# Deep Generative Models of Gravitational Waveforms via Conditional
Autoencoder

Chung-Hao Liao<EMAIL_ADDRESS>Department of Physics, National Taiwan
Normal University, Taipei 11677, Taiwan  Feng-Li Lin<EMAIL_ADDRESS>corresponding author. Department of Physics, National Taiwan Normal
University, Taipei 11677, Taiwan  Center of Astronomy and Gravitation,
National Taiwan Normal University, Taipei 11677, Taiwan

###### Abstract

We construct few deep generative models of gravitational waveforms based on
the semi-supervising scheme of conditional autoencoders and their variational
extensions. Once the training is done, we find that our best waveform model
can generate the inspiral-merger waveforms of binary black hole coalescence
with more than $97\%$ average overlap matched filtering accuracy for the mass
ratio between $1$ and $10$. Besides, the generation time of a single waveform
takes about one millisecond, which is about $10$ to $100$ times faster than
the EOBNR algorithm running on the same computing facility. Moreover, these
models can also help to explore the space of waveforms. That is, with mainly
the low-mass-ratio training set, the resultant trained model is capable of
generating large amount of accurate high-mass-ratio waveforms. This result
implies that our generative model can speed up the waveform generation for the
low latency search of gravitational wave events. With the improvement of the
accuracy in future work, the generative waveform model may also help to speed
up the parameter estimation and can assist the numerical relativity in
generating the waveforms of higher mass ratio by progressively self-training.

##  I Introduction

LIGO/Virgo has detected about a hundred of compact binary coalescence (CBC) up
to its O3 observations Abbott:2016blz ; LIGOScientific:2018mvr ;
Abbott:2020niy . This is remarkable achievement of modern science. Due to the
limitation of LIGO/Virgo’s sensitivity, these events are detected by the
method of matched filtering schutz_1991 ; Owen_1996 ; Owen_1999 , which
calculates the overlap between the whitening data and the theoretical
gravitational waveform templates. Similarly, the source properties of these
events are also extracted based on matched filtering to perform the Markov-
Chain-Monte-Carlo (MCMC) Bayesian parameter estimation (PE)
veitch2015parameter ; biwer2019pycbc . In both processes of detection and PE
of gravitational wave events, a huge number of theoretical waveform templates
are required for matched filtering, therefore the efficiency of evaluating
waveform templates is crucial for detection and to accelerate the PE
procedures. However, due to the nonlinear feature of Einstein gravity and the
unavoidable strong gravity regime for the mergers of two compact objects, it
is notoriously difficult to calculate the CBC dynamics and the associated
gravitational waveforms. For example, it is known Hinder:2010vn ;
Pfeiffer_2012  to require about hundred thousand CPU hours to obtain a state-
of-art CBC waveform by solving numerical relativity. The required computing
time will be increased by one or two order for the higher mass-ratio CBC
events, and is beyond what the current computing facility can afford. Thus, it
is impractical to adopt such ab initio waveforms directly for performing
either detection or PE.

To accelerate the generation of theoretical waveforms for practical
applications, some analytical waveform models are introduced with a few
parameters to be fitted by the results of numerical relativity. The well-known
examples are IMRPhenomP models Hannam:2013oca ; Khan:2018fmp , the synergy
models Buonanno_2007 ; Pan_2011 ; Ajith_2011  by combining the post-Newtonian
Einstein:1938yz ; Blanchet:2013haa , effective-one-body (EOB) formalism
PhysRevD.59.084006 ; Buonanno_2000 , black hole perturbation Kokkotas_1999 ;
Nollert:1999ji  and numerical relativity Centrella:2010mx ; Loffler:2011ay ,
and the reduced order models or surrogate models Blackman_2015 ; P_rrer_2016 ;
Williams_2020  which span the generic waveforms with some orthonormal basis.
However, it still takes few hundredths to few tenths of a second to evaluate a
single waveform based on the aforementioned analytical waveform models 111This
can be estimated by generating the waveforms from template library in either
PyCBC or GstLAL.. By this speed of waveform generation, it will usually take
weeks or even months to obtain the state-of-art PE results for a single event
based on the MCMC algorithm. One can then expect the overall computing power
cost or time-span for PE will increase rapidly for the latter operations of
LIGO/Virgo/KAGRA such as O4 or O5, for which the number of detection CBC
events will be increased by an order or more. Therefore, the speed-up of
waveform generation becomes a pressing issue even in the near future.

Besides, those aforementioned analytical waveform models are in nature
interpolating models by fitting the parameters with a known set of waveforms.
This implies that the model could become more complicated and cumbersome when
the range of the waveforms are extended, such as going to higher mass ratio.
The increase of the complexity will reduce the models’ efficiency of
generating the real-time waveforms for the detection or PE. Thus, it is
crucial to have some extrapolating models of waveform generation to resolve
the conflict between complexity and efficiency of the traditional analytical
waveform models.

Figure 1: A generative model of gravitational waveforms: Once the neural
network model is well-trained, it can generate the gravitational waveforms
with more than $95\%$ accuracy when providing just the source labels such
masses $(m_{1},m_{2})$ of the binary black holes. The accuracy rate of the
waveforms is defined in (8) below. As a preliminary study for the proof-of-
concept, in this work we mainly consider the inspiral-merger parts of the full
waveforms.

Motivated by the above discussions on the limitations of the known models of
waveform generation, we turns to the deep learning for the resolution. We aim
to construct some deep learning neural network to generate the CBC
gravitational waveforms of high accuracy by giving the source parameters such
as the masses, spins of the binary compact objects, as schematically depicted
in Fig. 1. Even the training time will be increased as the training set is
enlarged, the time of evaluating a new waveform with the trained machine will
not be increased much. This then resolves the conflict between the complexity
and efficiency for the real-time applications.

Moreover, we also hope this deep learning neural network to be generative so
that it can generate the waveforms which do not belong to the source parameter
ranges of the training set. For example, we can train the machine with
waveforms of only low mass ratio (LMR), and then generate the accurate
waveforms of higher mass ratio (HMR). This could help to efficiently obtain
the HMR waveforms, which will be computationally costly by numerical
relativity. However, in this work we will not explore this scenario but just a
toy version, for which we employ the training set containing small fraction of
HMR waveforms to demonstrate the possibility.

In view of the above target features, this deep learning machine should be the
supervised one when training with the given source parameters and the
associated waveforms. On the other hand, it is also better to be generative
and the unsupervised one so that it has the potential to turn into an
extrapolating model of generating the HMR waveforms. For this purpose, in this
paper we adopt the conditional (variational) autoencoder (cAE or cVAE)
Sohn2015LearningSO ; nguyen2017plug ; tonolini2020variational  to construct
various deep learning models to generate CBC gravitational waveforms 222The
cVAE framework is recently adopted as the generative model of posteriors for
the PE of the CBC events, see gabbard2019bayesian ; Green:2020hst ;
green2020complete .. This scheme belongs to the so-called semi-supervised
learning by combining both features of supervised and unsupervised learning
333For the basic discussion of supervised and unsupervised learning, please
see GoodBengCour16 ; Carleo_2019 .. It is built on a more basic scheme for the
unsupervised learning, the autoencoder (AE) 10.5555/2987189.2987190 , or its
generative extension, the variational autoencoder (VAE) kingma2019introduction
; yu2020tutorial . We will introduce the basics of these neural networks in
the next section.

As a preliminary study for the proof-of-concept, in this work we mainly
consider the inspiral-merger parts of the full waveforms but truncating the
ringdown part. We find that our best generative models can produce the
waveforms with accuracy higher than $97\%$ even for the generation of HMR
waveforms. Moreover, it can produce a single waveform within one millisecond,
which is about 10 to 100 times faster than producing an EOB waveform on the
same computing facility. To visualize the accuracy rate, in Fig. 2 we shows
some typical examples of the waveforms with different accuracy rates.

Figure 2: Some typical examples of the generated waveforms by well-trained
cVAE models with different accuracy rates. Top-left: a generated inspiral-
merger waveform of $99.50\%$ accuracy when comparing with the corresponding
EOB waveform by overlap match. Top-right: a generated inspiral-merger waveform
of $92.37\%$ accuracy. Bottom: A generated full inspiral-merger-ringdown
waveform of $99.74\%$ accuracy.

The rest of the paper is organized as follows. In the next section, we will
briefly sketch the basics of autoencoder and its extensions including VAE and
the conditional versions. In section III we describe the tomography of our
training data set, and how we prepare our training waveforms. Besides, the
fitting factor or faithfulness (FF) based on the overlap of matched filtering
is introduced to characterize the accuracy of the generative waveform models.
In section IV we consider four waveform models based on the cAE scheme, and
then summarize their accuracy and run-time in Table 2 and 4, respectively. By
comparing the accuracy, we pick up the best cAE waveform model and present its
detailed information. Finally, we conclude this paper in section V. In the
Appendix, we present the performance of the cVAE counterparts of the cAE
waveform models considered in the main text.

##  II Autoencoder and its extensions

Our goal is to construct some deep learning models of gravitational waveforms
as depicted in Fig. 1. The basic structure of this generative model is the so-
called autoencoder (AE) 10.5555/2987189.2987190  or it extension, the
variational auto encoder (VAE) kingma2019introduction ; yu2020tutorial . The
basic structure of AE and VAE is shown in Fig. 3, which contains two parts:
the encoder and the decoder. The encoder (denoted by $q_{\phi}(z|x)$ with
$\phi$ the abbreviation of biases and weights of the encoder’s neural network)
compress the input data $x$ into the latent layer $z$ of smaller dimensions
than the ones of $x$, and then the decoder (denoted by
$p_{\theta}(\tilde{x}|z)$ with $\theta$ the abbreviation of biases and weights
of the decoder’s neural network) un-compress the latent layer back to the
final result $\tilde{x}$ of the same dimension as $x$. One then use some
distance measure such as mean-squared error (MSE) between $x$ and $\tilde{x}$
as the reconstruction loss. The goal is to minimize the reconstruction loss to
optimize the biases and weights of the whole AE’s neural network. Since there
is no label for the input data, this is the unsupervised learning.

Figure 3: Schematic structure of a AE or VAE. It contains two components. (i)
An encoder $q_{\phi}(z|x)$ which transforms an input vector $x$ to a latent
vector $z$, which is deterministic for AE, but stochastic for VAE, i.e.,
$z=\mu_{\phi}(x)+\sigma^{2}_{\phi}(x){\cal N}(0,1)$. Here ${\cal N}(0,1)$ is
the unit normal distribution. (ii) An decoder $p_{\theta}(\tilde{x}|z)$ which
transforms $x$ to an output $\tilde{x}$. The loss function of AE is just the
reconstruction loss such as mean-squared error (MSE) between $\tilde{x}$ and
$x$. On the other hand, the loss function of VAE contains two parts: the
reconstruction loss and the Kullbac-Leibler (KL) loss as discussed in (1).

