Deep-learning continuous gravitational waves

Abstract

We present a first proof-of-principle study for using deep neural networks (DNNs) as a novel search method for continuous gravitational waves (CWs) from unknown spinning neutron stars. The sensitivity of current wide-parameter-space CW searches is limited by the available computing power, which makes neural networks an interesting alternative to investigate, as they are extremely fast once trained and have recently been shown to rival the sensitivity of matched filtering for black-hole merger signals [D. George and E. A. Huerta, Phys. Rev. D 97, 044039 (2018); H. Gabbard, M. Williams, F. Hayes, and C. Messenger, Phys. Rev. Lett. 120, 141103 (2018)]. We train a convolutional neural network with residual (shortcut) connections and compare its detection power to that of a fully coherent matched-filtering search using the Weave pipeline [K. Wette, S. Walsh, R. Prix, and M. A. Papa, Phys. Rev. D 97, 123016 (2018)]. As test benchmarks we consider two types of all-sky searches over the frequency range from 20 to 1000 Hz: an “easy” search using T=105 s of data, and a “harder” search using T=106 s. The detection probability pdet is measured on a signal population for which matched filtering achieves pdet=90% in Gaussian noise. In the easiest test case (T=105 s at 20 Hz) the DNN achieves pdet∼88%, corresponding to a loss in sensitivity depth of ∼5% versus coherent matched filtering. However, at higher frequencies and for longer observation times the DNN detection power decreases, until pdet∼13% and a loss of ∼66% in sensitivity depth in the hardest case (T=106 s at 1000 Hz). We study the DNN generalization ability by testing on signals of different frequencies, spindowns and signal strengths than they were trained on. We observe excellent generalization: only five networks, each trained at a different frequency, would be able to cover the whole frequency range of the search.

Full text

Deep-learning continuous gravitational waves
Christoph Dreissigacker,1,2,* Rahul Sharma,3,1,2 Chris Messenger,4Ruining Zhao,5,6,7 and Reinhard Prix1,2
1Max Planck Institute for Gravitational Physics (Albert-Einstein-Institute), D-30167 Hannover, Germany
2Leibniz Universität Hannover, D-30167 Hannover, Germany
3Birla Institute of Technology and Science, Pilani, Rajasthan 333031, India
4SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom
5Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100101, China
6University of Chinese Academy of Sciences, Beijing 100049, China
7Department of Astronomy, Beijing Normal University, Beijing 100875, China
(Received 6 May 2019; published 7 August 2019)
We present a first proof-of-principle study for using deep neural networks (DNNs) as a novel
search method for continuous gravitational waves (CWs) from unknown spinning neutron stars. The
sensitivity of current wide-parameter-space CW searches is limited by the available computing power,
which makes neural networks an interesting alternative to investigate, as they are extremely fast once
trained and have recently been shown to rival the sensitivity of matched filtering for black-hole merger
signals [D. George and E. A. Huerta, Phys.Rev.D97, 044039 (2018); H. Gabbard, M. Williams, F.
Hayes, and C. Messenger, Phys. Rev. Lett. 120, 141103 (2018)]. We train a convolutional neural network
with residual (shortcut) connections and compare its detection power to that of a fully coherent matched-
filtering search using the W
EAVE
pipeline[K.Wette,S.Walsh,R.Prix,andM.A.Papa,Phys. Rev. D 97,
123016 (2018)]. As test benchmarks we consider two types of all-sky searches over the frequency range
from20to1000Hz:an“easy”search using T¼105s of data, and a “harder”search using T¼106s.
The detection probability pdet is measured on a signal population for which matched filtering achieves
pdet ¼90% in Gaussian noise. In the easiest test case (T¼105s at 20 Hz) the DNN achieves
pdet ∼88%, corresponding to a loss in sensitivity depth of ∼5% versus coherent matched filtering.
However, at higher frequencies and for longer observation times the DNN detection power decreases,
until pdet ∼13% and a loss of ∼66% in sensitivity depth in the hardest case (T¼106sat1000Hz).
