Learnable Group Transform For Time-Series
Authors: Romain Cosentino
Behnaam Aazhang
Venue:  (2020)
Keywords: Group Transform
Time-Frequency Representation
Wavelet Transform
Group Theory
Representation Theory
Time-Series

Abstract: We propose to undertake the problem of representation learning for time-series by considering a Group Transform approach. This framework allows us to, first, generalize classical time-frequency transformations such as the Wavelet Transform, and second, to enable the learnability of the representation. While the creation of the Wavelet Transform filter-bank relies on the sampling of the affine group in order to transform the mother filter, our approach allows for non-linear transformations of the mother filter by introducing the group of strictly increasing and continuous functions. The transformations induced by such a group enable us to span a larger class of signal representations. The sampling of this group can be optimized with respect to a specific loss and function and thus cast into a Deep Learning architecture. The experiments on diverse time-series datasets demonstrate the expressivity of this framework which competes with state-of-the-art performances.

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LEARNABLE GROUP TRANSFORM FOR TIME-SERIES
Anonymous authors Paper under double-blind review
ABSTRACT
We undertake the problem of representation learning for time-series by considering Group Transform approach. This framework allows us to, first, generalize classical Lime-frequency transformations such as the Wavelet Transform, and second, to enable the learnability of the representation. While the creation of the Wavelet Transform flter-bank relies on affine transformations of mother filter, our approach allows for non-linear transformations. This achieved by sampling a subset of invertible maps on R. The subset considered contains strictly increasing and continuous functions. The transformations induced by such maps enable us to span larger class of signal representations, from wavelet to chirplet-like filters. We propose a parameterization of such a non non-linear map such that its sampling can be optimized for specific loss and signal. The Learnable Group Transform thus be cast into Deep Neural Network. The experiments on diverse time-series datasets demonstrate the expressivity of this framework, which competes with state-of-the-ar performances.
INTRODUCTION
The selection the fimeufreyynccy representation for analyzing, classifying, and predicting time-series has long been studied (Coifman & Wickerhauser 1992; Mallat & Zhang 1993 Gribonval
& Bacry 2003). To this day, the front-end processing of time-series remains a keystone toward the improvement of wealth of applications such health-care (Saritha al., (008)), environmental
sound (Balestriero et al. 2018 Lelandais & Glotin 2008). and seismic data analysis (Seydoux al. 2016). The common denominator of the recorded signals in these fields is their undulatory behavior. While these signals share this common behavior, two significant factors imply the need of learning the representation: 1) time-series are intrinsically different because of their physical nature, 2) the machine learning task can be different even within the same type of data. Therefore, the representation should be induced by both the signal and the task at hand.
An too common approach to performing inference on time-series consists of building a Deep Neural Network (DNN) that operates on spectral decomposition of the time-series such as Wavelet Transform (WT) or Mel Frequency Spectral Coefficients (MFSC). The selection of the judicious transform is either performed by expert in the signal hand, by considering the aforecited selection methods and their derivatives. However, an inherent drawback that the selection of the time-frequency transform the task. For instance, a selection based on the sparsity of the representation while the task is the classification of the signals. Besides, these selection methods and transformations require substantial cross-validations of large number of hyperparameters such as mother filter family, number of octaves, number of wavelets per octave, size of the window (Cosentino al. 2017).
often achieved with criteria that do not align with
To alleviate these drawbacks, Ravanelli & Bengio 2018): Balestriero et al. (2018); Cakir et al. (2016); Zeghidour et al. (0118) investigated the learnability filter is transformed by deterministic affine maps. These transformations constitute the filter-bank. The representation
mother filter. This learnable mother
the signal obtained by convolving the filter-bank atoms with the signals. Recently, Khan & Yener (2018) investigated the learnabllity of the affine transformations, that is, the sampling of the dilation parameter of the affine group inducing the wavelet filter- bank. Optimized jointly with the DNN, their method allows adaptive transformation of the mother filter. Another approach consists of building suuivarrinn-invarii representations. In Mallat (2012); |runa (2013) they propose ionsslationiivariaa representation, the Scattering Transform, which is stable under

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the action of small diffeomorphisms. In Oyallon et al. (2018: Cohen & Welling (2016), they focus on equivariant equivariant-invarian invariant representations for images, which reduces the sample complexity and endow DNN's 's layers with interpretability.
In this work, focus GT, which is achieved by taking the inner product between the filter-bank, which is built by taking the action of a transformation map on a mother filter, and the signal. Well-known GTs are the Short-Time Fourier Transform (STFT) and the Continuous Wavelet Transform (CWT). We propose to extend these GTss and improve their flexibility by introducing the Learnable Group Transform (LGT) by 1) generalizing the affine transformations of mother filter leading to wavelet filter-bank by introducing non-linear transformations (Section 3.1 ..1. 2) proposing parameterization such non-linear map such that can be learned efficiently and jointly with any DNN, (Sections 3.2 3.3 3) Replacing the affine transformations of CWT by non-linear maps allows for greater flcxibility in the learnable spectral decomposition which displays different equivariance properties (Section 3.4). This flexibility improves the lincarization capability of the representation as it eases the learning of a spectral decomposition that is able to discard intricate patterns in the time-series that nuisances. This specific transformation of filter induces filter with a non-linear instantaneous phase, which in turn, allows to span filters a la chirplets, which are of interest in variety of domains such as biology and medicine, mechanies and vibrations, and sonar systems (Flandrin 2001). Also, this generalization implies that for fixed network topology, replacing the learnable affine group with the continuous group leads larger class of spannable functions. which improves the approximation property of the DNN at hand (Winkler & Le 2017 [Balestriero & Baraniuk 2018). In order show the generality our approach, we apply our algorithm on two diverse time-series classification problems (Section
BACKGROUND AND NOTATIONS
We first highlight the properties of particular GTs by expressing their time-frequency tiling.