Since the AE is a deterministic machine so that it may lack the power of
extrapolations and could fail to be generative. To remedy this drawback, the
VAE is introduced by making the latent layer a stochastic one. This is done by
generating the means and variances of the Gaussian distributions as the output
of the encoder, from which one can sample a latent layer as the input to the
decoder, as shown in the middle of Fig. 3. The uncertainty of the layer make
the VAE to be able to “think out of the box”, thus a generative machine.
However, besides the reconstruction loss one should also consider the
regularization loss which characterizes how much the stochastic latent layer
deviates from ${\cal N}(0,1)$, i.e., the unit Gaussian with zero mean. This is
measured by their Kullbac-Leibler (KL) divergence. It turns out that the
combined loss is equal to upper bound of the negative of the log likelihood of
the input data distribution $p_{\theta}(x)$, i.e.,

$\displaystyle-\log p_{\theta}(x)$ $\displaystyle\leq$ $\displaystyle{\bf
E}_{z\sim q_{\phi}(z|x)}[-\log p_{\theta}(\tilde{x}|z)]\qquad$ (1)
$\displaystyle+$ $\displaystyle{\bf D}_{KL}[q_{\phi}(z|x)||{\cal
N}(0,1)]\qquad$

where the first term on the right-hand-side is the reconstruction loss and the
second term is the regularization loss.

Figure 4: The schematic structure of a cAE or cVAE as a generative waveform
model. Left panel: During the training period, it needs two encoders: (a) one
for training the input data such as strains/waveforms, and (b) one for
training the source labels associated with the input data such as
$(m_{1},m_{2})$. Right panel: After the training, the encoder (a) is removed
so that it becomes a generative model, namely it generates waveforms by
providing only the associated source labels.

When training the waveform models, the input $x$ from the training data set is
the theoretical waveform such as EOB waveform, and we call it strain for
short. We can choose the reconstruction loss to be the MSE between $x$ and
$\tilde{x}$. The training process is to optimize the biases and weights of the
neural network by minimizing the reconstruction loss such that the generated
$\tilde{x}$ can be as close to $x$ as possible. After the training, the
decoder can be turned into a generative model of strains, namely, given some
input latent vector, the decoder will output some strain. However, this
machine is not so useful in generating the strains with specific source
properties because the latent space may not correspond to the required
parameter space of the physical source properties, such as masses
$(m_{1},m_{2})$ of the binary black holes. For convenience we call the source
parameters the labels. To make the AE or VAE useful for our purpose, we adopt
the way of semi-supervised training by also conditioning the labels when
training the machine. After the training we will truncate the encoder part
associated with the strain input, the remaining one with the label as input
will then become the useful generative model of strains, namely, given the
label such as $(m_{1},m_{2})$, the machine will generate the associated
strain. The above scheme is called the conditional AE/VAE abbreviated as
cAE/cVAE Sohn2015LearningSO ; nguyen2017plug ; tonolini2020variational , and
the basic structure is depicted in Fig. 4 where an additional encoder for the
labels of input data is introduced.

Due to the additional encoder, we now have two latent vectors $z_{1}$ and
$z_{2}$ as shown in Fig. 4. We can then introduce the latent loss to measure
their difference. For AE, the latent loss can be MSE between latent vectors,
but for VAE it is the KL divergence between the Gaussian distributions
generated by the two encoders. On the other hand, the reconstruction loss is
the MSE for both AE or VAE.

##  III Waveform data preparation and overlap accuracy

Once we construct the code for cAE or cVAE, we prepare a set of strains to
train the machine. A strain is the linear combination of the two polarization
modes $h_{+}(t)$ and $h_{\times}(t)$, i.e.,

$h(t)=h_{+}(t)+ih_{\times}(t)\;.$ (2)

To be specific, we consider the inspiral-merger part of the CBC strains of
binary black holes with their masses $(m_{1},m_{2})$ as the only source
parameters, i.e., without spin and procession. We further divide the set into
two subsets, one is called the low-mass-ratio (LMR) set for $q=m_{2}/m_{1}\leq
5$ and the other one called the high-mass-ratio (HMR) set for $q>5$. These
strains are obtained from ENOBNR Buonanno_2007  of the PyCBC library
biwer2019pycbc , in which each strain is divided into $8192$ time segments. We
basically train the machine mainly with the LMR set but combining with about
$20\%$ of HMR strains. The latter is used as the tutor seed to train the
machine toward a generative model for the other $80\%$ of HMR strains. The
tomography of our training and test sets is shown in Fig. 5, and in Table 1 we
give the more details for the specs of this tomography. Moreover, the fraction
of the HMR templates in the total training set is only about $2.46\%$. This
tiny fraction is chosen on purpose to mimic as closely as possible the real
extrapolating model for which the training set contains no HMR waveform.

Figure 5: Tomography of data set for training, validation and test of a cAE model with the numeric as listed in Table 1. Top: Overview of the tomography. Bottom: Enlarged view of some portion circled in the Top figure, with more clear visual estimate.  | $m_{1}$ | $m_{2}$ | $q$ | $\Delta m$ | train | valid | test
---|---|---|---|---|---|---|---
$q\leq 5$ | [5.0,75.0] | [5.0,75.0] | [1,5] | 0.25 | 24865 | 3552 | 7104
$q>5$ | [5.0,75.0] | [5.0,75.0] | [5,10] | 0.25 | 682 | 36 | 2873
Table 1: The range of source parameters and the amounts of the data set. Here
the mass ratio is denoted by $q\equiv m_{2}/m_{1}$ with $1\leq q\leq 10$. The
corresponding percentages of (training, validation, test) is (70%, 10%, 20%)
for LMR (low-mass-ratio) data ($q\leq 5)$, and is (19%, 1%, 80%) for HMR
(high-mass-ratio) data ($q>5$). Note that the fraction of the HMR templates is
only about $2.46\%$ of the total training data set, including both training
and validation data.

Figure 6: A typical generative waveform (blue color) from a trained cAE or
cVAE by inputting a inspiral-merger parts of the CBC strains (orange color) in
a time series format. It cannot catch both the phase and amplitude at the same
time.

As a preliminary study to demonstrate that a generative model of gravitational
waveform is in principle possible, we will not consider the full CBC strain
but truncate the ringdown part which is far shorter than the other part of the
strain. The truncated waveform is denoted as the inspiral-merger strain. The
purpose of this truncation is to further reduce the complexity of the
frequency/amplitude part of a strain caused by the sudden change at the
merger, and will help to well train the machine with less efforts in tuning
the hyper-parameters.

Using the time series form of the inspiral-merger strains to train a cAE or
cVAE, the result turns out to be not good for reasonable machine size and
training, see Fig. 6 for a typical result. It implies that the model cannot
catch up the amplitude and phase correctly at the same time. This suggests
that this form of strain is still too complicated for a cAE or cVAE of
reasonable size to work properly. Motivated by this result, we then decide to
separate the amplitude and frequency parts of a strain, and then juxtapose
them as the input of the cAE or cVAE. To be specific, from the two
polarization modes we first obtain the instantaneous phase

$\theta(t)=\tan^{-1}\Big{(}{h_{\times}(t)\over h_{+}(t)}\Big{)}$ (3)

and the instantaneous frequency and amplitude are given respectively by

$\displaystyle\omega(t+{\delta t\over 2})={\theta(t+\delta t)-\theta(t)\over
2\pi\delta t}$ (4) $\displaystyle A(t)=\sqrt{h^{2}_{+}(t)+h^{2}_{\times}(t)}.$
(5)

A typical example showing the above decomposition is given in Fig. 7.

Figure 7: Decomposition of a time series strain $h(t)$ (top) into frequency
(bottom-left) and amplitude (bottom-right) by using (4) and (5). The amplitude
part has been multiplied by a factor of $10^{20}$ to have comparable scale
with the frequency one.

Even using this frequency/amplitude separated form of the strains to train the
cAE or cVAE model, the result is still not good because the magnitudes of the
input data have not been rescaled to avoid too small or too large values. This
however can be solved as in the usual deep learning process for neural network
by just normalizing the input data franoischollet2017learning . The way of
normalization we adopt is as follows:

$\hat{\omega}(t)={\omega(t)-\mu_{\omega}\over\sigma_{\omega}}\;,\qquad\hat{A}(t)={A(t)-\mu_{A}\over\sigma_{A}}$
(6)

where the normalization parameters $(\mu_{\omega},\sigma_{\omega})$ are
respectively mean and variance evaluated from the 8192 segments of $\omega(t)$
444Due to the nature of its definition from the difference between two
neighbor segments, there are only $8191$ segments for $\omega(t)$., and
similarly for $(\mu_{A},\sigma_{A})$. We call these four parameters the key
(to reconstruct the associated un-normalized strain).

Naively, we can juxtapose these four normalization parameters with the
normalized strain vector $(\hat{\omega}(t),\hat{A}(t))$ of $2\times 8192$
segments to form the input of cAE or cVAE. However, their dimensions are not
in proportion, the juxtaposition could suppress the significance of the key
during the training, which will induce the unbearable error in recovering the
full strain via (6). Therefore, we need to find the appropriate cAE or cVAE
schemes to well train both the normalizing strains and the associated keys to
the desirable accuracy.