We study the DNN generalization ability by testing on signals of different frequencies, spindowns
and signal strengths than they were trained on. We observe excellent generalization: only five networks,
each trained at a different frequency, would be able to cover the whole frequency range of the
search.
DOI: 10.1103/PhysRevD.100.044009
I. INTRODUCTION
Gravitational waves from binary mergers are now being
observed routinely [1–4] by the Advanced LIGO [5] and
Virgo [6] detectors. In contrast, the much weaker and
longer-lasting (days–months) narrow-band continuous
gravitational waves (CWs) from spinning deformed neu-
tron stars are yet to be detected, despite a multitude of
searches over the past decade (see Refs. [7–9] for reviews)
and continuing improvements in search methods (see e.g.,
Ref. [10] for a recent overview).
A key limitation of current search methods for CWs with
unknown parameters is the “exploding computing cost
problem”: give that a putative signal would be very weak,
one needs to integrate as much data as possible in order to
increase the signal-to-noise ratio (SNR). However, for a
fully coherent matched-filtering search (which is close to
statistically optimal [11]), the corresponding computing
cost grows as a high power ∼Tnof the data time span T,
with n≳5. This typically limits the longest coherent
duration to days–weeks before the computing cost would
become infeasible.
Therefore the class of semicoherent methods has been
developed, producing computationally cheaper searches.
They allow the analysis of more data, typically resulting in
*christoph.dreissigacker@aei.mpg.de
Published by the American Physical Society under the terms of
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Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
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PHYSICAL REVIEW D 100, 044009 (2019)
2470-0010=2019=100(4)=044009(11) 044009-1 Published by the American Physical Society

better sensitivity than a corresponding coherent search at
fixed computing cost (see e.g., Refs. [12,13]). Such
methods combined with massive amounts of computing
power, either via local computer clusters or via the
distributed public computing platform Einstein@Home
[14], currently yield the best state-of-the-art sensitivity to
CW signals (see e.g., Refs. [15–17] for recent examples).
In this work we investigate deep neural networks
(DNNs) [18–20] as a novel search method for CWs. The
field of DNNs, also referred to as deep learning, has
emerged as one of the most successful machine-learning
paradigms in the last decade, dominating wide-ranging
fields [20] such as image recognition, speech recognition
and language translation, as well as certain board [21] and
video games [22,23].
More recently DNNs have started to draw attention in
the field of gravitational-wave searches (i) as a classifier
for non-Gaussian detector transients (glitches)[24–27],
(ii) as a search method for unmodeled burst signals
[28,29] in time-frequency images produced by coherent
WaveBurst [30], and (iii) as a direct detection method for
black-hole merger signals in gravitational-wave strain
data [31–36].
This last approach (iii) is of particular interest to us, as
Refs. [31,32] illustrated for the first time that DNNs can
achieve a detection power comparable to that of (near-
optimal) matched filtering, at a fraction of the search time.
This is relevant for CW searches: while semicoherent
methods for wide-parameter-space searches are the most
sensitive approach currently known, they are by design less
sensitive than the statistical optimum achievable according
to the Neyman-Pearson-Searle lemma [37].
With DNNs the computationally expensive step is
shifted to the preparation stage of the search—the archi-
tecture tuning and “learning”of optimal network weights
(i.e., the training)—while the execution time on given input
vectors is very short (typically fractions of a second). The
determination of the noise distribution (for estimation of
the false-alarm level pfa) and measurement of upper limits
require many repeated searches over different input data
sets, with and without injected signals. The relative
search speed advantage of DNNs compared to traditional
search methods therefore accumulates dramatically over
these operations allowing very fast and flexible search
characterizations.
The plan of this paper is as follows. In Sec. II we define
and characterize our test benchmarks. In Sec. III we
describe our deep-learning approach to searching for
continuous gravitational waves, and explain the network
architecture and how it was trained. In Sec. IV we
characterize the performance our DNN achieves on the
test benchmarks in comparison to the matched-filtering
performance and how it generalizes beyond the bench-
marks’search parameters. And finally we discuss these
results in Sec. V.
