# Classification of the core-collapse supernova explosion mechanism with
learned dictionaries

Ainara Saiz-Pérez1, Alejandro Torres-Forné1, and José A. Font1,2
1Departamento de Astronomía y Astrofísica, Universitat de València, Dr.
Moliner, 50, Burjassot (València) E46100, Spain.
2Observatori Astronòmic, Universitat de València, Catedrático José Beltrán 2,
46980, Paterna (València) E46100, Spain.
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Core-collapse supernovae (CCSN) are a prime source of gravitational waves.
Estimations of their typical frequencies make them perfect targets for the
current network of advanced, ground-based detectors. A successful detection
could potentially reveal the underlying explosion mechanism through the
analysis of the waveform. This has been illustrated using the Supernova Model
Evidence Extractor (SMEE; Logue et al. (2012)), an algorithm based on
principal component analysis and Bayesian model selection. Here, we present a
complementary approach to SMEE based on (supervised) dictionary-learning and
show that it is able to reconstruct and classify CCSN signals according to
their morphology. Our waveform signals are obtained from (a) two publicly
available catalogs built from numerical simulations of neutrino-driven (Mur)
and magneto-rotational (Dim) CCSN explosions and (b) from a third ‘mock’
catalog of simulated sine-Gaussian (SG) waveforms. Those signals are injected
into coloured Gaussian noise to simulate the background noise of Advanced LIGO
in its broadband configuration and scaled to a freely-specifiable signal-to-
noise ratio (SNR). We show that our approach correctly classifies signals from
all three dictionaries. In particular, for SNR=15-20, we obtain perfect
matches for both Dim and SG signals and about 85% true classifications for Mur
signals. These results are comparable to those reported by SMEE for the same
CCSN signals when those are injected in only one LIGO detector. We discuss the
main limitations of our approach as well as possible improvements.

###### keywords:

gravitational waves – transients: supernovae

††pubyear: 2021††pagerange: Classification of the core-collapse supernova
explosion mechanism with learned dictionaries–2

##  1 Introduction

The detection of the gravitational-wave (GW) signal GW150914 by Advanced LIGO
(Aasi et al., 2015) on September 14, 2015 (Abbott et al., 2016) inaugurated
the field of GW astronomy. The observed waveform is in agreement with the
signals predicted by numerical-relativity simulations of binary black hole
(BBH) mergers. Two years later, the signal from a binary neutron star (BNS)
merger, GW170817, was jointly detected by Advanced LIGO and Advanced Virgo
(Acernese et al., 2015; Abbott et al., 2017a), an event accompanied by
electromagnetic radiation famously observed by dozens of telescopes worldwide
(Abbott et al., 2017b). The LIGO-Virgo-KAGRA (LVK) Collaboration has already
released three catalogs of GW transients comprising over 50 compact binary
coalescence (CBC) confirmed detections (Abbott et al., 2019, 2020a; Abbott et
al., 2021). GW astronomy is allowing for an entirely new way to observe the
Universe, one which could give us insight into astronomical phenomena which
are not yet fully understood. Among these, the detection of GW signals emitted
by core-collapse supernovae (CCSN) is highly anticipated. This is the type of
GW source we discuss in this paper.

GW data analysis has played a major role in the detection of all CBC events
observed so far. The inspiral phase of quasi-circular compact binary mergers
can be accurately described with generic waveform models derived from general
relativity. On the contrary, CCSN waveforms are insufficiently modeled, which
renders the strategy for their analysis entirely different to that of CBC
sources. Searches of unmodeled sources are based on coherent time-frequency
analysis of the data over a network of detectors, especially for long signals
lasting several seconds. Moreover, unlike for CBC sources, it is currently not
possible to relate the properties of CCSN progenitors (e.g. mass, rotation
rate, metallicity, magnetic fields, etc) with numerically-generated waveforms
due to the large parameter space and uncertainties involved. In addition,
owing to their intrinsic short duration, CCSN burst signals are likely to be
affected by detector noise transients or ‘glitches’. Specific pipelines based
on Bayesian inference have been developed by the LVK Collaboration to
differentiate between signals and noise transients (see e.g. Cornish &
Littenberg (2015); Kanner et al. (2016); Littenberg et al. (2016); Lynch et
al. (2017); Pannarale et al. (2019); Drago et al. (2021)).

Notwithstanding the intrinsic difficulties involved in the numerical modeling
of CCSN, steady progress on waveform generation has been accomplished. The
early bounce signal is the part of the waveform best understood since the
early works of Zwerger & Mueller (1997) and Dimmelmeier et al. (2002). The
frequency of this part of the signal ($\sim 800$ Hz) is directly related to
the rotational properties of the core (Dimmelmeier et al., 2008a; Abdikamalov
et al., 2014; Richers et al., 2017; Takiwaki & Kotake, 2018; Pajkos et al.,
2019). However, fast-rotating progenitors are not generic in all CCSN (Gossan
et al., 2016) and, moreover, the bounce signal is probably difficult to
observe by current detectors due to its high frequency and low amplitude.
However, the GW signal associated with the post-bounce evolution of the proto-
neutron star (PNS) is more promising. This signal is sourced by the dynamical
trigger of convection and oscillations due to downdrafts of accreting material
and by the standing accretion shock instability (SASI) (Murphy et al., 2009a;
Müller et al., 2013; Cerdá-Durán et al., 2013; Kuroda et al., 2016; Andresen
et al., 2017; Takiwaki & Kotake, 2018; Andresen et al., 2019; Radice et al.,
2019; Powell & Müller, 2019; Shibagaki et al., 2020; Powell & Müller, 2020).
Waveforms typically last for $\sim 500$ ms until the onset of explosion; they
may even extend beyond one second in the event of black hole formation. GW
frequencies rise monotonically with time due to the contraction of the PNS.
They can be as low as $\sim 100$ Hz, which turns them into a perfect target
for advanced ground-based detectors. Recent progress on how to infer PNS
properties from the analysis of GW data in CCSN are reported in Torres-Forné
et al. (2019); Bizouard et al. (2021).

The LVK Collaboration has recently conducted an optically-targeted search of
GW from CCSN within a distance of $\sim 20$ Mpc during the first and second
observing runs of Advanced LIGO and Advanced Virgo (Abbott et al., 2020c). No
significant GW candidate was detected in this search. Even a strong galactic
CCSN, at a distance of $\sim 10$ kpc, would likely result in a (dimensionless)
GW strain of $\sim 10^{-23}$ (Evans & Zanolin, 2017). For reference, GW150914
had a peak strain of $\sim 10^{-21}$ (Abbott et al., 2016). Outside of our
Galaxy, certain CCSN models suggest signals might be observed from sources up
to the distance of the Large Magellanic Cloud (Abbott et al., 2020c).