Once the training of a waveform deep learning model is done, we need to
evaluate their performance based on some criterion of accuracy by comparing a
machine-generated waveforms $h_{\rm ML}(t)$ with the corresponding waveform
$h_{\rm EOB}(t)$ obtained from EOBNR. To calculate the accuracy we adopt the
conventional overlap method used in gravitational waveform community. The
overlap method is motivated by the matched filtering schutz_1991 ; Owen_1996 ;
Owen_1999  for the signal detection or parameter estimation, in which the
overlap between two waveforms $h_{1}(t)$ and $h_{2}(t)$ is defined by

$\langle
h_{1}|h_{2}\rangle=4\;{\textrm{Re}}\int_{0}^{\infty}\frac{\tilde{h}_{1}(f)\tilde{h}_{2}(f)^{*}}{S_{n}(f)}df$
(7)

where $\tilde{h}_{i}(f)$ is the Fourier transform of $h_{i}(t)$ and $S_{n}(f)$
is the power spectral density (PSD) of detector’s noise. In practical, some
appropriate low and high frequency cutoffs will be imposed when performing the
integral. To evaluate the accuracy of a waveform model, the following fitting
factor (FF) or faithfulness Babak:2006uv ; Buonanno_2007 ; Williams_2020  is
adopted to compare $h_{\rm ML}(t)$ generated by our waveform model and the
standard EOB waveform $h_{\rm EOB}(t)$,

$\textrm{FF}=\max_{t_{0},\phi_{0}}\left[\frac{\langle h_{\rm EOB}|h_{\rm
ML}\rangle}{\sqrt{\langle h_{\rm EOB}|h_{\rm EOB}\rangle\langle h_{\rm
ML}|h_{\rm ML}\rangle}}\right]$ (8)

where $t_{0}$ and $\phi_{0}$ are respectively the initial time and inital
phase of $h_{\rm EOB}(t)$. Without being biased by the detector noise, below
we will choose flat PSD, i.e., $S_{n}(f)=1$ for evaluating the FF
Williams_2020 . To characterize the performance, we need to evaluate the FF of
each template in the test data set, i.e., $20\%$ of LMR and $80\%$ of HMR, and
find out the distribution of FFs, which can also be represented by its
maximum, median and minimum. However, for simplicity we can represent and
denote the accuracy simply by the average of the FFs over the test data set.
This may not be precise enough but is more convenient when comparing the
performances of different generative waveform model. Later on, we will give
the cumulative distribution function of FFs and the associated maximum, median
and minimum for the best model selected by comparing the average of FFs over
the test data set. Moreover, at current stage the initial phase is not
optimized when evaluating FF. Despite that, our best waveform models can be
shown to achieve more than $97\%$ accuracy even without optimizing the initial
phase. Once the initial phase is also optimized, the accuracy can be expected
to be further enhanced.

Note that it turns out that the cVAE models yield comparable but lower
accuracy than the cAE models. This is probably because the template data sets
considered in this paper are parameterized by two mass parameters, which cause
not much degeneracy in mapping the parameter space to the template space. The
degeneracy here means that the different set of parameters may yield quite
similar waveforms. Thus, the variational feature of latent space of cVAE
models may not be needed for such kind of deterministic training set. Despite
that, when considering more complicated template sets with more source
parameters, the variational feature could be helpful to disentangle the
degeneracy, which may occur more often. In this respect, it is still
interesting to consider the VAE type models as a preliminary study for the
future work. To not digress the main theme of this work, from now on we will
simply focus on the various models based on the cAE scheme in the main text.
As a comparison, in Appendix we present the performance of cVAE models with
the same schematic structures of the cAE models.

##  IV Conditional Autoencoder Waveform Models

Based on the cAE scheme we can construct various waveform models by different
arrangements of the encoders and decoders. Since the input data are separated
into keys and normalized strains, their associated cAE can be arranged to
share a common decoder or not. For either cases, we consider two waveform
models which further differ by how the labels are conditioned. In total, we
will consider four waveform models, and compare their performances. In this
way, we can understand the relevance of different arrangements to the
performance, so that such experiences could be helpful for further
constructions. Below we first consider the models in which the keys and
normalized strains do not share a common decoder, and then the models do.

12

Figure 8: Schematic structure of the cAE+NN waveform model.

Figure 9: Schematic structure of the 2cAE waveform model.

The first cAE waveform model as shown in Fig. 8 is what we call cAE+NN model,
in which the strains are trained with cAE and the associated keys are trained
with conventional supervised learning neural network (NN) because the
dimensions of the key are relatively small. After done the training, we can
drop the strain part from the model, and turn the remaining network (Right-
most part of Fig. 8) into a generative machine for waveforms. Since the keys
and the normalized amplitude and frequency parts of the strains are trained
separately, when generating a strain we need to combine them together to get
the un-normalized amplitude $A(t)$ and frequency $\omega(t)$. Finally, we need
to integrate the frequency to get the phase $\theta(t)$ and then combine with
the amplitude to get the strain $h_{\rm ML}(t)$. With the output strain
$h_{\rm ML}(t)$ we can use (8) to evaluate the FF for each waveform in test
data set, and then take the average to obtain the accuracy. The resultant
performance is $85.73\%$ for LMR, and $55.95\%$ for HMR. As expected, the
accuracy for HMR is lower than the one for LMR. The accuracy is not good
enough for the purpose of data analysis. This is because the simple NN for
training the key is not the optimal scheme as AE.

By the faith on the power of semi-supervised training, we can replace the NN
by AE to train the key, and the new scheme is shown in Fig. 9. As expected,
the performance for both LMR and HMR get improved. The resultant accuracy is
$89.92\%$ for LMR and $67.20\%$ for HMR. The accuracy of LMR is now barely
good for detection purpose, but not good for PE to extract the source
properties. Still the HRM part is not accurate enough for practical purpose.

Figure 10: Schematic structure of the 1c2E1D waveform model.

Figure 11: Schematic structure of the 2c2E1D waveform model.

A common feature of the above two models is that they train the keys and
strains separately, and the correlation is only through the common labels.
Instead, we can correlate the outputs of the different encoders into a common
decoder, and directly compare the decoder’s output to the corresponding un-
normalized frequency and amplitude of the input strain to obtain the
reconstruction loss. Intuitively, the additional correlation may improve the
accuracy of the generative model. We will consider two such kinds of models.
The first one is shown in Fig. 10, which we call 1c2E1D, i.e., one conditional
encoder, and two waveform encoders (one for the strain and one for the key),
and finally one common decoder. In this model, the latent sizes of all the
encoders should be the same. This may cause redundancy of the latent space for
training the key since the key’s dimension is far less than the strain’s. To
remedy this, we introduce the second model as shown in Fig. 11. We call this
model 2c2E1D, i.e., now we have two conditional encoders so that the latent
sizes of the encoders for the strain and the key can be different. Moreover,
since now the keys and the normalized strains share a common encoder, we can
in fact choose the target in Fig. 10 and 11 to be the un-normalized amplitude
and frequency, i.e., $A(t)$ and $\omega(t)$, but not the normalized
$\hat{A}(t)$ and $\hat{\omega}(t)$, to evaluate the reconstruction loss. For
simplicity, we choose the MSE for the reconstruction loss so that the
minimization of the reconstruction loss will make the machine to generate
$\tilde{x}$ being as close to the target $A(t)$ and $\omega(t)$ as possible.
In contrast to the cAE+NN and 2cAE waveform models, this will save the
procedure of converting the normalized strains into their un-normalized
counterparts.

| cAE+NN | 2cAE | 1c2E1D | 2c2E1D
---|---|---|---|---
accuracy (LMR) | 85.73% | 89.92% | 97.65% | 98.20%
accuracy (HMR) | 55.95% | 67.20% | 97.70% | 97.02%

Table 2: Summary of the accuracy of the low-mass-ratio (LMR) and high-mass-
ratio (HMR) waveforms for each cAE waveform model considered in this paper.
The accuracy is the average of the fitting factors (FFs) (see (8)) for all the
test data. We see that both 1c2E1D and 2c2E1D models can have accuracy more
than $97\%$ for both LMR and HMR.

FFs for 2c2E1D | LMR | HMR
---|---|---
Minimum FF | 82.49% | 74.13%
Median FF | 98.61% | 98.22%
Maximum FF | 100.0% | 99.99%

Table 3: Summary of the FFs of the LMR and HMR waveforms for 2c2E1D cAE
waveform model considered in this paper. The associated cumulative
distribution function of FFs is shown in Fig. 14. We see that the medians are
comparable with accuracy listed in Table 2.

| cAE+NN | 2cAE | 1c2E1D(cAE) | 2c2E1D(cAE)
---|---|---|---|---
Training time (strain)(sec) | 4042.5 | 3329.4 | 4536.6 | 4462.6
Training time (key)(sec) | 81.2 | 144.8
Generations/epochs (strain) | 8000 | 8000 | 10000 | 10000
Generations/epochs (key) | 10000 | 10000
Generation time per waveform (milli sec) | 0.8-1.0

Table 4: Summary of the training time, generation/epoch number and the
generation time of a single waveform for each cAE waveform model considered in
this paper. Note that it takes less than 1 millisecond to generate a single
waveform. This is about 10 to 100 times faster than the EOB running on the
same computing facility.

The resultant performance for the above two models are the following. For the
1c2E1D waveform model, the accuracy is $97.65\%$ for LMR, and $97.70\%$ for
HMR. For the 2c2E1D waveform model, the accuracy is $98.20\%$ for LMR, and
$97.02\%$ for HMR. Their performances are comparable, and are accurate enough
for both LMR and HMR, i.e., greater than $97\%$. Note that we have not
optimized the overlap accuracy over the initial phase yet, once it is done we
will expect higher overlap accuracy 555Our preliminary study shows that it can
achieve almost $99\%$ for LMR and $98\%$ for HMR.. The high accuracy of the
generated waveforms indicates that both models are good for the purpose of low
latency detection and for PE of gravitational wave events with improvement of
accuracy in the future work. The high accuracy for HMR part can also be
exploited for progressively self-training to generate waveforms of HMR.

We summarize in Table 2 the accuracy for each cAE waveform model considered in
this paper. The 1c2E1D and 2c2E1D model are the best in the accuracy rate.
Overall the 2c2E1D model is slightly superior. To characterize its detailed
performance, we also give the minimum FF, median FF and maximum FF of this
model in Table 3, from which we see the median FF is comparable with the
accuracy, i.e., the average of FFs. Later we will discuss more details for
this model. Note that, the above models are all implemented based on the cAE
scheme. We can also replace the cAE schemes in these models by the cVAE ones,
and obtain the corresponding cVAE waveform models. However, there is one
additional model called cVAE+cAE, see Fig. 15 in the Appendix, in which we use
cAE to train the keys and cVAE to train the strains. The performances of these
cVAE waveform models are listed in the Table 6 of the Appendix. It turns out
that the accuracy of the cVAE waveform models are comparable to their cAE
counterparts. However, by examining in more details it seems that cAE models
are superior than the cVAE ones, even for the HMR. This is a bit surprising
that the generative nature of VAE does not help to improve the accuracy.

Besides, we also summarize in Table 4 the training time and generation/epoch
number of the waveform models and the generation time of a single waveform for
each cAE waveform model considered in this paper. The run-times of the cVAE
waveform models are comparable with their cAE counterparts listed in Table 4,
thus are omitted for simplicity. We see that the training time is about $4000$
seconds for all the waveform models, it is quite modest and implies that the
extension to the full waveform models with more source parameters is
manageable in the near future. Furthermore, the generation time of a single
waveform is about one millisecond. Compared to the typical generation time for
a EOB waveform, which is about few hundredths to few tenths of a second on the
same computing facility, the speed enhancement is about $10$ to $100$ times.