II. COMPARISON TEST BENCHMARKS
A. Benchmark definitions
In order to characterize the detection power of the DNN
that we introduce in the next section, we define two
benchmark search setups and measure the corresponding
sensitivity achieved on them with a classical (near-optimal)
matched-filter search method described in Sec. II B.
We compare the sensitivity in the Neyman-Pearson
sense, also known as the receiver-operator characteristic
(ROC), using the “upper limit”conventions used in most
CW searches (cf. Ref. [10]): measure the detection prob-
ability pdet at a chosen false-alarm level pfa for a signal
population of fixed amplitude h0, with all other signal
parameters (i.e., polarization, sky position, frequency and
spindown) drawn randomly from their priors. In order to
characterize the signal strength in noise, we use the
sensitivity depth D[10,38], defined as
D≡
ffiffiffiffiffi
Sn
p
h0
;ð1Þ
where Snis the power spectral density of the (Gaussian)
noise at the signal frequency, and h0is the signal amplitude.
In particular we are interested in the sensitivity depth D90%
that corresponds to the signal amplitude h90%
0at which the
search yields a detection probability of pdet ¼90% at a
fixed false-alarm level, which here is chosen as pfa ¼1%
per 50 mHz frequency band.
We choose to use stationary white Gaussian noise, which
allows us to efficiently generate training data, and it
simplifies comparing against the sensitivity of idealized
matched filtering.
We consider two all-sky searches (parameters summa-
rized in Table I) over a range of frequency fand first-order
spindown _
f, one using T¼105s∼1.2days, and one
using T¼106s∼12 days of data assuming a single
detector (chosen as LIGO Hanford). These two searches
could realistically be performed with coherent matched
filtering. The required computing cost for the search and its
characterization (upper limits, false-alarm level) however
would still require a large cluster of, say, Oð1000Þcores for
over a month or so (see Table II). Therefore actually
performing these two full searches only for the purpose
of characterizing the matched-filtering sensitivity would be
TABLE I. Definition of the two benchmark searches.
Data span T¼105s=T ¼106s
Detectors LIGO Hanford
Noise Stationary, white, Gaussian
Sky-region All-sky
Frequency band f∈½20;1000Hz
Spindown range _
f∈½−10−10;0Hz=s
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infeasible. Instead we characterize the matched-filter
search on only five narrow frequency bands of width Δf¼
50 mHz starting at frequencies f0¼20, 100, 200, 500 and
1000 Hz, yielding a total of ten representative test cases.
B. W
EAVE
matched-filtering sensitivity
For the matched-filter search we use the recently
developed W
EAVE
code [39], which implements a state-
of-the-art CW search algorithm [40] based on summing
coherent Fstatistics [41,42] over semicoherent segments
on optimal lattice-based template banks [43,44]. This
code can also perform fully coherent (i.e., single-segment)
F-statistic searches, which we use for the present proof-of-
principle study. The benchmark search definitions in
Table Iare chosen in such a way that a fully coherent
search is still computationally feasible. This yields a
simpler and cleaner comparison than using a semicoherent
search setup, as we can easily design near-optimal search
setups (by using relatively fine template banks) without the
extra complication of requiring costly sensitivity optimi-
zation at fixed computing cost [13,40,45].
The W
EAVE
template banks are characterized by a
maximal-mismatch parameter m, which controls how fine
the templates are spaced in parameter space. These are
chosen as m¼0.1and m¼0.2for the two searches with
T¼105and T¼106s, respectively. The reason for
choosing the larger mismatch value (i.e., coarser template
bank) in the T¼106s case is to keep the computing cost
of the corresponding test cases still practically manageable,
as the coherent cost scales with the mismatch parameter as
∝m2for a four-dimensional template bank [see e.g.,
Eq. (24) in [43]].
