L EARNABLE G ROUP T RANSFORM F OR T IME -S ERIES
Anonymous authors
Paper under double-blind review
A BSTRACT
We 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 a 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.
I NTRODUCTION
The selection of the time-frequency 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 a wealth of applications such as health-care (Saritha et al., 2008; Cosentino et al., 2016)), environmental sound (Balestriero et al., 2018; Lelandais & Glotin, 2008), and seismic data analysis (Seydoux et al., 2016; Liu & Fomel, 2013).
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 all too common approach to performing inference on time-series consists of building a
Deep Neural Network (DNN) that operates on a 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 an expert in the signal at hand, or by considering the aforecited selection methods and their derivatives.
However, an inherent drawback is that the selection of the time-frequency transform is often achieved with criteria that do not align with 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 a large number of hyperparameters such as mother ﬁlter family, number of octaves, and number of wavelets per octave, size of the window (Cosentino et al., 2017).
To alleviate these drawbacks, we present a Learnable Group Transform (LGT) that can be learned jointly with the inference optimization problem and the data at hand.
Our methodology is developed through the lens of harmonic analysis , which generalizes the Fourier Transform using group theory.
We will study the learnability of a group transform .
Well-known group transforms are the Short-Time Fourier Transform (STFT) and the Continuous Wavelet Transform (CWT).
The theoretical building blocks of time-frequency analysis via group transforms are well developed in Coorbit Theory
Feichtinger & Gr¨ochenig (1989) and Generalized Coherent States Grossmann et al.
In order to build a group transform, one requires to select two elements; a mother ﬁlter and a group (Section 2).
The learnability of a mother ﬁlter has been already developed in Ravanelli & Bengio (2018);
Balestriero et al.
; Cakir et al.
; Zeghidour et al.
Recently, Khan & Yener (2018) investigated the learnability of the afﬁne transformations, that is, the sampling of dilation parameter of the afﬁne group inducing the CWT.
This ﬁlter-bank is then used to build a group transform of the signal.
In this work, we propose to extend the learnability of the afﬁne transform into strictly increasing and continuous functions enabling non-linear transformations of the mother ﬁlter (Section 3).
This generalization allows for greater ﬂexibility in the learnable spectral decomposition.
This ﬂexibility improves the linearization 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 are nuisances.
This speciﬁc transformation of a ﬁlter induces a ﬁlter with a non-linear instantaneous phase, which in turn, allows to span ﬁlters a la chirplets, which are of interest in a variety of domains such as biology and medicine, mechanics 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 to a larger class of spannable functions which improves the approximation property of the DNN at hand (Winkler & Le, 2017; Balestriero & Baraniuk, 2018).
In order to show the generality of our approach, we apply our algorithm on two diverse time-series classiﬁcation problems (Section 4).
B ACKGROUND AND N OTATIONS
We ﬁrst highlight the properties of certain group transforms by expressing their time-frequency tiling.
Then we develop the theoretical tools necessary to build a group transform and illustrate it via the wavelet transform.
2. T IME -F REQUENCY T ILING
the ”tiling” is induced by the learned group underlying the ﬁlter-bank decomposition.
The spread of a ﬁlter and its Fourier transform are inversely proportional as per the Heisenberg uncertanty principle (Mallat, 1999).
Following this principle, we can observe that in the case of STFT (respectively WT with a Gabor wavelet), at a given time τ , the signal is transformed by a window of constant bandwidth (respectively proportional bandwidth) modulated by complex exponential resulting in a uniform tiling (respectively proportional) on the frequency axis, Figure (1).
In the case of chirp-like ﬁlter, as proposed in Baraniuk & Jones (1996), each tile is a sheared rectangular, more generally, an afﬁnely transformed rectangular.
In this case, as well, the lower bound area of the sheared rectangular is constrained by the uncertainty principle.
As such, the understanding of the beneﬁts of various time-frequency 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 propose framework, the LGT allows for an adaptive tiling as illustrated in Figure (1) such that the trade-off between time and frequency precision depends on the task and data.
2. G ROUP A ND R EPRESENTATION
For further details on the group theoretical aspects described in this section, the reader should refer to
Vilenkin (1978).
Deﬁnition 1.
A group is a set G with a multiplicative operation (cid:12) that respects enclosure, identity element, inverse element, and associativity.
