# Towards a robust and reliable deep learning approach for detection of
compact binary mergers in gravitational wave data

Shreejit Jadhav<EMAIL_ADDRESS>Inter-University Centre for Astronomy and
Astrophysics (IUCAA), Post Bag 4, Ganeshkhind, Pune 411 007, India  Centre for
Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn,
Victoria 3122, Australia  ARC Centre of Excellence for Gravitational Wave
Discovery (OzGrav), Melbourne, Australia  Mihir Shrivastava
<EMAIL_ADDRESS>Indian Institute of Technology (IIT), Kharagpur, India
Sanjit Mitra<EMAIL_ADDRESS>Inter-University Centre for Astronomy and
Astrophysics (IUCAA), Post Bag 4, Ganeshkhind, Pune 411 007, India

###### Abstract

The ability of deep learning (DL) approaches to learn generalised signal and
noise models, coupled with their fast inference on GPUs, holds great promise
for enhancing gravitational-wave (GW) searches in terms of speed, parameter
space coverage, and search sensitivity. However, the opaque nature of DL
models severely harms their reliability. In this work, we meticulously develop
a DL model stage-wise and work towards improving its robustness and
reliability. First, we address the problems in maintaining the purity of
training data by deriving a new metric that better reflects the visual
strength of the “chirp” signal features in the data. Using a reduced, smooth
representation obtained through a variational auto-encoder (VAE), we build a
classifier to search for compact binary coalescence (CBC) signals. Our tests
on real LIGO data show an impressive performance of the model. However, upon
probing the robustness of the model through adversarial attacks, its simple
failure modes were identified, underlining how such models can still be highly
fragile. As a first step towards bringing robustness, we retrain the model in
a novel framework involving a generative adversarial network (GAN). Over the
course of training, the model learns to eliminate the primary modes of failure
identified by the adversaries. Although absolute robustness is practically
impossible to achieve, we demonstrate some fundamental improvements earned
through such training, like sparseness and reduced degeneracy in the extracted
features at different layers inside the model. Through comparative inference
on real LIGO data, we show that the prescribed robustness is achieved at
practically zero cost in terms of performance. Through a direct search on
$\sim 8.8$ days of LIGO data, we recover two significant CBC events from
GWTC-2.1 [1], GW190519_153544 and GW190521_074359. We also report the search
sensitivity obtained from an injection study.

###### pacs:

##  I Introduction

The first detection of a gravitational-wave (GW) signal originating from a
binary black-hole (BBH) merger [2] by the advanced LIGO detectors [3] marked
the beginning of a new era in astronomy. Collectively, more than ninety GW
events have been detected so far [4, 5, 6, 7, 8, 1, 9, 10] in the data
released [11, 12] by the advanced LIGO [3] and advanced Virgo [13] detectors.
These detections include a variety of compact binary coalescence (CBC) events
of great astrophysical significance [14, 2, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26]. Advanced LIGO detectors and KAGRA [27, 28] began the fourth
observing run on May $24$, $2023$ and Virgo is planning to join shortly [29].
In the latter half of the decade, a third LIGO detector in India is expected
to join the network [30, 31, 32], which will enable improvements in the event
significance, polarisation studies, sky localisation and overall duty factor
[33].

In the past few years, applications of machine learning algorithms have been
explored for a variety of tasks in GW data analysis, most notably, CBC
searches [34, 35, 36, 37], sensitivity improvements of existing search
pipelines [38, 39, 40, 41, 42, 43], non-linear noise modelling and subtraction
[44, 45] that is useful for early alerts for CBCs [46], parameter estimation
[47, 48], glitch classification [49, 50, 51, 52, 53, 54, 55], population
studies [56, 57] and searches for other interesting signals originating from
intermediate-mass black hole (IMBH) mergers [58], strongly-lensed events [59],
short gamma-ray bursts [43], continuous waves [60] and unknown astrophysical
events [61, 62].

This work focuses on the applications of deep learning (DL) for CBC searches.
Though template-based matched-filtering searches are presently the most
sensitive, their parameter space coverage is limited by the availability of
computational resources. This limitation also restricts the search sensitivity
for astrophysically interesting parameters like eccentricity, precession and
higher-order modes. While the unmodelled searches cover a larger parameter
space, they have a relatively lower sensitivity due to a loosely bounded
likelihood function and, more importantly, the presence of glitches in the
data. The strength of DL models in distinguishing glitches from CBC signals
makes them excellent alternative tools for searches [38, 34, 35, 36, 58].
Also, DL methods readily utilise accelerated computing hardware like graphics
processing units (GPUs), improving the speed by orders of magnitude.

The field of GW searches has witnessed promising developments on the machine
learning (ML) front in recent years. Some of the early works professing the
advantages of ML methods for GW searches focused on sensitivity comparisons
against matched-filtering primarily using simulated noise [63, 64] with a
limited extension to real data [65]. These works based their sensitivity
estimates on Receiver operating characteristic (ROC) curves obtained from
datasets consisting of discrete samples that placed CBC injections in specific
regions of the analysis window. Gebhard  _et al._ [66] critically examined the
suitability of such approaches for realistic searches by addressing various
issues like class imbalance, temporal localisation of candidate events and
assigning statistical significance. A more reliable approach of performing
trigger-based coincident analysis with the DL outputs on a stream of
continuous data was proposed in the literature [35, 34], which is followed in
this work as well. A systematic approach in constructing a proper ranking
statistic is also a crucial step towards implementing a coincident search [38,
67, 41, 34]. To enable comparative evaluation of different ML pipelines in a
realistic search setting and to provide a common platform, a mock data
challenge was organised recently [68].

The DL approaches come with a cost. The most critical hurdle that keeps them
from being considered for production-level deployment is the unreliability
originating from their black-box nature. It has been shown that tiny
perturbations to the input images achieved through targeted attacks can lead
to surprise failures of these models [69]. Also, generative adversarial
attacks can reveal many other failure modes. A generator trained to deceive a
DL model can yield many rubbish images which do not belong to any meaningful
class, although, the model classifies them into specific classes with high
probabilities. These failures highlight the ambiguity over the actual features
learnt by a DL model that otherwise shows an excellent performance in metrics
like accuracy, F1-score, and area under the ROC curve. It also highlights how
the performance of DL models on new data is unpredictable and fragile, making
them unreliable for the analysis of real data at the production stage. In the
context of GW searches, by obtaining adversarial examples through input
perturbations, Gebhard  _et al._ [66] first demonstrated the simple “failure
modes” of DL models, which the conventional approaches could handle easily. In
this work, we take a generative adversarial network (GAN)-based approach to
generate the adversarial samples. GANs have previously been used in GW data
analysis primarily for generating various transient signals [70, 71]. Here, we
take a novel approach of utilising the GAN framework for bringing robustness
to our DL model.

In this work, we explore several ways of making DL models more reliable.
Although the focus is on CBC searches, the methods presented here are
extendable to searches for signals from other sources and possibly to tasks
other than searches, such as glitch classification and parameter estimation.
In Section II, we address the issues related to the purity of the training
data, describe the development of a new metric for data generation and the
preparation of various datasets for different stages of training. Section III
elaborates the training of a variational auto-encoder (VAE) model to achieve a
reduced representation of the signal-specific features in the input.
Repurposing this information for the task of signal identification is detailed
in Section IV. Section V describes the scrutiny of the resulting classifier
model for robustness and its retraining in the GAN setup to increase
robustness. Lastly, the implementation of a direct search using the classifier
is provided in Section VI. We conclude with outlook and future directions in
Section VII.