Because the supernova explosion starts in the deepest parts of the star opaque
to electromagnetic radiation, only GWs and neutrino emission can help
understand its dynamics. Indeed, a neutrino burst was detected in association
with supernova 1987A, in remarkable agreement with CCSN models (Hirata et al.,
1987). This makes CCSN good targets for multimessenger astronomy, by combining
the search for GWs with other types of messengers such as neutrino emission or
electromagnetic signals (see Halim et al. (2021) and references therein).
Presently, two main possibilities are considered to explain the explosion,
namely the neutrino-driven mechanism and the the magneto-rotational mechanism.
Both lead to distinct features in the GW signal. As a result, a successful
detection could potentially reveal the underlying explosion mechanism through
the analysis of the waveform. This would also open the possibility to infer
properties of the progenitor stars as well as of the PNS. Attempts to work out
the feasibility of this proposal have already been reported. Logue et al.
(2012) performed a study based on principal component analysis (PCA) and
Bayesian model selection over CCSN numerical signals and built an analysis
algorithm called the Supernova Model Evidence Extractor (SMEE) (see also Brady
& Ray-Majumder (2004); Summerscales et al. (2008); Heng (2009); Röver et al.
(2009) for earlier investigations). Logue et al. (2012) found that the method
was able to associate a potential GW signal injected into Gaussian noise with
the magneto-rotational mechanism up to a distance of $\sim 10$ kpc and with
the neutrino mechanism up to only $\sim 2$ kpc, using a single detector. More
recently Powell et al. (2016) employed SMEE with CCSN signals in a three-
detector network. The signals were injected into real noise from the initial
Advanced LIGO and Advanced Virgo with their sensitivities altered to simulate
the nominal design sensitivity of those detectors. They found that it is
possible to determine if the explosion is driven by neutrino convection for
sources within the Galaxy and driven by rapidly-rotating core collapse for
sources up to to the Large Magellanic Cloud. Finally, Powell et al. (2017b)
and Powell (2018) further developed SMEE to incorporate the possibility of
determining whether a candidate CCSN signal is a glitch before identifying the
explosion mechanism, and Roma et al. (2019) extended SMEE to identify specific
features in the GW signals associated with g-modes of oscillation of the PNS
and SASI as well as using spectrograms of supernova waveforms rather than
time-series waveforms.

Those past studies have motivated the present work. Here, we investigate the
performance of an alternative approach to SMEE for the classification of the
CCSN explosion mechanism through a (supervised) machine-learning algorithm
based on learned dictionaries. We have recently applied this approach for
reconstruction and classification of simulated GW signals and actual detector
glitches, obtaining promising results (Torres-Forné et al., 2016a; Llorens-
Monteagudo et al., 2019; Torres-Forné et al., 2020). Our goal in this paper is
to develop a method to denoise numerically-generated CCSN GW signals and to
classify them depending on their morphology. To do so we use publicly
available GW catalogs of simulated CCSN to train dictionaries and use them to
denoise and reconstruct signals injected in Gaussian noise. For the sake of
comparison with the results of Logue et al. (2012) we employ the same waveform
catalogs. We also develop a method of classification to associate a given
signal to a certain type of explosion mechanism and check the dependence of
the results on the signal-to-noise ratio. On the whole our results are
comparable to those obtained with SMEE when the same CCSN signals are injected
in only one LIGO detector, as was done in Logue et al. (2012).

This paper is organized as follows: Section 2 presents a brief description of
the two prevailing CCSN explosion mechanisms along with an overview of the
types of waveform signals we use in our study. In Section 3 we discuss the
mathematical framework of our approach, providing details on dictionary-
learning, signal denoising and classification. Section 4 deals with the
process of training the dictionaries and selecting the optimal parameters of
the method. Specific technical details are reported on Appendix A. In Section
5 we show the main results of our study. Finally, Section 6 contains our
conclusions and outlines possible extensions of this work.

##  2 Gravitational waves from CCSN

CCSN are a prime source of GWs (see Abdikamalov et al. (2020) and references
therein). Their complex dynamics are characterized by the existence of a time-
dependent, mass-quadrupole moment which yields a non-zero GW strain. Such a
time-dependent quadrupole originates from a number of processes, namely
convection in the PNS and in the neutrino-heated hot bubble, anisotropic
neutrino emission, and non-radial instabilities (e.g. SASI). All of those are
generic to all CCSN while additional mechanisms such as rapid rotation and
magnetic fields are less generic. We next describe the two explosion
mechanisms that are believed to operate in most CCSN, the neutrino mechanism
and the magnetorotational mechanism, and discuss the two waveform catalogs we
consider in our study.

###  2.1 Neutrino-driven mechanism

Figure 1: Two examples of the numerically-simulated gravitational waveforms
used in this work. The left panel shows a Mur signal corresponding to a
neutrino luminosity of 6.0 $\times$ 1052erg s-1 and a 40$M_{\odot}$ progenitor
model. The numbers indicate the different phases of the core-collapse process:
prompt convection (1), PNS and postshock convection (2), non-linear SASI (3)
and explosion onset (4). The right panel shows Dim signal s40A3O09_LS,
corresponding to a 40$M_{\odot}$ progenitor with strong differential rotation.
The waveform displays the pre-bounce phase (1) and the post-bounce PNS
ringdown signal (2).

The idea of the stalled supernova shock being revived by the transfer of
energy by neutrinos emitted from the core was first proposed by Colgate &
White (1966) and Arnett (1966), and it was revisited into its modern form by
Bethe & Wilson (1985). During collapse and the subsequent evolution of the PNS
into a neutron star, about $\sim 10^{53}$ erg of energy is released (Bethe,
1990), mainly in the form of neutrinos. Most of the neutrino emission takes
place after bounce, when neutrino losses dominate in the ‘cooling layer’
surrounding the neutrinosphere. Neutrinos can deposit a considerable amount of
energy in the ‘gain layer’ between the cooling layer and the shock (Janka et
al., 2012). If a small fraction of this energy is reabsorbed, heating the
shock, a successful supernova explosion could ensue. Spherically-symmetric
simulations implementing neutrino heating have shown that the shock can only
be revived in the case of $8-10$ $M_{\odot}$ stars with O-Ne-Mg cores, the
lowest-mass massive stars (Kitaura et al., 2006). For more massive
progenitors, multidimensional effects, i.e. convection and SASI, are essential
for the success of the neutrino mechanism (Janka, 2012). Those nonradial
effects increase the efficiency of neutrino heating, and they dominate the GW
emission.

In this work we employ the neutrino-driven GW catalog of Murphy et al.
(2009b), which we refer to as the ‘Mur’ catalog. It comprises 16 waveforms
obtained through Newtonian-gravity 2D simulations accounting for a stalled
shock at a radius of $\sim 200$ km, a gain layer beginning at $\sim 100$ km,
prompt and postshock convection as well as PNS convection from $\sim 20$ to
$\sim 40$ km and SASI. The effects of neutrino transport are approximated and
the simulations account for neutrino cooling and heating as well as the
effects of electron capture and deleptonization. The waveforms of Murphy et
al. (2009b) correspond to progenitor models with 12, 15, 20 and 40 $M_{\odot}$
zero-age-main-sequence (ZAMS) masses.

The left panel of Figure 1 shows the waveform of a typical GW signal of the
Mur catalog. Before bounce, a spherical collapse results in zero GW strain.
After the initial bounce, which is marked in the plot as section 1, the
stalling shock leaves behind a negative entropy gradient, leading to a
convective instability in the gain layer where ‘prompt convection’ takes
place. This results in the initial GW burst with peak strains of $\sim
10^{-22}$ (for a source at 10 kpc). Convection takes place both within the PNS
and in the postshock region above the neutrinosphere, resulting in GW signals
with typical strains of $\sim 10^{-23}$. These are visible once prompt
convection settles down as the entropy gradient smooths out, as seen in
section 2 of the figure. In the Mur catalog this takes place $\sim 50$ ms
after bounce (Ott, 2009; Murphy et al., 2009b). After $\sim 550$ ms SASI
reaches a non-linear phase which dominates the signal and causes the amplitude
to increase as displayed in section 3 of Fig. 1. During this phase dense
plumes from the stalling shock stream down and strike the surface of the PNS,
resulting in spikes in the GW signal of several $\sim 10^{-22}$. After this,
the explosion begins in section 4. The high-frequency signal is replaced by a
low-frequency $\sim 10$ Hz waveform. While this signal would give us
information on the shape of the explosion, its detection is unlikely by
current GW detectors as those are sensitive to frequencies $\gtrsim 20$ Hz.
Therefore, the explosion signal would be an abrupt decrease of strain
amplitude (Murphy et al., 2009b). Most of the GW signal from the neutrino-
driven mechanism is emitted in the $\sim 100-1000$ Hz frequency range and it
has a duration of $\sim 0.3-1$ s.

