Galaxy Spectra Neural Networks (GaSNets). I. Searching for Strong Lens Candidates in eBOSS Spectra Using Deep Learning

Fucheng Zhong, Rui Li, and Nicola R. Napolitano

1 School of Physics and Astronomy, Sun Yat-sen University, Zhuhai Campus, Zhuhai 519082, China; lrui@bao.ac.cn, napolitano@mail.sysu.edu.cn

2 CSST Science Center for Guangdong-Hong Kong-Macau Great Bay Area, Zhuhai 519082, China

3 School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China

4 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China

Received 2022 March 23; revised 2022 April 14; accepted 2022 April 19; published 2022 May 26

Abstract With the advent of new spectroscopic surveys from ground and space, observing up to hundreds of millions of galaxies, spectra classification will become overwhelming for standard analysis techniques. To prepare for this challenge,we introduce a family of deep learning tools to classify features in one-dimensional spectra.As the first application of these Galaxy Spectra neural Networks (GaSNets), we focus on tools specialized in identifying emission lines from strongly lensed star-forming galaxies in the eBOSS spectra. We first discuss the training and testing of these networks and define a threshold probability,PL,of 95%for the high-quality event detection.Then,using a previous set of spectroscopically selected strong lenses from eBOSS, confirmed with the Hubble Space Telescope(HST),we estimate a completeness of ～80%as the fraction of lenses recovered above the adopted PL.We finally apply the GaSNets to ～1.3M eBOSS spectra to collect the first list of ～430 new high-quality candidates identified with deep learning from spectroscopy and visually graded as highly probable real events.A preliminary check against ground-based observations tentatively shows that this sample has a confirmation rate of 38%,in line with previous samples selected with standard (no deep learning) classification tools and confirmed by the HST.This first test shows that machine learning can be efficiently extended to feature recognition in the wavelength space,which will be crucial for future surveys like 4MOST,DESI,Euclid,and the China Space Station Telescope.

Key words: gravitational lensing: strong – surveys – techniques: spectroscopic

1. Introduction

Strong gravitational lensing (SGL) is a powerful tool to investigate a large variety of open questions in cosmology.The formation of the distorted images of background galaxies, the“sources,” depends on the total mass of the foreground gravitational systems acting as “deflectors” or “l(fā)enses.” In case the latter are galaxies, SGL provides us accurate constraints on different properties correlated to their total mass, like the mass-to-light ratio (Ghosh et al. 2021), the dark matter fraction (Auger et al. 2009; Tortora et al. 2010), the slope of the total density profile (Koopmans et al. 2006, 2009;Auger et al. 2009) and its relation with other parameters(Bolton et al.2012;Shu et al.2015;Li et al.2018).SGL is also used to constrain the evolution of galaxies via merging(Bolton et al. 2012; Sonnenfeld et al. 2013, 2014, 2015), the initial mass function in massive ellipticals (Spiniello et al. 2011;Barnabè et al. 2012), and study the dark substructures around large galaxies (Gilman et al. 2018; Schuldt et al. 2019).

Moving to more cosmological constraints, SGL is used to measure the Hubble constant (H0), and other cosmological parameters (Suyu et al. 2013, 2017; Sluse et al. 2019; Rusu et al. 2020; Wong et al. 2020).

Strong lenses are generally searched in imaging data, where one can clearly distinguish the lensing features in the form of arcs, or multiple images of compact sources, like galaxies or quasars(Bolton et al.2008;Brownstein et al.2012;Sonnenfeld et al. 2013; Suyu et al. 2013). Here, a great impulse to lens hunting has been recently provided by automatized tools for lens finding (Gavazzi et al. 2014). In particular, machine learning(ML) techniques have been lately found to be very powerful in collecting hundreds of high quality (HQ) candidates (Dark Energy Survey—DES:Jacobs et al.2019,Kilo Degree Survey—KiDS: Petrillo et al. 2019; Li et al. 2020, 2021b, Hyper Supreme-Cam—HSC: Sonnenfeld et al. 2018).

After the identification of HQ candidates, a spectroscopical follow-up is needed to confirm their gravitational lensing nature (Metcalf et al. 2019; Witstok et al. 2021). In practice,one needs to collect the spectra of the lens and the source and measure their relative redshift, confirming that the lens is located in front of the source as expected from ray-tracing lensing models (see Cornachione et al. 2018; Napolitano et al.2020). This is a severe bottleneck in the SGL studies and, so far, there have been only sparse programs dedicated to these follow-up observations (Spiniello et al. 2019a, 2019b;Lemon et al.2020;Nord et al.2020).However,future large sky spectroscopic surveys (e.g., Taipan: da Cunha et al. 2017 4MOST: de Jong et al. 2012, DESI: DESI Collaboration et al.2016) will provide an unprecedented opportunity for massive follow-ups of lensing candidates, e.g., by reserving them dedicated observing nieces in wide programs or by accommodating them as filler targets in large-sky, multi-purpose surveys.

More interestingly, these large spectroscopic sky surveys will offer a unique chance to be used as a playground for lens finding, e.g., by looking for blended emission lines of background “l(fā)ensed” galaxies, e.g., star-forming systems, in the spectrum of a forward massive systems. This method has been extensively used in the last years to produce tens of discoveries of new unknown lens candidates.

The first example of a search of this kind was presented by Bolton et al. (2004), within the Sloan Lens ACS (SLACS).They found 49 SGL candidates in 50,996 Sloan Sky Digital Survey (SDSS) spectra of luminous red galaxies (LRG). They used the principal-component analysis to subtract the main components of the foreground LRG spectrum and a Gaussian kernel to find the best emission lines in the residual flux.They mainly focused on [O II] (3728 ?), [O III] (4960 ?, 5007 ?),and Hβ(4863 ?) lines, hence exploring a redshift range of z=(0.16–0.49) for the lenses and z=(0.25–0.81) for the sources. Later analyses increased the number of SLACS candidates to 131 (Bolton et al. 2008, SLACS hereafter).Within the BOSS Emission-Line Lens Survey (BELLS,hereafter),Brownstein et al.(2012)extended the spectroscopic search, previously performed in SLACS, to higher redshift, by looking for lenses up to z=0.7 and the background sources up to z=1.4, with no color pre-selection. This allowed them to finally find 45 SGL candidates in 133,852 SDSS galaxy spectra. Along the same line of approaches, in the SLACS Survey for the Masses(S4TM)project,Shu et al.(2015,2017,S4TM hereafter) have extended the search for SGL candidates to lower masses and found 118 new lens candidates. On the other hand,Shu et al.(2016a,2016b,BELLS GALLERY)and Cao et al. (2020) looked for high-redshift Lyα emitters as background sources and found 361 candidates.