By repeated injections of signals in the data and recovery
of the loudest F-statistic candidate in the template bank,
one can measure the relative SNR loss μcompared to a
perfectly matched template. The resulting measured aver-
age mismatch hμiquantifies in some sense how close to
“optimal”the matched-filter sensitivity will be (compared
to an infinite computing cost search with m¼0), and is
found as hμi∼5% and hμi∼11%, respectively for the two
searches.
Using the template-counting and timing models
[39,46,47] for W
EAVE
and the resampling Fstatistic, we
can estimate the total number of templates and the
corresponding total runtime for these two benchmark
searches as ∼78 and ∼45000 days on a single CPU core,
respectively. Table II provides a summary of the W
EAVE
search parameters and characteristics.
In order to estimate the sensitivity for the ten test cases
defined in the previous section (i.e., five frequency slices of
Δf¼50 mHz for each search of T¼105and T¼106s,
respectively), we first determine the corresponding detec-
tion thresholds Fth on the Fstatistic corresponding to a
false-alarm level of pfa ¼1% for each case. This is done
by repeatedly (105times for T¼105s, and ∼104times for
T¼106s, respectively) performing each search over
Gaussian noise and thereby recording the distribution of
the loudest candidate, which yields the relationship
between the threshold and false-alarm level. The corre-
sponding detection probability pdet for any given signal
population of fixed Dis obtained by injecting signals into
Gaussian noise data and measuring how many times the
loudest candidate exceeds the detection threshold. By
varying the injected Dwe can eventually find D90% for
the desired pdet ¼90% (see e.g., Ref. [10] for more details
and discussion of this standard “upper limit”procedure).
By a final injection þrecovery Monte Carlo we can verify
that the achieved W
EAVE
detection probability for the
quoted thresholds and signal strengths D90% in Table III
is pdet ∼90–91%, which is sufficiently accurate for our
present purposes.
The sky template resolution grows as ∝f2as a function
of frequency f, resulting in a corresponding increase in the
number of templates at higher frequency. This increases the
number of “trials”in noise at the higher-frequency slices,
which results in a corresponding increased false-alarm
threshold (chosen in order to keep the false-alarm level
at pfa ¼1%) as well as an increased computing cost, shown
in Table III.
TABLE II. W
EAVE
parameters and characteristics for the two
searches.
Name T¼105sT¼106s
Mismatch parameter m0.1 0.2
Average SNR loss hμi5% 11%
Number of templates N4×1011 3×1014
Search time [single CPU core] 6.7×106s3.9×109s
TABLE III. W
EAVE
characteristics for the ten test cases, each
covering a frequency “slice”of Δf¼50 mHz, starting at f0,of
the full searches defined in Table I. The detection thresholds Fth
correspond to a false-alarm level of pfa ¼1% over the band Δf.
CPUΔfdenotes the search time in seconds for the respective Δf
band on a single CPU core.
f020 Hz 100 Hz 200 Hz 500 Hz 1000 Hz
T¼105s
NΔf5×1051×1075×1073×1081×109
CPUΔf[s] 0.1 4.9 19 2.3×1021.7×103
Fthðpfa Þ20.6 23.6 25.1 27.0 28.6
T¼106s
NΔf3×1088×1093×1010 2×1011 8×1011
CPUΔf[s] 45 3×1031.4×1041.6×1056.9×105
Fthðpfa Þ27.5 31.1 32.5 34.2 36.2
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III. DEEP-LEARNING CWs
Our general approach is similar to that of Refs. [31,32] in
that we directly use the detector strain data as our network
input, and train a simple classifier with two output neurons
for the classes “noise”and “signal (in noise).”However,
given that CW signals are long in duration and narrow in
frequency, instead of using the time-series input it makes
more sense in our case to use the frequency-domain
representation of that data. We therefore provide the real
and imaginary parts of the fast Fourier transform (FFT) of
the data as a two-dimensional input vector over frequency
bins, using the native FFT resolution of 1=T . We chose the
network input size to be sufficiently large to contain the
widest signal (signals get stretched in the frequency domain
by the spindown _
fand Doppler shifts) twice, so that we can
slide the network along the frequency axis in steps of half
the network input width, guaranteeing that one input
window will always contain the full signal.