2.1 TIME-FREQUENCY TILING
The spread of a filter and its Fourier transform are inversely proportional as per the Heisenberg uncertainty prin- ciple (Mallat) 1999). Following this principle, we can observe that in the case of STFT (respectively WT with Gabor wavelet), at given time the signal transformed by a window of constant bandwidth (respectively proportional bandwidth) modulated by complex exponential resulting in a uniform tiling (respectively propor- tional) on the frequency axis, Fig- ure (T. In the case of a chirp-like fil- ter, as proposed in Baraniuk & Jones (1996). each tile is a sheared rectangu- lar, more generally, an affinely trans- formed rectangular. In this case, well, the lower bound area of the sheared rectangular is constrained by the uncertainty principle. decompositions can be achieved by analyzing how they tile the time-frequency plane. For instance, in the case of WT, the precision in frequency degrades as the frequency increases while its precision in time increases (Mallat 1999). In the case of STFT, the uniform tiling implies that the precision is constant along the frequency axis. In our proposed framework, the LGT allows for an adaptive tiling. as illustrated in Figure ([] such that the trade-off between time and frequency precision depends the task and data.
Prequnay
SIFT
WT LGT
Figure 1: Time-Frequency Tilings given time 7: (left) Short-Time Fourier Transform, i... constant bandwidth, (middle) Wavelet Transform, i.e., proportional bandwidth, (right) Learnable Group Transform, i.e, adaptive bandwidth, the "tiling" is induced by the learned non-linear transforma- tion underlying the filter-bank decomposition.
such, the understanding the benefits of various time-frees fime-freuuency
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3 LEARNABLE GROUP TRANSFORM
To extend the filter-bank derivation as proposed in wavelet decomposition we introduce a learnable group transform. We now define subset of invertible maps IR enabling the transformation of mother filter Then, we provide parameterization of such functions and show how one can efficiently learn these parameters. Finally, we derive the equivariance properties of the induced group transform. The overall building block the LGT and its application on signal is depicted in Figure
Pinc ( (aa,
* Pinc (9 bb))
Figure 2: Learnable Group Transform: (left) generating the strictly increasing continuous func- tions Pine ( (((a))())))) b.)) which stands for the strictly increasing and continuous transformation operator with parameters (ag, by), vk E {l,..., Kj. where K denotes the number of filters in the filter- bank. (middle) Each generated operators Pine (9(n. are applied the mother filter denoted by z (presently Morlet wavelet), where the imaginary part is shown in red and the real part in blue. This transformation leads the filter-bank, Pine (9 bb))) where ((.) E Cine Then, the convolution between this generated filter-bank and the signal leads to the LGT of the signal. The black box on the LGT representation (right) corresponds to the convolution of the 1tth filter with the signal. The strictly increasing and continuous piece-wise linear functions can be learned efficiently by back pookppropgg the error induced by the generated GT.
3.1 STRICTLY INCREASING AND CONTINUOUS TRANSFORMATTIONS
In order to generalize the classical affine transformations used in WT, we propose the utilization of strictly increasing and continuous functions defined as
Cine (R) = {g E C(R)gg strictly increasing),
where C(R) defines the space of continuous functions defined on R. This function is composed of invertible maps which is crucial in order to derive invariance properties as well as avoid artifacts in the transformed filters.
We define the linear operator (ine(g) by
JPine (g)+b (g(t)), Vel (2) Vg (In),
where v denotes mother filter. We can see that the increasing and continuous group representation operator Pine induces a mapping which depends on the function g E (TR). If for instance g = i.e., the identity map, then we have Pine (e) = vb, it is in fact the identity operator in the space of the mother filter. Given a mother filter a E La (IR), Pinc ( (g)v, Vg E Cine CiRe(IR) induces a non- linear transformation of the mother filter which can be visualized in Figure (3). Note that in signal Kerkyacharian
processing, such a transformation called warping (Goldenstein & Gomes et al. 2004).
1999
g E Cine(R)
(g((t))
Affine
Wavelet
Quadratic Convex
Increasing Quadratic Chirplet
Quadratic Concave
Decreasing Quadratic Chirplet
Logarithmic
Logarithimic Chirplet
Exponential
Exponential Chirplet
Table Special cases of the function g inducing filters belonging to well-known filter-banks.
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Mother Filter
Affine Transformation
Non-Linear Transformation
For all the filters, the real part is shown blue and the imaginary in red. (left) Morlet wavelet mother filter. (middle) Transformation of the mother filter with respect to an affine transform: the dilation parameter O < < 1, i.e., contraction, and translation i.e., translation. (right) Increasing and continuous transformation of the mother filter for some randomly generated function E Cnee (IR) leading to chirplet-like filter.
Figure 3: Transformation of a Morlet Wavelet:
Among the possible transformations induced mother filter by the mapping g Cine (IR), some of them correspond to well-known filters (Table
3.2 SAMPLING THE TRANSFORMATION MAPS
In this work, we are specifically interested the learnability such an increasing and continuous map. As such, provide way to sample such space via its parameterization. We propose to use piece-wise wise affine functions constrained such that they belong to the class strictly increasing and continuous functions. This constrained piece-wise affine map is defined as
(aa (1) E(ai (all + (t),,,(), VIER. 1=1
(3)
s.t.: a1 > 0, E fl,......1 n1.
(4)
1/+1 (a: a++ 14+1)4+1 by AE (1.....,,
(5)
where (a1....,,n)) An). b (b.....,b,) bn). 1, is the indicator function of the intervals I [fi, t441), E .2......2 ......... 1} and 1 = (-00, t1). In = [tn. +x0). and a and by denote respectively the slope and offset of each piece of the function and is the number of pieces. As such, for each (a, b) satisfying the constraints and (5) the function ((a.b) a sample from the set Cine
Notice that this mapping be performed using 1-layer ReLU Neural Network (Arora al. 2016). This implementation implies a knot-free piece-wise affine mapping, providing more flexibility regarding the transformation map. The knot-free mapping defined such that the uniform support, i.e. the intervals I (3). replaced with varying support for different E nI...., As such, this flexibility induces better approximation property (Jupp) 1978).