As the 2c2E1D waveform model is the best among all the waveform models
considered in this paper, we look into some details of this model. First, we
list the hyperparameters of this model in in Table 5, and histogram of its
training losses in Fig. 12. Based on the information one can reproduce the
model quite easily. From Fig. 12 we see that the training and validation
losses match well and stop increasing around $8000$ generations/epochs. This
implies that our training is not over-fitted and stabilized.

| $E_{1}$ | $E_{2}$ | $E_{1}^{{}^{\prime}}$ | $E_{2}^{{}^{\prime}}$ | Decoder
---|---|---|---|---|---
Latent size | 8 | 8 | 3 | 3 | 8 & 3
CNN layers | None | 2 | None | None | 3
Filter size | None | [16,16] | None | None | [4,4,4]
conv features | None | [5,15] | None | None | [16,32,64]
Pool size | None | [4,4] | None | None | [4,4,4]
dilation rate | None | None | None | None | [1,2,2]
NN layers | 4 | 3 | 4 | 3 | 6
Neural size | 500 | 500 | 400 | 400 | 800

Table 5: Hyperparameters for the 2c2E1D model. Here “conv features” is the
abbreviation of the convolution features.

Figure 12: Histogram of the training losses for the 2c2E1D cAE waveform model.
The types of training/validation losses are denoted by (Loss/Eval$\\_$loss,
latent Loss(strain)/Eval latent Loss(strain), latent Loss(key)/Eval latent
Loss(key)) in the graphic illustrations, which literally mean the total loss,
the latent loss of the normalized strain, and the latent loss of the key,
respectively. The match of Loss and latent Loss implies no over-fitting. The
overall trends show that the training is stabilized around $10000$
generations/epochs.

Figure 13: Cumulative distributions functions of fitting factors (FFs) of
2c2E1D cAE waveform model for the LMR (blue) and HMR (orange) generated
waveforms. As expected, the HMR one has a broader tail. Overall, the outliers
are rare.

Figure 14: Distributions of FFs of 2c1E1D cAE waveform model as functions of
mass ratio (Upper row) and total mass (Lower row) for LMR (blue) and HMR (red)
test data set. Note that each dot represent one template in the test data set.

Further, we can understand more the tomography of the accuracy for the 2c2E1D
cAE waveform model by plotting the cumulative distributions function (CDF) of
the FF for both LMR and HMR. The results are shown in Fig.13. As expected, we
see that the HMR one has a broader tail, however,overall the FFs are
concentrated on the the side of high FF near $100\%$. This implies the
outliers are rare, and the generated waveforms can be reliably implemented for
the practical data analysis such as the detection and PE of the gravitational
wave events. For curiosity, it is also interesting to see how FF changes with
the mass ratio and total mass of the BBH. We present the distributions for
2c2E1D cAE waveform model in Fig. 14. We see that the low FFs appear more
often in the regime of lower mass ratio and smaller total mass for LMR ones.
The latter one could be due to the the fact that the templates in this regime
are less loud (small mass), but it is not clear about the cause for the former
one (lower mass ratio). On the other hand, the FF for the HMR does not behaves
the same way, which could be due to the insufficient training data.

Finally, the typical neural network structure of the above cAE models is given
in Fig. 16 in Appendix.

##  V Conclusion and Discussion

In this paper we construct various waveform models based on the neural network
of conditional autoencoders (cAE) and its variational extensions (cVAE). Their
accuracy and run-time have been summarized in Table 2 and Table 4 in the main
text for cAE models, and in Table 6 in Appendix for cVAE models, respectively.
For simplicity, we represent the accuracy of these generative waveform models
by the the average of fitting factor, which is based on the waveform overlap
of the matched filtering. Among these waveform models, the so-called 2c2E1D
cAE model is the best with more than $97\%$ for both the low-mass-ratio (LMR)
and high-mass-ratio (HMR) waveform generation. This demonstrates the viability
of our best waveform model to be implemented in the practical gravitational
wave data analysis and parameter estimation (PE). Especially, the generation
time of a single waveform is $10$ to $100$ times faster than the traditional
EOBNR method, it implies that the waveform generation for the low latency
detection can be accelerated by our waveform models. With the improvement of
the accuracy in the future work, the revised version of our generative
waveform model may also help to speed the parameter estimation.

Moreover, the impressive accuracy for HMR waveform generations is encouraging
because fraction of the HMR waveforms in the training and validation data set
is less than $3\%$. This implies that one may be able to generate higher mass-
ratio waveforms by a series of self-training with the generative outputs of
the lower-mass ratio machine as the training data for the higher mass-ratio
ones. This may open a new venue to generate the waveforms with intermediate
mass-ratio, say greater than $15$.

Despite that, there are still ample space to improve our waveform models. As a
proof-of-concept study, we only consider the inspiral-merger part of the full
waveforms. Although the ringdown part is quite short but it contains the
information of quasi-normal modes. We are currently training the waveform
models for the full waveform based on the similar cAE or cVAE scheme, and will
report our results in the near future. Moreover, to be more useful in the
practical data analysis tasks, we shall also include more source parameters
such as spins, precession and tidal deformabilities. Once the above goals are
achieved, we can incorporate our waveform models to the standard pipeline of
detection and PE, and help to accelerate the data analysis tasks in the coming
O4 and O5 observation runs of LIGO/Virgo/KAGRA.

Added-in-proof. When finalizing the draft, an eprint Lee:2021isa  with the
similar goal appears, in which a RNN framework is adopted to generate the
merger-ringdown parts of the waveform from the input associated inspiral one.

## Acknowledgement

We thank Kai-Feng Chen, Yao-Yu (Joshua) Lin and members of TGWG and CAG for
discussions and comments. In particular, we also thank Han-Shiang Kuo for the
discussions on cVAE structures, and Jie-Shiun Tsao for helping the sever
implementation. CHL would like to thank Wei-Ren Xu for his guidance into the
machine learning. This work is supported by Taiwan’s Ministry of Science and
Technology (MoST) through Grant No. 109-2112-M-003-007-MY3. We also thank NCTS
for partial financial support.

## Appendix: Structures and Results of cVAE waveform models

In this Appendix, we summarize the performance of the cVAE counterpart of the
cAE waveform models considered in the main text. These counterpart models are
simply obtained by replacing the cAE with cVAE in the associated cAE waveform
model. However, for the 2cAE model, we can in fact replace only the cAE for
the strains by cVAE, and still keep the cAE for the keys intact. In this way,
we have a new model called cVAE+cAE model as shown in Fig. 15. For all the cAE
and cVAE waveforms models considered in this work, the typical neural network
structure is shown in Fig. 16, which serve as the guideline for the readers to
implement the coding.

Figure 15: Schematic structure of the cVAE+cAE waveform model.

Figure 16: A typical machine structure with details of hyperparameters for the
cAE or cVAE used in this work. Each neural network (NN) or convolutional
neural network (CNN) is denoted by a box with its dimension specified. Top-
left: the encoder for the strain. Top-right: the encoder for the label.
Bottom: the decoder to reproduce the strain by inputting the latent vector and
the label.

The accuracy for the cVAE waveform models are shown in Table 6, from which one
can compare with Table 2 and finds that the accuracy are comparable for the
cVAE models and their counterparts. Besides, the run-times for these cVAE
models are comparable with their cAE counterparts, i.e., about $4000$ seconds
for training and $1$ milli second to generate a single waveform, thus for
simplicity we will not listed here. Finally, one more point is that after
training is done, we will just choose the mean value of the latent layer to
generate the waveform to avoid the stochastic feature of VAE.

| cVAE+NN | cVAE+cAE | 2cVAE | 1c2E1D | 2c2E1D
---|---|---|---|---|---
accuracy | 89.73% | 89.03% | 73.23% | 94.35% | 97.16%
(LMR)
accuracy | 65.56% | 70.26% | 73.65% | 79.11% | 91.92%
(HMR)

Table 6: Summary of the accuracy for both LMR and HMR for the cVAE counterpart
of each cAE waveform model considered in the main text.

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]The homology groups of finite cyclic covering of line arrangement complement

Yongqiang Liu
The Institute of Geometry and Physics, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui 230026 China

Wentao Xie
School of Mathematical Sciences, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui 230026 China

[2020]Primary 52C35, 32S55

In this paper, we study the first homology group of finite cyclic covering of complex line arrangement complement. We show that this first integral homology group is torsion-free under certain condition similar to the one used by Cohen-Dimca-Orlik <cit.>.  In particular, this includes the case of the Milnor fiber, which generalizes the previous results obtained by Williams <cit.> for complexified line arrangement to any complex line arrangement.

§ INTRODUCTION

§.§ Background

An arrangement of hyperplanes $\sA$ is a finite collection of hyperplanes in a finite dimensional complex vector (affine, or projective) space.  A fundamental problem in the theory of hyperplane arrangements is to decide whether various topological invariants of the complement of $\sA$ are determined by the combinatorial data (i.e., the poset of intersections of hyperplanes) of $\sA$. For instance, the homology groups and the cohomology ring of the hyperplane arrangement complement are combinatorially determined (e.g., see [27]), while the fundamental group of the complement are not [32].   It is an open question whether
the homology groups of  a finite cover  of a hyperplane arrangement
complement are combinatorially determined. This includes the Milnor fiber of a central hyperplane arrangement.
See Papadima-Suciu's work [29] for recent progress in this direction and also Suciu's survey paper [33] for an overview of the theory.

It is well-known that the homology groups of hyperplane arrangement complement are torsion-free. In fact, Dimca-Papadima [11] and Randell [30] independently showed that the complement of a hyperplane arrangement is homotopy equivalent to a minimal CW complex. On the other hand,  Cohen-Denham-Suciu <cit.> gave examples of arrangements with multiplicities whose first homology group of the
Milnor fiber has torsion. Later on Denham-Suciu <cit.> showed that for every prime $p\geq2$, there are examples of hyperplane arrangements whose higher degree homology of the Milnor fiber has $p$-torsion. This nice result resolved Dimca-Némethi <cit.> and Randell's question <cit.>  whether the Milnor fiber of central hyperplane arrangement always has torsion-free homology.
In particular, this gives examples of hyperplane arrangement whose Milnor fiber do not necessarily have a minimal CW structure. Yoshinaga <cit.> gave the first example of hyperplane arrangement whose first homology group of Minor fiber  has $2$-torsion recently.

Denham-Suciu raised the following question:
<cit.> Is the torsion in the homology of the Milnor fiber of a hyperplane
arrangement combinatorially determined?

Note that the Milnor fiber of central hyperplane arrangement is a finite cyclic covering of the complement of the associated projective hyperplane arrangement.
One can ask the same question for any finite cyclic covering of hyperplane arrangement complement.
In this paper, we focus on the line arrangement case regarding this question.