##  II What are we showing to the model?

In this section, we describe the development of a new metric for data
generation and the preparation of datasets. For efficient training of various
stages of our model, we construct four different datasets for training, viz.,
injections in simulated Gaussian noise (INJ), the same set of injections
without noise (CLEANINJ), the same Gaussian noise realisations without
injections (GN) and a glitch dataset (GL) obtained from GravitySpy [72, 73,
53, 55]. To ensure near absolute fidelity of the training data, we stick to
simulated Gaussian noise. However, we consider only real LIGO data for
testing. We prepare three classes of test datasets, viz., CBC injections (INJ-
TEST), PyCBC background triggers (TRIG-TEST) and glitches (GL-TEST). Our
experiments on the test datasets show that the features learnt by our model
are generalised enough to ensure the model’s applicability to real data.

###  II.1 New metric for INJ data

(a) $\mathcal{M}=9.0$; $\rho=8.2$

(b) $\mathcal{M}=54.3$; $\rho=8.1$

Figure 1: Example CWT maps of two injections highlighting how the chirp-masses
of CBC signals determine their _visibility_ in the time-frequency domain
representation even when the recovered SNR values are nearly the same.
However, in $\gamma$ metric (as described in Equation 2), these injections are
well separated with values $26.5$ and $44.9$, respectively.

Most of the past works considered a fixed lower cutoff on signal-to-noise
ratio (SNR) across the parameter space for generating training data either
directly or by fixing the upper cutoff on the chirp-distance111Chirp-distance
is the luminosity distance adjusted for the leading order chirp mass term in
the CBC waveform.. However, the duration of time the signal remains in the
sensitive band of a detector, which in turn depends on the component masses
and the power-spectral density (PSD), strongly determines the _visibility_ of
the signal222Here, _visibility_ refers to the relative strength of the pixels
corresponding to the signal features in the input image to those corresponding
to the noise features. It can also be imagined as a metric that captures
humans’ ability to distinguish signal features from noise features in the
image data.. Figure 1 illustrates this with the continuous wavelet transforms
(CWT) maps of two injections that have nearly the same recovered SNR values
but different chirp masses. Clearly, the low chirp-mass binary has a smaller
amplitude but spends a much longer time in the sensitive band of LIGO/Virgo.
Therefore, the total energy contained in its signal is distributed over a much
longer duration resulting in reduced visibility of its chirp features compared
to the high chirp-mass binary. This difference is expected to play a crucial
role for data representations which do not include the phase information,
e.g., time-frequency representations such as CWT maps, Q-transforms, and
short-time Fourier transforms saved as images with pixel values proportional
to the amplitude. To efficiently train the DL models to detect the weakest
possible signals across the parameter space, maintaining uniform visibility of
signal features across the training data is important. Therefore, instead of
using SNR-based thresholds, an approximate metric that better reflects the
signal’s visible strength could be the SNR divided by the duration of the
signal in the detector’s sensitive band. For a detector with flat frequency
response, the duration of the CBC signal can be captured in leading order
using the _chirp time_ $(\tau_{0})$. It is the Newtonian coalescence time
calculated starting from the frequency $f_{0}$ and is given by,

$\tau_{0}=\frac{5}{256\pi f_{0}}\bigg{(}\frac{\pi
G\mathcal{M}f_{0}}{c^{3}}\bigg{)}^{-5/3}\,.$ (1)

Considering that the detector noise is coloured, we can choose $f_{0}$ in such
a way that we operate in a frequency band where the response of the detector
is approximately flat and $\tau_{0}$ remains relevant even after whitening.
However, we found that the approximate metric constructed this way lost its
validity rapidly upon approaching higher masses where Newtonian approximation
fails to describe the waveform. To solve this issue, we numerically modelled
the intrinsic mass dependence of the visibility of the chirp features. We
obtained maximum amplitude values of the normalised whitened waveforms
generated at each point in the component mass space to capture the variations
in the maximum pixel values corresponding to the CBC chirp features in the CWT
maps. We uniformly divided the mass-ratio ($q$) - total mass ($M$) space,
generated waveforms at each point of the grid using IMRPhenomXPHM approximant
and normalised them (such that
$\rho(\hat{\mathrm{\textbf{h}}},\hat{\mathrm{\textbf{h}}})=1$). The $q$ and
$M$ values were sampled in the range $[1,8]$ and $[4,800]$ with step-sizes
$0.1$ and $10$, respectively. We whitened both ’$+$’ and ’$\crossproduct$’
polarisations of each waveform using the aLIGOZeroDetHighPower PSD, and
recorded the maximum amplitude of their quadrature sum as the intrinsic mass
dependence factor in the amplitude, $A_{\mathrm{intr}}(q_{i},M_{i})$ for the
$i^{\mathrm{th}}$ binary.

Figure 2: Value of the maximum amplitude of normalised whitened waveform
$A_{\mathrm{intr}}(q,M)$ as a function of the mass ratio $q$ and total mass
$M$. We use aLIGOZeroDetHighPower PSD for whitening. The relative variation of
$A_{\mathrm{intr}}(q,M)$ can be as large as $\sim 7$ in the extreme regions of
the mass space. The dashed line marks the region of parameter space used for
the training and testing.

We used linear interpolation to obtain a continuous function
$A_{\mathrm{intr}}(q,M)$ that is valid at all points within the range of
parameters considered for this numerical estimation. It can be observed from
Figure 2 that the maximum relative variation in $A_{\mathrm{intr}}(q,M)$ is as
high as $\sim 7$. Using this intrinsic mass dependence, we construct a new
metric that better reflects the _visible strength_ of the chirp features. We
call this metric $\gamma$, which is obtained as,

$\gamma(\bm{\theta})=A_{\mathrm{intr}}(\bm{\theta})\rho(\mathrm{\textbf{s}},\hat{\mathrm{\textbf{h}}}(\bm{\theta})),$
(2)

where, $\rho(\mathrm{\textbf{s}},\hat{\mathrm{\textbf{h}}}(\bm{\theta}))$ is
the SNR calculated with data s and template
$\hat{\mathrm{\textbf{h}}}(\bm{\theta})$ with $\bm{\theta}$ limited to only
the mass parameters.

Figure 3: Variation in the visible strength of chirp features with the metric
$\gamma(\theta)$, independent of the chirp mass. The images are sampled from
the INJ dataset with $M_{chirp}$ and $\gamma$ values lying in the
corresponding bins shown in the brackets.

We, therefore, obtain a metric that is a direct function of the visibility of
the signal features in the input image and is independent of the component
masses of the CBC. As an example, the injections shown in Figure 1 having
nearly the same SNR values are fairly separated in their $\gamma$ values which
come out to be $26.5$ and $44.9$, respectively. The non-inclusion of the spin
parameters in the $A_{\mathrm{intr}}$ function is expected to retain some
impurity in the training data. However, these impurities are expected to be
small enough that their effect on training remains minimal. In Figure 3, we
show how the visible strength of signals in the image changes as a function of
$\gamma$ alone and remains almost unaffected by chirp mass.