3.3 LEARNING THE PIECE- WISE AFFINE TRANSFORMATION MAPS
The parameters (Ak, ba), vk E (1...., K} are differentiable with respect to the filter and thus any deep learning pipeline using those filters can be used and optimized jointly with the other DNN parameters by stochastic gradient descend methods. Given a set of signals {s; E (L2))) N and given a task specific loss function L, aim solving the following optimization problem
min (a1 be., ..... (ax By EE
Σ L(F(W(s (g..))
(6)
where N denotes the number of signals, K the number of filters, F represents DNN, Qt {ax RT. by E RD (²(b(..+1)) = (a - 2(A.+11)44441. + b vk E {l,..., (l....,K, K), and W(si, ()
[W[s,, b)) W(s,, (9 bx): -)1"" and
W[s,, ((( (s: Pine ))) I(a,.be) Cin((B). Vk K}. (7)
where (() = (--)) and (.) corresponds to the time axis.
We propose different settings that will impact the type of filter-bank method can reach.
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First, We propose normalization of the frequeney of the transform filter (denoted in the result tables by nLGT). This normalization helps to reduce the aliasing induced by the filters. We propose to use f. the normalized frequency f with respect to the maximum slope the piece-wise affine mapping. For instance, in the case of Morlet wavelet, the normalization
follows
(a) (() = - exp 2m ( exp (-2
(I (a.
where f/ (1... 0t, is the imaginary unit, and a the width parameter defining the localization of the wavelet time and frequency. This normalization will be performed for each sample of the group, and thus for each generated filter & E {1,..., K} of the filter-bank.
Second, we constrain the domain of the piece-wiss wise affine map, derived in (3) (denoted in the result tables by cLGT). In the following experiments, we propose a dyadic constraint of the domain as the WT. The support of the filter is close the support of wavelet filter-bank. However, the envelope of the filter and the instantancous frequency still vary as in the Chirplet Transform (Baraniuk & Jones 1996).
3.4 EQUIVARIANCE PROPERTIES
The equivariance- invariance properties of signal representations play crucial role in the efficiency of the algorithm at hand (Mallat 2016). By considering the mapping Pine as a group action on the space of the mother filter, i.e., L2 IR, or the LGT. More details regarding the background
precisely, representation of group on (R)
develop the equivariance properties theoretical approach are given in Appendix A We consider the set Cine (R) with the operation @ consisting of the composition of function to form the group of strictly increasing and continuous maps denoted by Gine. This formulation eases the derivation of the equivariance properties of group transforms which can be defined for group G by
Wip(g')s;, (1 = W[s,, - g...), EG
(8)
That is, transforming the signal with respect to the group G and computing its representation is equal to computing the representation of the signal and then transforming the representation. If G corresponds to the affine group, the associated group transform is the WT the transformation which is equivariant to scalings and translations. One can already notice that since W(.,-) employs convolution, for any group G, the LGT is translation equivariant. We now focus on more specific cquivariance properties of the LGT by defining the local equivariance by
3r E R. Wip(g')s, s; Vl(.. r) = W[s,, (ig'') @ V.T)). G.
That is, the representation of a local transformation a signal a window centred at t is equals to the transformation of the representation filter. As a matter of fact, assuming that the representation of Gine is unitary, we have the following proposition.
The size of the window depends the support of the
Proposition 1. The LGT is locally equivariant with respect to the action of the group Gine-
Refer to Appendix for the proof.
4 EXPERIMENTS
For all the experiments and all the settings, i.e., LGT, nLGT, cLGT, enLGT, the increasing and continuous piece-wise affine map initialized randomly, and the optimization is performed with Adam Optimizer, and the number of knots of each piece-wise affine map is 256. The mother filter used for our setting a Morlet wavelet filter. The code of the LGT framework will provided on the Github page of the first author.
4.1 ARTIFICIAL DATA: CLASSIFICATION OF CHIRP SIGNALS
We present an artificial dataset that demonstrates how a specific time- frequency tiling might not be adapted or would require cross-validations for given task and data. To build the dataset,
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generate one high frequency ascending chirp and one descending fech-freuuecyy chirp of size 8192 following the chirplet formula provided in (Baraniuk & Jones (1996)). Then for both chirp signals, we add Gaussian noise samples (100 times for each class), see Figures in Appendix (C.I). The task aims at being able to detect whether the chirp is ascending or descending. Both the training and test sets are composed of 50 instances of each class. For all models, set the batch size to 10, the number of epochs to 50. Each experiment was repeated 5 times with randomly sampled train and test set, and the accuracy was the result of the average over these 5 runs. Each GT is composed with a non- non-linearity, and the inference is performed by a linear classifier. For the case of WT and LGT, the size of the filters 512. As we can observe in Table (2). the WT, as well as the STFT with few numbers of filters, perform poorly on this dataset. The chirp signals to be analyzed are localized close to the Nyquist frequency, and in the case WT, as illustrated in Figure | the wavelet filter-bank has poor frequency resolution in high frequency while benefiting from a high time resolution. In this experiment, we can see that this characteristic the WT time- freueeryynncy tiling implies that through time, the small frequency variations of the chirp are not efficiently captured.
In the case of STFT, as the num- ber of filter decreases, the fre- quency resolution gets altered. Thus, this frequency variation not captured. Using large window for the STFT increases the frequency resolution of the tiling and thus enables to capture the difference between the two classes. In the LGT setting, the tiling has adapted to the task and produces good performances cept for the cLGT model. In fact, the domain of the piece-wise linear map is constrained to be dyadic, and thus the adaptivity of the filter bank is reduced, which not suitable for this specific task. For all settings, the visualization be found in Appendix (C.1.2 of the filters, well the representations of the signals, This experiment shows an example of signals that are not easily classified by neither the proportional- bandwidth nor the bondwiht-bandwidth without considering gross-validation of hyperparameters.