A. Network architecture
We started experimenting with DNN architectures sim-
ilar to those described in Refs. [31,32], but eventually by
trial and error converged on a ResNet architecture [48],
which showed better performance for our problem cases.
We have chosen slightly different networks for the two
searches (T¼105and T¼106s) of Table I, as these
correspond to signals with rather different widths in the
frequency domain: the network in the T¼105s cases
contains six instances of a residual block, while in the
T¼106s cases the network uses 12.
The network layers can be separated into three parts: the
stem block, a block of multiple residual blocks, and an end
block; see Fig. 1. The stem block consists of a standard
convolutional layer, while each of the residual blocks is
built according to Ref. [48]. The end block contains a dense
softmax layer with two final output neurons, corresponding
to the estimated probability psignal that the input contains a
signal, and pnoise ¼1−psignal for a pure noise sample. The
DNNs are implemented in the K
ERAS
framework [49] on
top of a T
ENSORFLOW
[50] backend.
B. DNN training and validation
Training the network is performed on a synthesized data
set of input vectors, where half contain pure Gaussian
noise, and half contain a signal added to the noise. One full
pass through this training set is commonly referred to as a
training epoch. Using a precomputed set of 10 000 signals,
each signal is added to 24 dynamically generated noise
realizations, which are also used as pure-noise inputs.
The number of signals in the training set is chosen by
observing that in all our cases we find that going beyond
10 000 signals yields only negligible further improvements
in detection probability, as shown in Fig. 2for the example
of T¼105s, f0¼1000 Hz.
The signals are scaled to a fixed depth D90%
training for each
test case and randomly shifted in frequency within the
network input window. These training depths were esti-
mated semianalytically using the method of Refs. [10,47],
and differ slightly from the final measured values D90% of
Table IV, which were not available at the time of training.
When testing the network on signals of different depths, the
detection probability behaves as expected; see Sec. IV D.
Furthermore, we found that using a different choice of
training depth did not significantly affect training success.
Every few epochs of training, we perform a validation
step, where the detection probability of the network is
measured on an independent data set. This validation set
contains another 20 000 input vectors, with half containing
signals in noise (of fixed depth D90% ), and half containing
noise only.
In order to compute the network’s detection probability
pDNN
det , we treat the output neuron psignal as a statistic, and
follow the usual “upper limit”procedure described in
FIG. 1. Illustration of the general network architecture used in
this study.
FIG. 2. Validation detection probability for T¼105s, f0¼
1000 Hz for training with training sets containing 10,100,
1000,10 000 and 50 000 signals.
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Sec. II B: we repeatedly run the network on Gaussian noise
inputs in order to determine the pfa ¼1% detection thresh-
old. We then run the network on the signal set and measure
for what fraction of signals the statistic exceeds that
threshold.
The evolution of the detection probability as a function
of training epoch (or similarly, as a function of training
time) is presented in Fig. 3, illustrating the progress of
learning. In order to test the variability and dependence of
the learning success on the random initialization of the
network, we train a “cloud”of ∼100 differently initialized
network instances. We use the network at its point of best
validation performance from each test case for the further
test results presented in the next sections.
TABLE IV. Measured W
EAVE
“upper limit”sensitivity D90% at
a false-alarm level of pfa ¼1%.
D90% [Hz−1=2]f0¼20 Hz 100 Hz 200 Hz 500 Hz 1000 Hz
T¼105s11.4 10.8 10.4 9.9 9.7
T¼106s29.3 28.2 27.6 26.8 26.0
(a) (b)
(c) (d)
FIG. 3. Validation detection probability pDNN
det of the DNN versus training time (or mean trained epoch) for 100 different network
instances trained for each of four test cases: (a) T¼105s;f
0¼20 Hz, pbest
det ¼85.7%;(b)T¼105s;f
0¼1000 Hz, pbest
det ¼71.9%;
(c) T¼106s;f
0¼20 Hz, pbest
det ¼68.3%; and (d) T¼106s;f
0¼1000 Hz, pbest
det ¼8.3%, all trained on an Nvidia GTX 750. The
solid horizontal line denotes the matched-filtering detection performance of pdet ¼90%. The red circles indicate the networks with the
best detection probability pbest
det . The error bars on each of the 100 curves are smaller than the widths of the lines.