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 at hand.
The representation of a group can be thought as a farreaching generalization of the exponential function property, exp( x + y ) =
exp( x ) exp
( y ) , ∀ x, y ∈ R
(Baraniuk, 1993).
In fact, it is deﬁned as,
Deﬁnition 2.
A linear continuous representation ρ of a group G on the linear space H is deﬁned as
[Equation]
where GL ( H ) is the the group of linear map in H such that ∀ g, g
[Equation]
For instance, let H be a vector space such as R 3 , the representation of the group is induced by 3 ×matrices.
In this case, the operation on the right of (2) is a matrix multiplication, where each matrix depends on the group elements g and g (cid:48) .
This concept extends to linear operators acting on functional spaces.
As such, multiple transformations of a 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 a 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 a 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 is provided in Appendix A. 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.
L EARNABLE G ROUP T RANSFORM
We now develop the generalization of the afﬁne group and its representation on L 2 ( R ) .
Its application on a signal is depicted in Figure (2).
3. T HE G ROUP OF I NCREASING AND C ONTINUOUS F UNCTIONS
In order to generalize the classical afﬁne group, we propose the group of strictly increasing and continuous functions.
We deﬁne its multiplicative operation to be the function composition operation and denote this group by G inc = ( C inc ( R ) , (cid:12) ) , where
[Equation]
is strictly increasing
[Equation] [Equation]
and
[Equation]
where C ( R ) deﬁnes the space of continuous functions deﬁned on R .
The identity element of this group is the identity function on R , and the inverse element of g is g − 1 the inverse of the function g .
As this group allows non-linear transformations of the mother ﬁlter, the ﬁlter-bank derived based on this group has a higher adaptation capability for the pattern of interest in the time-series.
3. R EPRESENTATION OF THE G ROUP
In this work, we are interested in the action of this group on a prototype function.
As discussed earlier, the group transform is derived from the representation of the group on the mother ﬁlter space.
In this section, we present the representation of group G inc on square-integrable functions.
The transformation of square-integrable prototype functions based on this group allows us to span a wide range of group transforms as illustrated in Figure (1) and to generalize the wavelet transform.
Let’s deﬁne ρ inc ( g ) : G inc → GL ( L 2 ( R )) by [Equation]
[Equation]
where ψ denotes the mother ﬁlter.
Proposition 1.
ρ inc is a group representation of G inc on L 2 ( R ) .
The proof is given in Appendix E.1.
We can see that the increasing and continuous group representation operator ρ inc induces a mapping which depends on the group element g ∈ G inc .
For instance, if g = e , i.e., the identity element of the group, then we have ρ inc ( e ) ψ = ψ , it is in fact the identity operator in the space of the mother ﬁlter.
Given a mother ﬁlter ψ ∈ L 2 ( R ) , ρ inc ( g ) ψ, ∀ g ∈ G inc is a transformation of the mother ﬁlter with respect to the group element g belonging to the strictly increasing and continuous group which can be visualized in Figure (3).
This representation implies a transformation of the mother ﬁlter through a time transformation.
Note that in signal processing, such a time transformation is called warping (Goldenstein & Gomes, 1999; Kerkyacharian et al., 2004).
3. S AMPLING THE G ROUP
Sampling the group G inc can be achieved by a parametrization of strictly increasing and continuous functions.
In the present case, we propose to build a piece-wise afﬁne mapping constrained such that it belongs to the class of strictly increasing and continuous functions.
This constrained piece-wise afﬁne mapping is deﬁned as
[Equation]
where a = ( a 1 , . . . , a n ) , b = ( b 1 , . . .
, b n ) , 1 I l is the indicator function of the intervals
I l
=
[ t l , t l +1 ) , ∀ l ∈ { 2 , . . .
, n − 1 } and I 1 = ( −∞ , t 1 ) , I n =
[ t n , + ∞ ) , and a l and b l denote respectively the slope and offset of each piece of the function and n is the number of pieces.
As such, for each ( a , b ) satisfying the constraints (7) and (8) the function g ( a , b ) is a sample from the group G inc .
3. L EARNING THE C ONTINUOUS P IECE - WISE A FFINE I NCREASING M APS
The goal of learning the group transform leads to an optimization problem where the objective is to ﬁnd samples of the group that will produce the representation that minimizes a speciﬁc loss function, thus depending on the signals and the task.