Representation + Non-Lineurity + Linear Classifier
Accuracy
Wavelet Transform (64 Filters)
53.01 + 5,1
Short-Time Fourier Transform (64 Filters)
65.1 1 11,9
Short-Time Fourier Transform (128 Filters)
86.6 = 9.8
Short-Time Fourier Transform (512 Filters)
100 0.0
LGT (64 Filters)
92.9 1 4.0
nL.GT (64 Filters)
95.7 + 3.3
cLGT (64 Filters)
56.8 + 1.6
cnI.GT (64 Filters)
100.0 + 0.0
Table 2: Testing Accuracy for the Chirp Signals Classification Task
4.2 SUPERVISED BIRD DETECTION
Representation + Non-Linearity + Deep Network
AUC
MFSC (80 Filters)
77.83 + 1.34
Conv. Filter init, random (80 Filters)
66.77 1: 1.04
Conv. Filter init. Gabor (80 Filters)
67.67 1 0.98
Spline Conv. init. random (80 Filters) (Balestriero et al. (2018))
78.17 + 1.48
Spline Cony. init. Gabor (80 Filters) (Balestriero et al. (2018)
79.32 :: 1.52
LGT (80 Filters)
78.41 = 1.38
nL.TT (80 Filters)
75.50 + 1.39
cLGT (80 Filters)
79.14 = 0.83
cnI.GT (80 Filters)
79.68 + 1.35
Table 3: Testing AUC for the Bird Detection Task
We now propose large scale dataset to validate the suitability of our model in noisy and realistic setting. The dataset is extracted from the Freesound audio archive Stowell Plumbley (2013). This dataset contains about 7, 000 field recording signals of 10 seconds sampled at 44 kHz, representing slightly less than 20 hours of audio signals. The content of these recordings varies from water sounds to city noises. Among these signals, some contain bird songs that are mixed with different background sounds having more energy than the bird song, see Appendix (C2.I). The given task is a binary classification where one should predict the presence or absence of a bird song. As the dataset is unbalanced, we use the Area Under Curve (AÚC) metric. The results we propose for both the benchmarks and our models evaluated test set consisting of 33% of the total dataset. In order

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to compare with previously used methods, the batch size, i.e., 10, and the learning rate cross-valiee validation grid as in (Balestriero et al. (2018). For each model, the best hyperparameters models with early stopping, the results are shown Table (3.. While the first layer of the architecture has model-dependent representation (i.e., MFSC, LGT,, Conv. filters..... we the state-of-the-an architecture (Grill & Schlüter (2017)) for the DNN architecture, described in Appendix (B.2). Notice that this specific DNN architecture has been designed and optimized for MFSC representation. As we can see in Table 3 the case without constraints (LGT) reaches better accuracy than the domain expert tile the time-frequency plane
the same seeds to sample the train and test set,
selected, and train and evaluated randomly 10-times the
benchmark (MFSC), showing the ability of such transformation according to the task and data at hand. Besides, including more constraints on the model (cnL.GT) reduces overfitting and further improve results to outperform the other benchmarks.
Figure 4: Learnable Group Transform Visualisation of a sample containing bird song (cLGT),
where (left) at the initialization and (righr) after learning. For each subfigure, the z-axis corresponds to time and the y-axis to the different filters. Notice that the y-axis usually corresponds to the scale
or the center-frequency of the filters. Other representations are displayed in Appendix (C.2.3) We can observe that compared to the initialization, the learned representation is sparser and the SNR is increased. Besides, the representation is less redundant in the frequency axis.
Figure 5: Learnable Group Transform Filters for the Bird Detection Data Each row displays two selected filters (left and right sub-figure) for different settings: (from top to bottom) LGT, nLGT, cL.GT. For each subfigure, the left part corresponds to the filter before training and the right part to the filter after training. The blue and red denote respectively the real and imaginary part of the filters.
One can notice that all the learned filters in Figure contain either an increasing chirp decreasing chirp, corresponding respectively to the convexity or concavity of the instantancous phase of the filter and thus of the picce-wise linear map. Such feature being used and is crucial in the detection and analysis of bird song (Stowell & Plumbley 2012).
4.3 HAPTICS DATASET CLASSIFICATION
The Haptics dataset is classification problem with five classes and 155 training and 308 testing samples from the UCR Time Series Repository Chen et al. (2015). where each time-series has 1092 time samples. As opposed the bird dataset where features of interests are known, and competitive methods have been established, there is no expert knowledge regarding the specific signal features (see Table 4
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One that method outperforms other approaches in the cLGT setting while perform- ing the classification with a lin- classifier opposed other
Representation + Classifie
Accuracy
DTW (Al-Naymat et al. (2009)
37.7
BOSS (Schäfer (2015))
46,4
Residual (Wang et al. (2017)
50.5
COTE (Bagnall al. (2015)
51.2
methods using DNN algorithms.
Fully Convolutional NN (aagg et al. (2017))
55.1
This demonstrates the capability of our method to transform the
WD+ Convolutionl NN (Khan & Yener (2018))
57.5
LGT (96 Filters)+ Non-Linearity Linear Classifier
53.5
data efficiently while not requir- ing a further change of basis. Be- sides, even a small dataset set- ting, our approach is capable of learning an efficient transforma- tion of the data. We provide in Figure the visualization of some sampled filters for each setting of the LGT model. As opposed to the supervised bird dataset, we are commonly used in signal processing. This is an example of an application where the features of interest in the signals unknown, and requires learnable representation.
nLGT (96 Filters)+ Non-Lincarity + Linear Classifier
50.4
cLGT (96 Filters)+ Non-Linearity + Linear Classifier
58.2
enLGT (96 Filters)+ Non-Linearity Linear Classifier
54.3
Table 4: Testing Accuracy for the Haptics Classification Task
that the filters do not coineide with well-known filters that
54
128
128
44
128
64
128
MOM W JIMMON
228
320
Figure 6: Learnable Group Transform Filters for the Haptics Data Each row displays two selected filters (left and right sub- sugffigure for different settings: (from top to bottom) LGT. nLGT, cLGT, enLGT. For each subfigure, the left part corresponds to the filter before training and the right part to the filter after training. The blue and red denote respectively the real and imaginary part of the filters.