Figure 2: Three randomly chosen sine-Gaussian signals from our generated
catalog of 200 signals. They have been rescaled by an order of $10^{-21}$.

###  2.2 Magneto-rotational mechanism

MHD phenomena in the context of CCSN were first explored in the 1970s
(Bisnovatyi-Kogan et al., 1976; Meier et al., 1976). Simulations showed that
pre-collapse iron cores rotating with periods under $\sim 10$ s lead to
neutron stars of $\sim 1$ ms periods, indicating that conservation of angular
momentum leads to a spin-up of the collapsing core by a factor of about
$10^{3}-10^{4}$. The inner parts of the core collapse further than the
supersonically falling outer parts, resulting in a differential rotation.
Simulations suggest these rapidly rotating PNS could, $\sim 200$ ms after
bounce, have a rotational energy of around $10^{52}$ erg, ten times greater
than the energy of a typical CCSN (Ott et al., 2006).

The magneto-rotational mechanism was first proposed by Bisnovatyi-Kogan (1971)
who suggested that a mechanism involving magnetic fields would be able to
efficiently transfer angular momentum so that the core’s rotational energy
could be used to revive an explosion. Simulations later found that this
mechanism was indeed efficient enough to extract enough energy to trigger a
CCSN explosion (Moiseenko et al., 2005). The non-linear magneto-rotational
instability (MRI) was suggested as the dominant mechanism to amplify magnetic
fields to the necessary magnitudes to drive an explosion (Akiyama et al.,
2003). Indeed, in simulations where MRI was not present and magnetic field
amplification ocurred solely due to collapse and wrapping of the field lines
it was found that only progenitors with initially strong magnetic fields
($\sim 10^{12}-10^{13}$ G) could launch an explosion (Sawai et al., 2005).
Simulations accounting for MRI, however, suggest that MRI after bounce is the
main driver of field amplification, working for even initially weak fields
(Moiseenko et al., 2006; Obergaulinger et al., 2006; Kotake et al., 2004;
Kuroda et al., 2020; Jardine et al., 2021; Obergaulinger & Aloy, 2020). These
simulations all found collimated, jet-like explosions.

Our rapidly-rotating waveform signals were extracted from the waveform catalog
of Dimmelmeier et al. (2008b), which we will refer to as the ‘Dim’ catalog. It
consists of 128 signals obtained through 2D simulations accounting for general
relativity, two different equations of state (EoS), approximated
deleptonization and neutrino pressure and different initial rotation values
and degrees (uniform and differential). The catalog comprises progenitor
models with 11.2, 15, 20 and 40 $M_{\odot}$ ZAMS masses.

The right panel of Figure 1 shows a representative waveform of the Dim
catalog. Simulations have shown GW from rotating CCSN have a generic
morphology, known as Type I signals (Dimmelmeier et al., 2008b): a steady rise
pre-bounce, marked as section 1 in the plot, followed by a strong negative
peak at bounce, corresponding to the rapidly changing quadrupole moment of the
inner core which is deformed by centrifugal forces. Milliseconds after bounce,
convection sets in, resulting in a ringdown signal shown in section 2
(Dimmelmeier et al., 2008b; Ott et al., 2007; Dimmelmeier et al., 2007). The
frequency of most of the GW signals emitted are in the range $\sim 500-800$ Hz
and the signals have a duration of several $\sim 10$ ms after bounce.

###  2.3 Sine-Gaussian signals

In addition to the two types of signals we have just discussed, we will also
employ a third ‘mock’ catalog. This catalog comprises simulated signals
modeled as sine-Gaussian (SG) waveforms, which serve the purpose of estimating
detection sensitivity to generic, short-duration GW signals. The main
motivation in introducing this catalog is to expand the number of signal types
we can apply our classification approach to, which allows us to further assess
the algorithm. We also note that ad hoc signals such as this type of waveforms
have been used by the LVK in optically-targeted CCSN searches to estimate the
energy constraints of GW emitted by such source (Abbott et al., 2020c). We
build this catalog, dubbed ‘SG’, by creating an array of 200 sine-Gaussian
signals, varying in duration from 0.05 to 0.1 s and in frequency from 20 to
200 Hz, as well as having a varying phase. Figure 2 shows three illustrative
examples of waveforms from this catalog.

##  3 Signal reconstruction and classification

We turn now to explain the two-step process we have developed to reconstruct
and classify GW signals. The interested reader is addressed to Torres-Forné et
al. (2016b) for further details.

###  3.1 Fundamentals

In our approach we define $\textbf{D}\in\mathbb{R}^{l\times p}$ as the so-
called dictionary, a matrix composed of $p$ atoms of length $l$, and
$\boldsymbol{\alpha}\in\mathbb{R}^{p}$, which is a vector of coefficients that
represents the reconstructed signals. This vector should be as sparse as
possible, that is, it should contain a small amount of non-zero coefficients.
A variational problem balances the reconstruction of the original signals u
while ensuring that vector $\boldsymbol{\alpha}$ contains as many zeros as
possible. This variational problem is known as basis pursuit (Chen et al.,
2001) or LASSO (Tibshirani, 1996), and reads

$\boldsymbol{\alpha}=\operatorname{\mathop{argmin}_{\alpha}}{(\lVert\textbf{D}\boldsymbol{\alpha}-\textbf{f}\rVert^{2}_{2}+\lambda\lVert\boldsymbol{\alpha}\rVert_{1})}\,,$
(1)

where f is a vector of noisy signals resulting from injecting signals u into
Gaussian noise n, and $\lVert\cdot\rVert_{1}$ and $\lVert\cdot\rVert_{2}$ are
the L1-norm and the L2-norm, respectively. The L1-norm term introduces a
Lagrange multiplier $\lambda$, or regularization parameter, which helps ensure
a sparse $\boldsymbol{\alpha}$ and avoids overfitting. Large values of
$\lambda$ would return a coefficient vector with zero L1-norm. If $\lambda=0$,
the variational problem would only contain the first term and would simply
correspond to a least-square regression problem.

In this work we use two types of dictionaries. First, we separate each
waveform catalog into two sets, roughly split in a percentage 80/20. While the
smaller sets consist of the signals we will attempt to denoise and
reconstruct, the bigger sets will be directly used as dictionaries without any
prior training (see below), so that each atom is one of the original catalog
waveforms. We refer to these dictionaries as $\textbf{D}^{\rm cat}$ and they
will be used to classify signals by their type. The other class of dictionary,
$\textbf{D}^{\rm train}$, is trained using patches extracted from the same 80%
set. These will be the dictionaries used for signal denoising. The dictionary
training method is explained in Section 4.

Each signal will undergo two reconstruction processes. First, the denoising is
done over learned dictionaries, an effective method to solve signal-processing
problems (see Mairal et al. (2009) and references therein). This begins by
extracting patches from an initial noisy signal f and solving the LASSO
problem for $\boldsymbol{\alpha}$ over a previously trained dictionary,
$\textbf{D}^{\rm train}$. Then, a denoised signal y is built out of the
patches obtained from the product of $\textbf{D}^{\rm train}$ and
$\boldsymbol{\alpha}$. The second reconstruction process, pertaining to the
signal classification process, is done over the waveform dictionaries
$\textbf{D}^{\rm cat}$. Each denoised signal y is reconstructed whole, rather
than out of patches, as we aim to find out which waveforms best reconstruct a
denoised signal.