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Most of the training was performed on Nvidia GTX 750
GPUs. We see in Fig. 3that for most cases the improve-
ments in detection probability seem to level off after the
training time (about one day in the T¼105s cases, and
about 10 days in the T¼106s cases). However, in the case
of T¼106s;f
0¼1000 Hz seen in Fig. 3(d) [and also for
T¼106s;f
0¼500 Hz (not shown)], there still seems to
be a slowly increasing trend in detection probability at the
end of training time. Therefore we trained a single network
instance for these two cases again on a more powerful
Nvidia TITAN V GPU for many more epochs, until the
validation detection probability seemed to level off, which
is shown in Fig. 4.
Overall we observe a dramatic increase in the “diffi-
culty”the DNN has in learning the different test cases along
the direction of increasing data span Tand frequency f,
also seen clearly in Table V. In the easiest case of
T¼105s;f
0¼20 Hz the DNN achieves a detection
probability of pDNN
det ∼88%, nearly rivaling matched-
filtering performance, while in the hardest case of T¼
106s;f
0¼1000 Hz it only manages pdet ∼13% (also see
Table V). This may not be very surprising, given that the
cases become increasingly more computationally intensive
(more templates) along the same axis for matched filtering,
as seen in Table III. In the frequency-domain input vectors
of the DNN, this would manifest by the signals being more
widely spread out due to increased frequency drift _
fT and
Doppler stretching.
IV. TESTING DNN PERFORMANCE
After the training and validation steps, we perform a
series of tests on the best DNN found for each test case (i.e.,
with the highest pDNN
det over all validation steps), in order to
more fully characterize its performance as a CW detection
method. In these tests we simulate the signals and noise
directly for any given depth using the standard CW
LALS
UITE
[51] machinery, in order to independently verify
the network performance. Hence we are not using a
traditionally fixed testing set but generate it on demand.
A. Verifying detection probabilities
As a sanity check we use an independent test pipeline to
confirm the detection probabilities pDNN
det for the ten cases.
These results, given in Table V, are seen to be between
0.5–2 percentage points higher than the corresponding
validation pbest
det originally observed in Figs. 3and 4. This
can be understood as follows: in order to speed up training,
in the validation step we do not slide the network window
over the search frequency Δfin the signal case, but instead
use only one network window fully containing the signal.
In the test case, on the other hand, we perform a more
accurate simulation of a real search by sliding (see Sec. III),
which can only increase the detection probability.
A second interesting question is how the detection
probability depends on the false-alarm level pfa (commonly
referred to as ROC curve) for a fixed signal population. This
isshowninFig.5in comparison to the matched-filter ROC.
B. Generalization in frequency f
If we want to perform a search over the whole frequency
range (e.g., as defined in Table I) using DNNs, we would
need to determine how many different networks we have to
train in order to cover this range with a reasonable overall
sensitivity. Alternatively we can also train a single DNN
with signals drawn from the full frequency range of the
search and compare its performance.
The results of these tests are shown in Fig. 6, which show
how the five DNNs, trained at their respective frequencies
f0, perform over the full frequency range of the search. In
addition we show the performance of another network that
has been trained directly over the full frequency range.
We see that the “specific”networks trained only on a
narrow frequency range still perform reasonably well over a
fairly broad range of frequencies, and especially that
networks trained at higher frequencies generalize well to
lower frequencies. This result shows that a small number of
networks Oð5Þwould be able to cover the whole frequency
FIG. 4. Validation detection probability pDNN
det of the DNN
versus training time for a single network trained on an Nvidia
TITAN V for the case T¼106s;f
0¼1000 Hz. The best
network, indicated by the red circle, achieves a detection
probability of pbest
det ¼11.5%. The error bars are smaller than
the widths of the lines.