The main disadvantage of these spectroscopy-selected samples is the missing information from images. Indeed, even if spectra can provide the evidence of two different emitting sources along the line of sight located at different redshifts,they cannot guarantee that they represent an SGL event.Hence,high-resolution imaging from space telescopes or adaptive optics is needed to have a visual confirmation of the lenses.Currently, there are 135 confirmed lenses with Hubble Space Telescope(HST)observations of the 294 selected using optical lines(70/131 from SLACS,25/45 from BELLS,40/118 from S4TM), and 17/21 Lyα candidates from BELLS GALLERY.

With the lesson learned from SDSS/BOSS, other experiments have combined the spectroscopic selection and imaging:Chan et al. (2016) matched 45 spectra from the Galaxy And Mass Assembly (GAMA) survey and confirmed 10 of them with Hyper Suprime-Cam (HSC) imaging; Holwerda et al.(2022) selected lens candidates in AAOmega spectra and followed up 56 of them with HST to find nine confirmations.

The discovery power of this approach will be pushed to unprecedented limits by future surveys combining spectroscopy and imaging from space (e.g., Euclid mission and China Space Station Telescope (CSST)) and produce a revolution in the lensing searches.However,this revolution will stand on the ability to effectively analyze gigantic spectroscopic data loads,which will imply the inspection of millions of spectra and the identification of (sometimes very faint) emission lines from background lensed systems. This is a prohibitive task for standard human-driven analyses, unless one adopts severe selections to maximize the number of detection but reduce the spectra to visually inspect.

Machine learning techniques can provide, instead, fast and efficient methods to overcome these difficulties and systematically search for lensing features in spectra. For instance,Convolutional Neural Networks (CNNs) have been previously applied for lens searches(see Li et al.2019,Li+19,hereafter).In particular, they have focused on the identification of Lyα emitters at higher redshift (2 ＜z ＜3) in the spectra of lower redshift early-type galaxies (z ＜0.6), and showed that these techniques can be efficiently used as a classifier for galaxy spectra.

In this paper, we expand this approach and develop a new CNN tool to look for SGL in the Baryon Oscillation Spectroscopic Survey(BOSS)spectroscopic database(Dawson et al. 2016). Because these sources are usually star-forming galaxies,we plan to use machine learning techniques to search for higher redshift emission lines such as[O II],[O III],Hα,Hβand Hδmixed in the foreground galaxy spectra.To do that,we build three CNN models: a classifier, to search for reliable emission lines in spectra, and two regression models, to measure the foreground galaxy and the background source redshifts,respectively.Then,we combine the predictions of the three CNNs to provide a list of high probability events that we visually inspect to select HQ candidates.

Finally, we compare this first deep learning spectroscopicselected sample with the most complete spectroscopic sample of SGL candidates in BOSS observations from Talbot et al.(2021, T+21 hereafter), obtained with standard cross-correlation techniques. This catalog consists of 838 likely, 448 probable,and 265 possible strong lens candidates,for a total of 1551 objects.They have also obtained a preliminary confirmation of 477 of them with low-resolution imaging.

The paper is organized as follows.In Section 2,we introduce the whole idea of this project and the details of the new CNN models.In Section 3,we introduce the modeled emission lines and the construction of training data.In Section 4,we show the training and testing results of the new CNNs. In Section 5, we apply the new CNNs to the BOSS spectra and derive a list of candidates that we qualify via visual inspection of their spectra,finally providing a catalog of HQ candidates. In Section 6, we discuss the results and estimate a tentative confirmation rate based on the match with ground-based imaging. We also discuss some avenues for improvement of future CNNs. In the final Section 7, we draw some conclusions.

2. Methodology

In this work we want to apply ML techniques to spectroscopic data. In particular, we want to use the ability of these techniques to perform feature recognition and classification.ML has been widely used in astronomical data analysis: (1) to find or classify different astronomical target candidates, like active galactic nuclei (AGNs) (Teimoorinia & Keown 2018;Chang et al. 2021; Zhu et al. 2021), quasars (Khramtsov et al.2019; Yang et al. 2021), star clusters (He et al. 2021; Jadhav et al.2021)or(2)to measure or predict physical parameters of astronomical targets, like redshifts (Ball et al. 2007; Han et al.2021), masses (Ntampaka et al. 2016; Bonjean et al. 2019), or velocity profiles (MacBride et al. 2021). Here we want to test the possibility to use ML techniques to efficiently search for strong lenses on vast amounts of spectra and predict the redshift of their lenses and sources.

2.1. The Challenge of Searching for Strong Lenses in Spectra

Next generation spectroscopic surveys will target tens of millions of galaxies (Mandelbaum et al. 2019). These huge samples will allow us to systematically search for highprobability candidates from integrated spectra,as the number of expected events is noticeable, given the large number of background galaxies potentially giving rise to lensing events.

Using the set of predictions from Collett (2015),5https://github.com/tcollett/LensPopwe have estimated that the number of lenses with a 1″ Einstein radius,RE, producing lensed images of the source, observable with a spectroscopic survey with a 2″diameter fiber,over 15,000 deg2of the sky, is of the order of 7000. This is obtained assuming that the source is bright enough in some visual band (e.g.,Euclid visual mag = 24.5) to make also the signal-to-noise ratio of the emission lines high enough to be detected from the ground for typical spectroscopic surveys (e.g., 4MOST or DESI). This estimate is subject to different factors, including some flux loss, but it also excludes the contribution from sources with slightly larger REs that might eventually scatter part of their light into a 2″ fiber. Hence, combining all these effects, this forecast is possibly not far from realistic. This is a wealth of data extremely valuable because it provides,for free,the information on the lens and the source redshifts, which are crucial for the lensing modeling.Standard techniques based on sophisticated selection criteria (T+21) still require rather timeconsuming visual inspection. Hence, a more practical solution to perform a systematic search of lens candidates in these data sets is mandatory.

This is possibly true also for current spectroscopic surveys.For instance, using the same set of predictions for the BOSS area (～10,000 deg2), and assuming a fainter limiting magnitude for the sources, r=23.5, we get ～920 lenses within a 2″fiber, which become ～1470 within a 3″fiber (e.g.,the one available for SDSS releases earlier than 12).Currently,the largest collection of candidate lenses with BOSS spectra consists of 477 objects with lensing evidence from lowresolution images (T+21). Taking this sample as a bona fide high-completeness sample, this is rather far from the expected number of discoverable lenses, meaning that there might be more lenses to find in the full BOSS data set.Given the full set of BOSS spectra available, i.e., ～2.6M items (Ahumada et al.2020), this means that we should expect one real blended emission line object every ～3500 spectra.