Carrying out these two processes, from calculating the best denoising
parameters to determining which catalog best reconstructs the signals,
necessitates some way of quantifying which reconstructions actually correspond
to the best results. We will define the best results as those that minimize
the Wasserstein distance (WD), a metric function which measures the minimum
amount of ‘work’ required to transform u into the reconstructed signal (Ramdas
et al., 2017). To compute WD in specific ranges of the time series waveforms
must be aligned. All 128 Dim signals are aligned at core bounce, that is, the
large negative peak immediately after the pre-bounce rise. As can be seen in
the right panel of Figure 1, this happens at 0.24 s, though this will later be
changed to 0.74 s as we will pad the signals to be a second longer. Most of
the Mur signals are contained in the 0.15$-$1.29 s range. Correspondingly, SG
signals are centered at 1 s and vary between 0.05 and 0.1 s in duration, as
shown in Figure 2.

A flowchart of the entire two-step process we use is shown in Fig. 3. First,
the three catalogs are split into bigger sets, $\textbf{D}^{\rm cat}$, and
smaller sets of u signals. From the former we extract training patches and
find the optimal parameters to train $\textbf{D}^{\rm train}$ dictionaries
through the LASSO variational problem. Those optimal parameters are determined
by (i) injecting the u signals into Gaussian noise to obtain f signals, (ii)
denoising the latter with all trained dictionaries, (iii) computing the WD
between each original signal u and the corresponding denoised signal y, and
(iv) choosing the parameters that minimize said WD. The dictionary-training
part of the process assumes that we know to which class the original signals
belong to, so that e.g. only $\textbf{D}^{\rm train}_{\rm Dim}$ is used to
denoise $\textbf{f}_{\rm Dim}$ signals and so on. Further details of each of
the two steps are outlined next.

###  3.2 Reconstruction step

As mentioned before, each of the Dim, Mur, and SG catalogs is separated into
two 80/20% sets. We denote the larger sets $\textbf{D}_{\rm Dim}^{\rm cat}$,
$\textbf{D}_{\rm Mur}^{\rm cat}$ and $\textbf{D}_{\rm SG}^{\rm cat}$, as they
can themselves be used as dictionaries. The first thing we have to do with
them is to extract $m$ training patches from all the signals available in the
sets. From these patches we train dictionaries $\textbf{D}_{\rm Dim}^{\rm
train}$, $\textbf{D}_{\rm Mur}^{\rm train}$ and $\textbf{D}_{\rm SG}^{\rm
train}$, a process mediated by the regularization parameter $\lambda^{\rm
train}$ (see Section 4).

The signals u of the smaller sets are injected into Gaussian noise n to
produce signals $\textbf{f}=\textbf{u}+\textbf{n}$. The first operation we
perform on these noisy signals is to denoise (reconstruct) them using the
previously trained dictionaries $\textbf{D}^{\rm train}$. Therefore, denoising
f with $\textbf{D}^{\rm train}_{\rm Dim}$ will yield a $\textbf{y}_{\rm Dim}$
signal and so on. This reconstruction step is done following the LASSO problem
where $m$ patches will have been extracted from the noisy signals. We denote
the regularization parameter of this problem $\lambda^{\rm denoise}$. One of
the preliminary steps of our procedure is therefore to find an optimal value
of $\lambda^{\rm denoise}$ for each signal type as well as the optimal
parameters of the learned dictionaries. This corresponds to the ‘Denoising’
section of the flowchart in Fig. 3.

Figure 3: Flow diagram of the two-step process to reconstruct and classify GW
signals using learned dictionaries.

###  3.3 Classification step

The classification step is carried out using the three denoised signals
$\textbf{y}_{\rm Dim}$, $\textbf{y}_{\rm Mur}$, and $\textbf{y}_{\rm SG}$.
This allows us to identify whether u is a signal resulting from a magneto-
rotational-driven CCSN (Dim), a neutrino-driven CCSN (Mur), or an ad-hoc sine-
Gaussian signal (SG). For this purpose we work with the entire denoised signal
y instead of working with patches as done during the denoising step. We denote
the regularization parameter of this process $\lambda^{\rm class}$ and the
dictionaries we employ will now be the 80% sets $\textbf{D}^{\rm cat}$. As
$\boldsymbol{\alpha}$ is now a vector of size $p$, where $p$ is the number of
waveforms present in $\textbf{D}^{\rm cat}$, each non-zero coefficient
$\boldsymbol{\alpha}_{i}$ indicates the degree to which y is reconstructed
from the $i$-th waveform in the catalog. We will show below some results to
illustrate this.

Signals $\textbf{y}_{\rm Dim}$, $\textbf{y}_{\rm Mur}$ and $\textbf{y}_{\rm
SG}$, obtained using each of the three trained dictionaries, are our first set
of reconstructions. Our second set is obtained by carrying out the regression
step. Thus, reconstructing $\textbf{y}_{\rm Dim}$ over $\textbf{D}^{\rm
cat}_{\rm Dim}$ will yield $\textbf{y}_{\rm Dim-Dim}$ (and similarly for
$\textbf{y}_{\rm Mur-Mur}$ and $\textbf{y}_{\rm SG-SG}$). While a more in-
depth analysis can be carried out through cross reconstructions, such as
$\textbf{y}_{\rm Mur-Dim}$, here we will only work with signals reconstructed
with matching dictionary types. We will then compute the WD between
$\textbf{y}_{\rm Dim}-\textbf{y}_{\rm Dim-Dim}$, $\textbf{y}_{\rm
Mur}-\textbf{y}_{\rm Mur-Mur}$ and $\textbf{y}_{\rm SG}-\textbf{y}_{\rm SG-
SG}$, as in a realistic situation we will not have access to the original
signal u. Our classification criterion considers that the smallest WD value
will determine whether u is either a Dim, Mur, or SG signal. When presenting
our results we will go over the ranges we choose to compute the WD since, as
we shall see, those may affect the results, particularly in the case of Mur
waveforms.

By doing this classification process with a number of different signals and
contrasting the results with the true nature of said signals, we are able to
build a confusion matrix that serves to assess the effectiveness of our
algorithm.

##  4 Dictionary training

###  4.1 Separating signals and training the dictionaries

Waveforms from the catalogs are separated into signals used to train the
dictionaries and signals to validate the approach through reconstruction and
classification. For the Dim catalog we choose 102 signals to train and 26
signals for validation, for the Mur catalog 11 and 5 signals, respectively, to
account for their small number, and for the SG catalog 160 and 40 signals,
respectively. While the SG simulated signals are generated with an amplitude
between $-1$ and 1, the 102 and 11 sets of Dim and Mur signals have to be
first rescaled between $-$1 and 1. This is necessary to avoid the difficulties
of training dictionaries out of patches with $10^{-22}-10^{-21}$ amplitudes.

Once this is done we extract $m$ patches of length $n$ from each of the three
larger sets. The function designed to do this lets us specify how much overlap
is allowed, which in this case we set to maximum as to allow us to extract a
sufficiently high number of patches. It is also designed to avoid extracting
patches of zero module, which would interfere with the dictionary-training
process. Furthermore, we normalize these patches to zero mean and unit
standard deviation. For all cases we work with $m=3000$ training patches and
four different values of $n$. Specifically we work with patches of size
$n=64,128,256$ and 512 for the Dim and Mur dictionaries while for SG signals
we extract patches with $n=128,256,512$ and 1024 instead, as patch sizes
smaller than that resulted problematic when it came to training the
dictionaries for this particular type of signals.