Given a set of signals { s i ∈
L 2 ( R ) } Ni =1 and given a task speciﬁc loss function L , we aim at solving the following optimization problem [Equation]
[Equation]
where N denotes the number of signals, K the number of ﬁlters, F represents a DNN, Ω k = { a k ∈ R n + , b k ∈ R n | b ( k,l +1) =
( a ( k,l ) − a ( k,l +1) )
t ( l +1 ,k ) +
b ( k,l ) }∀ k ∈ { 1 , . . .
, K } , and W [ s i , ψ ]( g , . ) =
[ W [ s i , ψ ]( g ( a 1 , b 1 ) , . ) , . . .
, W [ s i , ψ ]( g ( a K , b K ) , . )]
T , and
[Equation] a k ,
[Equation] where ¯ ψ ( t ) = ψ ( − t ) and ( . ) corresponds to the time axis.
To solve this optimization problem for diverse time-series, we propose different settings that might be more adapted depending on the type of data and task at hand.
We ﬁrst propose a normalization of the frequency 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 ˆ f , the normalized frequency f with respect to the maximum slope of the piece-wise afﬁne mapping.
For instance, in the case of a Morlet wavelet, the normalization is as follows
[Equation]
where ˆ f = f/ max l ∈{ 1 ,...,n } a l , j is the imaginary unit, and σ is the width parameter deﬁning the localization of the wavelet in time and frequency.
This normalization will be performed for each sample of the group, and thus for each generated ﬁlter k ∈ { 1 , . . .
, K } of the ﬁlter-bank.
The second setting is a constraint on the domain of the piece-wise afﬁne map as derived in (6) (denoted in the result tables by cLGT).
In the following experiments, we propose a dyadic constraint of the domain as in the WT.
The support of the ﬁlter will thus be close to the support of a wavelet ﬁlter-bank.
However, the envelop of the ﬁlter and the instantaneous frequency will still vary as in the
Chirplet Transform (Baraniuk & Jones, 1996).
3. G ROUP E QUIVARIANCE
The equivariance-invariance of data representations plays a crucial role in the efﬁciency of the algorithm at hand (Mallat).
Exploiting this property can for instance help to reduce the sample complexity of the dataset Cohen & Welling (2016). We propose to characterize the equivariance properties of the developed group transform by considering local equivariance of the representation. Since W ( ., . ) employs convolution, for any group G , this representation is always translation equivariant. We now focus on others equivariance properties.
We recall that the equivariance of a transform is
[Equation]
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 = G aff , i.e., WT, the transformation is equivariant to scaling and translation.
Then, we deﬁne local equivariance by,
[Equation]
That is, the representation of a local transformation of a signal in a window centred at τ is equals to the transformation of the representation at τ .
The size of the window depends on the support of the ﬁlter.
As a matter of fact, assuming that the representation of G inc is unitary, we have that,
Theorem 1.
The group transform induced by G inc is locally equivariant.
Refer to Appendix E.2 for the proof.
For all the experiments and all the settings, i.e., LGT, nLGT, cLGT, cnLGT, the increasing and continuous piece-wise afﬁne map is 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 is a Morlet wavelet ﬁlter.
The code of our LGT framework will be provided on the Github page of the ﬁrst author.
4. A RTIFICIAL D ATA : C LASSIFICATION OF C HIRP S IGNALS
We present an artiﬁcial dataset that demonstrates how a speciﬁc time-frequency tiling might not be adapted or would require cross-validations for a given task and data.
To build the dataset, we generate one high frequency ascending chirp and one descending high-frequency 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.1).
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.
For the case of WT and LGT, the size of the ﬁlters is 512 .
As we can observe in Table (1), 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 of WT, we know from Figure 1 that it has a poor frequency resolution in high frequency.
Therefore, through time, the small frequency variations of the chirp are not efﬁciently captured.
[Table]
In the case of STFT, if the number of ﬁlters is too small, the resolution in frequency is also limited, and thus this variation is not captured as well.
This illustrated the fact in some case neither the proportional-bandwidth nor the constant-bandwidth are suitable.
However, using a 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 case of the LGT, the tiling has adapted to the task and produces good performances except in the cLGT setting.
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 is not suitable for this speciﬁc task.