CONCLUSION
We proposed to extend the WT by introducing GT based on non-linear transformations of the mother filter. We restrain the transformation to be in the space of strictly increasing and continuous functions enabling its connection with well known time-frequency filters as well as the derivation of equivariance properties. We also shown tractable way to learn to sample these transformations. From bird detection to haptics classification, our approach competes with state-of-the-an methods without priori knowledge the signal power spectrum and outperform classical fime-frey representations. Interestingly. in the bird detection experiment, we recover chirplet filters that known be crucial their detection, while in the of the haptic dataset where important features to be captured to perform the classification of the signals accurately are unknown, the filters learned are very dissimilar classical time-frey filters.
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S. Mallat. Understanding deep convolutional networks. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374(2065):2050220,, 2016
S. Mallat and Z. Zhang. Matching pursuits with time-frequency dictionaries. Signal Processing, IEEE Transactions on, 41(12):3397-3415, 1993.
E. Oyallon, S. Zagoruyko, G. Huang, N. Komodakis, S. Lacoste-Julien, M. B. Blaschko, and E. Belilovsky. Scattering networks for hybrid representation learning. IEEE transactions pattern analysis and machine intelligence, 2018.
M. Ravanelli and Y. Bengio. Interpretable convolutional filters with sincnet. arXiv preprint 181 2018.
C. Saritha, V. Sukanya, and Y. N. Murthy, Eeg signal analysis using wavelet transforms. Bulg. Phys, 35(1):68-1777 77, 2008.
P. Schäfer. The boss is concerned with time series classification in the presence of noise. Data Mining and Knowledge Discovery, 29(6):1505-1555 2015.
L. Seydoux, N. M. Shapiro, J. de Rosny, F. Brenguier, and M. Landès. Detecting seismic activity with covariance matrix analysis of data recorded on seismic arrays. Geophysical Journal International, 204(3):1430-1444 2016.
D. Stowell and M. D. Plumbley. Framewise heterodyne chirp analysis of birdsong. In 2012 Proceedings of the 20th European Signal Processing Conference (EUSIPCO), pp. 2694-2698. IEEE, 2012.
D. Stowell and M. D. Plumbley. An open dataset for research audio field recording archives: freeficld1010. CoRR, abss/1309.520 2013. UR.attpp.//axxi...cc///////
B. Torrésani. Wavelets associated with representations of the affine weyl-heisenberg group. Journal of Mathematical Physics, 32(5):1273-127) 1279, 1991.
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Z. Wang, W. Yan, and T. Oates. Time series classification from scratch with deep neural networks: A strong baseline. In 2017 international joint conference on neural networks (IJCNN) pp. 1578-1585 IEEE, 2017.
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N. Zeghidour, N. Usunier, I. Kokkinos, T. Schaiz, G. Synnaeve, and E. Dupoux. Learning filterbanks from raw speech for phone recognition. In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5509-5513. IEEE, 2018.
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A GROUP TRANSFORM: A GROUP REPRESENTATION APPROACH
A.1
BACKGROUND
For further details the group theoretical aspects described in this section, the reader should refer Vilenkin 1978).
Definition 1. group is a set G with a nultiplicatiiv operation 3 &that respects enclosure, identity element, inverse element, and associativiry.
The representation of the group determines its action on a function space and bridges the gap between group theory and linear algebra, allowing to compute the transformation of a function following the rules induced by the specific group hand. The representation of group can be thought as far- reaching generalization of the exponential function property, exp(x + y) = (x) exp(y), Vz, E (Baraniuk) 1993). fact, it is defined as,
Definition 2. A linear continuous representation p of a group G the linear space Hi is defined as
p G GL(H),
(9)
where GL(H) is the the group of linear map in H. such that Vg, 9 E G
plg @ g = p(g)p(g')
(10)
For instance, let H. be vector space such as R the representation of the group is induced by 3 x 3 matrices. In this case, the operation on the right of (TO) a matrix multiplication, where each matrix depends on the group elements g and g'. This concept extends to linear operators acting on functional spaces.
As such, multiple transformations of function by different elements of the group is equal to the representation of the combination of the group elements applied to the function.
This structure-preserving map defines the action of group on elements of function spaces. Group transforms such as STFT and CWT can be expressed in such a way by selecting a mother filter space and group. The representation of the group in the mother filter space provides an operator that takes as input an element of the group and acts on the filter to transform it. A filter-bank can thus be created by iterating this process with different group elements. Therefore, the selected group carries the characteristics of the filter-bank and consequently, the group transform and its time-frequency tiling. The building blocks of the WT through representation theory provided in Appendix A.2 Notice that further properties such as the invariant measure of the group and the resolution of the identity can be develop using the representation of the group.