Before training the dictionaries from these patches, the smaller sets of
signals are injected into Gaussian noise. Both the Dim and Mur catalogs have a
sampling frequency of 16384 Hz but they respectively have a duration of 1 s
and 2 s. As our final objective is to reconstruct a signal of unknown origin,
we pad the Dim signals with zeros so that every signal we work with consists
of 32768 samples. Once all signals have the same length we generate Gaussian
noise by specifying the duration and sampling frequency of the signals as well
as a cutoff frequency of 20 Hz, according to the $\sim$10$-$20 Hz sensitivity
lower limit of the Advanced LIGO and Advanced Virgo detectors (Martynov et
al., 2016; Abbott et al., 2020b). We also employ the sensitivity curve of
Advanced LIGO in the form of the power spectral density (PSD) of noise sources
(Barsotti et al., 2018).

Figure 4: Examples of several atoms from three trained dictionaries. The top
panel corresponds to a Dim dictionary trained with $n=128$ and $\lambda^{\rm
train}=1.2$. The plots in the central panel show atoms from a Mur dictionary
trained with $n=512$ and $\lambda^{\rm train}=1.7$. The bottom panel
corresponds to a SG dictionary trained with $n=256$ and $\lambda^{\rm
train}=0.7$. All of the atoms in these examples have been extracted from
dictionaries with $p=100$.

To inject each signal into Gaussian noise we need to scale its amplitude to a
specific signal-to-noise ratio (SNR). The SNR is defined as

$\textrm{SNR}=\sqrt{4\Delta t^{2}\Delta
f\sum_{k=1}^{N_{f}}\frac{|\tilde{h}(f_{k})|^{2}}{S(f_{k})}},$ (2)

where $\tilde{h}$ is the Fourier transform of the signal strain $h$, $f_{k}$
are the elements of the frequency vector, $S(f_{k})$ is the PSD of the noise,
$N_{f}$ is the number of frequencies, and $\Delta t$ and $\Delta f$ are the
time and frequency steps, respectively. In this work we will study the
dependence of our signal classification procedure with the SNR. We will use
the values SNR=20, 15, 10 and 5, the latter being a fairly low SNR scenario.

We turn now to discuss the training of the dictionaries using the large sets
of signals from the Dim, Mur, and SG catalogs. A key parameter of the LASSO
problem is the regularization parameter. We start analysing how to set the
lower and upper limits of this parameter, $\lambda^{\rm train}$, to train
dictionaries, referring to the next section the discussion of which of the
parameters best denoise signals. If $\lambda^{\rm train}$ is too large then
the minimization problem of training a dictionary will prioritize solutions
with zero L1-norm, resulting in noisy dictionary elements which would
difficult the reconstruction of signals (Torres-Forné et al., 2016b). On the
other hand, if $\lambda^{\rm train}$ is too small then the dictionary training
problem will turn into a least-square regression problem, resulting in an
overfitting of the dictionary elements which would come with a loss of
generality.

Figure 5: The orange curves show the results of denoising pure Gaussian noise
(blue; no signal injected) using the $\textbf{D}^{\rm train}$ dictionaries,
Dim (left), Mur (centre), and SG (right).

For every signal type we generate dictionaries with four different values of
$n$, as previously mentioned. In addition, for each $n$ we generate two
dictionaries, each with a different number of atoms, $p=100$ and $p=200$,
which gives eight different combinations of $n$ and $p$ for each signal type.
Furthermore, once the lower and upper limits of $\lambda^{\rm train}$ are
selected for each signal type and $n$, we train each of these dictionaries
with 4 different values of $\lambda^{\rm train}$. Therefore, in total we build
32 dictionaries for each signal type. As an illustration, several atoms from
three of these dictionaries are shown in Figure 4 for the three types of
waveform signals used in this study. Technical details on the computation and
optimization of the parameters of the dictionaries are presented in Appendix
A. In particular, the optimal values of $\lambda^{\rm train}$ we obtain, i.e.
those that correspond to the smallest WD, for signals with SNR=20, are
reported in Table 1.

If the denoising is carried out over pure Gaussian noise, all three trained
dictionaries create spurious signals where there should be none. This is
illustrated in Figure 5 where the spurious reconstruction is compared with the
original noise. Even though a fake reconstruction appears, its amplitude is
small. These artificial signals could be reduced by increasing the value of
$\lambda^{\rm denoise}$. However, doing so in general might lead to missing
actual low SNR GW injections. Therefore, our final denoising and
classification results, discussed in Section 5, are obtained with the optimal
values of $\lambda^{\rm train}$ and $\lambda^{\rm denoise}$ reported in Table
2 of Appendix A.

###  4.2 Waveform dictionaries

We now discuss the dictionaries that will be used for signal classification,
$\textbf{D}^{\rm cat}$. The classification is done by solving the LASSO
variational problem over previously denoised signals. Figure 6 shows the
result of reconstructing signals using $\textbf{D}^{\rm cat}$ instead of
$\textbf{D}^{\rm train}$. The left panels compare the original and
reconstructed signals and the right panels indicate the non-zero values of the
coefficient vector ${\boldsymbol{\alpha}}$. The upper row shows a Dim signal
which is contained in the dictionary, namely signal 97 of $\textbf{D}^{\rm
cat}_{\rm Dim}$. Therefore, ideally the reconstruction should result in only
one non-zero $\boldsymbol{\alpha}$ coefficient which exactly matches the
signal. This is what happens when reconstructing a signal without noise, as
can be seen in the right panel of the first row. The situation is, however,
quite different when reconstructing that same signal embedded in noise, as
shown in the second row for the particular case SNR=20 (note that we do not
plot the noise in the left panel of the figure). In this case, if we force the
algorithm to accept only a reconstruction with one non-zero coefficient, it
does not always return the correct one, as seen in the right panel. If the
previous condition is not enforced, the reconstruction results in many non-
zero coefficients.

This last outcome is also what happens in a situation more similar to what can
be found in practice, namely when the waveform dictionaries are used to
reconstruct signals that are not contained in them. In this case we allow the
signals to be reconstructed with the number of non-zero coefficients which
optimizes the WD. The results are plotted in the lower three rows of Fig. 6
for Dim, Mur, and SG signal (from top to bottom). As a test, the results shown
in this figure are performed over noiseless signals. The reconstructions shown
yield the following values of our metric estimator: WD${}_{\rm Dim}=8.99\times
10^{-3}$, WD${}_{\rm Mur}=1.36\times 10^{-2}$ and WD${}_{\rm SG}=1.37\times
10^{-3}$. As can be seen both by these values and qualitatively (in the signal
comparison in the left panels), the SG and Mur reconstructions give,
respectively, the best and worst results. This is to be expected as catalog
$\textbf{D}^{\rm cat}_{\rm SG}$ has the largest number of signals to draw
from, at 160, while $\textbf{D}^{\rm cat}_{\rm Mur}$ has the smallest, at 11.

Figure 6: Left panels: reconstructions of several representative signals using
$\textbf{D}^{\rm cat}$ dictionaries. Right panels: corresponding values of
coefficient vector $\boldsymbol{\alpha}$. The first row shows the
reconstruction of the noiseless signal no. 97 of the Dim catalog using the
$\textbf{D}^{\rm cat}_{\rm Dim}$ dictionary at its optimal $\lambda^{\rm
class}$ , while the second row showcases the reconstruction of that same
signal while embedded in noise with SNR=20 at a $\lambda^{\rm class}$ which
forced one non-zero coefficient. The next three rows show the optimized
reconstructions of noiseless Dim, Mur, and SG signals (from top to bottom)
which are not contained in their respective $\textbf{D}^{\rm cat}$
dictionaries.

