage to compare the methods on 28 of the 30 UEA datasets.

Original: In Figure 3 we can see the results.
Revised: In Figure 2 we can see the results.

Original: Although lower in rank, LightWaveS belongs in the same statistical group with (MINI)ROCKET, which have the best accuracy.
Revised: Although lower in rank, LightWaveS belongs in the same statistical group with (MINI)ROCKET, which have the best accuracy.

Original: In addition, it is among the ranks of more complex DL methods, such as TapNet and MOS-CNN.
Revised: In addition, it is among the ranks of more complex DL methods, such as TapNet and MOS-CNN.

Original: Out of the three variants, LightWaveS-L2 performs the worst, without the beneﬁt of faster execution that L1 has.
Revised: Out of the three variants, LightWaveS-L2 performs the worst, without the beneﬁt of faster execution that L1 has.

Original: We can focus on (MINI)ROCKET for a more detailed comparison, and also include the 5 additional datasets.
Revised: We can focus on (MINI)ROCKET for a more detailed comparison, and also include the 5 additional datasets.

Original: Since the aim of LightWaveS is to approach their state-of-the-art accuracy with fewer features, not necessarily surpass it, we place the LightWaveS results  in four accuracy bins compared to (MINI)ROCKET :  higher or equal, lower within 0.05, lower within 0.05-0.10 andlower more than 0.1.
Revised: Since the aim of LightWaveS is to approach their state-of-the-art accuracy with fewer features, not necessarily surpass it, we place the LightWaveS results for all datasets in four accuracy bins compared to (MINI)ROCKET : one for higher or equal accuracy, and three bins for lower accuracy, with difference less than 0.05, between 0.05 and 0.1 and more than 0.1 respectively.

Original: We see in Figure 4 that in  the majority of the datasets the accuracy stays in the ﬁrst 3 categories for all variants apart from L2, and the datasets with large accuracy deviation are few.[Figure]
Revised: In Figure 3 we see the amount of datasets in each bin. For the majority of the datasets the accuracy stays in the ﬁrstcategories for all variants apart from L2, and the datasets with large accuracy deviation are few.

Original: In addition, we can see that just increasing the number of features to 1500 leads tosigniﬁcant improvement of the comparison results, leaving potential for successfully tackling the harder datasets.
Revised: In addition, we can see that just increasing the number of features to 1500 leads to improvement of the comparison results, especially in the case of MINIROCKET, showing the potential of the method to successfully tackle the harder datasets as well.

Original: 4. C HANNEL AND NODE SCALING OF TRAINING TIME
Revised: 4. C HANNEL AND NODE SCALING OF TRAINING TIME

Original: As a point of reference, ROCKET processes the whole UEA set (apart from InsectWingbeat ) on a single node in approximately 9 minutes, MINIROCKET in 5 and LightWaveS (L1L2) in 14.
Revised: As a point of reference, ROCKET processes the whole UEA set (apart from InsectWingbeat ) on a single node in approximately 9 minutes, MINIROCKET in 5 and LightWaveS (L1L2) in 14.

Original: (MINI)ROCKET has complexity O ( kernels · samples · tslength ) .
Revised: (MINI)ROCKET has complexity O ( kernels · samples · tslength ) .

Original: Since LightWaveS uses one more convolution per kernel it adds a ﬁxed factor of two to the complexity, essentially keeping linear scalability with these variables.
Revised: Since LightWaveS uses one more convolution per kernel it adds a ﬁxed factor of two to the complexity, essentially keeping linear scalability with these variables.

Original: Thus, we can focus on the scalability of the LightWaveS training time with the additional channel dimension and with the number of distribution nodes. 
Revised: Thus, we can focus on the scalability of the LightWaveS training time with the additional channel dimension and with the number of distribution nodes. [Figure]

Original: We select PEMS-SF from UEA which has 963 channels, and we increment the channels by 100, starting from 100, training LightWaveS on 2 nodes.
Revised: We select PEMS-SF from UEA which has 963 channels, and we use subsets of the total dataset to train LightWaveS on 2 nodes, starting from 100 channels and incrementing by 100.