In this work, we want to tackle the problem of systematic searches of lens candidates in spectra with deep learning and use the BOSS data set to test the efficiency of this approach.

2.2. Convolutional Neural Networks as Lens Classifiers in 1D Spectra

When searching for strong lens candidates in one-dimension(1D) spectra, one needs to identify two main features: (1) the potential emission lines from the background sources, to determine the redshift of the source, and (2) the absorption or emission lines of the foreground galaxies, to determine the redshift of the lens and compare this with one of the putative sources to possibly qualify the whole system as a lensing candidate. In most of the current and planned surveys, the redshift of the main galaxy(the lens)is a standard data product,hence this can be assumed to be a label of the spectroscopic catalog. This can be either used as a first guess for the lens classifier, to estimate the lens redshift itself or kept fixed,asking the CNN to identify tentative background lines (see below).

2.3. Galaxy Spectra Convolutional Neural Networks(GaSNets)

In this work, we present the first set of Galaxy Spectra convolutional neural Networks (GaSNets) for Lensing (-L).These are CNNs trained to identify strong lensing event candidates in 1D galaxy spectra.To perform this task,we have built three different GaSNet models:

Figure 1. The CNN model adopted for the GasNet-L1, L2, and L3. The network structure is the same for the three GaSNets, except for the activation and loss functions as reported in Table 1.

1. GaSNet-L1. This CNN is a classifier, trained to look for the presence of emission lines blended in the features of the foreground galaxy and give the probability to be a lens (PL). In doing this, we do not assume any specific morphology for the lens, which can be either a standard early-type galaxy (ETG), dominated by absorption line features, or a late-type galaxy (LTG), with ongoing star formation. The GaSNet-L1 will learn whether, in the spectra of either kind, there are higher-redshift emission lines, to finally give the PL.

2. GaSNet-L2. This CNN is a regression algorithm, trained to identify potential emission lines, among a list of standard features from star-forming galaxies,overlapping a foreground galaxy spectrum and predict their redshift(zPE).

Table 1 CNN “My Activation” and “Loss Function”

3. GaSNet-L3. This CNN is also a regression algorithm,trained to predict the redshift of the foreground galaxy(zPG) from the combination of continuum plus (a)classical absorption features from ETG spectra or (b)emission lines of LTGs.Having such an output will make the overall Network general enough to be applied to spectroscopic databases, regardless these have gone through a pipeline to estimate galaxy redshifts. In our analysis below, even though we can assume that the redshift of the lenses is given (as they are provided with the BOSS spectra, see Section 3.1), we opt to use the redshift predictions of our GaSNet-L3 for the candidate selection and use the BOSS redshifts as ground truth to assess the accuracy of the deep learning estimates.

The three CNN models have the same structure. They are built by six convolution layers and three total connected layers(see Figure 1), assembled by Python modules TensorFlow and Keras.In the last layer in Figure 1,due to the different tasks to perform (classification versus regression), for GaSNet-L1 we use a “sigmoid” activation function (labeled as “my activation”),while for GaSNet-L2 and L3 we need no activation.For the same reason, we also use different loss functions. For GaSNet-L1 we adopt a “binary cross-entropy” loss, which is commonly used for a binary classifier. For GaSNet-L2 and GaSNet-L3, which are two regression models, instead of the commonly used MAE and MSE loss functions, we apply the“Huber” loss. This is defined as

where a=ytrue-ypred, ytrueis the real redshift, ypredis the predicted redshift by the CNNs,and δ is a parameter that can be preset(0.001 in this work).The choice of the“Huber”Loss has been made because, as shown in CNN regression models for galaxy light profiles (i.e., the GaLNets, Li et al. 2021a), it can achieve higher accuracy than MAE and MSE and better convergence. Both “activation” and “l(fā)oss functions” are summarized in Table 1.

From Figure 1 we see that the CNNs all accept a 1D spectrum (i.e., a vector of wavelength and fluxes) as input and produce as predicted parameters, either a probability (PLfor GaSNet-L1) or a redshift (i.e., zPEand zPGfor GaSNet-L2 and GaSNet-L3, respectively).

Figure 2. Flow-chart describing the process to obtain the HQ candidates combining the output of the three GaSNets. The final step is the visual inspection of the candidates selected using the probability criterion,PL ＞Pthresh, combined with the presence of background emission lines,zPE ＞zPG+0.1.

2.4. Decomposing a Complex CNN Model

To conclude this section, we briefly discuss the choice to combine the outcome of three CNNs to improve the accuracy of the identification of high-quality (HQ) candidates and minimize the chance of false detection.

This task involves two steps: (1) the identification of different kinds of features that can suggest the presence of a lensing event,i.e.,the coexistence of absorption and emissions lines from different objects along the line-of-sight, and (2) the verification that (some of) the emission lines come from the background system. This is a complex classification task that can be more efficiently performed by combining different CNNs with different specializations. Indeed, GaSNet-L1 is designed to identify a specific series of emission lines at a higher redshift overlying a lower redshift spectrum characterized either by a continuum plus absorption lines typical of ETGs or continuum plus emission lines from LTGs.Even if the training sample is made of real galaxy spectra, where the simulated emission lines from mock background sources are randomly redshifted with respect to the main galaxy (see Section 3), GaSNet-L1 can only give a probability of the coexistence of a lens and a source at different redshifts, but cannot predict by how much the emissions of the source are misplaced. Since this process can be uncertain, we cannot exclude that GaSNet-L1 can confuse a lensing event with other“l(fā)ocal” emission processes (e.g., AGNs, ongoing star formation,gas outflows,etc.),and vice versa.On the other hand,the GaSNet-L2 and GaSNet-L3 are able to predict the redshift of the tentative source and lens,independently,meaning that they cannot predict, individually, if there is another object at a different redshift, compatibly with a lensing event.

Only using the outputs of these three GaSNets together, we can both give a “high probability” that there are two different systems contributing to the spectrum and establish that the closer one is a galaxy, with redshift zPG, and the background one is a fainter line emitter, with redshift zPE. In particular, to qualify a spectrum as a candidate, we use the following conditions: (1) PL＞Pthresh, and (2) zPE＞zPG, where Pthreshis an appropriate lower probability threshold that will be chosen later to define the high-probability candidates that will be further visually investigated to assemble the list of candidates to pass to the visual inspection,which finally produces the HQ candidate list.The full process for the selection and grading of the HQ candidates is schematized in Figure 2.