###  II.2 Preparation of datasets

For generating the INJ dataset, we consider a region of CBC parameter space
covering masses that are most likely to be observed with LIGO/Virgo detectors,
and precession. A large number of injections are made in the simulated
coloured Gaussian noise generated from aLIGOZeroDetHighPower PSD. The hyper-
parameter choices for these injections are listed in Table 1. We whiten and
band-pass the data between $10-512$ Hz and collect $3$ second long slices
containing injections. Further, we obtain the time-frequency representations
of these data slices using CWT with Morlet wavelet. We then crop these $2$-D
arrays to $1$ second long CWT maps in such a way that the merger times of the
binaries are uniformly distributed between $0.7-0.85$ second of the slice.
Such a range is chosen to allow a reasonable coverage of the inspiral and
post-merger parts of the low-mass and high-mass binaries, respectively. To
discard the filter artefacts, we manually set the values corresponding to
frequencies less than $2$ Hz in the CWT maps to zero. For better
visualisation, the frequency scale is set to logarithmic. The same procedure
is repeated for generating the CLEANINJ dataset using the same injection set,
except without noise this time. The GN dataset is prepared by collecting the
Gaussian noise realisations from the INJ dataset while omitting the CBC
injections. Therefore, the INJ dataset is effectively the direct addition of
data slices from CLEANINJ and GN datasets. During training, as the visibility
of the CBC signals weakens, the model tends to over-fit the noise features.
Our data preparation eliminates the possibility of over-fitting the training
data, as the features corresponding to noise are the same in INJ and GN
datasets. This forces the model to focus on the uncommon features that
correspond to the CBC signal exclusively. To prepare the GL dataset, we obtain
the class and GPS time information from Gravity Spy [72, 73, 53, 55], cluster
the GPS times within $0.1$ second, and choose the data slices such that the
glitches are uniformly distributed between $0.25-0.75$ second of the slice.
Since no specific morphology is to be captured in the case of glitches, this
choice of range is made while ensuring complete coverage. The same procedure
mentioned above is repeated to prepare the CWT maps. For the datasets
involving CBC injections (INJ and CLEANINJ), we consider samples with
$\gamma>30$ and $\rho>5$. Note that we use the _recovered_ SNR instead of
_injected_ SNR while calculating the $\gamma$ values to account for variations
due to noise realisation, thereby reducing the error in the estimations of
$\gamma$. This is especially helpful in maintaining faithfulness of the
injection data even at lower values of $\gamma$. A total of $50000$ samples
were collected with these cuts, of which $95\%$ were used for training and the
rest for validation. We also generate separate test datasets using real data
as detailed in Section V.1.1.

We also prepare three separate test datasets using real LIGO data. For the
signal class (INJ-TEST), we make injections with the same parameter choices
described in Table 1. Though, unlike the INJ dataset, these were made in the
real data from the entire third observing run of LIGO detectors at Hanford
(H1) and Livingston (L1), and with a new seed value to ensure new samples of
parameters. For the noise class, we consider two cases of direct relevance for
actual searches: the glitches (GL-TEST) and the background triggers collected
through matched filtering (TRIG-TEST). The specification of GL-TEST dataset is
kept the same as that of GL while ensuring no overlap between the two. For
TRIG-TEST dataset, random samples were obtained from the triggers collected by
the PyCBC offline search of $\sim 7.24$ days long chunk of data between May
$29$ to June $5$, $2019$ UTC. As the CBC events are sporadic, we do not expect
any real events to be present in this dataset. We collect $\sim 5000$ samples
for each of the datasets.

Parameter | Value
---|---
Chirp-distance | Uniform in ($5$, $400$) Mpc
Component masses |  $M_{chirp}$ uniform in ($5$, $60$) $\text{M}_{\odot}$
|  $q$ uniform in ($1$, $4$)
Component spins |  $|S_{1}|,|S_{2}|$ uniform in ($0$, $0.998$)
| randomly orientated over sphere
Approximant | IMRPhenomXPHM
Table 1: Choices of hyper-parameters for injections used in INJ and INJ-TEST
datasets. Additional cutoffs of $\gamma>30$ and $\rho>5$ are applied to ensure
uniform visible strength of chirp features across the parameter space.

##  III Where is the model looking?

One of the primary hurdles that limit the reliability of DL methods is their
tendency to over-fit the data in undesirable ways. Here, we explore an
approach that significantly mitigates this issue and helps our model focus
only on the relevant features in the input data. Also, the signal-specific
information is minimal compared to the actual amount of data fed to the model.
Therefore, a reduced representation of the input data that captures the
complete signal information is particularly beneficial for any analysis.
Specifically, for the case of modelled signals, this reduced parameter space
can, in principle, be written in terms of only the non-degenerate set of
intrinsic and extrinsic parameters of the signals. Therefore, a model that
reads the complete signal information from the data should ideally be able to
represent the signal in a reduced parameter space that can be transformed back
to the full signal without any loss333Here, the assumption is that the input
data representation is obtained using a transformation of the strain data that
is lossless for the signal.. However, the presence of additive noise
inevitably introduces statistical errors in estimating source parameters in
any space. Though the resulting errors in the inferred parameters can be
theoretically characterised using the statistical noise model.

For the signal model, an extensive matched-filter-based search uses maximum
likelihood estimation (MLE) analytically over the extrinsic parameters and
numerically over the intrinsic parameters. MLE essentially achieves the
reduced representation of the input strain in the parameter space of CBC
signals. However, such searches are computationally expensive, and some
parameters, such as eccentricity, component spins, etc., must be omitted.
Otherwise, the computation cost could increase by orders of magnitude, which
is presently impossible to accommodate. Also, the noise model considered for
constructing the log-likelihood function for the search assumes stationary
Gaussian noise for calculating the signal-to-noise ratio [74]. In general,
search models built with this assumption (matched-filter-based searches or DL
approaches trained against Gaussian noise alone) prove inadequate when the
noise has non-Gaussianities in the form of glitches. Including glitches in the
construction of the search log-likelihood function can play a crucial role in
separating signal and noise more effectively. However, this is not achievable
in practice due to the lack of an analytical description of the glitches.
Rather, various signal consistency tests are often employed as additional
measures to mitigate the adverse effect of glitches on search sensitivity [75,
76]. DL models being excellent approximators of non-linear features can learn
the broad set of morphologies followed by these glitches. In the presence of
Gaussian noise alone, the performance of a DL model can be compared against
that of matched filtering. However, their added ability to learn artefacts
originating from glitches allows them to surpass matched-filter-based searches
in principle. However, these arguments are based on theoretical assumptions of
the capabilities of DL models and achieving such performance in practice is
often very challenging.