##  5 Results

###  5.1 Waveform classification

For each SNR we build four different confusion matrices. We work with 180
signals for each matrix, 60 from each catalog, extracted from the small sets
of (validation) signals not included in the dictionaries. As we only have 26,
5 and 40 Dim, Mur and SG signals available, respectively, many of them will be
repeated, although they are chosen randomly and they are injected each time in
newly generated noise realizations.

We start by denoising these signals with all three different $\textbf{D}^{\rm
train}$ using the parameters of Table 2 and then we reconstruct these denoised
signals using the waveform dictionary of the same type. Therefore, each matrix
is created out of 540 reconstructions: 180 signals reconstructed first with
$\textbf{D}^{\rm train}_{\rm Dim}$ and then with $\textbf{D}^{\rm cat}_{\rm
Dim}$, and the same amount for the Mur and SG catalogs. The second LASSO
regression is performed with the value of $\lambda^{\rm class}$ that optimizes
the WD between the reconstructed and the denoised signals for each individual
case. As explained in Section 3.3, we limit our study to reconstructions
created with dictionaries of the same type. For illustration purposes Figure 7
shows the outcome of denoising one example of each type of signal with either
$\textbf{D}^{\rm train}$ for SNR=20. As expected, each signal type is denoised
best with its respective dictionary.

Figure 7: Results of denoising three representative SNR=20 signals, Dim (top),
Mur (centre), and SG (bottom), using all three learned dictionaries
$\textbf{D}^{\rm train}$.

Therefore, each original signal f will result in three final reconstructions,
$\textbf{y}_{\rm Dim-Dim}$, $\textbf{y}_{\rm Mur-Mur}$ and $\textbf{y}_{\rm
SG-SG}$. Associated with each one of these there is a WD obtained by the
comparison with the denoised signals $\textbf{y}_{\rm Dim}$, $\textbf{y}_{\rm
Mur}$ and $\textbf{y}_{\rm SG}$. We choose the best reconstruction as the one
that minimizes the WD and by comparing that reconstruction with the actual
signal u, we are able to build the components of a confusion matrix.

It is necessary to explain the criterion we use to choose the best
reconstruction based on the WD. While the optimization process to calculate
the regularization parameters computes the WD in the ranges where most of the
signals are located, that does not produce satisfactory results in the
classification process. At SNR=20, 92% of the Dim signals are correctly
identified even when qualitatively those 8% misidentified signals show a
successful reconstruction of a Dim signal, as depicted in the top panels of
Figure 8. Even more of a concern is the fact that at SNR=10 the percentage of
correctly identified SG signals falls to only 20%. The bottom panels of Figure
8 show a clear reconstruction of a SG signal being misidentified as a Mur
signal (shown in blue).

Figure 8: Reconstructions of Dim signal 99 at SNR=20 (top panels) and SG
signal 23 at SNR=10 (bottom panels). Computing the three WD in the ranges
pictured in the plots results in both signals being misclassified as Mur.

Focusing on the top panels of Figure 8, $\textbf{y}_{\rm Mur}$ (shown in blue)
does show a large peak corresponding to the bounce peak of Dim signals in the
0.71$-$0.84 s range. This is not a surprise as Figure 7 already shows that
$\textbf{D}^{\rm train}_{\rm Mur}$ does an acceptable job of denoising a Dim
signal. Despite this, calculating the WD of the Dim reconstruction in the
0.71$-$0.84 s range (top-left plot of Fig. 8) and of the Mur reconstruction in
the 0.15$-$1.29 s range (top-centre plot), leads to WD${}_{\rm Dim}=0.062$ and
WD${}_{\rm Mur}=0.022$. Therefore, the signal is misclassified as Mur by
having a smaller WD. This could be due to the fact that Mur signals are
stochastic and they are the longest of the three catalogs, which may impact
the way the WD quantifies the work needed to turn one signal into another. We
explored the possibility of using other similarity indices, such as the mean-
squared error. In this case, the classification of Dim and SG signals was
robust but only about 40% of Mur signals were correctly identified at SNR=20.

To mitigate those issues we employ the WD in the following way: First, we
compute WDDim and WDMur in the temporal range of Dim signals (i.e. in the
0.71-0.84 s range in the example of Fig. 8). By doing this we compare two
signals of equal length and we give prominence to identifying whether the
distinctive bounce peak of Dim signals is present. If WD${}_{\rm Dim}<{\rm
WD}_{\rm Mur}$, we compare WDDim to WDSG in the temporal range of SG signals
(namely, in the 0.79-1.19 s range in the example of Fig. 8). If instead
WD${}_{\rm Dim}>{\rm WD}_{\rm Mur}$, we recalculate WDMur in the SG signals
range to compare it to WDSG. In either case we choose the smallest value of
the WD. This method correctly classifies the two signals of Fig. 8. We note
that while the procedure has been illustrated with only a few signals we apply
it to the full 180 signals of each confusion matrix.

Figure 9: Normalized confusion matrices for 180 signals (60 per catalog) at
four different SNR values.

Using this procedure over four different SNR values we obtain the
classification results summarized in the confusion matrices of Figure 9. The
classification of Dim and SG signals is entirely successful for SNR=20 and 15,
while around 85% Mur signals are correctly classified. The results worsen
significantly for SNR=10, with Dim signals remaining the best identified ones,
at 75%. At SNR=5, the algorithm tends to classify a majority of signals as Mur
irrespective of their actual class (most hits are in the horizontal middle row
in the bottom-right panel of Fig. 9). While this is an undesirable result it
is, however, not surprising as our method does not incorporate any criterion
to classify a signal as noise and by their stochastic nature Mur signals are
the most difficult ones to separate from noise. A possible reason for the
misclassification could be a failed denoising step in which the algorithm is
unable to identify anything similar to the original signal. Another reason may
be the generation of spurious signals from noise during the denoising step, as
illustrated in Section 4.2. Such spurious signals could be classified as Mur
if matched closely those of the catalog. While higher values of
$\boldsymbol{\lambda}^{\rm denoise}$ could avoid generating those, the
tradeoff would be for the denoising to miss actual GW signals in low SNR
conditions.

In an attempt to improve the performance of our algorithm when it comes to
separating Mur signals from noise, we run 60 different realizations of pure
noise “signals” through our algorithm. We then use the resulting pure noise
reconstructions to develop a way to classify signals as noise. By calculating
the mean value of the WD of the pure noise classifications and subtracting the
mean standard deviation, we set maximum values for WDDim, WDMur and WDSG over
which a signal would be classified as noise. We find that this method still
correctly classifies most Dim and SG signals at SNR=20 and SNR=15 without
dismissing them as noise, while at SNR=10, 27% of Dim and 50% of SG signals
are dismissed as noise. However, out of the remaining signals, 93% of Dim and
84% of SG signals are correctly identified. For SNR=5, around 70% of both
types of signals are dismissed as noise, while 50% of the remaining
classifications are successful. However, with this new method, around 60-80%
of Mur signals are classified as noise irrespective of the SNR. Again, these
results indicate that distinguishing Mur signals from pure noise represents a
challenge to our algorithm.

###  5.2 Comparison with SMEE

It is worth comparing our findings with those obtained by SMEE (Logue et al.,
2012; Powell et al., 2016; Powell et al., 2017b; Powell, 2018; Roma et al.,
2019). As mentioned in the introduction SMEE classifies CCSN signals using
principal component analysis and Bayesian model selection. To make a fair
comparison, we contrast our results with those of SMEE in the case of one LIGO
detector only, rather than with those of a three-detector network.

