KCL-PH-TH2022-41 Dictionary learning a novel approach to detecting binary black holes in the presence of Galactic noise with LISA

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KCL-PH-TH/2022-41

Dictionary learning: a novel approach to detecting binary black holes in the presence

of Galactic noise with LISA

Charles Badger,1Katarina Martinovic,1Alejandro Torres-Forn´e,2, 3 Mairi Sakellariadou,1and Jos´e A. Font2, 3

1Theoretical Particle Physics and Cosmology Group, Physics Department,

King’s College London, University of London, Strand, London WC2R 2LS, United Kingdom

2Departamento de Astronom´ıa y Astrof´ısica, Universitat de Val`encia,

Dr. Moliner 50, 46100 Burjassot (Val`encia), Spain

3Observatori Astron`omic, Universitat de Val`encia,

Catedr´atico Jos´e Beltr´an 2, 46980 Paterna (Val`encia), Spain

(Dated: October 17, 2022)

The noise produced by the inspiral of millions of white dwarf binaries in the Milky Way may pose a

threat to one of the main goals of the space-based LISA mission: the detection of massive black hole

binary mergers. We present a novel study for reconstruction of merger waveforms in the presence of

Galactic confusion noise using dictionary learning. We discuss the limitations of untangling signals

from binaries with total mass from 102Mto 104M. Our method proves extremely successful for

binaries with total mass greater than ∼3×103Mup to redshift 3 in conservative scenarios, and

up to redshift 7.5 in optimistic scenarios. In addition, consistently good waveform reconstruction of

merger events is found if the signal-to-noise ratio is approximately 5 or greater.

Introduction.— The LIGO/Virgo interferometer net-

work [1, 2] has already detected gravitational waves

(GWs) from almost one hundred compact binary coa-

lescence (CBC) events [3–5]. These detections populate

GW transient catalogues and reveal information about

properties of the underlying black hole and neutron star

populations [6]. The high frequency range probed by the

current terrestrial detectors, at around (10 −1000) Hz,

is sensitive to stellar-mass binaries that mostly lie be-

low the pair-instability supernova mass gap1. Mergers of

supermassive black holes, however, are expected to emit

low-frequency (mHz) gravitational waves.

The space-based GW interferometer LISA, anticipated

to be launched in the mid-2030s, will be sensitive to GWs

in the mHz range [8]. Other GW sources will be de-

tectable at these frequencies: Galactic white dwarf bi-

naries, inspiraling binaries with extreme mass-ratio, or

colliding true vacuum bubbles formed at the electroweak

phase transition [9–11]. The tens of millions of double

white dwarf binaries in the Galaxy could have an impact

on detectability of massive black hole binaries coalescing

in the LISA frequency band [12]. LISA will observe con-

tinuous GWs from inspiraling white dwarfs, and although

it may be sensitive to individual sources, most will remain

unresolved and these are referred to as Galactic confusion

noise [13–15]. It has been shown that modulation of the

Galactic foreground from the LISA orbit could lead to

a reduction in signal-to-noise ratio (SNR) of other GW

sources by a factor of 4 [16]. A LISA Data Challenge2

is underway to study the impact of overlapping Galactic

1GW190521 is the only exceptional event where the mass of the

primary black hole is unambiguously in the mass gap [7].

2https://lisa-ldc.lal.in2p3.fr.

sources on the sensitivity to massive black hole merg-

ers [17], and attempts to separate the foreground from

other GW sources have been conducted [18–22].

In this Letter we apply a dictionary learning method

to separate CBCs from the Galactic foreground in the

LISA frequency band. Such a method has been success-

fully applied in GW data analysis to classify and denoise

Advanced LIGO’s “blip” noise transients [23] and eﬀec-

tively improve the performance of the detector. More

precisely, we assess the suitability of the dictionary learn-

ing method for the classiﬁcation and reconstruction of

massive binary black hole merger signals in the presence

of Galactic noise.

Previous studies focused on the inspiral of loud CBC

sources and demonstrated that SNR accumulated over

time is suﬃciently large to overcome the noise from

Galactic binaries [24]. In other literature, detectability of

CBCs was investigated for equal-mass and non-spinning

binaries [25–27], conﬁrming the largest SNR is expected

from binaries with combined mass ∼(105−106)M. In

particular, Fig. 3 in [27] presents two mass ranges with

low SNR that could be aﬀected by Galactic foreground,

namely (102−104)Mand (107−109)M.