A.2 EXAMPLE: THE WAVELET TRANSFORM
As an introductory example, we consider the creation of a wavelet filter-bank utilizing transformation group. Let's denote by Gaff the affine group, the so called "ax + group, where the elements (X, E I2 x R, where R1- (0, +x)), where the multiplicative operation of the group @ is defined by
(11)
Let's define by Par the representation of the affine group in (RR), i.e., Paft Gaft - GL (R)). such that Parr is homomorphism as per Definition defined as
2 Its action square integrable function và is
u " te X
(parlg) wi (t)
(12)
where (a, b) are respectively the dilation and translation parameters. The wavelet filter-bank is built by transforming mother filter, &i by the representation Par for specific elements of the group. A visualization of this approach for a Morlet wavelet filter can be seen in Figure The wavelet transform of a signal s; L2 is achieved by
W(s,, v)) = (si, Par Pan( Vg(A.)) E Gull,
(13)
(ss Part (a Vg(1.)) E Gaff,
(14)
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as the translation one. In the case where the translation Under review as a conference paper at ICLR 2020
where db(t) cb(-t), (...) denotes the inner product, the convolution, and (ar((())r)) the action of the operator Patr- evaluated at the group element ((,,)) on the mother filter v0 as per (12). In practice, the filter-bank is generated by sampling few elements of the group. For instance, the case of the dyadic wavelet transform, the dilation parameters follow a geometric progression of common ratio equals to 2. In general, the translation parameter is sampled according the scaling one (Daubechies) 1992). Notice that in the convolution expression (14), the translation parameter in fact the convolution operator parameter depends on the scaling one, a specific stride is used to perform the discrete convolution.
Note that the STFT can be constructed similarly utillizing the Weyl--eeienbbrrg group (Feichtinger et al. 2009), whose representation on (2 consists of frequency modulations and translations. More intricated group representations can be built as Torrésani (191]) where the combination of the affine group and Weyl-Heisenherg group is considered.
B ARCHITECTURE DETAILS
B.1 ARTIFICIAL DATA
Group Transform + Complex Modulus + Log
Dense Layer (1 sigmoid)
After the Group Transform, batch-normalization is applied.
B.2 SUPERVISED BIRD DETECTION
Group Transform + Complex Modulus + Log + Average-Pooling (stride:(1, 5125)) size:(1, 1024))
Conv2D. layer (16 filters 3 3) and Max-Pooling (3 x 3) and ReLU
Conv2D. layer (16 filters 3 3) and Max-Pooling (3 x 3) and ReLU
Conv2D. layer (16 filters 3 x 1) and Max-Pooling (3 x 1) and ReLU
Conv2D. layer (16 filters 3 1) and Max-Pooling (3 x 1) and ReLU
Dense layer (256) and ReLU
Dense layer (32) and ReLU
Dense layer (1 sigmoid)
At each layer a batch-normalization is applied and for the last three layers a 50% dropout is applied as in (Grill & Schlüter (2017)). The dimension of the input the DNN presented is the same for the different benchmarks.
B.3 HAPTICS DATA
Group Transform + Complex Modulus + Log + Average-Pooling (stride:(1,64 64) size: (1, 128))
Dense Layer (5 softmax)
After the Group Transform, batch-normalizatiio is applied.
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C
ADDITIONAL FIGURES
C.I ARTIFICIAL DATA
C.I.1 DATA
Figure 7: Artificial Dataset: (Top Left) Ascending Chirp, (Top Right) Descending Chirp, i.c class (Bottom Left) Ascending Chirp plus Gaussian noise, (Bottom Right) Descending Chirp plus Gaussian noise, i.e., class 1. The samples contained in the training and testing set are higher in frequency and close to the Nyquist frequency.
C.1.2 FILTERS
128
216
128
Figure 8: Learnable Group Transform Filters for the Artificial Data Each row displays two selected filters (left and right sub-figure) for different settings: (from top to bottom) LGT, nLGT, cLGT, enLGT. For each subfigure. the left part corresponds to the filter before training and the right part to the filter after training. The blue and red denote respectively the real and imaginary part the filters.
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C.1.3 GROUP TRANSFORM
Figure 9: Learnable Group Transform- Visualisation of an ascending chirp sample, where for each row (left) at the initialization and (right) after learning. Each row displays different setting: (from top to bottom): LGT, nLGT, cLGT, CnLGT.
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Figure 10: Learnable Group Transform Visualisation of a descending chirp sample, where for each row (left) at the initialization and (right) after learning. Each row displays a different setting: (from top bottom): LGT, nLGT, cLGT, CnLGT.
C.2 SUPERVISED BIRD DETECTION
C.2.1 DATA
Figure 11; Bird Detection Dataset- Sample containing bird song. The red boxes the locations of the bird song.
Each data sample, normalized, centered and subsampled by two before experiment.
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C.2.2 FILTERS
295
Figure 12: Learnable Group Transform Filters for the Bird Detection Data Each row displays two selected filters (left and right sub-figure) for different settings: (from top to bottom) LGT, nLGT. cLGT. For each subfigure, the left part corresponds to the filter before training and the right part to the filter after training.
C.2.3 GROUP TRANSFORM
Figure 13: Learnable Group Transform Visualisation of sample containing bird song, where for each row (left) at the initialization and (right) after learning. Each row displays different setting: (from top to bottom): LGT, nLGT, cLGT.
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Figure 14: Learnable Group Transform Visualisation of sample without a bird song, where for each row (leff) at the initialization and (right) after learning. Each row displays different setting: (from top bottom): LGT, nLGT, cLGT.
C.3 HAPTICS DATA
C.3.1 DATA
Figure 15: Haptic Dataset Sample of each class of the Haptic dataset.
Each data is centered and normalized, For the experiments, the number of epochs is set to 1000 and we perform early-stopping and obtain the testing accuracy at this specific epoch as in Khan & Yener (2018), the batch size set to 64. In order to avoid overfitting, perform different asymmetric zeros- prrdong on the training samples. For the testing samples, we perform symmetric zeros-padding (512 zeros each side of the signals).
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C.3.2 FILTERS
120
128
128
Figure 16: Learnable Group Transform Filters for the Haptics Data Each row displays two selected filters (left and right sub-figure) for different settings: (from top to bottom) LGT, nLGT, cLGT, enLGT. For each subfigure. the left part corresponds to the filter before training and the right part to the filter after training. The blue and red denote respectively the real and imaginary part the filters.