SMEE correctly identifies signals that are already included in their model,
namely 100% of the magneto-rotational signals and around 95% of the neutrino
signals (Powell et al., 2017a). If the signals are not included in the model,
over 80% of the neutrino signals and over 95% of the magneto-rotational
signals are correctly identified by SMEE. These numbers are comparable to our
best results for SNR=20 and 15.

We note that besides the different methodological approach to reconstruct and
classify signals between SMEE and our work, there are other aspects to
consider that are relevant in the comparison. First, the majority of magneto-
rotational signals classified by SMEE correspond to SNR values in the range
20-70, with some reaching up to 140, i.e. much higher values than ours. Most
of the neutrino signals, however, have SNR of 5 to 20, comparable to our
values, with only a small set of signals reaching up to 30. Furthermore,
Powell et al. (2017a) incorporate a criterion based on Bayesian analysis to
identify an incoming signal as pure noise, something which is missing in our
approach. The percentages of correctly classified signals mentioned above for
SMEE take this into account and they correspond to signals identified as GW
signals (those identified as noise were discarded). For signals included in
the model, 73% of neutrino signals and 9% of magneto-rotational signals were
identified as noise by SMEE. For those not included in the model, it amounted
to 80% and 32%, respectively.

From this comparison we conclude that learned dictionaries yield satisfactory
results when identifying CCSN signals at high-enough SNR (15-20) and behave as
expected when lowering the SNR.

##  6 Conclusion

In this paper we have presented an algorithm based on learned dictionaries to
reconstruct and classify core-collapse supernova signals according to their
morphology. CCSN are a prime source of gravitational waves with frequencies
that lie within the sensitivity band of the current network of advanced,
ground-based detectors (Abdikamalov et al., 2020). The analysis of those
waveforms could potentially reveal the underlying explosion mechanism as has
been shown with the so-called Supernova Model Evidence Extractor in a number
of investigations (Logue et al., 2012; Powell et al., 2016; Powell et al.,
2017b; Powell, 2018; Roma et al., 2019), a method based on principal component
analysis and Bayesian model selection. Those works motivated us to find out
whether the results of SMEE could be reproduced with a supervised dictionary-
learning algorithm as a suitable complementary approach.

To carry out this comparison we have used the same waveform signals employed
by SMEE, namely waveforms from two publicly available catalogs obtained from
numerical simulations of neutrino-driven CCSN explosions (Murphy et al.,
2009a) as well as magneto-rotational explosions (Dimmelmeier et al., 2008a).
In addition, we have used a third ‘mock’ catalog of simulated sine-Gaussian
waveforms not only to increase the types of signals in our study but also to
enlarge the complexity for the algorithm. All signals have been injected into
coloured Gaussian noise to simulate the background noise of Advanced LIGO in
its broadband configuration and scaled to a freely-specifiable SNR. By using a
quantitative criterion based on the Wasserstein distance we have constructed
confusion matrices at four different SNR. Those matrices have shown that the
behavior of the algorithm at SNR=20 and 15 is fairly satisfactory, with nearly
all Dim and SG signals being correctly identified, while only around 15% of
the Mur signals being misclassified.

It is not surprising that Mur signals are harder to classify than the rest.
They intrinsically resemble noise (characterized by a stochastic time series),
they involve a large range of frequencies, their shapes are less generic, the
signals are longer, and, perhaps most importantly, the amount of signals in
the catalog is quite small (only 11) which impacts the training and
classification. We note that the algorithm could be improved by devising a
criterion for when a signal can be classified as ‘Noise’. In our current setup
and at our lowest SNR there is a tendency for around 70% of signals to be
classified as Mur. Discarding those signals which are identified as noise
could possibly improve the success rate of our classifications. This is of
particular interest after comparing our results with those of Powell et al.
(2017a), which do include a ‘Noise’ category. Using a one-detector-only setup
and signals not included in the model, which is comparable to our own setup,
SMEE classified 32% magneto-rotational signals as pure noise. As most of those
magneto-rotational signals have SNR$\geq 20$, this is an encouraging
comparison for our method, which correctly identified all Dim signals at
SNR=15-20.

A drawback of our algorithm is the appearance of spurious signals when working
in pure-noise conditions, which could lead to both misclassification of
legitimate signals and false positives. While in an ideal scenario a LASSO
regression would report no signal when applied to an only-noise time series,
in our case it does return a signal, and this depends on the value of the
regularization parameter. While working with higher values of $\lambda^{\rm
denoise}$ might reduce the number of false signals, it could also carry the
tradeoff of increasing false negatives, particularly for more noisy scenarios.
Furthermore, the classification of CCSN signals relies on having access to
simulated waveforms such as the Dim and Mur catalogs employed in this work,
which are computationally costly to obtain. The scarcity of available
numerical waveforms (i.e. data to train the algorithm) is an obvious limiting
factor of our method. Despite possible future improvements, our current
results show it is possible to use LASSO regression to extract gravitational-
wave signals injected in Gaussian noise using trained dictionaries, and to
classify them depending on their morphology after being reconstructed out of
pre-existing waveforms.

On a final note, the Dimmelmeier et al. (2008a) and Murphy et al. (2009a)
waveforms employed in this study, while useful for the comparison with SMEE,
would not be used to determine the explosion mechanism in an actual case of a
supernova detection candidate. Among the reasons to discard them is the fact
that the Dimmelmeier et al. (2008a) waveforms are both axisymmetric and fairly
short, since they only include the early bounce part of the signal, while the
Murphy et al. (2009a) waveforms are also only 2D and there are currently more
accurate 3D neutrino-driven waveforms available. Performing a further
comparison study using a larger set of improved CCSN waveforms is an
interesting idea to pursue as a follow-up of this work,

## Acknowledgements

We thank Jade Powell for her careful reading of the manuscript and useful
comments. This work has been supported by the Spanish Agencia Estatal de
Investigación (grant PGC2018-095984-B-I00) and by the Generalitat Valenciana
(grant PROMETEO/2019/071).

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##  Appendix A Optimization of the dictionary parameters

This appendix summarizes the procedures through which we determine the best
parameters of the dictionaries. First, we choose an optimal $\lambda^{\rm
train}$ for each of the dictionaries created. We qualitatively choose lower
and upper limits to generate each dictionary following the criteria detailed
in Section 4.1. Then, we train four Dtrain dictionaries for each $n$, $p$, and
signal type, with four different values of $\lambda^{\rm train}$. Finally, we
use all the dictionaries to denoise signals of their corresponding type
injected in noise with SNR=20, and we compute the WD from each denoised signal
y to the original signal u. The optimal values of $\lambda^{\rm train}$ for
which the smallest WD are found are reported in Table 1.