###  III.1 Obtaining a reduced representation of the input

A specific class of neural networks, known as auto-encoders, offer encoding of
the data into an efficient coding [77]. The architecture of auto-encoders
involves an encoder $\mathcal{E}_{\phi}$ that encodes the input data
$\mathcal{X}$ into a latent space coding $\mathcal{Z}$ and a decoder
$\mathcal{D}_{\theta}$ that learns to regenerate the required output data
$\mathcal{X^{\prime}}$ from it, where, $\mathcal{X^{\prime}}$ can be same as
or a transformed version of $\mathcal{X}$. The reduced coding $\mathcal{Z}$
offers a condensed representation of the data. Going a step beyond, a VAE
offers the representation of each input sample $x$ as a multivariate posterior
distribution $q_{\phi}(z|x)$ for $x\in\mathcal{X}$ and $z\in\mathcal{Z}$ [78].
Typically, the posterior $q_{\phi}(z|x)$ is modelled as a Gaussian and is
regularised to be as close as possible to the prior of $\mathcal{D}_{\theta}$
using Kullback-Leibler (KL) divergence. This prior is denoted by
$p_{\mathrm{model}}(\textbf{z})$ and is employed for generating new data using
$\mathcal{D}_{\theta}$. It is generally taken as the Normal distribution
$\mathcal{N}(0,1)$. As values of $z$ are drawn from the distribution
$q_{\phi}(z|x)$ at the latent layer, performing back-propagation becomes
problematic. This issue is solved using the re-parametrisation trick, which
models the distribution $q_{\phi}(z|x)$ as a multivariate Gaussian whose mean,
$\mu$ and standard deviation, $\boldmath\sigma$ are predicted as latent space
outputs from the encoder. Now, $q_{\phi}(z|x)$ can be written as,

$q_{\phi}(z|x)=\mu+\epsilon\cdot\sigma,$ (3)

where the random samples $\epsilon$ are separately obtained from
$\mathcal{N}(0,1)$, which remains fixed. Therefore, the back-propagation is
made possible with the gradient descent performed over the parameters $\mu$
and $\sigma$ that govern the distribution $q_{\phi}(z|x)$ instead of the
sampled values $z$. The final loss function to train a VAE, given by Equation
4, has two components. First, the reconstruction loss that compares output
against $\mathcal{X^{\prime}}$, and second, the KL divergence-based loss.

$\begin{split}\mathcal{L}_{\phi,\theta}(\mathcal{X},\mathcal{X^{\prime}},q)&=\mathcal{L}_{\mathrm{rec}}(\mathcal{X},\mathcal{X^{\prime}})+\sum_{j}\mathcal{D}_{\mathrm{KL}}\big{(}q_{j}(z|x)\parallel\mathcal{N}(0,1)\big{)}\\\
&=\sum_{j}\Big{[}\big{\|}x^{\prime}_{j}-\mathcal{D}_{\theta}(\mathcal{E}_{\phi}(x_{j}))\big{\|}^{2}+\mathcal{D}_{\mathrm{KL}}(\mathcal{N}(\mu_{j},\sigma_{j})\parallel\mathcal{N}(0,1))\Big{]}.\end{split}$
(4)

Training of the VAE model with such a loss function instils bounded, smooth
and meaningful parametrisation of the latent code $\mathcal{Z}$.

Figure 4: The schematic of the VAE model, along with its input (INJ) and
target output (CLEANINJ) used for training. The sampling layer at the latent
space learns to represent the CBC features extracted from the input as a
multivariate Gaussian, thereby obtaining their reduced and smooth
representation. Figure 5: Evolution of losses over $600$ epochs of training.
The KL divergence saturates to an almost constant value after $\sim 200$
epochs while the reconstruction loss continues to reduce slowly. Note that the
y-axis is in log scale. The VAE model reaches a satisfactory level of
reconstruction ability at the end of training.

We build a VAE for the task of _denoising_ the CWT maps. However, the
denoising task here is not the literal pixel-level denoising of the image.
Instead, it is transforming the input CWT maps from INJ into the corresponding
CWT maps of CLEANINJ, which demands preserving of features belonging to the
injection exclusively and ignoring other artefacts belonging to the noise
class as shown in Figure 4. Therefore, during training, the model learns to
_look for_ only the signal features, encode them into the latent space and
decode a pure estimate of the signal. Since our final classification task is
only signal-specific, any lack of retention of other non-signal-specific
information from the input, such as glitch or Gaussian noise features, is not
a concern. Rather, the insensitivity of the encoder to some of the glitch
features may, in fact, be desirable for eliminating the respective glitches
right at the first step of inference. We construct the encoder part of the VAE
model using sub-modules consisting of parallel branches of convolutions of
different kernel sizes as shown in Figure 19. The parallel modules are
inspired by Inception modules which help extract features at varying scales
[79]. Similarly, the architecture of the decoder part is shown in Figure 20.
We use L$1$ regularisation with $l_{1}=0.001$ for kernel and bias parameters
in all convolution layers of the encoder and decoder. The weights of
convolutional kernels were initialised using Xavier uniform initialiser [80]
while biases were initialised with a constant value of $0.2$. We use a
cyclical learning rate varying in the range $10^{-5}-10^{-4}$, which resulted
in faster convergence and better performance at the same time [81]. Since,
such a scheme results in oscillating losses, we use a callback function to
monitor the total loss during training and stored the best checkpoint with
minimum loss. The choice of optimiser was stochastic gradient descent (SGD)
with momentum $0.01$. We also set the gradient clipping at $0.5$ to deal with
the exploding gradient problem. We trained our VAE model on four NVIDIA A$100$
Tensor Core GPUs.

Figure 5 shows the evolution of losses during model training for $600$ epochs
with an overall asymptotically saturating behaviour with oscillations
originating from cyclical learning. The best-collected checkpoint had a
reconstruction loss value of $19.3$ and the KL divergence loss value of
$15.3$. To showcase the model’s ability to reconstruct clean signals from
input injections, in Figure 6, we show $4$ randomly selected images from the
validation INJ data, their reconstructions using the trained VAE model and the
corresponding target CLEANINJ images for comparison. Once trained, we discard
the decoder part of the model and only use the encoder, which gives the
reduced latent code of the input image as an output.

Figure 6: Reconstructed clean maps output by the trained VAE model for some of
the input samples from the validation INJ dataset. The corresponding target
samples from the CLEANINJ dataset are also shown for comparison.

(a)

(b)

Figure 7: Latent space representations of the validation data from INJ, GN and
GL datasets along with the TRIG-TEST dataset visualised using t-SNE at the
encoder output. (a) Encoder output taken from the trained VAE model. Though
the VAE was trained on INJ and CLEANINJ data alone, the encoder can already
separate the classes very well. (b) Encoder output from the final classifier
after fine-tuning. Since the classifier was trained on INJ, GN and GL
datasets, an improved class separation can be observed after fine-tuning.

We use t-distributed Stochastic Neighbour Embedding (t-SNE) to visualise the
latent space encoding at different stages of model development and make
qualitative observations on their performance. t-SNE is a statistical method
for visualising high-dimensional data by giving each data point a location in
a two or three-dimensional map [82]. Unlike Principal Component Analysis
(PCA), a linear dimensionality reduction algorithm, t-SNE is based on
probability distributions with a random walk on neighbourhood graphs to
interpret the complex polynomial relationships between features and find
structure within the data. We generate the t-SNE visualisations of the latent
space coding obtained from the encoder at two separate stages. The first being
the encoder output from the trained VAE model, while the latter one is
obtained after the classifier fine-tuning is accomplished (described in
Section IV). The prime difference between the two is that the first model was
trained with the INJ dataset alone, while the second one was trained with INJ,
GN and GL datasets. In order to obtain the latent space required for t-SNE, we
analyse the validation data samples from INJ, GN, GL and test data samples
from TRIG-TEST with both the encoder models. Before proceeding with t-SNE
visualisation, we reduce the dimensions of the latent space from $256$ to $10$
using PCA. The benefits are two-fold, it helps suppress some noise in the
final representation and speeds up the algorithm without losing much
information. After performing PCA, we find that the explained variance ratio
for the encoder output after VAE training was $0.46$ compared to $0.94$ after
the fine-tuning stage. From the t-SNE representations shown in Figure 7, it is
evident that even without training on the GN, GL and TRIG-TEST datasets, the
initial encoder is already able to separate them satisfactorily. The class
separation improves further after fine-tuning.