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C.3.3 GROUP TRANSFORM
Figure 17: Learnable Group Transform Visualisation of a sample belonging to class 1, where for each row (left) the initialization and (right) after learning. Each row displays different setting: (from top to bottom): LGT, nLGT, cLGT, CnLGT.
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Figure 18: Learnable Group Transform Visualisation of sample belonging class 2, where for each row (left) at the initialization and (right) after learning. Each row displays a different setting: (from top bottom): LGT, nLGT, cLGT, CnLGT.
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Figure 19: Learnable Group Transform Visualisation of sample belonging class 3, where for each row (left) at the initialization and (right) after learning. Each row displays a different setting: (from top bottom): LGT, nLGT, cLGT, CnLGT.
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Figure 20: Learnable Group Transform- Visualisation of a sample belonging to class 4, where for each row (left) at the initialization and (right) after learning. Each row displays a different setting: (from top bottom): LGT, nLGT, cLGT, CnLGT.
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Figure 21: Learnable Group Transform Visualisation of sample belonging to class 5, where for each row (left) at the initialization and (right) after learning. Each row displays a different setting: (from top bottom): LGT, nLGT, cLGT, cnLGT.
D GROUP PARAMETER OPTIMIZATION
In order learn the group transform module, we can use the back-propagation algorithm and gradient-based optimization technique such that the parameters of the group transform module, denoted by g. can be learned jointly with the parameters of the DNN, or any other differentiable algo- rithm taking as input the learnable time-fregue teeefreuuency representation. Using the notations of Section 3..] where L denotes a loss function and DNN, the learnability of the optimal group transform leading to the most suitable time- freeuucyy representation is performed by the chain rule,
ØL
aL 2 F(W, (gE,
OWF ss))
Vi E (l,..., NJ. Vk E fl,..., K}. Ih. E Gines
Oyk
Oy:.
where (g 8;)] is the convolution of the signals s; with the transformed filter Pine Pine(gr ) v as defined in (14)
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E PROOFS
Proposition 2. Pine is a group representation of Gine on (2(
E.1 PROOF THEOREM 2
Proof. Let 9,9 E Gines then
[pnn ())( v'lg' g)(())
(og((o(t))
and,
[Pine(g') (gne)g()c (t) = P (g) ( (a((t)
= <bggg(g (g(())
which verifies the homogeneity property. The linearity implied by.
[Pine (1) + (t)) (Not/ + (z))) (g(()) vr(g(()), Vt e R.
where 1/1, the (RR and K E R. is in fact Koopman operator Korda & Mezic 2018
E.2 PROOF THEOREM
Proof. Let T E I and g,g' E Gine.
Wi )s, (g. r) (g st, Pinc (g)())
(85, Pine (g)) ( Pinc (g))/)
(Sis Pinc (g'-1) ) Pinc (g)b=)
(si, Pino (i (g' g)b,))
W[ss,, Vg((g
g, T),
where #, denotes the filter v centered at position Then, there is not guarantee that this can be extrapolated to all 7 E R. ie.. the convolution case, except in the affine case where the global transformation matches the iteration of a local one.
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Reviews:
[{"review_id": "r1xHWpnatr", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "A typical Wavelet Transform is built through the dilation and/or rotation of a mother wavelet, which can been viewed as a group action on a mother wavelet. This work proposes to extend this construction beyond the Euclidean group, and to supervisedly learn operators that will be applied on a mother wavelet. Competitive numerical performances are obtained.\n\nOverall, I think that re-thinking the way a Wavelet Transform is designed, is an interesting direction of research, but I think some of the theoretical tools developed in this paper are not dedicated to achieve this purpose. In particular, the group/representation properties seem to not be used, and the authors could simply consider a specific subset of invertible mapping on $\\mathbb{R}^2$ which would be applied on the mother wavelet and lead to a Wavelet Transform. In other words, the overall formulation could be simplified.\n\n\nPros:\n- In general, the numerical experiments are at the level of the state of the art.\n- Parametrizing a subset of the group of increasing function and its application to signal processing tools is novel, to my knowledge.\n\nCons:\n- Some very relevant elements in the literature review are missing. Learning or using an underlying group of symmetry that will be combined with a deep neural network is not novel, cf: https://arxiv.org/abs/1601.04920 ; in particular for reducing the number of parameters, filters or samples: https://arxiv.org/abs/1809.06367 ; http://proceedings.mlr.press/v48/cohenc16.pdf ; https://arxiv.org/abs/1809.10200 ; https://arxiv.org/abs/1605.06644 - I think the authors should discuss at least one or two of those papers, if not all.\n- The performance on the bird detection task is good but the improvement compared to other work is not clear, given that some supervision in the first layer is incorporated.\n- Subsections 2.2 and 2.3 are difficult to parse because the authors introduce a lot of equations or notion that are not useful to understand their algorithm/method. The equation (6) seems wrong to me (one should consider t->s_i(-t) and not t->s_i(t) and b seems missing in the second line).\n- Figure 2 is difficult to read because of the illustrative graphics. Maybe a block schema would be easier to parse.\n- It seems to me that no-where the group properties are used, such as the stability to composition. In this paper, the authors simply try to parametrize a diffeomorphism to dilate the mother wavelet. From my understanding of 3.3, the subset of function used to approximate $G_{inc}$ do not form a subgroup as well, contrary to the Euclidean case, where for instance discrete rotations in the case of images are a finite group.\n- In subsection 4.1(Table 1), a comparison with a wavelet transform followed by a linear operator is compared with the proposed method. I find this result surprising : LGT/nLGT/cLGT/cnLGT and the WT are some linear methods whereas the STFT is non linear. As the WT should be unitary, if the linear classifier method is reasonably trained, then both methods should lead to the same result, except if the data are poorly conditioned. In which case, this experiment would not be meaningful. I think the authors should comment more this result because it is surprising.\n- I slightly disagree with the sentence \"in the case of WT, the precision in frequency degrades as the frequency increases\". Actually, the heisenberg principle is optimally optimized by wavelets, meaning that the area of the frequency/spatial support on a spectrogram is constant. On the contrary, the STFT has a lack of localisation (and thus the \"precision\" is not constant along frequencies). Maybe this could be rephrased slightly.\n- Given the filter learned in Figure 5, one can wonder if a foveal approach (i.e., Foveal Wavelets) could perform similarly? It would be interesting to display the littlewood-paley plot(i.e., the sum of the modulus of the filters in the Fourier domain) of this representation to understand the nature of this operator in the Fourier domain.