Here we consider all of the mass ranges, along with

varying spins and redshifts, and we study their wave-

forms around coalescence time. The dictionary learning

method reconstructs CBC signals with ease in the trivial

case where the CBCs are above the Galactic noise, i.e.

for the (105−106)Mmass range. We ﬁnd the dictionary

learning method to be too computationally expensive for

very heavy mergers in the range (107−109)M. However,

our method succeeds in separating low-SNR binaries in

the range (102−104)Mfrom the Galactic noise. Hence,

the dictionary learning method could signiﬁcantly assist

the detection of this prime LISA source [28].

arXiv:2210.06194v2 [gr-qc] 14 Oct 2022

Dictionary learning.— Any CBC signal in the LISA

band will be overlaid with continuous waves from the

inspiral of double white dwarfs. Therefore, we can model

the detector strain, y(t), as a superposition of the CBC

signal u(t) and the Galactic confusion noise n(t):

y(t) = u(t) + n(t).(1)

We express the loss function as

J(u) = ||y−u||2

L2+λR(u),(2)

and search for a solution uλthat minimises J(u), where

||·||L2is the L2norm [29, 30]. The ﬁrst term in the loss

function, often referred to as the error term, measures

how well the solution ﬁts the data, while the regularisa-

tion term R(u) captures any imposed constraints. The

regularisation parameter λtunes the weight of the regu-

larisation term relative to the error term; it is a hyper-

parameter of the optimisation process.

The goal of the dictionary learning method [31] is to

ﬁnd the sparse vector αthat reconstructs the true signal

uas a linear combination of columns of a dictionary D,

u∼Dα, (3)

with Da matrix of prototype signals (atoms) trained to

reconstruct a given set of signals, which for our study

is CBCs. Sparsity of the vector αis imposed via the

regularisation term R(u) = ||α||L1, using the L1norm.

Therefore, the constrained variational problem in (2)

reads

αλ= argmin

α||y−Dα||2

L2+λ||α||L1,(4)

and is called “basis pursuit” [32] or “least absolute

shrinkage and selection operator” (LASSO) [33].

The basis pursuit can be improved signiﬁcantly if, in-

stead of using a predeﬁned dictionary, we apply a learning

process where the dictionary is trained to ﬁt a given set

of signals. The procedure starts by selecting templates

of CBC waveforms and whitening the data. The wave-

forms are aligned at the strain maximum and divided

into patches, with the number of patches (p) much larger

than the length of each patch (d). To train the dictionary

we solve (4) considering both the sparse vector αand the

dictionary Das variables:

αλ,Dλ= argmin

α,D(1

i=1 ||Dαi−xi||2

L2+λ||αi||L1),

(5)

with xidenoting the i-th training patch. This problem

is not jointly convex unless the variables are considered

separately as outlined in [34].

In our study we create training signals that contain

CBC waveforms only and no noise. The dictionary cre-

ated is then tested on signals that include new CBC wave-

forms combined with Galactic noise. We describe brieﬂy

Parameter Distribution

Total mass (M) logU[102,104]

Mass ratio U[1,10]

Primary spin U[-1,1]

Secondary spin U[-1,1]

Redshift 2

Luminosity Distance (Mpc) 15975

TABLE I. Parameters used to construct training CBC signals

with the IMRPhenomD waveform approximant. We choose val-

ues randomly from the uniform distributions indicated in the

right column, keeping redshift and luminosity distance ﬁxed.

the massive black hole and white dwarf binary waveforms

used in our datasets below.

Training and testing datasets.— We utilise the

IMRPhenomD approximant [35] provided by the LISA Data

Challenge to model waveforms of binary black holes de-

tectable by LISA, capturing inspiral, merger and ring-

down of the signal. Binaries with total mass (105−

106)Mare expected to have SNR ≥150, making sep-

aration from the Galactic foreground a trivial problem.

CBCs in the mass range (107−109)Mhave lower fre-

quencies, making it diﬃcult for the dictionary learning

to reconstruct their sinusoidal behavior. We thus study

reconstruction capabilities of binary black holes with to-

tal mass ranging from (102−104)M. The dictionary

is trained on a set of 100 noiseless CBC signals, simu-

lated over one day with cadence3∆t= 2 s. Table I lists

the relevant parameters of the IMRPhenomD waveform and

the corresponding ranges of values we choose for the CBC

sources. We simulate the data by drawing randomly from

the probability distribution of the parameters. The red-

shift for all sources is ﬁxed to z= 2, since changing the

redshift leads to a simple rescaling of the amplitude that

has no impact on our whitened data in the training set.

Note that the same does not hold for the testing data,

since changing redshift would change the relative ampli-

tude of the CBCs to the Galactic foreground.

Consider two white dwarfs of mass M1and M2on a

quasi-circular orbit with inclination ιat a distance R.

They emit GWs with amplitude [36]

A+(Mc, R, fGW, ι) = 2G5/3M5/3

c4R(πfGW)2/3(1+cos2ι),

A×(Mc, R, fGW, ι) = −4G5/3M5/3

c4R(πfGW)2/3cos ι,

(6)

3The cadence was chosen low enough to have a high sampling

rate that avoids aliasing, but high enough to allow for reasonable

computational time.

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KCL-PH-TH/2022-41Dictionarylearning:anovelapproachtodetectingbinaryblackholesinthepresenceofGalacticnoisewithLISACharlesBadger,1KatarinaMartinovic,1AlejandroTorres-Forne,2,3MairiSakellariadou,1andJoseA.Font2,31TheoreticalParticlePhysicsandCosmologyGroup,PhysicsDepartment,King'sCollegeLondon,Univer...

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KCL-PH-TH2022-41 Dictionary learning a novel approach to detecting binary black holes in the presence of Galactic noise with LISA

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