3. Data

The construction of the training sample is a critical step of any supervised ML algorithm. Indeed, to avoid biased predictions and fictitious performances, the training samples need to be as close as possible to real observations.In our case,we build our training sample starting from real spectra from BOSS,over which we simulate the presence of emission lines.Here below, we first introduce the data set we use for our analysis. Then, we describe the way we have constructed the training set. This is constituted by two samples.

First, the negative sample, which represents a catalog of galaxy spectra with no background sources blended in. As mentioned earlier, we do not make any selection of galaxy types and we include ETGs and LTGs.

Second, the so-called positive sample, which represents a simulated sample of spectra that emulates the presence of emission lines from a background source. This is made of the same galaxy spectra of the negative sample, but with the addition of artificial emission lines, redshifted with respect to the “foreground” galaxies.

Table 2 Model Parameters of Equation (2)

3.1. Data Selection and Predictive Sample

The Sloan Digital Sky Survey (SDSS, see York et al. 2000)has observed over 10,000 deg2of the sky, performing multiband photometry and spectroscopy (Szalay et al. 1999). In 2009, before the start of the Baryon Oscillation Spectroscopic Survey (BOSS, Schlegel et al. 2009), in the third stage of the project(SDSS-III),the spectrograph operating the observations has been upgraded. Compared to SDSS-I/II, the number of fibers was increased from 640 to 1000, and the fiber diameter has been reduced from 3″to 2″(Ahn et al.2012).The extended version of the BOSS survey,eBOSS(Dawson et al.2016),has overall produced spectra for around 2.6 million galaxies,in the wavelength range 361–1014 nm. These are publicly available through the latest data release 16(DR16,Ahumada et al.2020).This is the data set that we use in this work,6For convenience we will address this as eBOSS or DR16.over which we operate a series of selections to ensure the quality of the spectra to analyze. In particular, we select only: (1) plates labeled as“good” quality, (2) “Object” flags labeled as “galaxy,” (3)spectroscopic redshift between 0.05 and 0.8, (4) spectra with SNR ＞2, (5) wavelength range 3700–9200 ?.

This latter criterion is applied to avoid a rather noisy region of the spectra, at λ ＞9200 ?, where the residuals from the telluric line subtraction might be a source of spurious detections. This is a problem that we expect to deal with in future developments, but we wanted to avoid in this first test.We stress, though, that the reduced wavelength range will allow us to train tools that can be straightforwardly applicable to the SDSS-I/II spectra, whose wavelength range is also limited to 3700–9200 ?. Of course, this makes our tools less sensitive to higher-z systems,as many of the emission lines we want to detect will fall out of the range at redshift z ?1.4 (see below).

Criterion 3 is dictated by the line observability. Indeed, we will assume that typical sources in the SGL events are starforming galaxies characterized by emission lines as reported in Table 2. Here, for each line, we list the central wavelength(s),the maximum redshift the emission line can reach below 9200 ?,zmax, and an intensity parameter, he, that will be used in Section 3.3 for the simulated spectra. According to this list,for redshift zE?1.4, all lines would fall out of the eBOSS wavelength range, while with zE?1.2, we can still retain two emission lines, i.e., [O II] and Hγ. In order to select lenses that are compatible with the visibility of the background lines and with a reasonable lens-source distance to guarantee an SGL event, we collect spectra in the range zG=(0.05 ～0.8).

Criterion 4, on the other hand, is an optimistic lower limit that we have chosen to increase the completeness. We have considered that the emission lines from background sources have an SNR which is not necessarily correlated to the SNR of the whole spectrum and, thus, can be seen also in a noisy galaxy spectrum.

The final selected sample consists of 1,339,895 spectra: in the following, we will refer to this as the DR16-predictive sample. In Figure 3 we show the SNR distribution of the selected spectra and the redshift of the central galaxy.

3.2. Construction of the Negative Sample

The first step to produce our training data set is the selection of the negative sample. This is chosen to make the CNN as general as possible, hence assuming that every type of galaxy can work as a lens,with no particular restrictions in luminosity or color,as it is typically done to contain the predictive samples in imaging classifiers (see e.g., Petrillo et al. 2019; Li et al.2020).

For this purpose, we select 140,000 galaxies spectra from the DR16-predictive sample, with a wavelength range of 3700–9200 ?. We take particular care that the selected spectra uniformly cover, in number, the full zGrange, by counting the spectra in redshift bins of 0.05.This is crucial to avoid any bias in the prediction of zGfrom the poor sampling of one redshift bin with respect to the close ones.

To mimic the presence of emission lines from local processes, for 1/5 of the negative sample, we add artificial emission lines with the same redshift of the galaxy, while the remaining 4/5 of the negative sample is left unchanged. In particular, for this simulated “l(fā)ocal emissions,” we use the same lines,reported in Table 2,that will be used to simulate the background source emissions,which we expect the GaSNets to distinguish from the local ones (if any).

3.3. Artificial Emission Line Model

In this section, we give more details about the artificial emission lines we want to add to the original eBOSS spectra to emulate both some local and higher-z emissions in the negative and positive samples, to be used for the CNN training.Following Li+19, we use a one-dimensional “double Gaussian” profile, defined as:

Figure 3. The redshift and SNR distribution of the DR16-predictive sample(see text for details).

where F is the flux,λe,1and λe,2are the central wavelengths of the emission lines, and h1, h2, σ1, σ2are the four model parameters. λe,1, λe,2, h1, and h2are listed in Table 2. These parameters are further defined to satisfy the following conditions:

where σ1is uniformly selected in the interval [0.8, 1.6], the amplitude parameter, he, is given in Table 2, and zEis the redshift of the emission line we want to simulate,assumed to be uniform in the range [zG+0.1, 1.2].7As we will discuss in Section 3.4,because zG is assumed to be also uniform for the positive sample,this condition produces a final zE distribution which is pseudo log-normal with a cut at 1.2. For the negative sample, discussed in Section 3.2, this condition produces a zE distribution that follows the distribution of zG as in Figure 3.

The σ1range is determined under the assumption that the emission lines from sources are enlarged by rotation. Hence,the line broadening in wavelength can be written as Δλ ≈2λ0vr/c, where λ0is the central wavelength of the emission line,vris the max velocity along the line-of-sight,and c is the speed of light. Then, we can approximate 2σ1≈Δλ,which, for a rotation vr～100 km s-1and λ0≈370 nm, gives σ1≈1.2 ?.Taking into account a larger wavelength range and rotation spectrum, we can reasonably make σ1vary over a further±0.4 ? range, i.e., the one we have assumed in Equation (3). Finally, we remark that the absolute amplitude of the emission lines, depending on he, is not of major importance, as the final SNR of the line strongly depends on the continuum of the spectrum the lines are added to. On the other hand, two other important features are (1) the relative distance of the line central wavelengths (λe,1and λe,2) and (2)their relative full width half maximum,connected to the σ1and σ2parameters.