Original: We see in Figure 5a that all variants show (sub)linear scaling with the number of channels, which is expected since each additional channel results in a ﬁxed amount of additional convolutions, and there are no combinations or permutations of channels.
Revised: We see in Figure 4a that all variants show (sub)linear scaling with the number of channels, which is expected since each additional channel results in a ﬁxed amount of additional convolutions, and there are no combinations or permutations of channels.

Original: For the node scaling, we also select FaceDetection , with 144 channels.
Revised: For the node scaling, we also select FaceDetection , with 144 channels and we measure the training time across 1,2,4 and 8 nodes.

Original: As we see in Figure 5b, with PEMS-SF the gained training speedup is more close to ideal, since there are sufﬁcient channels for each node to work on.
Revised: As we see in Figure 4b, with PEMS-SF the gained training speedup is more close to ideal, since there are many channels for each node to work on and the distributed computation constitutes the majority of the execution time.

Original: In contrast to that, with FaceDetection that has fewer channels, the computational cost starts being dominated by the operations on the main node (application of kernels, feature selection, ﬁnal transformation), so the gained speedup falls more quickly. 
Revised: In contrast to that, with FaceDetection that has fewer channels, the computational cost starts being dominated by the operations on the main node (application of kernels, feature selection, ﬁnal transformation), so the gained speedup falls more quickly. [Figure]

Original: 4. I NFERENCE RESULTS
Revised: 4. I NFERENCE RESULTS

Original: We manage to achieve signiﬁcant speedup during inference due to the small amount of features, which heavily reduces the amount of convolutions required to transform a test sample.
Revised: We manage to achieve signiﬁcant speedup during inference due to the small amount of features, which heavily reduces the amount of convolutions required to transform a test sample.

Original: We perform a qualitative analysis, focusing on measuring the inference on the machinery datasets, since they could correspond to a scenario of edge deployment of a model for e.g. fault detection of a machine.
Revised: We perform a qualitative analysis, focusing on measuring the inference on the machinery datasets, since they could correspond to a scenario of edge deployment of a model for e.g. fault detection of a machine.

Original: However, our conclusions hold for all datasets.
Revised: However, our conclusions hold for all datasets.

Original: We can see in Figure 6 the relation between the accuracy difference of the LightWaveS variants with (MINI)ROCKET and the speedup achieved.
Revised: We can see in Figure 5 the relation between the accuracy difference of the LightWaveS variants with (MINI)ROCKET and the speedup achieved.

Original: This ﬁgure frames clearly the trade-off that we propose: it is up to the end user to select their preferred balance of speedup and accuracy, in the cases where no ideal case is available.
Revised: This ﬁgure frames clearly the trade-off that we propose: it is up to the end user to select their preferred balance of speedup and accuracy, in the cases where no ideal case is available.

Original: The MF dataset in the MINIROCKET comparison acts as an outlier in the graph due to its large length in combination with the optimizations of MINIROCKET.
Revised: The MF dataset in the MINIROCKET comparison acts as an outlier in the graph due to its large length in combination with the optimizations of MINIROCKET.

Original: Our method of applying the kernels matches
Revised: Our method of applying the kernels matches

Original: ROCKET’s, but could also beneﬁt from these optimizations, restoring the discrepancy.
Revised: ROCKET’s, but could also beneﬁt from these optimizations, restoring the discrepancy.

Original: [Figure]

Original: [Figure]
Revised: [Figure]

Original:  D ISCUSSIONS AND C ONCLUSION
Revised:  D ISCUSSIONS AND C ONCLUSION

Original: Summarizing, LightWaveS offers comparable to state-of-the-art accuracy on the datasets tested, with training time similar to (MINI)ROCKET and inference time 8x-30x faster on an edge device.
Revised: Summarizing, LightWaveS offers comparable to state-of-the-art accuracy on the datasets tested, with training time similar to (MINI)ROCKET and inference time 9x-65x faster on an edge device in the case of ROCKET and 1.3x - 14x for MINIROCKET.

Original: Apart from the advantages we have mentioned, a signiﬁcant advantage of LightWaveS is that it is based on wavelet theory, which has a strong theoretical base.
Revised: Apart from the advantages we have mentioned, a signiﬁcant advantage of LightWaveS is that it is based on wavelet theory, which has a strong theoretical base.