§.§ Main results

Let $\sA$ be an essential  line arrangement of $n$ lines in $\CP^{2}$ with complement $\U$.
Let  $Q(\sA)\in \C[z_0, z_1,z_2]$ be the reduced defining homogeneous  polynomial for $\sA$, defined up to multiplication by a non-zero scalar. The Milnor fiber $F$ of the projective arrangement $\sA$ is defined by $Q(\sA)=1$, which is a smooth hypersurface in $\C^3$. Then $F$ is a $n$-fold cyclic covering of $\U$.
We recall the following result due to Williams, which gives a partial answer to Question <ref>.
<cit.>  Let $\sA$ be a complexified real arrangement of $n$ lines in $\CP^2$ with the Milnor fiber $F$.  Taking any line $H\in\sA$, let  $\{P_1,\cdots,P_s\}$ denote the set of multiple points on $H$ and let $m_i$ denote  the  multiplicity of $P_i$ for $1\leq i\leq s$.
Then, for any field $\K$, the first Betti number of F with respect to $\K$ satisfies

b_1(F,)  ≤n-1+∑^s_i=1(m_i-2)((m_i,n)-1).
Furthermore, if $\gcd(m_i,n)=1$ for every $m_i>2$ (e.g. $n$ is a prime number), then $H_1(F,\Z)\cong \Z^{n-1}$ is torsion-free.

For a full generalization beyond the Milnor fiber case, we use the following setup. Let $\sA$ be an essential complex arrangement of $n$ lines in $\CP^2$ with complement $\U$. Fix any line $H\in \sA$,  let  $\{P_1,\cdots,P_s\}$ denote the set of multiple points on $H$ and let $m_i$ denote  the  multiplicity of $P_i$ for $1\leq i\leq s$.  Here since $\sA$ is essential, we have $s>1$.
We label the other lines according to their intersections with $H$ by two indices: the
first indicating the intersection point and the second indicating the position of the hyperplane with respect to the intersection point. Note that the multiplicity of each intersection point $P_i$ is given by $m_i$, and that $H$ is always the first hyperplane with respect to each intersection point. We then have

$$ \sA = \{H, H_{1,2}, H_{1,3},\cdots, H_{1,m_1}, H_{2,2},\cdots, H_{s,m_s} \}.$$

Fix an epimorphism $\varphi:\pi_1(\U)\twoheadrightarrow \Z$ with

$\varphi(\alpha)=\epsilon\in \Z$ and   $\varphi(\alpha_{i,j})=\epsilon_{i,j}\in \Z$ for all $1\leq i\leq s, 2\leq j\leq m_i.$

Here $\alpha$ and $\alpha_{i,j}$ are the meridians associated to $H$ and $H_{i,j}$ respectively. Set $\varepsilon_i=\epsilon+\sum^{m_i}_{j=2} \epsilon_{i,j}$ for all $1\leq i \leq s$.

With the above notations and assumptions,   consider the composed map $\pi_1(\U)\overset{\varphi}{ \twoheadrightarrow} \Z \twoheadrightarrow \Z_N$ and let $\U^{\varphi, N}$ denote the associated $N$-fold covering.
If  $\epsilon=1$ and $\varepsilon_i\neq 0$ for all $m_i>2$, then
for any field $\K$ we have

b_1 (^φ, N,) ≤(n-1)+∑^s_i=1(m_i-2)((ε_i,N)-1) .
Furthermore, if  $\gcd(\varepsilon_i,N)=1$ for all $m_i>2$,
$H_1(\U^{\varphi, N},\Z)\cong \Z^{n-1}$ is torsion-free.
The setup we used here is to consider all finite cyclic covering determined by $\varphi$ at once. This setup is convenient for our proof and can be used to prove a divisibility result for the Alexander polynomial of $\U$ (see Proposition <ref>), which partially generalizes
Elduque's result <cit.>. If one only focuses on the $N$-fold covering, the assumption can be changed to

$\epsilon \equiv 1 \mod N $ and $\varepsilon_i\neq 0 \mod N$ for all $m_i>2$.

When $N=2$, this corresponds to the CDO-condition considered by Sugawara in <cit.>. In particular, it is already proved by the first author joint with Liu <cit.> that the 2-fold covering of the complement has torsion-free homology groups using the results proved by Sugawara for complexified real arrangement case <cit.> and the first author joint with Maxim and Wang <cit.>.

In fact, the two conditions $\epsilon\neq 0$ and $\varepsilon_i\neq 0$ for all $m_i>2$ are corresponding to the conditions used by Cohen-Dimca-Orlik in <cit.>. We have the condition $\epsilon=1$ in Theorem <ref>  to make the computation manageable.
The following result is a direct application of Theorem <ref>.
With the same assumptions and notations as in Theorem <ref>,
if $m_i=2$ for every  multiple point $P_i$ in $H$ and $\epsilon=1$,  $H_1(\U^{\varphi, N},\Z)\cong \Z^{n-1}$ is torsion-free.

Since the Milnor fiber is a special case satisfying the assumptions in Theorem <ref>, we extend Theorem <ref> to any complex line arrangement.
Theorem <ref> holds for any complex line arrangement.

As noted by Williams in <cit.>, the inequality (<ref>) is quite similar to the one obtained by Cohen-Dimca-Orlik in <cit.>, which is linked to the direction of divisibility results for Alexander polynomials of hypersurface complements. This topic is initiated by Libgober [19, 20] and developed by many other authors, see [8, 26, 9, 1, 21].
In particular, the recent work by Elduque [13] gives a divisibility result for twisted Alexander polynomial for line arrangement complement. She constructed a tubular neighborhood along one line in loc. cit., which
plays a key role in the proof of Theorem <ref>. The key point in the proof is to compute the fundamental group of the boundary manifold of this neighbourhood.  The computation is done by a method due to Hirzebruch [18]. For related works,  Westlund [36]  computed the fundamental group of the boundary manifold of the complement $\U$. Cohen-Suciu [5] found a minimal representation for such fundamental group and computed their twisted Alexander polynomials. Hironaka [17] and Florens-Guerville-Marco [14]  studied relationships between the fundamental groups of a line arrangement complement and that of the boundary manifold of such an arrangement. The computation in our case is similar but much simpler, since
our boundary manifold is only part of the boundary manifold $\U$.

§.§ Summary

The paper is organized as follows.
In section 2, we recall the construction of the boundary manifold of the tubular neighbourhood of $H$ and compute its fundamental group. Then we use Fox calculus to compute the Alexander matrix.
Section 3 is devoted to the proof of Theorem <ref>, Corollary <ref> and the divisibility results we mentioned earlier.

Convention. We will use the following convention throughout the paper:

* $\Z_N$ denotes the cyclic group  $\Z/N\Z$.

* For a space $X$, the $i$-th Betti number with field coefficients $\K$ of $X$ is denoted by $b_i(X,\K)$.

* A covering always means an unbranched covering.

* We will assume that the base point $x_0 \in X$ is fixed when we consider the fundamental group $\pi_1(X) = \pi_1(X, x_0)$.

Acknowledgements. Both authors are  supported by National Key Research and Development Project SQ2020YFA070080, the starting grant from University of Science and Technology of China, NSFC grant No. 12001511, the Project of Stable Support for Youth Team in Basic Research Field, CAS (YSBR-001),  the project “Analysis and Geometry on Bundles" of Ministry of Science and Technology of the People's Republic of China and  Fundamental Research Funds for the Central Universities.

§ THE BOUNDARY MANIFOLD OF THE TUBULAR NEIGHBOURHOOD

In this section, we focus on the boundary manifold of the tubular neighbourhood along a line $H\in \sA$.

§.§ Construction

We follow Elduque's construction presented in <cit.>.

Let $\sA$ be a line arrangement in $\CP^2$ with $n$ lines.
For a fixed line $H\in \sA$, recall that $H$ has the set of  multiple points $\{P_1,\cdots, P_s\}$ with the  multiplicity $m_i$ for $P_i$ and

$$\sA=\{H,H_{1,2},\cdots,H_{1,m_1},\cdots,H_{s,2},\cdots,H_{1,m_s} \}.$$

For each singular point $P_i$, we take a  small enough ball $\B_i$ centered at the point $P_i$.
Then we take a tubular neighbourhood of $H\setminus\cup_{i=1}^s \B_i(\frac{1}{2})$  with a even smaller radius, denoted by $\Sigma$. Here $\B_i(\frac{1}{2})$ is the ball  with the same center as  $\B_i$  but half radius.
$$W=\Sigma \cup (\cup_{i=1}^s \B_i).$$
By rounding corners, its boundary manifold $X=\partial W$  is a real 3-dimensional manifold, which can be described as follows.

Set $N=\big(H\setminus \cup_{i=1}^s \B_i\big)\times S^1$, which should be thought of as the boundary of $\Sigma$ around the non-singular part of $H$.
We have that
$$\partial N= \partial \big(H \setminus\cup_{i=1}^s \B_i\big)\times S^1.$$
Since $\partial(H\setminus \bigcup_{i=1}^s \B_i)$ is a union of disjoint $S^1$'s (one from every disk $H\cap \B_i$ removed), then $\partial N$ is a union of disjoint tori $T_{i}$.
Let $L_i$ be the link of the singularity at the point $P_i$ (which is a Hopf link with $m_i$ components), and let $S_i^3$ be the boundary of $\B_i$. Consider the space

$$X= N\cup_{\sqcup _i T_i }  (\bigsqcup_{i=1}^s S^3_i \setminus L_i) ,$$

where the gluing is done as follows.
Note that  $L_i=\bigcup_{j=1}^{m_i} L_{i,j}$, where $L_{i,j}$ is the Hopf link which corresponds to $H_{i,j}$ (here we regard $H$ as $H_{i,1}$).  A meridian around $L_{i,1}$ is glued to $\text{\{point\}}\times S^1 \subset  N$ and $L_{i,1}$ is glued to the $S^1$ corresponding to the boundary of $H \cap \B_i$ for all $1\leq i\leq s$.

§.§ The fundamental group of $X$

In this subsection, we compute $\pi_1(X)$.