For all settings, the visualization of the ﬁlters, as well as the representations of the signals, can be found in Appendix (C.1.2,C.1.3).
4. S UPERVISED B IRD D ETECTION
[Table]
The proposed Learnable Group Transform is applied to a challenging supervised bird detection task.
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 (C.2.1).
The given task is a binary classiﬁcation where one should predict the presence or absence of bird song.
As the dataset is unbalanced, we use the Area Under Curve (AUC) metric.
The results we propose for both the benchmarks and our models are evaluated on a test set consisting of 33% of the total dataset.
In order to compare with previously used methods, we use the same seeds to sample the train and test set, the batch size, i.e., 10 , and the learning rate cross-validation grid as in Balestriero et al.
For each model, the best hyperparameters are selected, and we train and evaluated randomly 10 -times the models with early stopping, the results are shown in Table (4.2).
While the ﬁrst layer of the architecture has a model-dependent representation (i.e., MFSC, LGT, Conv. ﬁlters,...), we use the state-of-the-art architecture (Grill & Schl ¨ uter (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 4.2, the case without constraints (LGT) reaches better accuracy than the domain expert benchmark (MFSC), showing the ability of such transformation to tile the time-frequency plane according to the task and data at hand.
Besides, including more constraints on the model (cnLGT) reduces overﬁtting and further improve results to outperform the other benchmarks. One can notice that all the learned ﬁlters in Figure C.2.2 contain either an increasing chirp or a decreasing chirp, corresponding respectively to the convexity or concavity of the instantaneous phase of the ﬁlter and thus of the piece-wise linear map. Such a feature is being used and is crucial in the detection and analysis of bird song (Stowell & Plumbley, 2012) [Table]
4. H APTICS D ATASET C LASSIFICATION
The Haptics dataset is a classiﬁcation problem with ﬁve classes and 155 training and 308 testing samples from the UCR Time Series Repository Chen et al [Table]
[Table] where each time-series has 1092 time samples.
As opposed to the bird dataset where features of interests are known
[Table]
and competitive methods have been established, there are no hand-crafted features that can perform accurately (see Table 3)
[Table]
Accuracy for the Haptics Classiﬁcation Task One can see that our method outperforms other approaches in the cLGT setting while performing the classiﬁcation with a linear classiﬁer as opposed to other methods using DNN algorithms.
This demonstrates the capability of our method to transform the data efﬁciently while not requiring a further change of basis.
Besides, this shows that even in a small dataset setting, our approach is capable of learning an efﬁcient transformation of the data.
C ONCLUSION
We developed a novel group transform introducing the group of strictly increasing and continuous functions as well as a tractable way to sample it.
From bird detection to haptics classiﬁcation, our approach competes with state-of-the-art methods without a priori knowledge on the signal power spectrum and outperform classical hand-crafted time-frequency representations.
Interestingly, in the bird detection experiment, we recover chirplet ﬁlters that are known to be crucial to their detection, while in the case of the Haptic dataset where important feature to be captured to perform accuratly the classication of the signals is unknown, the ﬁlters learned are very dissimilar to classical timefrequency transfoms.
While we have considered only the use of a Morlet mother wavelet, a future approach would be to explore the learnability of the group with the learnability of the mother wavelet. A A G ROUP T RANSFORM : T HE W AVELET T RANSFORM
As an introductory example, we consider the creation of a wavelet ﬁlter-bank utilizing transformation group.
Let’s denote by G aff the afﬁne group, the so called ” ax + b ” group, where the elements ( λ, τ ) ∈ R (cid:63) + × R , where R (cid:63) + = (0 , + ∞ ) , where the multiplicative operation of the group (cid:12) is deﬁned by
[Equation]
[Equation] Let’s deﬁne by ρ aff the representation of the afﬁne group in L 2 ( R ) , i.e., ρ aff : G aff → GL ( L 2 ( R )) , such that ρ aff is a homomorphism as per Deﬁnition 2. Its action on square integrable function ψ is deﬁned as [Equation]
(14) where ( a, b ) are respectively the dilation and translation parameters.
The wavelet ﬁlter-bank is built by transforming a mother ﬁlter, ψ by the representation ρ aff for speciﬁc elements of the group.
A visualization of this approach for a Morlet wavelet ﬁlter can be seen in Figure (3).