TABLE V. Detection probabilities in %of the best networks for
each case at a false-alarm level pfa ¼1% and 90% matched-
filtering depth.
p90%
det f0¼20 Hz 100 Hz 200 Hz 500 Hz 1000 Hz
T¼105s87.6þ0.7
−0.685.4þ0.7
−0.784.1þ0.7
−0.780.2þ0.8
−0.873.0þ0.9
−0.9
T¼106s68.8þ0.9
−0.950.0þ1.0
−1.038.7þ0.9
−1.025.4þ0.8
−0.913.1þ0.6
−0.7
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range at a similar detection performance that was obtained
on the individual training frequencies. Furthermore, for the
T¼105s search, it seems quite feasible to train a single
network over the full frequency range directly, achieving
similar (albeit lower) performance to the “specialized”
networks trained on narrow frequency bands. On the
contrary for the T¼106s search the detection probability
of the “full-range”network drops up to 20 percentage
points against the “specialized”networks.
C. Generalization in spindown _
f
A further interesting aspect to consider is how far in
spindown _
fthe performance network extends beyond the
range that it was trained on, i.e., _
f∈½−10−10;0Hz=sas
given in Table I. This is shown in Fig. 7. We see that the
DNN detection probability remains high even for
spindowns that are 1–2 orders of magnitude larger than
the training range. In particular, networks trained at higher
frequencies seem to have a wider generalization range in
spindown, which makes sense as they would have learned
to recognize signal shapes with larger Doppler broadening,
a qualitatively similar effect to having more spindown.
D. Generalization in signal strength
Another important issue is how well the DNN general-
izes for signals of different strength D, given that we only
trained each network at one specific depth D90%
training,an
estimate of the matched-filtering depth. The results of this
test are shown in the efficiency plots of Fig. 8. We see that
generally the dependence of pdetðDÞfor the DNNs seems to
be quite similar to that of matched filtering, but shifted to its
overall (lower) performance level.
(a) (b)
FIG. 5. ROC curve: Detection probability pdet versus pfa for the 105s search (left) and the 106s search (right). The solid red lines
indicate the measured ROC curves for matched filtering.
(a) (b)
FIG. 6. Detection probability pdet versus injection frequency ffor networks trained at five different frequencies and for a network trained
with signalsdrawn from the full frequency range (solid black line). The dashed vertical lines markthe respective trainingfrequencies for the
five “specialized”networks. The horizontal dashed line represents the coherent matched-filtering detection performance.
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(a) (b)
FIG. 7. Detection probability pdet versus injected spindown _
ffor networks trained at five different frequencies. The green shade in the
middle marks the 10−10 Hz=s wide spindown band the networks were trained on. The xaxis is plotted as a symmetric logarithm, i.e.,
logarithmical for the larger negative values, linear for j_
fj<−10−10 Hz=s and logarithmical for the larger positive values. The red shades
at the edges illustrate where we start losing SNR purely by the network input window being smaller than the widest signals.
(a) (b)
(c) (d)
FIG. 8. Detection probability pdet versus injection depth Dfor networks trained on the respective matched-filtering depth D90%
(indicated by the vertical solid line with the diamond at 90%). The second vertical line with the star at 90% gives the sensitivity depth for
the DNN at 90% detection probability.
CHRISTOPH DREISSIGACKER et al. PHYS. REV. D 100, 044009 (2019)
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Conversely we also calculated the “upper limit”sensi-
tivity depth D90%
DNN where the network achieves 90%
detection probability (see Table VI). These values corre-
spond to a sensitivity loss of 5–21% (as a function of
frequency) for the T¼105s search, and 26–66% for the
T¼106s search.
E. Timing
The total amount of computational resources needed, is
another interesting point of comparison to a matched-filter
search. The total search times for using the matched-filter
W
EAVE
method on the two benchmark searches can be
found in Table II.