Table 1: Optimal values of the regularization coefficients $\lambda^{\rm train}$ for dictionaries trained at different $n$ and $p$. The Wasserstein distances shown are the mean from all denoisings of their corresponding signal type with SNR=20. The left and right values shown in the last two columns indicate the results for $p=100$ and $p=200$, respectively. | Signal
---
type
n | $\boldsymbol{\lambda}^{\rm train}_{\rm min}$ | $\boldsymbol{\lambda}^{\rm train}_{\rm max}$ |  | $\boldsymbol{\lambda}^{\rm train}_{\rm opt}$
---
100$|$200
|  WD${}_{\rm opt}\times 10^{-2}$
---
100$|$200
| 64 | 0.2 | 0.8 | 0.4$|$0.2 | 2.08$|$1.85
Dim | 128 | 0.3 | 1.2 | 1.2$|$1.2 | 2.28$|$2.04
| 256 | 0.6 | 1.5 | 1.2$|$1.5 | 2.71$|$2.25
| 512 | 0.6 | 2.0 | 2.0$|$1.5 | 3.09$|$2.56
| 64 | 0.2 | 0.8 | 0.2$|$0.2 | 1.38$|$0.86
Mur | 128 | 0.4 | 0.9 | 0.4$|$0.6 | 2.01$|$1.62
| 256 | 0.4 | 1.1 | 1.1$|$0.4 | 3.25$|$2.54
| 512 | 0.6 | 1.7 | 1.0$|$1.4 | 4.05$|$3.67
| 128 | 0.2 | 0.3 | 0.25$|$0.30 | 9.55$|$10.4
SG | 256 | 0.55 | 0.7 | 0.70$|$0.65 | 7.42$|$10.4
| 512 | 0.7 | 0.85 | 0.80$|$0.85 | 6.49$|$8.95
| 1024 | 1.2 | 1.9 | 1.20$|$1.70 | 4.87$|$4.11

We note that the WD are not computed over the entire duration of the signals
but rather over three specific ranges for each signal type. Those are chosen
to avoid the optimization process from evaluating signals containing mostly or
entirely zeros. The specific ranges are 0.71-0.84 s for Dim signals, 0.15-1.29
s for Mur signals, and 0.79-1.19 s for SG signals. These are the ranges for
which we optimize both $\lambda^{\rm train}$ and $\lambda^{\rm denoise}$.

Once the optimal $\lambda^{\rm train}_{\rm opt}$ is chosen for each dictionary
we have to decide the optimal values of $n$ and $p$. For this we carry out
both a quantitative analysis through the WD and a qualitative one to check the
results. We begin by comparing the WD calculated for every signal with each
dictionary. We start with the Dim dictionaries and signals, displaying the
results of the reconstructions in Figure 10. We see that there is no unique
value of $n$ that yields the best reconstruction for every single signal. From
this plot it appears that $n=64$, $p=200$ gives on average the best results,
as can be seen on Table 1 as well. In general larger values of $p$ will result
in better reconstructions, though at a higher computational cost.

Figure 10:  Histograms of the WD for the 26 Dim validation signals injected in
noise at SNR=20, denoised with different dictionaries as indicated by the
legends. The top and bottom panels show the results for $p=100$ and $p=200$,
respectively.

However, a qualitative analysis shows that $n=64$ dictionaries build more
oscillatory signals, as the shorter length of the atoms results in the
introduction of larger amounts of noise. Larger atom lengths could reduce this
at the cost of missing smaller (physical) oscillations in the signals.
Therefore, despite the WD computes the $n=64$ signals as the best scenario,
minimizing the amount of noise in the denoising process is preferable for the
subsequent reconstructions. As our aim is to develop an algorithm that
classifies signals by their type, we thus select a dictionary by requiring it
to be able to identify the overall shape of a given signal without introducing
too much noise that might interfere with the classification process. With
these considerations in mind we decide on the $n=256$, $p=200$ dictionary.

The next parameter that must be evaluated is $\lambda^{\rm denoise}$. Up to
this point in the algorithm for each denoising we have optimized every signal
and dictionary so that it would find the value of $\lambda^{\rm denoise}$ that
minimizes the WD. Moving forward, we want to find a fixed value $\lambda^{\rm
denoise}_{\rm opt}$ which will be used to denoise every signal (one per
catalog) as, in a realistic setup, we do not have access to the original
signal u to optimize the WD. Figure 11 shows $\lambda^{\rm denoise}_{\rm opt}$
for the 26 Dim signals. The values are bounded between around 0.1 and 0.8,
with very few of them going above 0.3. Therefore, we discard the three highest
values and choose the mean value of $\lambda^{\rm denoise}_{\rm Dim}=0.2$,
which we expect will produce good reconstructions for most Dim signals.

Figure 11: Histogram of the optimal values of the regularization parameter
$\lambda^{\rm denoise}_{\rm opt}$ for the Dim signals. The values correspond
to SNR=20 and to a D${}^{\rm train}_{\rm Dim}$ dictionary with $n=256$ and
$p=200$.

The optimization process we have just described for the Dim signals has to be
repeated for the other types of signals, Mur and SG. Figure 12 displays the
results for the Mur dictionaries. From this figure we can safely rule out the
cases $n=256$ and $n=512$ as the corresponding values of the WD are too large
to yield acceptable reconstructions. The tradeoff between $n=64$ and $n=128$,
as discussed earlier, is a better reproduction of the oscillations for smaller
atom sizes against the presence of more noise throughout the signal. Figure 12
shows that the $n=64$, $p=200$ dictionary is the best choice for practically
all signals. We therefore choose this dictionary as the results can be more
easily identified with Mur signals despite the extra noise. In general, we
expect the reconstructions of Mur signals to yield the worst results of all
signal types considered. In addition, Figure 13 depicts the histogram of the
optimal values of $\lambda^{\rm denoise}_{\rm opt}$ for the Mur signals. We
choose $\lambda^{\rm denoise}_{\rm Mur}=0.16$, as the values are bounded
between 0.15 and 0.175.

Figure 12:  Histograms of the WD for the 5 Mur validation signals injected in
noise at SNR=20, denoised with different dictionaries as indicated by the
legends. The top and bottom panels show the results for $p=100$ and $p=200$,
respectively. Figure 13:  Histogram of the optimal values of the
regularization parameter $\lambda^{\rm denoise}_{\rm opt}$ for the Mur
signals. The values correspond to SNR=20 and to a D${}^{\rm train}_{\rm Mur}$
dictionary with $n=64$ and $p=200$.

Finally, the estimation of the parameters for the SG signals is displayed in
Figure 14 and Figure 15. For clarity we only display in Fig. 14 the results
for half of the validation signals (i.e. only 20 SG signals). Nevertheless,
the conclusion that $n=1024$ and $p=200$ produces the best reconstructions
applies to the entire set. On the other hand, Figure 15 reveals that the
values of $\lambda^{\rm denoise}_{\rm opt}$ show larger variations for SG
signals than for the previous two sets of signals. As a result we choose the
mean value of the entire sample, namely $\lambda^{\rm denoise}_{\rm SG}=0.17$.

Figure 14:  Histograms of the WD for the half the SG validation signals
injected in noise at SNR=20, denoised with different dictionaries as indicated
by the legends. The top and bottom panels show the results for $p=100$ and
$p=200$, respectively. Figure 15:  Histogram of the optimal values of the
regularization parameter $\lambda^{\rm denoise}_{\rm opt}$ for the SG signals.
The values correspond to SNR=20 and to a D${}^{\rm train}_{\rm Dim}$
dictionary with $n=1024$ and $p=200$.

As a summary Table 2 reports the denoising parameters we use in the signal
classification process, namely the values of $n$ and $\lambda^{\rm train}$ to
train our three $\textbf{D}^{\rm train}$ dictionaries and the corresponding
$\lambda^{\rm denoise}$ values for the LASSO regression. Although finding the
best value of $\lambda^{\rm denoise}$ for each separate SNR would yield better
results, and even more so optimizing $\lambda^{\rm denoise}$ for each
individual signal, our method has been developed to be applicable to CCSN
signals of unknown parameters, including its SNR.

Table 2: Optimal values of the parameters of the algorithm inferred from the denoising process for each signal type. All dictionaries used will have $p=200$. Type | n | $\boldsymbol{\lambda}^{\rm train}$ | $\boldsymbol{\lambda}^{\rm denoise}$
---|---|---|---
Dim | 256 | 1.5 | 0.2
Mur | 64 | 0.2 | 0.16
SG | 1024 | 1.7 | 0.17