Simulated lines are first randomly generated at zE=0 and then randomly redshifted to zE＞zG+0.1, where zGis the redshift of the negative spectrum from which the positive is generated (see Section 3.4).

The flux at the redshift zEis then defined according to the standard equation:

where the function F is the rest frame emission line flux function (Equation (2)).

The central wavelength of FZ, λcz, is defined as

The interval of λ is equal

where λc0is the central wavelength of the rest frame.According to the equations above, λc0shifted to (1+zE)λc0,and the interval of λ in the rest frame will broaden to(1+zE)dλ.

Figures 4 and 5 show how typical simulated emission lines from Table 2 are simulated according to the random parameters from Equation (3) and shifted to 0.5 and 1 redshifts.

Figure 4. Detail of the line profiles of the simulated emissions. Artificial Oxygen, O II (3727.1 ? and 3729.9 ?) lines (top) and Hα (6562.8 ?) lines(bottom), as observed at different redshifts. The line fluxes are redshifted and dimmed according to Equations (4)–(6).

Figure 5. Full spectrum of artificial emission lines added to the negative sample (see Table 2 and text for details). The line fluxes are redshifted and dimmed according to Equations (4)–(6).

Figure 6. The spectra after adding artificial emission lines. We add the mock emission lines in several different redshift positions(zE)in the real spectrum,in order to simulate the positive sample.

3.4. Simulating the Positive Sample

The next step is to build a positive sample by adding simulated emission lines to the negative sample.As anticipated,we use the same lines as in Table 2, this time with lines redshifted to zE＞zG+0.1, with the condition that zE?1.2.

In Figure 6 we show three simulated positive spectra for a single moderate SNR (～7) negative spectrum (SDSS-468-51912). Here, we have marked the location of the simulated emission lines at different redshifts,on top of the continuum of the real eBOSS galaxy spectrum.Looking at these spectra,one can visually figure out what are the major challenges to identify the“ground truth”emission lines in them.First,the SNR of the lines,as this depends not only on the h1and h2but also on the intrinsic spectrum noise. Second, the contamination from residual sky lines, e.g., at λ ＞8000 ?. Third, the effect of the source redshift, which can shift most of the relevant emission lines from Table 2 out of the spectral range (at λ ＞9200 ?,e.g., in Figure 6(c)). This latter issue could be in principle solved by including more emission lines in our reference catalog.We will consider this option for the next developments of GaSNets. However, we stress here that adding more lines,which in most of the cases have much lower SNRs in real galaxies, might introduce more uncertainties in the predictions of the GaSNet-L2, as they might be easier confused with random noise, especially in low-SNR spectra.

As a comparison with real lensing events, in Figure 7 we report some spectra of confirmed lenses from Bolton et al.(2008,see also Section 3.5).Here the locations of the emission lines of the background sources are marked, again, as red vertical lines.In particular,in this figure,we show spectra with different SNRs to visualize the impact of the spectra quality on the recognizability of the lines. In SDSS-393-51794 all lines are visible and show a pattern similar to the simulated lines in Figure 5. In SDSS-436-51883, despite the spectrum’s SNR being comparable to the one above, the lower signal of the background lines makes some of them embedded in the noise,although some others still stick out rather clearly. Here, the number of visible lines is reduced by the higher redshift of the source (zE=0.452). Finally, in SDSS-391-51782, the redshift of the source(zE=0.931)permits the observations of only two lines, which are yet rather easy to spot because of the decent SNR of the spectrum and the high signal of the lines. Overall,these examples show the kind of features the CNN needs to be trained on identifying in the spectra and the impact of the spectra quality and SNR of the background emission on the final line detection and redshift determination.

Similarly, these examples provide textbook cases of HQ candidates we will visually grade among the high probability candidates provided from the GaSNets (see Section 5.2).

3.5. Confirmed Lenses from Previous Spectroscopic Searches

As anticipated in the previous section, we also collect candidate/confirmed lenses from previous spectroscopic searches in SDSS/BOSS, using standard techniques, to be used as a real test sample for our deep learning tools. In particular, we have collected 131 objects from Bolton et al.(2008), 45 from Brownstein et al. (2012), and 118 from Shu et al. (2017), that have secure confirmation based on HST follow-up.This“test sample”made of real systems is useful for two main purposes: (1) to measure the completeness of our tool, by checking how many of these lenses are recovered by GaSNet-L1; (2) to test how accurate the GaSNet-L2 and GaSNet-L3 are in determining zEand zG,respectively.We will also compare our final catalog of HQ candidates versus the latest highly complete sample of spectroscopic selected candidates in eBOSS from T+21. This will allow us to check the presence of candidates missed by standard techniques, and compare the different approaches.

4. Implementation

To proceed with the construction of the training and test samples, we collect 140,000 positives and the same number of negatives. These samples are further split into the three data sets:100,000 for training,20,000 for validation,and 20,000 for testing.The first two samples are used to train the GaSNets and evaluate how well the model predicts the ground truth targets based on the unseen data during the training process. The last sample is used to qualify the final performance of the GaSNets.Finally, we also test the performance against real candidates from literature, as discussed in Section 3.5.

Figure 7.Spectra of confirmed gravitational lenses in Bolton et al.(2008),the red lines are the locations of the identified emission lines.

4.1. Training the Networks

According to the description in Section 2.3 and the tasks they are expected to fulfill,during the training,the GaSNets are fed with the training spectra to produce accurate predictions of the “target” quantities. For GaSNet-L1 the inputs are the spectra of the positives and negatives as well as their labels to give as output the probabilities (PL) to be lens candidates. For GaSNet-L2, the inputs are the simulated positive spectra with their labels, while the outputs are the predicted redshifts of the emission lines zE. For GaSNet-L3, the inputs are the labeled spectra of positives and negatives and the output are the redshifts of the foreground spectra(zG).The full process of the training sample building and labeling is summarized in Figure 8.