We first recall the fundamental group of the local complement $\U\cap \B_i$, which is homotopy equivalent to $\U_i= \U\cap S^3_i$. Let $\alpha_{i,j}$ denotes the meridian associated to $H_{i,j}$ and $\beta_i$ denote the generator of the  fundamental group of the fiber for the Hopf map over $\U_i$. Then we have that

\begin{equation}\label{eq1}
\begin{aligned}
\pi_1(\U_i) &\cong \left\langle \beta_i,\alpha_{i,1}, \alpha_{i,2},\cdots,\alpha_{i,m_i} | \beta_i=\alpha_{i,1}\alpha_{i,2}\cdots \alpha_{i,m_i}, [\beta_i,\alpha_{i,j}] , 1\leq j\leq m_i \right\rangle \\
&\cong \left\langle \beta_i,\alpha_{i,1},\alpha_{i,2},\cdots,\alpha_{i,m_i-1} | [\beta_i,\alpha_{i,j}], 1\leq j\leq m_i-1 \right\rangle
\end{aligned}
\end{equation}

Abusing notations, we will look at  the $\alpha_{i,j}$'s and $\beta_i$'s  inside of $\pi_1(X)$ via the maps $\pi_1(\U_i)\to \pi_1(X)$ induced by inclusion.
Since we regard $H$ as $H_{i,1}$, hence $\alpha_{i,1}=\alpha$ for all $1\leq i \leq s$, where $\alpha$ is the meridian along $H$.  In fact,  all $\alpha_{i,1}$'s represent the same element in $\pi_1(X)$ up to homotopy equivalence, since moving the meridian of $H$ along $H\setminus \cup_{i=1}^s \B_i$ does not change its homotopy class.

With the above assumptions and notations, we have
\begin{equation}\label{eqX}
\begin{aligned}
\pi_1(X)
& \cong
\left\langle\begin{array}{cc|cc}
\alpha, &  & \alpha^{1-s}\beta_1\cdots\beta_s=1, & \\
\beta_i, & 1\leq i\leq s,   & \beta_i=\alpha\alpha_{i,2}\cdots \alpha_{1,m_i}, & 1\leq i\leq s\\
\alpha_{i,j},&  2\leq j\leq m_{i} & [\alpha_{i,j},\beta_i] ,[\alpha,\beta_i], & 2\leq j\leq m_{i}
\end{array}\right\rangle
\\
\left\langle\begin{array}{cc|cc}
\alpha,\beta_i, & 1\leq i\leq s & \alpha^{1-s}\beta_1\cdots\beta_s=1 ,& 1\leq i\leq s\\
\alpha_{i,j}, & 2\leq j\leq m_i-1 & [\alpha_{i,j},\beta_i] ,[\alpha,\beta_i], & 2\leq j\leq m_i-1
\end{array}\right\rangle \\
\end{aligned}
\end{equation}

We use Van-Kampen Theorem to compute $\pi_1(X)$. Note that $S^3_i \setminus L_i$ is isomorphic to $S^1\times (\CP^1 - \{ m_i \text{ points}\})$, which is a trivial bundle. So $X$ is constructed by pasting certain $S^1$-bundles along their boundaries, which are some disjoint tori.
Using Van-Kampen Theorem, we see that $\pi_1(X)$ can be generated by the elements $\alpha,\beta_i,\alpha_{i,j}$. We have the following commutation relation due to the fundamental group of $\U_i$:

$$[\alpha,\beta_i]=1,[\alpha_{i,j},\beta_i]=1, 1\leq i\leq s, 2 \leq j\leq m_i.

To have a better understanding of  $\pi_1(X)$, we follow the approach by Hirzebruch [18]. Consider the blow up of $\CP^2$ at all points $P_i$ for $1\leq i \leq s$, denoted by $\widetilde{\CP^2}$. Let $\widetilde{H}$ denote the strict transform of $H$ in $\widetilde{\CP^2}$.

Consider the $S^1$-bundle over $\widetilde{H}$ associated to the conormal bundle of $\widetilde{H}$ in $\widetilde{\CP^2}$. Since $\widetilde{H}=\CP^1$, this is a $S^1$-bundle over $\CP^1$ with Euler number $1-s$. For a detailed computation of this Euler number, see <cit.> and <cit.>.

Since the blow-up map does not change $\U$, we can consider $X\subset \U \subset \widetilde{\CP^2}$.
The left relation relies on the Euler number of $ S^1$  bundle over $\widetilde{H}$. In fact, considering the product of the generators (total boundary) in the $ S^1$  bundle over $\widetilde{H}$ gives the following relation
For a detailed discussion, see  <cit.> by Hirzebruch and <cit.> by Westlund.

Using the commutation relations, $\alpha^{1-s}\beta_1\cdots\beta_s=1$ is equivalent to the relation
$$\alpha \alpha_{1,2}\cdots \alpha_{1,m_1}\cdots \alpha_{s,2}\cdots \alpha_{s,m_s}=1.$$
This phenomenon is already explored by Cohen-Suciu <cit.>. It is compatible with the fact that the map $\pi_1(X)\to \pi_1(\U)$ induced by inclusion is an epimorphism (see section 3.1) and the same relation holds for $\pi_1(\U)$, see e.g. <cit.>.

§.§ Fox calculus and the Alexander matrix

In this subsection, we use Fox calculus to compute the Alexander matrix.
We first briefly recall the basic properties of Fox calculus.
For more details, see [15] and <cit.>.

Given a finitely presented group $G=\left\langle x_1,\cdots,x_m|r_1,\cdots,r_l\right\rangle$, let $\tilde{G}=\left\langle x_1,\cdots, x_m\right\rangle$ be a free group with $m$-generators. Let $\xi:\tilde{G}\xrightarrow{} G$
and $\rho\colon G\xrightarrow{} \dfrac{G}{[G,G]}$ denote  the natural quotient morphisms, where $[G,G]$ is the commutator subgroup of $G$.  Note that $\rho$ can be uniquely  extended to a homomorphism on the corresponding group rings, still denoted by $\rho$ by abuse of notations.

Define the $i$-th Fox derivative  $\dfrac{\partial}{\partial x_i}: \tilde{G} \rightarrow \Z G$ for all $1\leq i \leq m$ such that

$$\dfrac{\partial}{\partial x_i}(x_j)=\delta_{ij}$$

$$\dfrac{\partial}{\partial x_i}(g_1g_2)=\dfrac{\partial}{\partial x_i}(g_1)+ \xi(g_1)\dfrac{\partial}{\partial x_i}(g_2),$$

where $\delta_{ij}$ is Kronecker symbol, $g_1,g_2$ are elements of $\tilde{G}$ .

The Alexander matrix of group $G$ is a matrix $M=(c_{ij})$, where the size of $M$ is $l\times m$ and the entries are given by $c_{ij}=\rho (\dfrac{\partial r_i}{\partial x_j})$.

Now let us do the Fox calculus for $\pi_1(X)$. For the relations in $\pi_1(X)$, we have the  following formulas:

$$\dfrac{\partial}{\partial \alpha}(\alpha^{-1})=-\alpha^{-1} ,$$

$$\dfrac{\partial}{\partial \alpha}(\alpha^k)=\dfrac{\alpha^k-1}{\alpha-1},  $$

$$\dfrac{\partial}{\partial \alpha_{i,j}}( [\alpha_{i,j},\beta_i])
=1-\alpha_{i,j}\beta_i \alpha_{i,j}^{-1}
\overset{\rho}{\mapsto} 1-\beta_i,

$$\dfrac{\partial }{\partial \beta_i}( [\alpha_{i,j},\beta_i])
=\alpha_{i,j}-\alpha_{i,j}\beta_i \alpha_{i,j}^{-1} \beta_i^{-1}
\overset{\rho}{\mapsto} \alpha_{i,j}-1,

$$\dfrac{\partial }{\partial \beta_i}(\alpha ^{1-s}\beta_1\cdots \beta_i)
=\alpha^{1-s}\beta_1\cdots \beta_{i-1}
\overset{\rho}{\mapsto} \alpha^{1-s}\beta_1\cdots \beta_{i-1}.

Then we get the Alexander matrix of $\pi_1(X)$:

\begin{equation}
\begin{pNiceMatrix}
\dfrac{\alpha^{(1-s)}-1}{\alpha-1}       & Q_1  & Q_2  & \cdots & Q_s  \\
V_1    & M_1  &      &        &      \\
V_2    &      & M_2  &        &      \\
\vdots &      &      & \ddots &      \\
V_s    &      &      &        & M_s

\end{pNiceMatrix},
\end{equation}

where the empty entries are zero and $M_i,V_i,Q_i$ are matrices with the following form:

\begin{equation}
\begin{aligned}
\begin{pNiceMatrix}[first-row,first-col]
& \beta_i & \alpha_{i,2} & \cdots & \alpha_{i,m_i-1} \\
[\alpha,\beta_i] &  \alpha-1  \\
[\alpha_{i,2},\beta_i]  &\alpha_{i,2}-1 & 1-\beta_i \\
\vdots &  \vdots &  & \ddots\\
[\alpha_{i,m_i-1},\beta_i]  &\alpha_{i,m_i-1}-1 & & & 1-\beta_i\\
\end{pNiceMatrix}_{(m_i-1)\times (m_i-1)}, \\
V_i&=(1-\beta_i,0,\cdots,0)^{\mathsf{T}}_{1\times (m_i-1)} , \\
Q_i&=(\alpha^{1-s}\beta_1\cdots \beta_{i-1},0,\cdots,0)_{1\times (m_i-1)}.
\end{aligned}
\end{equation}

Here  $\mathsf{T}$ denote the transpose of  the matrix.

§ PROOF OF THE MAIN RESULTS

In this section, we give the proof of Theorem <ref>.
First we list two equalities that will be used later:
ϵ+ ∑_i=1^s ∑_j=2^m_i ϵ_i,j= (1-s)ϵ+∑_i=1^s ε_i=0
n=1 +∑_i=1^s (m_i-1)=1-s+∑_i=1^s m_i.

§.§ Divisibility results for Alexander polynomial

Consider the inclusion map $X\to \U$. Note that $X$ is connected, hence the map induced by inclusion $\pi_0(X)\to \pi_0(\U)$ is an isomorphism. By a Lefschetz hyperplane section theorem type argument, the map  $\pi_1(X)\to \pi_1(\U)$ induced by inclusion is an epimorphism. In fact, we can take a generic line $L$ in $\CP^2$ with respect to $\sA$, which is close enough to the line $H$ such that $L$ is contained in the tubular neighborhood we constructed in section 2. Then
the map $\pi_1(L\cap \U)\twoheadrightarrow \pi_1(\U)$ is surjective by Lefschetz hyperplane section theorem (see <cit.>).   Note that $L\cap \U \subset  W\setminus H$ and $W\setminus H$ is homotopy equivalent to $X$. Putting all together, we get that the
the map $\pi_1(X)\to \pi_1(\U)$ induced by inclusion is an epimorphism.

Consider the composed maps $\pi_1(X)\to \pi_1(\U) \overset{\varphi}{\to} \Z$.
Let $X^\varphi$ denote the corresponding infinite cyclic covering space and $X^{\varphi,N}$ denote the $N$-fold cyclic covering for any integer $N>1$ induced by further composing with $\Z \to \Z_N$. Then we have the following result simply due to the fact that $\pi_1(X)\to \pi_1(\U)$ is an epimorphism.