##  IV Utility of the latent code for classification

Figure 8: A hybrid classifier model obtained by appending densely connected
layers on top of the trained VAE’s encoder. The network is trained to map the
latent space coding to the binary classification output corresponding to
signal (INJ) and noise (GL and GN with losses weighed down by half).

(a)

(b)

Figure 9: Performance metrics for the hybrid classifier are shown for its two
stages of training, viz., with encoder frozen and fine-tuning of the entire
model. During both stages, the classifier is trained to separate the INJ class
from the noise classes (GN and GL). (a) Evolution of binary cross-entropy
losses for training and validation data. Initially, with the encoder frozen,
both training and validation losses saturated to a value of $42.4$. During
fine-tuning, the best checkpoint was collected with a training loss of $0.025$
and a validation loss of $0.031$. (b) Evolution of training and validation
accuracies. At the end of the training, both the accuracies for the hybrid
model with frozen encoder were close to $99.6\%$. The best checkpoint had a
training accuracy of $99.99\%$ and a validation accuracy of $99.55\%$.

We develop a hybrid classifier model by appending the encoder with additional
densely connected layers that map the latent space to a single output node
predicting the presence of a CBC signal in the input (refer to Figure 8). The
input to the densely connected layers is the mean-vector $\mu$ of the
multivariate Gaussian distribution from the latent space. We train this hybrid
model in two stages. Initially, we froze the encoder and only allowed the
appended dense layers to learn the mapping between the latent space and the
binary outputs. The training data was composed of signal class (INJ dataset)
and noise class (GN and GL datasets) with $1$’s and $0$’s as targets,
respectively. The losses for GN and GL datasets were weighed down by half to
maintain class balance. The data were shuffled before the start of training.
We train the hybrid network for $600$ epochs with a cyclical learning rate
varying between $10^{-5}-10^{-4}$ and binary cross-entropy as the loss
function with label smoothing set to $0.005$. We used Adam optimiser with
$\beta_{1}=0.9$, $\beta_{2}=0.999$ and $\epsilon=10^{-7}$. The gradients were
clipped at a maximum value of $50$. At the end of the training of the first
stage, both training and validation accuracies were close to $99.6\%$ while
the losses saturated around $42.4$. After the parameters of the densely
connected layers were learnt, we further fine-tuned the network for $600$
epochs by making the entire model, including the encoder, trainable. We used a
callback function that monitored the validation loss during training and
stored the best checkpoint version of the network. The best checkpoint had a
training accuracy of $99.99\%$ and a validation accuracy of $98.55\%$ while
the corresponding loss values are $0.025$ and $0.031$, respectively. The
evolution of loss and accuracy for the training of hybrid classifier for both
frozen and fine-tuning stages are shown in Figure 9.

##  V Bringing robustness to the model

---
Figure 10: Examples generated by performing adversarial attacks on the fine-
tuned classifier model. Despite having no discernible CBC features, the
classifier identified all of these examples as CBC signals with
$\mathrm{P}_{\mathrm{cbc}}>99\%$.

Large DL models typically show an over-sensitivity to certain features over
others [83]. A class of techniques called _adversarial attacks_ exploits this
weakness of DL models and finds their failure modes. This can either be
accomplished by systematically perturbing the input data with _gradient
ascent_ or training a separate generative adversarial DL model. The first
kind, that of perturbative attack on the regular input, can be as simple as a
single pixel attack [84] or a more generic attack [69], leading to a
dramatically degraded model performance. These perturbations in the data are
typically insignificant when perceived by a human. The second kind of attack
can be achieved through generative adversarial frameworks which involve
training generative DL models which can create images that deceive the
discriminator successfully. Initialised randomly, these models learn to
generate images that do not contain any practically meaningful information,
yet are able to mislead the classifier [85, 86].

Our fine-tuned classifier shows very promising performance provided the
inference is made on data that belong to the same input space as the training
data. However, as mentioned above, the model may show a dramatically degraded
performance if the data for inference deviates from this space. Such
deviations could occur due to changes in the data characteristics or the pre-
processing scheme. Such changes can render the model inapplicable, and
retraining the model with an updated dataset may be required each time they
occur. This can become especially challenging in certain cases, for example,
new types of glitches that may keep arising with changing operating conditions
of the detectors.

In recent years, multiple strategies have been explored to make DL models more
robust [69, 87, 88]. Particularly, including adversarially perturbed images in
training has been shown to improve the robustness of the models to small
perturbations [69]. However, it leads to robustness against perturbations only
in the vicinity of the input space covered by training data. As we intend to
explore the failure modes in a parameter space that is not necessarily close
to the input image space, we take a somewhat different approach of using the
generative adversarial framework to bring robustness to our classifier, as
described in the next section.

###  V.1 Using GAN for robustness

Figure 11: Schematic of the GAN setup with the hybrid classifier being trained
as the discriminator alongside $5$ adversarial generator models tasked with
finding its failure modes. The training data was the same as that used for
training the hybrid classifier earlier. The initially rubbish-looking fake
images produced by the generators eventually converge to images similar to
those from the INJ dataset.

A GAN is composed of two DL models, a generator $\mathcal{G}_{\phi}$ and a
discriminator $\mathcal{D}_{\theta}$, playing a minimax game and being trained
in the process [85]. The generator uses a latent space input $z$ to generate
fake data $\mbox{{$\mathcal{G}_{\phi}$}}(z)$. The discriminator analyses this
data and gives the output predictions
$\mbox{{$\mathcal{D}_{\theta}$}}(\mbox{{$\mathcal{G}_{\phi}$}}(z))$ of them
being real. In such a setup, $\mathcal{D}_{\theta}$ is trained for correctly
classifying the training data $x$ from the fake data
$\mbox{{$\mathcal{G}_{\phi}$}}(z)$ generated by $\mathcal{G}_{\phi}$. In other
words, it is trained with $1$’s and $0$’s as targets for
$\mbox{{$\mathcal{D}_{\theta}$}}(x)$ and
$\mbox{{$\mathcal{D}_{\theta}$}}(\mbox{{$\mathcal{G}_{\phi}$}}(z))$,
respectively. Therefore, for $\mathcal{D}_{\theta}$, the training objective is
the minimisation of a loss function given by,

$\mathcal{L}(\theta)=-\bigg{[}\mathrm{log}\mbox{{$\mathcal{D}_{\theta}$}}(\mkern
1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern
1.5mu)+\mathrm{log}(1-\mbox{{$\mathcal{D}_{\theta}$}}(\mbox{{$\mathcal{G}_{\phi}$}}(\mkern
1.5mu\overline{\mkern-1.5muZ\mkern-1.5mu}\mkern 1.5mu)))\bigg{]},$ (5)

through gradient descent. Simultaneously, $\mathcal{G}_{\phi}$ is trained to
generate fake data that can mislead the discriminator into giving positive
outputs. The loss function that is to be minimised in this case is,

$\mathcal{L}(\phi)=-\mathrm{log}(\mbox{{$\mathcal{D}_{\theta}$}}(\mbox{{$\mathcal{G}_{\phi}$}}(\mkern
1.5mu\overline{\mkern-1.5muZ\mkern-1.5mu}\mkern 1.5mu))).$ (6)

Provided $\mathcal{G}_{\phi}$ and $\mathcal{D}_{\theta}$ have enough capacity,
it can be proved that the optimisation algorithm saturates when the generator
learns to produce data that is indistinguishable from the real data, i.e., the
posterior of $\mathcal{G}_{\phi}$, $p_{g}$ converges to that of data,
$p_{\mathrm{data}}$ which is the global optimum of the minimax game. At this
point, the output of $\mathcal{D}_{\theta}$ saturates to
$\mbox{{$\mathcal{D}_{\theta}$}}^{*}(x)=\frac{p_{\mathrm{data}}}{p_{\mathrm{data}}+p_{g}}$,
which, for equal proportions of training and generated data, is $\frac{1}{2}$.