The wavelet transform of a signal s i ∈ L 2 ( R ) is achieved by [Equation] [Equation]
where ¯ ψ ( t ) = ψ ( − t ) , (cid:104) .
(cid:105) denotes the inner product, (cid:63) the convolution, and ρ aff ( g ( λ,τ ) )
ψ the action of the operator ρ aff , evaluated at the group element g ( λ,τ ) , on the mother ﬁlter ψ as per (14).
In practice, the ﬁlter-bank is generated by sampling a few elements of the group.
For instance, in 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 to the scaling one (Daubechies, 1992).
Notice that in the convolution expression (16), the translation parameter τ = 0 , in fact the convolution operator (cid:63) acts as the translation one.
In the case where the translation parameter depends on the scaling one, a speciﬁc stride is used to perform the discrete convolution. Note that the STFT can be constructed similarly utilizing the Weyl-Heisenberg group (Feichtinger et al., 2009), whose representation on L 2 ( R ) consists of frequency modulations and translations. More intricated group representations can be built as in Torr´esani (1991) where the combination of the afﬁne group and Weyl-Heisenberg group is considered. B A RCHITECTURE D ETAILS B. A RTIFICIAL D ATA Group Transform + Complex Modulus + Log Dense Layer ( 1 sigmoid)
After the Group Transform, a batch-normalization is applied.
B. S UPERVISED B IRD D ETECTION
Group Transform + Complex Modulus + Log + Average-Pooling (stride: (1 , 512) size: (1 , 1024) )
Conv2D. layer ( 16 ﬁlters 3 × 3 ) and Max-Pooling ( 3 × 3 ) and ReLU Conv2D. layer ( 16 ﬁlters 3 × 3 ) and Max-Pooling ( 3 × 3 ) and ReLU
Conv2D. layer ( 16 ﬁlters 3 × 1 ) and Max-Pooling ( 3 × 1 ) and ReLU
Conv2D. layer ( 16 ﬁlters 3 × and Max-Pooling ( 3 × 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 & Schl¨uter (2017)).
The dimension of the input of the DNN presented is the same for the different benchmarks.
[Figure]
Group Transform + Complex Modulus + Log + Average-Pooling (stride: (1 , 64) size: (1 , 128) )
Dense Layer ( 5 softmax)
After the Group Transform, a batch-normalization is applied.
C A DDITIONAL F IGURES
[Figure] [Figure]
[Figure] [Figure]
[Figure] [Figure]
Each data is centered and normalized.
For the experiments, the number of epochs is set to and we perform early-stopping and obtain the testing accuracy at this speciﬁc epoch as in Khan & Yener (2018), the batch size was set to 64 .
In order to avoid overﬁtting, we perform different asymmetric zeros-paddings on the training samples.
For the testing samples, we perform a symmetric zeros-padding ( 512 zeros on each side of the signals).
[Figure] [Figure]
D G ROUP P ARAMETER O PTIMIZATION
In order to learn the group transform module, we can use the back-propagation algorithm and a 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 algorithm taking as input the learnable time-frequency representation.
Using the notations of Section 3.
where L denotes a loss function and F a DNN, the learnability of the optimal group transform leading to the most suitable time-frequency representation is performed by the chain rule,
[Equation]
N
[Equation] [Equation]
. . . , K } , g k ∈ G inc
[Equation]
where [ W ψ ( g k , s i )] is the convolution of the signals s i with the transformed ﬁlter ρ inc ( g k ) ψ as deﬁned in (16).
E P ROOFS E. P ROOF T HEOREM
Proof.
Let g, g (cid:48)
∈ G inc , then
[Equation]
and,
[Equation]
which veriﬁes the homogeneity property.
The linearity is implied by,
[Equation]
where ψ 1 , ψ 2 ∈ L 2 ( R ) and κ ∈ R .
It is in fact a Koopman operator Korda & Mezi´c (2018). E. P ROOF T HEOREM
Proof.
Let τ ∈ R and g, g (cid:48) ∈ G inc ,
[Equation]
where ψ τ denotes the ﬁlter ψ centered at position τ .
Then, there is not guarantee that this can be extrapolated to all τ ∈ R , i.e., in the convolution case, except in the afﬁne case where the global transformation matches the iteration of a local one.