With the above assumptions and notations,   the induced map $H_1(X^{\varphi,N},\K)\to H_1(\U^{\varphi,N},\K)$ is surjective for any field $\K$.

For any field $\K$, by lifting the cell structure of $X$ to $X^\varphi$,
we obtain a free basis for the cellular chain complex of $X^\varphi$ as $\K[t^{\pm}]$-modules.
Then the cellular chain complex $C_*(X^\varphi,\K)$ is a bounded complex of finitely generated free $\K[t^{\pm}]$-modules.
The first Alexander module $H_1(X^\varphi,\K)$ is the first homology of the complex   $C_*(X^\varphi,\K)$, hence $H_1(X^\varphi,\K)$  is a finitely generated $\K[t^{\pm}]$-modules. Note that $\K[t^{\pm}]$ is a PID.

For any field $\K$, the first Alexander polynomial of $X$ with respect the covering space $X^\varphi $ is defined
as the order of the torsion part of $H_1(X^\varphi,\K)$, denoted by $\Delta_1(X^\varphi,\K)$.

Consider the covering space $\U^\varphi$ associated to the map $\pi_1(\U) \overset{\varphi}{\to} \Z$.
For any field $\K$, the first Alexander polynomial $\Delta_1(\U^\varphi,\K)$ of $\U$ with respect to the covering space $\U^\varphi$ can be defined similarly.

With the above assumptions and notations, if $\epsilon\neq 0$ and $\varepsilon_i\neq 0$ for all $m_i>2$,  then for any field $\K$, $H_1(\U^\varphi,\K)$ is a torsion $\K[t^{\pm}]$-module and

$$\Delta_1(\U^\varphi,\K) \mid \ (t-1)\cdot(t^{\epsilon}-1)^{s-2}
\cdot \prod^s_{i=1}(t^{\varepsilon_i}-1)^{m_i-2}.$$

We use Fox calculus to compute the first Alexander polynomial $\Delta_1(X^\varphi,\K)$.
In fact, it can be computed by the following Alexander matrix associated to  $X^\varphi$:

\begin{equation}
\begin{pNiceMatrix}[first-row,first-col]
& \alpha & \beta_1 & \alpha_{1,2} & \cdots & \alpha_{1,m_1-1} & \cdots  &\beta_s & \alpha_{s,2} & \cdots & \alpha_{s,m_s-1}  \\
\alpha^{1-s}\beta_1 \cdots \beta_s &\dfrac{t^{\epsilon (1-s)}-1}{t^\epsilon-1} & t^{\epsilon (1-s)} & 0 & \cdots & 0 & \cdots &  t^{\epsilon (1-s)+\sum_{i=1}^{s-1} \varepsilon_i} & 0 & \cdots & 0\\
[\alpha,\beta_1] & 1-t^{\varepsilon_1} & t^{\epsilon}-1 \\
[\alpha_{1,2},\beta_1] &0 &t^{\epsilon_{1,2}}-1 & 1-t^{\varepsilon_1}\\
\vdots &\vdots & \vdots & & \ddots\\
[\alpha_{1,m_1-1},\beta_1] &0 & t^{\epsilon_{1,m_1-1}}-1 & & & 1-t^{\varepsilon_1}\\
\vdots &\vdots &  & &  & &\ddots\\
\vdots &\vdots &  & &  & &\\
[\alpha,\beta_s] & 1-t^{\varepsilon_s} & & & & & & t^{\epsilon}-1 \\
[\alpha_{s,2},\beta_s] & 0 & & & & & & t^{\epsilon_{s,2}}-1 & 1-t^{\varepsilon_s} \\
\vdots  & \vdots  &  & &  & & & \vdots& & \ddots \\
[\alpha_{s,m_s-1},\beta_s] & 0 & & & & & & t^{\epsilon_{s,m_s-1}}-1 & & & 1-t^{\varepsilon_s}\\
\end{pNiceMatrix}.
\end{equation}

First, one can show the determinant of this matrix is 0 by cofactor expansion of the first row. Then
it is easy to see that this matrix has rank $n-1$ if $\epsilon\neq 0$ and  $\varepsilon_i\neq 0$ for all $m_i>2$ (if $m_i=2$, one can see that the column for $\alpha_{i,2}$ does not appear, hence the value of $\varepsilon_i$ does not effect the rank in this case).
In particular, $H_1(X^\varphi,\K)$ is a torsion $\K[t^\pm]$-module.

Using some elementary column and row transformations, we get the following new matrix:

\begin{equation*}
\begin{pNiceArray}[first-row]{ccccc|ccccccc}
\alpha & \beta_1 & \beta_2 &\cdots & \beta_s & \alpha_{1,2} & \cdots & \alpha_{1,m_1-1}& \cdots  & \alpha_{s,2} & \cdots & \alpha_{s,m_s-1} \\
\dfrac{t^{\epsilon(1-s)}-1}{t^{\epsilon}-1} &  t^{\epsilon(1-s)} & t^{\epsilon(1-s)+\varepsilon_1} & \cdots & t^{\epsilon(1-s)+\sum_{i=1}^{s-1}\varepsilon_i}\\
1-t^{\varepsilon_1} & t^{\epsilon}-1\\
1-t^{\varepsilon_2} & &t^{\epsilon}-1\\
\vdots & & & \ddots \\
1-t^{\varepsilon_s} & & & & t^{\epsilon}-1\\ \hline
& t^{\epsilon_{1,2}}-1 & & & & 1-t^{\varepsilon_1} \\
& \vdots & & & & & \ddots \\
& t^{\epsilon_{1,m_1-1}}-1 & & & & & & 1-t^{\varepsilon_1} \\
&&&\ddots&&&&&\ddots \\
&&&&t^{\epsilon_{s,2}}-1 &&&&& 1-t^{\varepsilon_s} \\
&&&& \vdots  &&&&&& \ddots \\
&&&& t^{\epsilon_{s,m_s-1}}-1 &&&&&&& 1-t^{\varepsilon_s} \\

\end{pNiceArray}.
\end{equation*}

$$f(t)=(t^{\epsilon}-1)^{s-2} \prod^s_{i=1}(t^{\varepsilon_i}-1)^{m_i-2}.$$
We consider some minors with size $n-1$.
Removing the row for $[\alpha,\beta_i]$ and the first column, the column for $\beta_i$ and the column for $\alpha_{i,j}$ with $1\leq i \leq s$ and $2\leq j \leq m_i-1$ respectively, the corresponding  minors are

$(t^{\epsilon}-1)\cdot f(t)$, $(t^{\varepsilon_{i}}-1)\cdot f(t) $ and $(t^{\epsilon_{i,j}}-1)\cdot f(t) $

up to a unit of $\K[t^{\pm}]$. For  the computation of these minors, one needs  the following equality

$$0=1-t^{\epsilon(1-s)}+(1-t^{\varepsilon_1})t^{\epsilon(1-s)}+\cdots+(1-t^{\varepsilon_{s}})t^{\epsilon(1-s)+\varepsilon_1+\varepsilon_{2}+\cdots+\varepsilon_{s-1}}, $$

which follows from (<ref>).

Since $\varepsilon_i=\epsilon+\sum^{m_i}_{j=2}\epsilon_{i,j}$, we have the following equality

$$1-t^{\varepsilon_i}=(1-t^{\epsilon})+(1-t^{\epsilon_{i,2}})t^{\epsilon}+\cdots +(1-t^{\epsilon_{i,m_i}})t^{\epsilon+\epsilon_{i,2}+\cdots+\epsilon_{i,m_i-1}} .$$

Putting all these together, we have that the Alexander polynomial of $X^\varphi$ divides

$$\gcd(t^{\epsilon}-1,t^{\epsilon_{1,2}}-1,\cdots, t^{\epsilon_{1,m_i}}-1, t^{\epsilon_{2,2}}-1, \cdots, t^{\epsilon_{s,m_s}}-1)f(t).$$

Note that $\varphi: \pi_1(\U)\twoheadrightarrow \Z$ is surjective,
which implies  $\gcd(\epsilon,\epsilon_{1,2},\cdots,\epsilon_{s,m_s})=1$.
This fact implies that

$$\gcd(t^{\epsilon}-1,t^{\epsilon_{1,2}}-1,\cdots, t^{\epsilon_{1,m_i}}-1, t^{\epsilon_{2,2}}-1, \cdots, t^{\epsilon_{s,m_s}}-1)=t-1,$$

which can be proved by the similar trick for above equality of $1-t^{\varepsilon_i}$.
Hence the Alexander polynomial of $X^\varphi$ divides $(t-1)f(t)$. One should check that the proof still works even if some $\epsilon_{i,j}=0$.

On the other hand, since $\pi_1(X) \to \pi_1(\U)$ is surjective, it follows that the inclusion map induces an epimorphism

$$H_1(X^\varphi,\K)\twoheadrightarrow H_1(\U^\varphi,\K), $$

see e.g. <cit.>. Putting all together, we get that $H_1(\U^\varphi,\K)$ is a torsion $\K[t^\pm]$-module
and the Alexander polynomial of $\U^{\varphi}$ divides the Alexander polynomial of $X^\varphi$. Then the claim follows.

This proposition is compatible with <cit.> and <cit.> for $\K=\C$.
One may also compare this proposition with the one obtained by Elduque <cit.>. Elduque studied the boundary manifold for the affine tubular neighborhood which is slightly different from ours. She computed the Alexander polynomial by Reidemeister torsion, while we did it by Fox calculus after computing the fundamental group of $X$. In particular, once can check that Proposition <ref> implies <cit.> for the non-twisted version. For example, to have $H_1(\U,\K)$ being a torsion $\K[t^\pm]$-module, <cit.> needs the assumption $\epsilon_{i,j}>0$ for all $1\leq i \leq s$ and $2\leq j \leq m_i$. Using equality (<ref>) and $s>1$, this certainly implies the assumption in Proposition <ref>.

§.§ Proof of Theorem <ref>

To prove Theorem <ref>, we need the following lemma.
Consider the integer matrix $C_N$ of size $N\times N$:

\begin{pNiceMatrix}
0 & 1 & 0 & \cdots & 0\\
0 & 0 & 1 & \cdots & 0\\
\vdots & & & & \vdots\\
0 & 0 & 0 & \cdots & 1\\
1 & 0 & 0 & \cdots & 0\\
\end{pNiceMatrix}.
For any positive integer $k$ and any field $\K$, $\rk_\K \big(C_N^k-I_N\big)=N-\gcd(k,N)$, where $I_N$ is identity matrix.