Regarding the training step, for GaSNet-L1 and GaSNet-L3 we use the 120,000 positive (training+validation data) and 120,000 negative samples, i.e., a total of 240,000 spectra.Because GaSNet-L2 only predicts zE, in this case the training+validation sample is made by 120,000 spectra from the positive sample only. For each GaSNet, we use the training data to train 30 epochs with a learning rate of 0.0001 and use validation data to evaluate the performance. We have found that this produces rather stable validation results. During the training process,we optimize the three GaSNets with the Adam optimizer (Friedman 1999).

Figure 8.Summary of the training data building process.After constructing the negative and positive samples, they are labeled before they are fed into the training process of the three GaSNets. In this scheme, we illustrate the steps made to add the label to the two training samples.

4.2. Testing on Simulation Data

After training,we first test the GaSNets’performances using the simulated “test” spectra. As anticipated, the test sample is made of 20,000 positive and 20,000 negative samples for GaSNet-L1 and GaSNet-L3 and 20,000 positive samples for GaSNet-L2.

In Figure 9 we first show the results of the training run for the three GaSNets to have a first evaluation of their performances. In particular, we plot the first 30 training epochs. The solid lines in Figure 9 represent the accuracy reached on the training data as the average deviation of the predictions from the ground truth (loss). The dotted–dashed lines represent the same quantity on the test data.

For each GaSNet, we set a different evaluation function: for GaSNet-L1,being a classifier giving a probability as output,we use the “accuracy” (acc) as loss variable; for GaSNet-L2 and GaSNet-L3 as they predict zEand zG,we set the mean absolute error (MAE) as loss variable. From Figure 9, GaSNet-L1 and GaSNet-L3 both show good convergence at about the same epoch toward the end of the training,while GaSNet-L2 shows a larger loss because of the degeneracy between noise and emission lines (see comment above). One possibility to improve this result might be the adoption of some spectra pre-processing,e.g.,via smoothing.However,this would imply an incursion on the data characterization that is beyond the purposes of this paper, and we rather plan to address this in next analyses.Here,we just stress that the accuracy reached by GaSNet-L2 is high enough to separate the background emission lines from the foreground spectral features in lens candidates, hence more than sufficient for its actual purposes(see also Section 6.2).

In Table 3, we report some statistical estimators to measure the GaSNets’ performances. Besides the standard MAE and MSE, we add other three estimators.

First, the R-squared (R2) is used to evaluate the linear relationship between prediction and true values.It is defined as

where zPis the predicted value and zTis the true value,and ˉzTis the average value of zT. The closer the R2is to 1, the better the prediction. In Table 3 we see that for the test sample,R2is close to 1 for both zPGand zPE, meaning that both GaSNet-L2 and GaSNet-L3 are expected to produce accurate results.

Second, the outlier fraction, which is defined as the fraction of predicted redshifts scattering more than 15% from the true values:

For the test sample, in Table 3 we show that the outlier fractions are ?1%,implying a very small fraction of anomalous predictions.

Third, the normalized median absolute deviation (NMAD),which is defined as:

It gives the absolute deviation of the predicted value from the central value of δZ.As seen in Table 3,the NMAD for the test sample is close to zero, meaning again a very small deviation from the true values, i.e., very accurate predictions.

In Figure 10 we also show the receiver operating characteristic (ROC) curve where we plot the true-positive rate(TPR)against the false-positive rate(FPR).The TPR is the fraction of lenses that are correctly classified with respect to the total number of “ground truth” lenses, while the FPR is the fraction of non-lenses that are misclassified as lenses with respect to the total number of non-lenses. The ROC curve can be used to decide the probability threshold to adopt as a tradeoff between true detection and contaminants from false positives. In the same figure, we report the TPR-FPR for different PLs.We can see that for a PL=0.95,we almost reach 90% completeness with a negligible false positive rate. We stress here that this result derived from simulated spectra is in rather ideal conditions. Hence, both the TPR and, most of all,the FPR might be just upper and lower limits, respectively, as compared to the real cases. However, the PL=0.95 occurs before the slope of the ROC becomes flatter, meaning that the gain in the number of true detections, at lower thresholds,increases at the cost of a larger number of contaminants. We will come back to these results later when we will discuss the threshold to adopt to select HQ candidates in real data.

Figure 9. GasNets training results. Left: Accuracy and the loss of GasNets-L1, the training and evaluation curves both converge to the same point, and show high accuracy.Middle:MAE and loss of GasNets-L2,which show a worse but yet acceptable convergence,with a reasonably low MAE.Right:MAE and loss of GasNets-L2, which well converge to the same values, because zG is easier to predict than zE.

Table 3 Statistical Properties of the Predicted Parameters

Finally, in Figure 11 we detail the results obtained for the test sample.In the left panel,we show the distribution of the PLfrom GaSNet-L1 for both the negative and the positive samples. As expected the former tends to cluster more toward a peak at PL=1,but with a rather long tail toward the PL=0,meaning that, statistically, there is a significant fraction of true positives to which GaSNet-L1 has given a low probability.We have checked the latter cases and found no correlation with the overall SNR of the spectra. Instead, we have found a correlation of the low PLobjects with the zE, in the sense that the larger the zE, the bigger the number of the object with PL＜0.5.This suggests that either the lower number of lines or the intrinsically lower SNR of the lines suppress the PLand makes the classification of the lensing event more difficult at higher-z.

In the central panel of the same figure,we show the output of the GaSNet-L2 by comparing the predicted zPE, against the ground truth values, zE. Overall, the majority of the predicted values are tightly distributed around the one-to-one relation,as also quantified by the large R2values found in Table 3(Test/zPE).Numerous predictions scatter quite largely from the perfect correlation, because of the degeneracy of noise and background emission lines, as mentioned above. However,these are statistically irrelevant as the estimated outlier fraction in Table 3 is close to 1%.

Figure 10.ROC curve of training samples where the true-positive rate(TPR)is plotted against the false-positive rate (FPR) (see text for details).

Finally, in the right panel of Figure 11, we show the predicted zPGfrom GaSNet-L3 against the ground truth values,zG. In this case, the correlation is quite perfect and the outlier fraction is negligible(＜0.1%see Table 3—Test/zPG),both for the positive and the negative sample.Indeed,for this latter test,we have also input the negative sample to check the performance of GaSNet-L3, as a pure automatic spectroscopic redshift tool,in absence of artificial emission lines.This shows that the ability of GaSNet-L3 to predict the galaxy redshift is not driven by the emission lines, easier to spot, but by the overall features of the spectrum (i.e., continuum and absorption/emission lines).

Figure 11.GaSNets results on the test sample.Left:The PL distribution from GaSNet-L1.Middle:Predicted emission line redshifts from GaSNet-L2,zPE,vs.ground truth values, zE. Right: Predicted lens galaxy redshifts from GaSNet-L3, zPG, vs. ground truth values, zG.