For the DNN the total computation time consists of two
parts: training time and prediction time (i.e., calculating
one output statistic psignal for one input data vector). The
training time for the two network architectures is ∼1and
∼10 d per network for the T¼105and T¼106s cases,
respectively. Only part of this time is actually spent on
training the network; another part is spent calculating the
detection probability of the network every few epochs in
order to monitor the progress of training.
The prediction time in comparison is almost negligible.
The smaller networks for the T¼105s cases require
∼3ms to process one input window. The larger networks
for the T¼106s cases need ∼10 ms per prediction. Each
search requires a different number of sliding input windows
to cover the whole frequency range, and the total search
time can be found in Table VII.
An important detail to note in a direct comparison
between matched filtering and a pure classifier “signal”
versus “noise”DNN search is that matched filtering yields
far more information on outlier candidates that cross the
threshold. In particular, its signal parameters will be well
constrained already, allowing a follow-up search to be
performed in a small region of parameter space. The DNN
classifier, on the other hand, would flag input windows
(of width ΔfIW) in frequency as outliers to be followed up.
Assuming we follow up two input windows per candidate,
one can estimate the total expected follow-up cost (using
matched filtering) as a fraction 2ðΔfIW=ΔfÞpfa of the total
matched-filtering cost (see Table II), where pfa ¼1% is the
false-alarm probability per Δf¼50 mHz band.
Therefore even including all the training time and
assuming a matched-filter follow-up, the DNN search would
still seem to require less computing power. At the present
stage, however, we cannot realize this potential benefit given
that our DNN search so far is far less sensitive overall.
V. DISCUSSION
In this work we have shown that deep learning (DNNs)
can in principle be used to directly search for CW signals in
data, at substantially faster search times than matched
filtering. For the hand-optimized network architecture
studied here, the DNN detection probability (at fixed false
alarm) is found to be somewhat competitive (88–73% over
the full frequency range) with matched filtering (90%) for
short data spans of T∼1day, while the detection perfor-
mance falls short (69–13%) for a longer data span of
T∼12 days. On the plus side, the DNN search shows a
surprising ability to extend further in frequency and spin-
down than it was trained for, and is generally much faster in
search performance than matched filtering.
In order to make this a competitive search method, we
can identify a few necessary next steps:
(1) Extend to a multidetector search.
(2) Find better networks with a comparable detection
probability to existing methods in Gaussian noise.
(3) Train for parameter estimation in addition to pure
classification in order to reduce follow-up cost of
candidates.
(4) Test how a network trained on Gaussian noise
performs on real detector data. Given the network’s
ability to generalize, one might expect problems if
non-Gaussian artifacts are identified as signals. On
the other hand, training a network on real detector
noise should alleviate that problem.
Overall we think that deep learning has the potential to
become a useful CW search tool, but there is substantial
further research and development effort required in order to
achieve this.
ACKNOWLEDGMENTS
We thank Sin´ead Walsh, Maria Alessandra Papa, Marlin
Schäfer and the AEI CW group for helpful comments. All
W
EAVE
Monte Carlo simulations and all DNN training and
testing were performed on the ATLAS computing cluster of
the Albert-Einstein Institute in Hannover. C. M. is sup-
ported by the Science and Technology Research Council
(Grant No. ST/L000946/1) and the European Cooperation
in Science and Technology (COST) action CA17137.
TABLE VI. Sensitivity depths D90%
DNN at a false-alarm level of
pfa ¼1% achieved by the network for the ten test cases. The
respective matched-filter depths can be found in Table IV.
D90%
DNN [Hz−1=2]f0¼20 Hz 100 Hz 200 Hz 500 Hz 1000 Hz
T¼105s10.8 10.0 9.5 8.6 7.7
T¼106s21.6 16.5 14.3 11.1 8.9
TABLE VII. DNN computing cost (in seconds) for training,
search and follow-up (using matched filtering). The respective
matched-filtering cost can be found in Table II.
Cost [s] Training Search Follow-up Total
T¼105s4.3×10558.8 2.2×1044.5×105
T¼106s4.3×106196 6.5×1076.9×107
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