\n- I think group actions could be considered instead of representations: it would be simpler to understand for a potential reader.\n\nTypos: \n- abstract \"in order to transform the mother ..\" > \"in order to transform a mother..\"\n- page 7 \"this variation is as not captured as well\" > \"this variation is not captured as well\".\n", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory."}, "extra_text": {}}, {"review_id": "r1gr5RqnYH", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "*Paper summary* \n\nThe authors design a learnable time-series pre-processing, which they refer to as a learnable group transform (LGT). This is a generalization of the wavelet transform, which maps a time-series signal onto the affine group. In the wavelet transform, multiple scaled and shifted versions of a mother wavelet are \"inner-producted\" with a signal; the resulting coefficents are the output of the transform. In the LGT, a more flexible transform that just scaling and shifting is applied to the shape of the mother wavelet, which is piece-wise linearly stretched. This elegantly encompasses time-warping and many other wavelet-style transforms into one learnable preprocessing step.\n\n*Paper decision* \n\nI would like to recommend this paper be accepted. It is clearly written and the idea is simple; that said, the idea is an elegant generalization of the wavelet transform. (I admit my own expertise is not in time-series data, so I may be mistaken). The experiments are also simple, but straightforward and easy to reimplement.\n\n*Supporting arguments* \n\nWhat I like about this paper is that the idea is a straightforward generalization of the wavelet transform through the lens of group theory. Furthermore, other transforms, such as the short-time fourier transform, or methods such as time-warping are easily encompassed by the method. By making the transform learnable, the authors reduce the number of in-built assumptions in the problem. \n\nI believe the idea is novel; although, since this is not my area of expertise, I defer to other reviewers who may wish to contest this.\n\nI do wonder, however, what the interpretation is behind some of the learned LGTs. Typically, a group transform is favored because of certain symmetry properties of the task at hand. For instance, wavelet transforms are shift-covariant, reflecting the shift-covariance of underlying signal statistics.\n\nExperimentally, I think there was a nice selection of toy and harder examples. I found the visualized filters and group transforms in the appendix really interesting. I think it would have been nicer to see some more analysis of the interpretation of the learned LGT in the main text though. Further, it would have been nicer to see some ablation studies conducted, such as varying the number of degrees of freedom in the LGT and seeing how that affects performance.\n\n\n*Smaller questions/notes for the authors*\n\n- The explanation of what a group is is quite high-level and I think if you did not know what it was beforehand, it would be hard to understand what one is. For instance, just after introducing groups, homomorphisms are mentioned. I would guess that someone who has never seen a group beforehand, would have no idea about homomorphisms.\n\n- In equation 6, is \\star a convolution or a cross-correlation? It looks like a cross-correlation to me.\n\n- I really liked the connection drawn between the group transform and time-warping. This is an aspect I personally had never considered.\n\n- Would it be possible to move some of the material on the learned filter transformations to the main text and write some analysis, even if it is only qualitative? I would find that fascinating.\n", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory."}, "extra_text": {}}, {"review_id": "BJgjzdW3KH", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper defines a set of learnable basis functions and a joint learning algorithm to estimate them. It is based on the premise that the common learning approach in time-series is to first represent them in some spectral domain; thus, the main problem is to define and estimate the basis functions. However, this premise is not accurate and many learning algorithms operate just in the actual time domain (even in speech).\n\nThe paper advocates for a special group of strictly increasing transformations of the basis functions. While in Section 3.2 the authors do explain the connections to the time-warping operation, they never explicitly justify why such a group is interesting. At least, it is not clear why they are not interested in a semi-group of (non-strictly) increasing functions, which is more flexible and used in DTW.\n\nThe authors implement the more general group using piece-wise linear functions, which makes enforcing the strictly increasing property easier. This idea is nice, however I am concerned with speed of such transformations in practice. Unfortunately the authors do not report runtime numbers in the experiments.\n\nWhile the paper has verbose and wordy writing, it also spends a major part of the paper describing the common knowledge about group theory and wavelets. Instead, some crucial design decisions are not well-justified. For example, the description of the constraints in cLGT suddenly appears.\n\nThere are several major problems in the experiments. First, the results in Tables 1 and 2 do not seem to be statistically significant and Table 3 does not have the intervals. Second, the Haptics dataset is too small. Also, choosing a single dataset from the UCR sets only sends the message that this dataset is one of the few datasets on which the algorithm has worked better.\nOverall, the key message out of the experiments is that if you project the data to a parameterized basis set and constrain the parameters, you will get better generalization, which is not very novel. \n\nOther than improving the experiments, the authors can significantly improve the writing by decreasing the emphasis on the background knowledge and elaborating more on the new proposed ideas.", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper received two weak rejects (3) and one accept (8).  In the discussion phase, the paper received significant discussion between the authors and reviewers and internally between the reviewers (which is tremendously appreciated).  In particular, there was a discussion about the novelty of the contribution and ideas (AnonReviewer3 felt that the ideas presented provided an interesting new thought-provoking perspective) and the strength of the empirical results.  None of the reviewers felt really strongly about rejecting and would not argue strongly against acceptance.   However, AnonReviewer3 was not prepared to really champion the paper for acceptance due to a lack of confidence.  Unfortunately, the paper falls just below the bar for acceptance.  Taking the reviewer feedback into account and adding careful new experiments with strong results would make this a much stronger paper for a future submission.
Decision: Reject