####  V.1.1 Implementation

Firstly, we demonstrate the fragility of our fine-tuned model in its original
state by identifying its failure modes. We trained five adversarial generators
with the loss function given in Equation 6 against the classifier, which was
frozen. Training in this setup is equivalent to training only the adversaries
in a GAN to find the failure modes of the discriminator. The adversaries were
initialised with random weights at the start of training. Hence, their output
images are rubbish in the initial epochs. However, we observed that the
adversaries learn quickly, within $\sim 5$ epochs, and produce images
classified as signals by the classifier with very high output probabilities.
Figure 10 shows a few sample images generated by the adversaries at the end of
$5^{\mathrm{th}}$ epoch. It can be seen that these images do not belong to any
meaningful category, and yet, are classified as CBC signals with output values
$>0.99$.

To train our classifier against its failure modes, we trained the
discriminator alongside the five adversaries in the same GAN framework. The
motivation for such training is to identify and exhaust all the possible
failure modes of the classifier by simultaneously besetting it with several
adversaries while the training is still in progress. It would lead to an
ultimate saturation point when the adversaries start producing images very
close to the signal space in the training data and exhaust all other
possibilities.

However, there are several pitfalls in this proposal. We need to modify the
regular GAN framework to achieve desirable performance for the required task
of robust classification. GANs are typically used for obtaining a competent
generator model that generates realistic images similar to the training data.
Therefore, the usual training data of GAN comprises of samples belonging to
only one category. The biases inherited by the discriminator through such
training are not a concern, as the discriminator usually gets discarded at the
end of training. In our case, we wish to retain the discriminator as a robust
classifier and discard the generator instead. Thus, retaining the capabilities
of the original classifier is crucial. In principle, the generator is expected
to exhaust all the failure regions of the input space, including those
corresponding to GN and GL. However, since the input space of the model can be
considered infinitely large for all practical purposes, a generator model
starting from a random initialisation of parameters can only attack the
discriminator from _one side_ with respect to the training data. During
training, it typically covers only a limited region that lies along the path
explored by it and cannot leap to a vastly different region of input space.
Our experiments also confirmed that training a GAN with only the INJ dataset
results in a bias towards the signal class and a loss of performance in the
other noise classes viz., GN and GL. To make the training setup more complete
and retain the performance of our classifier on all three classes of data,
viz., INJ, GN and GL, we use the same composition of training data that was
used for training the classifier as described in Section IV.

(a) GAN Discriminator Losses

(b) GAN Generator Losses

Figure 12: (a) Evolution of discriminator losses given by Equation 5 in the
GAN setup during training of $400$ epochs. Loss on fake data generated by the
generators is shown along with the losses for INJ, GN and GL data. (b)
Progress of generator losses given in Equation 6 through the training. Five
generators were trained simultaneously alongside the discriminator. The
initial incapability of generators is overcome quickly in the first $\sim 30$
epochs.

As mentioned before, covering the entire input space is practically
impossible; Therefore, absolute robustness is unachievable. Nevertheless, to
exhaust as much space around the input space as possible, we create a setup
where our fine-tuned hybrid model is simultaneously beset with $5$
generators444The number of generators considered here is limited by the
computational capacity. initialised independently. The schematic of the setup
is shown in Figure 11. We use three loss terms to constitute the total loss
for training the discriminator. The first loss term is constructed from the
fake generated samples as an average of losses obtained from each generator.
The other two are INJ data loss and an average of GN and GL losses. The
detailed architecture of the generator is shown in Figure 21. We train the GAN
setup for $400$ epochs with a cyclical learning rate as used previously. The
evolution of generator and discriminator losses through the course of training
is shown in Figure 12. During the initial epochs of training, the incapability
of the generators leads to their relatively higher losses. However, within a
few epochs, the generators start producing images with features the
discriminator is possibly sensitive to. Through simultaneous training, the
discriminator learns to eliminate the sensitivity to these artefacts.
Eventually, the samples generated by the generators start resembling the INJ
data present in the training data.

(a)

(b)

Figure 13: Comparative performance of the classifier before and after GAN
training is shown with ROC curves for INJ-TEST vs GL-TEST (left) and INJ-TEST
vs TRIG-TEST (right). Though we did not use real data, except for GL, during
any stage of training, the models show high AUC values on the test datasets
that were generated from real data. Also, the variations in AUC values due to
GAN training are small enough to be ignored.

To test the performance of the classifier on real data and study the possible
deterioration in its capabilities after GAN training, we ran the inference on
the test datasets with the states of the classifier before and after GAN
training. Figure 13 shows the separate sets of ROC curves with GL-TEST and
TRIG-TEST as the noise class for classifier before and after GAN training. The
area under the curve (AUC) corresponding to the ROC curve of CBCs vs glitches
reduces from $0.989$ to $0.984$, while for CBCs vs PyCBC triggers it improves
from $0.976$ to $0.989$ after GAN training. This change is relatively
insignificant and can be attributed to statistical fluctuations in the
performance. Therefore, we can say that the improvement in robustness comes at
a negligible cost in terms of the model performance.

Since it is impossible to achieve absolute robustness using this approach, the
very need for such an additional training effort may naturally be questioned.
To thoroughly examine the actual changes the GAN training brought about, we
inspect the transformations of sample input data inside our classifier at
various levels with and without GAN training. Specifically, we focus on the
changes in the operations of $3\crossproduct 3$ convolutional filters from the
second and third parallel modules, and the one before the flattening operation
towards the end. We pass a sample image from INJ and GN data each through the
classifier states before and after GAN training and obtain the transformed
maps at the outputs of these convolutions as shown in Figure 17 and Figure 18.
For the classifier state before GAN training, the transformed maps display a
high degree of similarity in terms of the features collected from the input
image. Whereas, in the case of the robust classifier obtained after the GAN
training, this degeneracy is largely eliminated. Fewer and more helpful
features are recovered that can be translated to the binary output more
reliably.