Original: A promising future direction is to explore expert tuning of the framework, by preparing and including in the base set well tested wavelets, with different properties such as padding or dilations depending on the use case.
Revised: A promising future direction is to explore expert tuning of the framework, by preparing and including in the base set well tested wavelets, with different properties such as padding or dilations depending on the use case.

Original: Anothersecondary beneﬁt is that due to the feature selection, LightWaveS can ﬁlter the channels to a subset of the originals.
Revised: Another beneﬁt is that due to the feature selection, LightWaveS can ﬁlter the channels to a subset of the originals.

Original: In the UEA, this reduction ranged from 0 to 94%, with an average of 20% across all datasets.
Revised: As seen in Table 5, this reduction ranged from 0 to 92%, with an average of 15% across all datasets and the larger datasets beneﬁting more.

Original: This has multiple advantages: it can give insights into which channels contain useful information for the problem, leading to knowledge extraction.
Revised: This has multiple advantages: it can give insights into which channels contain useful information for the problem, leading to knowledge extraction.

Original: On edge devices, where resources are valuable, it can free up incoming signal channels.
Revised: On edge devices, where resources are valuable, it can free up incoming signal channels.

Original: Finally, LightWaveS can act as an initial fast channel ﬁltering method that precedes another deep learning solution, reducing the training data required.
Revised: Finally, LightWaveS can act as an initial fast channel ﬁltering method that precedes another deep learning solution, reducing the training data required.

Original: LightWaveS can also beneﬁt from, and is indeed orthogonal to, recent works on feature selection, since additional descriptive features can be used on the scattering coefﬁcients, improving accuracy.
Revised: LightWaveS can also beneﬁt from, and is indeed orthogonal to, recent works on feature selection, since additional descriptive features can be used on the scattering coefﬁcients, improving accuracy.

Original: Future work can include an initial, more informed selection of the wavelets, so that the wavelet scattering network can extend to more paths with wavelet combinations, as well as explicit, informed combinations of different channels.
Revised: Future work can include an initial, more informed selection of the wavelets, so that the wavelet scattering network can extend to more paths with wavelet combinations, as well as explicit, informed combinations of different channels.

Original: R EPRODUCIBILITY S TATEMENT
Revised: R EPRODUCIBILITY S TATEMENT

Original: The code of LightWaveS, as well as the code used to train (MINI)ROCKET, run the inference experiments and preprocess the machinery datasets is provided as supplementary material.
Revised: The code of LightWaveS, as well as the code used to train (MINI)ROCKET, run the inference experiments and preprocess the machinery datasets is provided as supplementary material.

Original: All accuracy and time experiments were repeated multiple times (30-100), with different seeds and the average of metrics was taken for a fairer representation of the results. 
Revised: All accuracy and time experiments were repeated multiple times (30-100), with different seeds and the average of metrics was taken for a fairer representation of the results. The detailed metrics are provided in the Appendix.

Original: All datasets used are publicly available.
Revised: All datasets used are publicly available.

Original: There has been an effort to explicitly set the random seed for all methods that accept such a parameter, in order to enable reproducibility of the results as closely as possible
Revised: There has been an effort to explicitly set the random seed for all methods that accept such a parameter, in order to enable reproducibility of the results as closely as possible. A A PPENDIX A. (M INI )R OCKET ANALYSIS

Revised: Using the sktime repository version of the code, we can see in Figure 6a that even keeping the top half features of MINIROCKET using an unsophisticated chi-squared test does not result in a statistically signiﬁcant difference in the classiﬁer performance (rather, it slightly increases the rank).

Revised: As for the channel inclusion, as an extreme case, we can add noise channels to the UEA datasets (10% of the original channel number) and we see that both solutions are affected negatively when trained on them, belonging in a statistically different group.

Revised: [Figure]

Revised: A. D ETAILED EXPERIMENT METRICS

Revised: Below are the detailed metrics on accuracy, inference speedup and channel reduction.

Revised: [Table]

Revised: Heartbeat 0.

Revised: JapaneseVowels 0.

Revised: LSST 0.

Revised: Libras 0.

Revised: MotorImagery 0.

Revised: NATOPS 0.

Revised: PEMS-SF 0.

Revised: PenDigits 0.

Revised: Phoneme 0.

Revised: RacketSports 0.