4.3. Test on HST Confirmed Samples

Previous analyses of the SDSS/BOSS spectra have brought to the collection of 294 strong lenses candidates: 131 from SLACS,45 from BELLS,and 118 from S4TM(see Section 1).Being candidates based on spectroscopic features, these samples contain both real lenses and contaminants. Indeed,space imaging follow-ups have confirmed 70/131 SLACS candidates (the Grade-A objects in Table 3 of Bolton et al.2008), 25/45 BELLS candidates (Grade-A objects in Table 3 of Brownstein et al. 2012), and 40/118 S4TM (Grade-A in Table 1 of Shu et al. 2017). As also commented in Section 1 these correspond to an average confirmation rate of 46%.Note,though, that the HST samples often tend to optimize the confirmation rate by pre-selecting targets with low-resolution imaging (see e.g., Bolton et al. 2004; Shu et al. 2016a), hence this can be considered an optimistic upper limit estimate.These are the main statistical samples that have been systematically followed up to collect space imaging confirmations of spectroscopically selected SGL candidates, using optical lines.As such, these represent the most secure sample to check our results against. These data can be used for two main purposes:(1)to compare the classification of the GaSNets against human selection and help us set a reasonable threshold to optimize the chance of finding real lenses with the minimal contamination from false positives; (2) to forecast the success rate we might expect from our set-up because we have a reference sample of“candidates” and “confirmed” events. Being this literature sample far from complete (see Section 2.1), it cannot be fully used to draw firm conclusions about the completeness of the GaSNets, however, this is the only sample we can use to benchmark the GaSNets’performances,with a necessary grain of salt.On the other hand,the large sample from T+21,having no space observations cannot be used for the same purpose as the ones above.As anticipated,we will use it for an a posteriori test to assess the differences(if any)between standard and deep learning approaches.

To proceed with the test of the HST confirmed catalogs against GaSNets, we first select the literature spectra that are located in the predictive range of our CNNs (i.e.,0.05 ＜zG＜0.8 and 0.15 ＜zE＜1.2). These are 264/294 candidates and 121/135 confirmed objects. In Figure 12 we show the probability predicted from the GaSNet-L1 (left panel), the redshift of the source predicted from GaSNet-L2(central panel), and the redshift of the lens galaxies predicted by the GaSNet-L3 (right panel), for the candidates and confirmed literature objects face-to-face.

In particular, we see that GaSNet-L1 predicts high probabilities for most of the lenses:e.g.,69%of the candidates and 80% of the confirmed objects have PL＞0.95, which becomes 81% of the candidates and 90% of the confirmed objects for PL＞0.8.

More importantly, the ratio of the confirmed/candidates increases dramatically from 0.8 ＜PL＜0.95 to PL＞0.95, as we have 12/33,i.e.,36%for the former and 97/182,i.e.,53%,for the latter, versus the overall 46% estimated for the full sample (see above). On the other hand, for PL＜0.8 the confirmation rate drops to 12/49, i.e., 24%, which is too low for successful space observations and still anti-economical for lens search in spectra. Indeed, as discussed in Section 4.2, at PL＜0.8 the FPR becomes prohibitive,producing massive false detections in large samples that should be cleaned with tedious visual inspections. Interestingly enough, for PL＞0.95 the fraction of true SGL events recovered (80%) is rather close to the TPR (～89%) predicted by the ROC curve (Figure 10) for an idealized mock population of strong lenses.This means that the performances of the GaSNets on the real data might be not far from the expectations from simulated data.

However, in Figure 12 (left) a misalignment between the deep learning and human filtered selections is further demonstrated by the fact that some confirmed lenses have received a small probability by GaSNet-L1.As discussed in the previous section, these are mainly low-SNR emission line spectra or higher-z systems that, even if accounted for in the training sample,are difficult to be highly scored by GaSNet-L1 but might have been picked by the human eye with higher confidence. Hence, we conclude that a PL=0.95 threshold is very likely to produce effective completeness higher than the 80% obtained above over a complete and unbiased true SGL sample.

Figure 12.Results of the GaSNets applied to spectra of strong gravitational lens candidates(in blue)and HST-confirmed events(in red)from SLACS,BELLS,and S4TM samples(see text for details).Left:The PL distribution from GaSNet-L1.Bottom:Predicted emission line redshifts from GaSNet-L2,zPE,vs.literature redshifts,zE,for the candidate objects(top)and the HST confirmed(bottom).Right:Predicted lens redshifts from GaSNet-L3,zPG,vs.literature redshifts,zG,for the candidate objects (top) and the HST confirmed (bottom).

The middle and right panels of Figure 12 show that both GaSNet-L2 and GaSNet-L3 can make good predictions on the redshift of the emission lines and the lens galaxies. In general,GaSNet-L3 performs better than GaSNet-L2 (see Table 2),possibly because the spectra of the lens galaxies can provide more information,both from the continuum and the absorption or emission lines, while GaSNet-L2 relies only on a few emission lines, which provide intrinsically less information.We also see that the confirmed objects generally show a smaller scatter and outlier fraction than the candidates, especially in zPE, and also that the highest probability objects show tighter one-to-one predictions. This demonstrates that misclassifications of SGL events might be related to uncertainties on the redshift of the background sources, which tend to be placed further away than sometimes they are, i.e., confusing “l(fā)ocal”emissions with background ones.However,the chance of such misclassification is reduced for PL＞0.95 systems.

In summary,Figure 12 indicates that the PL＞0.95 sample is accurate enough to produce reliable lens candidates from the DR16-predictive sample.

Figure 13.Predictions of the three GaSNets on the DR16-predictive sample(～1.3 M objects):the PL from GaSNet-L1(left),zPE-zPG(center)and zPG(right),from GaSNet-L2 and GaSNet-L3 outputs, vs. zG, the galaxy redshift from eBOSS/DR16 catalog (see text for the details). In the bottom and right panels we show the PL ＞0.95 sample (in red) and the PL ＜0.95 in blue.

5. Results

In this section, we apply the trained GaSNets to the DR16-predictive sample, introduced in Section 3.1. This is made of 1,339,895 galaxy spectra and represents the sample among which we want to find new strong lens candidates and, for them, determine the redshift of the background source, zE.