##  VI Search

Though improving the sensitivity of CBC searches was not the target of this
work, to assess the capabilities of our model to identify events in real data,
we implement a direct search on the GW strain data from H1 and L1555Though we
have not used the data from Virgo in this work, our analysis is extendable to
include data from multiple detectors.. For inference, we analyse the GW strain
data in $512$ seconds long chunks. The data were first whitened with a PSD
generated from the same chunk. We analyse $1$ second-long slices of whitened
strain by obtaining their CWT maps. For analysis, we stride the data with
fixed steps, generate CWT maps and save them for inference with GPU later.
Taking a shorter stride leads to wasteful computation of image generation and
GPU inference, whereas taking longer strides leads to the possibility of
missing out on signals. To systematically decide a reliable stride value, we
need to identify a sensitive time window of the classifier. One can anticipate
the sensitive time window as a function of chirp mass since the chirp
morphology is expected to affect such a window. Once this window function is
obtained, we can choose an optimal stride step such that no detectable signal
is missed while the number of images generated and analysed remains as low as
possible.

###  VI.1 Estimating the sensitive time-window

Figure 14: The sensitivity of the classifier as a function of the location of
the merger in the CWT map and the chirp-time of the CBC signal. The injections
are stridden in steps of $0.05$ second across the analysis window of $1$
second (shown by dotted lines) and the average $\mathrm{P}_{\mathrm{CBC}}$
values are shown with a colour bar. The sensitive window can be seen to become
relatively narrower for injections with shorter duration.

To obtain the sensitive time window of the classifier, we produce a set of
injections generated by uniformly sampling the chirp time $\tau_{0}$ between
$0.01-2$ seconds. To calculate $\tau_{0}$, we take the starting frequency as
$30$ Hz. The component masses are constrained to lie in the range $2-400\
\text{M}_{\odot}$ with mass ratio $q<8$. The injections were generated without
noise, in a setting similar to the preparation of CLEANINJ dataset as
described in Section II.2. Next, the analysis window was stridden over the
injection so that the merger of the signal traversed from the left to right
end of the window in steps of $0.05$ second. The complex CWT map of the clean
injection was obtained at each step and added to the CWT maps of $10$
different simulated noise realisations. An average of the predictions on this
stack of CWT maps gives a more reliable output score which was recorded for
the corresponding injection at the respective step. The injections were
generated with optimal SNR values such that the corresponding $\gamma$ values
came out to be $50$ based on the respective component masses. The
aLIGOZeroDetHighPower PSD was used for generating simulated noise and
calculating the optimal SNR. The resulting response function obtained from the
classifier is shown in Figure 14. It can be observed that the sensitive window
is $\sim 0.2$ second for signals with higher $\tau_{0}$ values, whereas it
reduces to $\sim 0.1$ second for the shortest signals. Based on this plot, we
heuristically chose the stride value to be $0.1$ second.

###  VI.2 Search on real LIGO data

Parameter | Value
---|---
Chirp-distance | Uniform in ($5$, $400$) Mpc
Component masses | log($m_{1,2}$) uniform in $(2.5,150)\ \text{M}_{\odot}$
Component spins | Aligned, $S_{i,z}$ uniform in ($-0.998$, $0.998$)
inclination | Uniform in cos${}^{-1}\iota$
Sky location | Uniform over sky
Approximant | SEOBNRv4_opt
Injection times | Uniform in a $25$ seconds interval; Average step of $100$ seconds
Low frequency cut-off |  $19$ Hz
Table 2: Parameter choices for the injections in the real LIGO data used for
sensitivity estimation of the direct DL search. Figure 15: The sensitive
distance of the DL search as a function of FAR for three chirp-mass bins.
Figure 16: Cumulative rate of foreground and background events (inclusive and
exclusive of foreground triggers) plotted against the coincident ranking
statistic. The direct DL search was performed on $8.8$ days of real data from
LIGO Hanford and Livingston. Two CBC events, viz., GW190519_153544 and
GW190521_074359 were recovered with FAR $<1$ per $20350.4$ years.

To test the independent performance of our classifier, we perform a CBC search
on $8.8$ days long chunk of real data from LIGO Hanford and Livingston
detectors between May $12-21$, $2019$ UTC. This data includes six
gravitational wave candidates with $\mathrm{P}_{astro}\geq 0.5$ as listed in
GWTC-2.1 [1]. For estimating the sensitivity of the search, we also create an
injection set whose parameters are described in Table 2. The image data for
CWT maps were generated as described in the previous section with a stride of
$0.1$ second. We ran a multi-GPU inference on these images to obtain the model
predictions as time-series data. We call these predictions
$\mathrm{P}_{\mathrm{CBC}}$ scores which correspond to how the classifier
ranks the triggers for them to belong to the CBC class. However, note that
this is not a realistic statistical probability of the trigger belonging to
the CBC class. Triggers were generated using peak finding and clustering over
a $1$ second window. Also, a lower cut-off of $10^{-5}$ was used for trigger
collection. By dividing all the triggers by the maximum trigger value of the
entire run, we normalised them to the range $[0-1]$. We used the PyCBC
workflow for trigger collection and post-processing of these triggers for
coincident analysis. The coincident foreground events were generated by
collecting triggers that lie in a time window of $0.4$ second to
conservatively collect triggers captured at an offset of $\sim 0.2$ second
from the actual merger time in each detector. The $\mathrm{P}_{\mathrm{CBC}}$
data from H1 were time-shifted in $0.9$ second steps with respect to those
from L1, and the same coincidence collection procedure was repeated to obtain
the background events. We build the coincident ranking statistic as a simple
multiplication of the $\mathrm{P}_{\mathrm{CBC}}$ values of the coincident
triggers, which essentially expects the triggers to belong to the CBC class in
both detectors at the same time. We do not consider single detector events in
this work. We perform injection recoveries by looking for coincident
foreground events lying within $2$ seconds on either side of the injections.
From the significance of these recovered injections, the sensitive distance
for the search can be found [89]. The sensitive distance as a function of FAR
is plotted in Figure 15 for different chirp-mass bins. The sensitive distance
at the FAR of $1$ per month is found to be $140$ Mpc, $996$ Mpc and $2173$ Mpc
for chirp mass bins $[1,10]$, $[10,40]$ and $[40,80]$, respectively.

The foreground and background events obtained from the search are shown in
Figure 16. Our search recovered two significant CBC events, viz.,
GW190519_153544 and GW190521_074359 with FAR $<1$ per $20350.4$ years. There
are four additional events in the analysed chunk that were found by other
search pipelines in GWTC-2.1, viz., GW190513_205428, GW190514_065416,
GW190517_055101 and GW190521. Initial investigations suggest that there could
be several reasons our search did not find them, viz., one of the detectors
having a weaker signal or the presence of a glitch artefact in addition to
signal (GW190514_065416 and GW190513_205428), signal lying outside the
parameter space considered for training the model (GW190521) and lack of
almost any tail feature in the chirp morphology (could be due to high spin
modulations in the signal) leading to resemblance with blip glitches
(GW190517_055101). However, it should be noted that due to the inherent
opaqueness of the DL models, it is very difficult to provide a definitive and
complete explanation of the case-to-case performance. It is also worth
mentioning that the search results could be further improved by optimising the
hyper-parameters of the search. However, we did not pursue this aspect as
improving the search sensitivity is not the goal of this work.

##  VII Conclusions

DL algorithms hold much promise due to their ability to learn and model vast
amounts of complex data - a task that is formidably difficult to perform using
classical methods. Their human-like capability to generalise the learned
features, along with the deployability on accelerated computing hardware like
GPUs, make them better placed for solving many data-intensive problems across
domains. However, these algorithms lack transparency over the actually learned
features and a prescription of where they could possibly fail. The fragility
of large DL models has been exposed using various methods, and improving their
robustness is an area of active research [69, 87, 88].