Revised: SelfRegulationSCP 0.

Revised: SelfRegulationSCP 0.

Revised: SpokenArabicDigits 0.

Revised: StandWalkJump 0.

Revised: UWaveGestureLibrary 0.

Revised: FD

Revised: [Table]

Revised: [Table]

Revised: FD

Revised: [Table]

Revised: FD

Revised: [Table]

Revised: FD

Revised: [Table]

Revised: Mafaulda

Revised: [Table]

Revised: [Table]

Revised: A. N UMBER OF FEATURES SELECTION

Revised: In Figure 7 we can see the evolution of the accuracy with the increasing number of features for the L1L2 method, as measured in our randomly selected development subset of datasets.

Revised: We see that a common pattern is that the accuracy initially increases, and reaches a plateau after a number of features (different for each dataset).

Revised: Another pattern is the oscillation of accuracy around an initial value.

Revised: The selection of 500 as the default number of features is based on this graph, and is a reasonable number which balances accuracy and method speed, as well as ensures that most development datasets are close to or in the plateau region.

Revised: [Table]

Revised: JapaneseVowels 0.

Revised: LSST 0.

Revised: Libras 0.

Revised: MotorImagery 0.

Revised: NATOPS 0.

Revised: PEMS-SF 0.

Revised: PenDigits 0.

Revised: Phoneme 0.

Revised: RacketSports 0.

Revised: SelfRegulationSCP 0.

Revised: SelfRegulationSCP 0.

Revised: SpokenArabicDigits 0.

Revised: StandWalkJump 0.

Revised: UWaveGestureLibrary 0.

Revised: FD 0.

Revised: FD 0.

Revised: FD 0.

Revised: FD 0.

Revised: Mafaulda 0.

Revised: [Table]

Revised: HandMovementDirection 61.

Revised: Handwriting 28.

Revised: Heartbeat 67.

Revised: JapaneseVowels 25.

Revised: LSST 21.

Revised: Libras 14.

Revised: MotorImagery 79.

Revised: NATOPS 37.

Revised: PEMS-SF 283.

Revised: PenDigits 9.

Revised: Phoneme 47.

Revised: RacketSports 19.

Revised: SelfRegulationSCP 55.

Revised: SelfRegulationSCP 59.

Revised: SpokenArabicDigits 37.

Revised: StandWalkJump 54.

Revised: UWaveGestureLibrary 36.

Revised: FD 23.

Revised: FD 21.

Revised: FD 30.

Revised: FD 28.

Revised: Mafaulda 55.

Revised: [Table]

Revised: HandMovementDirection 4.

Revised: Handwriting 4.

Revised: Heartbeat 13.

Revised: JapaneseVowels 12.

Revised: LSST 8.

Revised: Libras 6.

Revised: MotorImagery 4.

Revised: NATOPS 17.

Revised: PEMS-SF 258.

Revised: PenDigits 7.

Revised: Phoneme 7.

Revised: RacketSports 9.

Revised: SelfRegulationSCP 2.

Revised: SelfRegulationSCP 2.

Revised: SpokenArabicDigits 10.

Revised: StandWalkJump 1.

Revised: UWaveGestureLibrary 3.

Revised: FD 14.

Revised: FD 15.

Revised: FD 17.

Revised: FD 20.

Revised: mafaulda 2.

Revised: [Table]

Revised: AWR

Revised: AtrialFib

Revised: BasicMotions

Revised: ChTrajectories

Revised: Cricket

Revised: DuckGeese

Revised: ERing

Revised: EigenWorms

Revised: Epilepsy

Revised: EthanolC

Revised: FaceDetection 106.

Revised: FingerMov

Revised: HMD

Revised: Handwriting

Revised: Heartbeat

Revised: JapVowels

Revised: LSST 4.

Revised: Libras

Revised: MotorImagery

Revised: NATOPS

Revised: PEMS-SF

Revised: PenDigits

Revised: Phoneme

Revised: RacketSports

Revised: SelfRegSCP

Revised: SelfRegSCP

Revised: SpArabicDigits 8.

Revised: SWJump

Revised: UWaveGL

Revised: FD  11.

Revised: FD  21.

Revised: FD  22.

Revised: FD  19.

Revised: Mafaulda