Since $C_N^N=I_N$, we can assume that $1\leq k < N$ without loss of generality (the case $k=N$ is clear).
Since $(C_N^k-I_N)^\mathsf{T}= C_N^{N-k}-I_N$ (here $*^\mathsf{T}$ denotes the transpose matrix), we only focus on the case $N-k\leq k$ in the rest of the proof. Then  $C_N^k-I_N$ equals to

\begin{pNiceArray}[first-row,first-col]{ccc|ccccc}
& 1 & \cdots      &       &       &  \cdots     & k+1     &  \cdots      &    \\
1     & -1 &       &       &       &       & 1     &       &    \\
\vdots &    &\ddots &       &       &       &       &\ddots &    \\
N-k   &    &       & -1    &       &       &       &       & 1  \\ \hline
N-k+1 & 1  &       &       & -1    &       &       &       &    \\
\vdots &    &\ddots &       &       &\ddots &       &       &    \\
\vdots &    &       &\ddots &       &       &\ddots &       &    \\
\vdots &    &       &       &\ddots &       &       &\ddots &    \\
N     &    &       &       &       & 1     &       &       & -1
\end{pNiceArray}

where all the entries that are not on the three skew lines are zero.

Using row operations, we can kill the lower-left sub-matrix by adding the $i$-th row to the $(N-k+i)$-th row for all $1\leq i\leq N-k$. Next, using column operations, we can kill the upper-right sub-matrix.
Then $C_N^k-I_N$ becomes in the form:

I_{N-k} & 0 \\
0       & C_{k}^{\ell} -I_{k}
\end{pNiceMatrix}$$

for some positive integer $\ell$.

Note that $C_{k}^{\ell} -I_{k}$ corresponds to $(t^\ell-1)(C_k)$. Since $k<N$, using induction on $N$  we can prove that the Smith normal form of $C_N^k-I_N$ is a matrix with entries on the diagonal being either $1$ or $0$. The starting step for $N=2$ can be checked easily.
So the rank of $ (C_N^k-I_N)$ does not depend on the choice of $\K$ and we only need to compute $\rk_\C (C_N^k-I_N)$.

Note that the characteristic polynomial of $C_N$ is $t^N-1$.
Then it is easy to see that $\rk_\C (C_N^k-I_N)=N-\gcd(k,N)$.

We proceed the proof by two steps.

Step 1: We first simplify the Alexander matrix of $X$ associated to $X^\varphi$ by elementary transformations.

Recall the following the Alexander matrix $A_\varphi$ associated to $X^\varphi$ by furthermore assuming $\epsilon=1$, where we identify the group ring $\Z[\Z]$ as $\Z[t^\pm]$:

& \alpha & \beta_1 & \alpha_{1,2} & \cdots & \alpha_{1,m_1-1} & \cdots  &\beta_s & \alpha_{s,2} & \cdots & \alpha_{s,m_s-1}  \\
\alpha^{1-s}\beta_1 \cdots \beta_s &\dfrac{t^{1-s}-1}{t-1} & t^{ 1-s} & 0 & \cdots & 0 & \cdots &  t^{ (1-s)+\sum_{i=1}^{s-1} \varepsilon_i} & 0 & \cdots & 0\\
[\alpha,\beta_1] & 1-t^{\varepsilon_1} & t-1 \\
[\alpha_{1,2},\beta_1] &0 &t^{\epsilon_{1,2}}-1 & 1-t^{\varepsilon_1}\\
\vdots &\vdots & \vdots & & \ddots\\
[\alpha_{1,m_1-1},\beta_1] &0 & t^{\epsilon_{1,m_1-1}}-1 & & & 1-t^{\varepsilon_1}\\
\vdots &\vdots &  & &  & &\ddots\\
\vdots &\vdots &  & &  & &\\
[\alpha,\beta_s] & 1-t^{\varepsilon_s} & & & & & & t-1 \\
[\alpha_{s,2},\beta_s] & 0 & & & & & & t^{\epsilon_{s,2}}-1 & 1-t^{\varepsilon_s} \\
\vdots  & \vdots  &  & &  & & & \vdots& & \ddots \\
[\alpha_{s,m_s-1},\beta_s] & 0 & & & & & & t^{\epsilon_{s,m_s-1}}-1 & & & 1-t^{\varepsilon_s}\\
\end{pNiceMatrix}.

Next we do some elementary transformations for $A_\varphi$.
First, we add the sum of the column for $\beta_i$ multiplying $\dfrac{t^{\varepsilon_i}-1}{t-1}$ to the first column.
Then the entry $(1,1)$ becomes to

+\sum^s_{i=1} t^{1-s+\varepsilon_1+\cdots+\varepsilon_{i-1}}
\dfrac{t^{\varepsilon_i}-1}{t-1}
=\dfrac{t^{1-s+\varepsilon_1+\cdots +\varepsilon_s}-1}{t-1}

where the second equality follows from (<ref>).

Note that the entry $([\alpha_{i,j},\beta_i],1)$ is $\dfrac{(t^{\varepsilon_i}-1)(t^{\epsilon_{i,j}}-1)}{t-1}$, which can be killed by the entry $t^{\epsilon_i}-1$ in  $\alpha_{i,j}$ column for $2\leq j \leq m_i-1$.
Here if $\epsilon_{i,j}=0$, this entry automatically equals to $0$, and if $m_i=2$, the row for $[\alpha_{i,j},\beta_i]$ does not appear.
Then the matrix become in the following form:

\begin{pNiceMatrix}
0 & t^{ 1-s} & 0 & \cdots & 0 & \cdots &t^{ (1-s)+\sum_{i=1}^{s-1} \varepsilon_i}  & 0 & \cdots & 0\\
0  & t-1 \\
0 &  & 1-t^{\varepsilon_1}\\
\vdots & & & \ddots\\
0 &  & & & 1-t^{\varepsilon_1}\\
\vdots &  & &  & &\ddots\\
\vdots & &  & &  & &\\
0 & & & & & & t-1 \\
0 & & & & & &  & 1-t^{\varepsilon_s} \\
\vdots    &  & &  & & & & & \ddots \\
0 & & & & & &  & & & 1-t^{\varepsilon_s}\\
\end{pNiceMatrix}.

By adding the first row multiplying  $-(t-1)t^{s-1}$  to the second row, the entry $(2,2)$ becomes 0,  and the other non-zero entries on the new second row can be killed by the row corresponding to $[\alpha,\beta_i]$, whose only non-zero entry is $(t-1)$ on the diagonal.

Finally by these procedures $A_\varphi$ becomes to a diagonal matrix

A'_φ=diag(0, t^1-s, t-1,⋯,t-1^s-1,  1-t^ε_1^m_1-2,⋯, 1-t^ε_s^m_s-2).

Step 2:  We compute $H_1(X^{\varphi,N},\K)$ for any field $\K$.

Since we focus on the first homology which is determined by $\pi_1(X)$, we can assume that $X$ is a 2-dimensional CW complex with its 2-cells corresponding to the relations in $\pi_1(X)$.
The group of deck transformations of the covering space $X^\varphi$ is isomorphic to $\Z$ and acts on it.
By lifting the cell structure of $X$ to $X^\varphi$,
we obtain a free basis for the cellular chain complex of $X^\varphi$ as $\K[t^{\pm}]$-modules.
Then the cellular chain complex $C_*(X^\varphi,\K)$ is a bounded complex of finitely generated free $\K[t^{\pm}]$-modules:

$$0\rightarrow C_2(X^\varphi,\K) \xrightarrow{\partial_2} C_1(X^\varphi,\K) \xrightarrow{\partial_1} C_0(X^\varphi,\K) \rightarrow 0.$$

With the  right cell structure of $X$ with respect to the presentation of $\pi_1(X)$ in section 2.2, $\partial_2$ can be written down as $A_\varphi$. Note that one can compute $H_1(X^{\varphi,N},\K)$ by taking the first homology (as $\K$-vector spaces) of the chain complex
$C_*(X^\varphi,\K) \otimes_{\K[t^{\pm}]} \K[t^{\pm}]/(t^N-1) $.
The map $\partial_2$ becomes $A_\varphi(C_N)$, which  we replace $t$ by $C_N$. The elementary transformation of $\Z[t,t^{-1}] $-coefficient on $A_\varphi$ corresponds to elementary transformation of block matrix on $A_{\varphi}(C_N)$. So we can work on the diagonal matrix $A'_\varphi$ directly.

By Lemma <ref>, we have that

$$ \begin{aligned}
\rk_\K A_\varphi(C_N)
\end{aligned}

where the second equality follows from (<ref>).

On the other hand, since the composed map $\pi_1(X)\to \pi_1(\U) \to \Z_N$ is surjective, $X^{\varphi,N}$ is connected. Hence

$$\dim_\K H_1(X^{\varphi,N},\K)= nN-(N-1)-\rk_\K A_\varphi(C_N)=n-1+\sum^s_{i=1}(m_i-2)(\gcd(\varepsilon_i,N)-1)$$

Then the inequality (<ref>) follows from Lemma <ref>.

For any field $\K$, by  the Universal Coefficients Theorem and <cit.>  we have

$$\dim_\K H_1(\U^{\varphi,N},\K)\geq \dim_\C H_1(\U^{\varphi,N},\C) \geq \dim_\C H_1(\U,\C) =n-1.$$
Therefore, if  $\gcd(\varepsilon_i,N)=1$ for all $m_i>2$,
$\dim_\K H_1(\U^{\varphi,N},\K) = n-1$ for any field $\K$, hence $H_1(\U^{\varphi_N},\Z)\cong \Z^{n-1}$ by the Universal Coefficients Theorem.

We only need to show that the Minor fiber satisfies the assumption in Theorem <ref>. The Milnor fiber is associated to the map $\pi_1(\U) \to \Z_n$, which sends every meridian of lines in $\sA$ to $1\in \Z_n$. By Remark <ref>, it is clear that the assumption holds.

Consider the icosidodecahedral arrangement consisting of 16 lines in $\CP^2$ introduced by Yoshinaga <cit.>.
Every line in this arrangement has only multiple points with multiplicity 2 or 4. In particular, the line $H_{16}$ has exactly 5 multiple points with multiplicity all being 4. So torsion in the homology of the Milnor fiber is possible according to Theorem <ref>. In fact, Yoshinaga showed that $H_1(F,\Z)$ has 2-torsion <cit.>.

On the other hand, we have a lot of 16-fold covering of $\U$, whose first homology group is torsion-free by Theorem <ref>. For example,
consider the 16-fold covering by sending $\alpha_{1}\to 1$, $\alpha_2,\alpha_5,\alpha_8 \to 2$, $\alpha_3 \to -2$ and other lines to $1$.
Then the first integral homology group of this 16-fold cover is torsion-free by applying Theorem <ref> on the first line $H_1\in \sA$ as in <cit.>.

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