In the field of GW astronomy, identifying true GW signals in the data against
the spurious noise transients of terrestrial origin is one of the leading
applications of DL that has invoked a lot of interest. However, the lack of
transparency and trustworthiness of these models has been a significant hurdle
in deploying them for production-level analysis and attestation of their
results. In this work, we take the first steps towards having better control
over the performance of the DL models and improving their reliability to
perform CBC searches in the GW data.

We started by addressing ways to maintain the purity of training data,
especially for the CBC class across the parameter space. Due to varying chirp-
time of the signals based on their component masses, the CBC injection data
generated with a constant lower threshold on SNR leads to variations in the
_visibility_ of their chirp-features. We built a metric that accounts for
variations in the peak intrinsic amplitude of the CBC signals and enables the
generation of large amounts of data with uniform visibility of chirp features
across the parameter space. Next, to better control which features of the
input image the model learns to focus on, we trained a VAE to reproduce only
the chirp features from the input data and ignore everything else. Such a VAE
encodes the signal-specific information from the input into a reduced, smooth
latent space representation. Using densely connected layers, we mapped this
information-rich latent coding to the binary output that corresponds to the
signal or noise class. The signal class consisted of CBC signals sampled from
a focused BBH parameter space injected into the simulated LIGO noise. Whereas,
the noise class consisted of simulated Gaussian noise and glitches obtained
from real LIGO data. We fine-tuned the model further to achieve high training
and validation accuracies. Going beyond just careful model building, we tested
this model for robustness by subjecting it to adversarial attacks. These
attacks revealed several simple failure modes of the model which did not have
any features similar to CBC signals. These results also underlined the crucial
fact that despite various advantages, DL approaches can be extremely fragile.
As the first step towards bringing robustness to our DL model, we employed a
novel GAN setup to retrain the model. We used our classifier as a
discriminator and trained it alongside $5$ adversarial generators. We tested
the model’s performance before and after GAN training on separate test
datasets. These datasets were composed of injections in real data, glitches
and random triggers collected from matched-filter searches. We found that the
GAN training had no significant effect on the model’s performance on real
data, while it brought some fundamental improvements in terms of sparseness
which could contribute to enhanced robustness. We obtained the transformations
of sample inputs at different layers of the classifier and found that a large
amount of degeneracy in the extracted features was removed after GAN training
and much fewer channels were used to make inferences. Also, the outputs from
the layers were better localised, which could help in enabling a more robust
mapping to the binary output.

Lastly, we examined the stand-alone capability of the model to search for CBC
signals in $8.8$ days of real LIGO data. We chose an optimum stride for the
generation and analysis of CWT maps by systematically investigating the
model’s response to injections as a function of time. This investigation
helped in optimising the computational cost while ensuring recovery of all
underlying signals to which the classifier is sensitive. We also performed an
injection study to estimate the sensitive distance of the CBC search. At a FAR
of $1$ per month, the sensitive distance for chirp mass bins $[1,10]$,
$[10,40]$ and $[40,80]$ was $140$ Mpc, $996$ Mpc and $2173$ Mpc, respectively.
Importantly, our search recovered two CBC events, viz., GW190519_153544 and
GW190521_074359 with high significance. The analysed data also contained four
additional events, viz., GW190513_205428, GW190514_065416, GW190517_055101 and
GW190521 found by various pipelines in GWTC-2.1 [1]. We discuss the potential
causes of our classifier missing these signals in Section VI.2. As search
sensitivity was not the focus of this work, we did not pursue possible
improvements in the results by optimising the hyper-parameters of the search.
Also, the overall search sensitivity can be considerably improved if our
classifier is used in conjunction with an existing offline search using MLStat
[38]. In future, we intend to analyse the real LIGO/Virgo data with this
classifier in the framework of MLStat and study the improvement in the
significance of events from GWTC-2 and GWTC-3 [8, 9].

Apart from providing speedy inference, the fast-growing field of DL
applications in GW astronomy is expected to surpass conventional algorithms in
terms of sensitivity and parameter space coverage in the near future. Through
this work, we have addressed a crucial aspect of this field: the reliability
of DL approaches. While the novel proposal to increase the robustness
presented here is still in its nascent stages and under further development,
we believe these techniques will pave the way for the successful integration
of DL methods in production-level analyses. We hope this work goes beyond the
field of GW astronomy and proves to be a valuable stepping-stone for other
areas of science and industrial applications.

###### Acknowledgements.

Authors express thanks to the members of LIGO-Virgo-KAGRA Collaboration for
their valuable comments and suggestions. SJ acknowledges support of Council
for Scientific and Industrial Research (CSIR), India and the Australian
Research Council Centre of Excellence for Gravitational Wave Discovery
(OzGrav), project number CE170100004. SM acknowledges support from the
Department of Science and Technology (DST), India, provided under the Swarna
Jayanti Fellowships scheme. This material is based upon work supported by
NSF’s LIGO Laboratory which is a major facility fully funded by the National
Science Foundation. We acknowledge the use of GWA and the LDG clusters at
Inter-University Centre for Astronomy and Astrophysics (Sarathi) for the
computational work. We also acknowledge National Supercomputing Mission (NSM)
for providing computing resources of ‘PARAM Siddhi-AI’, under National PARAM
Supercomputing Facility (NPSF), C-DAC, Pune and supported by the Ministry of
Electronics and Information Technology (MeitY) and Department of Science and
Technology (DST), Government of India. This document has been assigned IUCAA
preprint number IUCAA-03/2023 and LIGO document number LIGO-P2200351.

Figure 17: Comparison of image transformations of a sample CWT map taken from
the INJ database with the model-state before (left column) and after the GAN
training (right column). Outputs of the model extracted at the end of the
$3\crossproduct 3$ convolutional layers from second (top) and third parallel
modules (middle), and the one before flattening operation (bottom) are shown.
The extracted features on the left show many similarities. On the right, these
degeneracies disappear, and only a few more helpful sets of features are
extracted, which can be mapped to the binary output more reliably.

Figure 18: Comparison of image transformations similar to Figure 17 with a
sample CWT map taken from the GN database. A similar difference in feature
extraction can be observed between the two models.

(a) Structure of the parallel module.

(b) Sampling layer at the latent space.

(c) Overall encoder architecture.

Figure 19: Architecture of the encoder part of the VAE model along with its
sub-components. Outputs of different layers, colour coded for distinction, are
shown. The shapes of these outputs are given at the top with the number in the
bracket denoting the size of the square kernel. The encoder consists of three
parallel modules, which have convolutions in parallel and in series, inspired
by the inception architecture [79]. The latent space has a sampling layer for
the multi-variate Gaussian distribution represented by the mean and variance
vectors.

(a) Branch structure.

(b) Overall decoder architecture.

Figure 20: Architecture of the decoder part of the VAE model along with its
branch structure. The depictions are similar to those shown in Figure 19
except for the difference in the types of layers. The inputs to the decoder
are the latent space mean vector codings of the INJ dataset and the outputs
are the corresponding images from the CLEANINJ dataset.

(a) Branch structure.

(b) Overall generator architecture.

Figure 21: Architecture of the generators used for adversarial attacks and in
GAN training with their branch structure. The input to the generator is a
noise vector of size $256$. The generator outputs are the fake images
deceiving the classifier model.

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