5.1. Predictions on the eBOSS Spectra

According to the workflow described in Figure 2, the first step to perform is the classification of candidates using GaSNet-L1. In Figure 13 (left) we report the probability PLdistribution obtained from GaSNet-L1 for the DR16-predictive sample. From this histogram, we see that using a PL＞0.8,which, according to the ROC curve, would return almost 95%of the true lenses, would produce a list of about 10,000 candidates. This is a sample hard to handle for two main reasons:(1)it is time-consuming to visually inspect and(2)it is foreseen to be severely contaminated from false detections.The latter case has been confirmed by randomly inspecting 100 candidates with 0.8 ＜PL＜0.95 to find that about 90%are very poor candidates. On the other hand, choosing PL＞0.95,which,for the true lens cases,allowed to recover of 80%of the confirmed lens known in SDSS/BOSS,would produce a more manageable sample of ～4000 candidates.Hence,at the cost of some acceptable incompleteness,for this first test,we decide to adopt a more conservative approach and search for high-quality candidates among the ones with PL＞0.95. We can now look into the predictions of the GaSNet-L2 and GaSNet-L3 to finalize the sample to visually inspect.In Figure 13(center)we report the redshift gap between the lens and the source,ΔZ=zPE-zPGas a function of the lens redshift zGfor the full predictive sample. Here we highlight the objects with PL＞0.95, from all the other spectra in the predictive sample.We can distinguish a few features: (1)the upper limit imposed on the zEproduces a zone of avoidance on the up-right side of the image; (2) there is a crowded sequence of high PLin the box defined by zG=[0.5,0.6] and Δz=[0.4,0.6]. This is due to the presence of rather redundant residual emission lines from sky subtraction in the SDSS pipeline at λ ～5600 ? (see Figure 14)that is very often ignored by GaSNet-L2 but that in many cases is confused as a real emission.As we will see later,this sequence is easily filtered out by the visual inspection,but it has to be better accounted for in the training sample to reduce its impact in future analyses.

Figure 14.HQ candidate spectra.The SDSS/BOSS spectra are plotted with highlighted their main spectral features.Red vertical lines indicate the emission lines of the background source at the redshift eyezE(i.e.,the one corrected during the visual inspection).Blue vertical lines indicate the spectral features of the lens at redshift zG.Green lines show the location at rest frame(z=0)of the emission lines from sky).On the top of each spectrum from left to right,we report the probability from GaSNet-L1,the average SNR of the spectrum,the cataloged redshift of the galaxy from SDSS/BOSS,the predicted redshift from GaSNet-L3,the corrected redshift from visual inspection, the predicted redshift from GaSNet-L2, and finally, the R.A., decl., and ID of the target.

A similar effect is produced by the residual sky lines at λ ＞8000 ?, which also produce a sequence of spurious zEpredictions (see zG～0.2 and ΔZ ～0.9). These have a small PL, according to GaSNet-L1, and thus they do not bother, as they are excluded by the following analysis.

Overall,the PL＞0.95 sample looks rather unbiased,as seen by the zGestimates from GaSNet-L3 in the right panel of Figure 13, where the predicted zPGis extremely tightly correlated to the eBOSS catalog values (see also the statistical estimators in Table 3).

However,before proceeding with the visual inspection of the background emissions estimated by the GaSNet-L2, to minimize the heterogeneity in the human grading, we preselect the spectra that show an average SNR, computed at the expected positions of the reference lines from Table 2,〈SNRlines〉, to be larger than one. This further selection gives us 931 potential candidates pass to the visual inspection.

5.2. Visual Inspection of Spectra

The 931 candidates are visually inspected by the three authors, according to an ABCD ranking scheme, being A=“sure positive,” B=“maybe positive,” C=“maybe not a positive” and D=“sure negative.” To combine the human grading with the PL, we have turned the ranking above into a score according to the conversion A=10,B=7,C=3,D=0(see also Li+21). We finally select the spectra for which we have obtained an average score ≥7, as the final high-quality candidate sample. This is made of 497 objects in total.

Figure 15. zPE-zPG vs. zPG distribution of visual inspection 497 good potential candidates and all PL ＞0.95 spectra.

Some spectra of this “high quality” sample are plotted in Figure 14. Here we clearly see the emission lines, marked as red vertical lines, from background lensed star-forming galaxies.

During the visual inspection process, besides grading, we also check that the predicted values, zPEand zPG, given by GaSNets, are perfectly aligned with visible spectral features.This is not often the case as the prediction process has some intrinsic uncertainty. For instance, the two GaSNets need to interpolate across a grid of training spectra that have been shifted with a coarse sampling (i.e., 0.05 in redshift, see Section 3.2). However, other sources of errors are possibly causing even more significant shifts,as we will discuss in more detail in Section 6.2. Using an interactive GUI developed by one of us (ZF), we then determine by eye the needed shift to obtain a perfect visual alignment and a“corrected”redshift for the zPE, assuming the zGfrom the eBOSS catalog as an unbiased estimate of the main galaxy redshift.

Finally, to qualify a spectrum as a lensed galaxies candidate we check that (1) the emission lines do not belong to the sky lines (green lines in Figure 14) and (2) that the identified emission lines, i.e., red lines in Figure 14, having redshift zPEfrom GaSNet-L2,do not correspond to any line from the galaxy(i.e., blue lines in Figure 14 at redshift zPGfrom GaSNet-L3).In other words,the ΔZ=zPE-zPGhas to be larger than 0.1,as shown in Figure 15, where it is plotted as a function of the estimated zPG.Here we also see that Δz is decreasing with zPGbecause the further the lenses, the smaller the difference in redshift with the background source.From Figure 15 it is clear that this is mainly a selection effect due to our condition on zPE＜1.2,however,because the high-quality candidates do not cluster toward the upper bound of the zone of avoidance, we conclude that the candidate distribution becomes incomplete when zPE～1.2. This is consistent with the correlation of the low PLwith the higher-zPEwe have discussed in Section 4.2.An encouraging feature, in the same figure, is that the combination of〈SNRlines〉＞1 and the visual inspection,allows us to drop the stripe of spurious detection from residual sky lines discussed in Section 5.1.

5.3. Deep Learning versus Traditional Methods

We end this section by comparing our HQ catalog,based on deep learning, with the catalog of 1551 candidates selected with the rest-frame optical bands from T+21, using traditional selection methods. They used the complete eBOSS/DR16 database and applied the standard spectroscopic detection method introduced in the eBOSS Emission-Line Lens Survey(BELLS) and added Gaussian fit information, grading,additional inspection observables, and additional inspection methods to improve the BELLS selection method.They used a total of 2 million objects with no selection on the redshift of the lenses. Furthermore, they used a larger database of reference lines, including also [N II]a/b and [S II]a/b: these are best suited for low-redshift detections being all placed at λ ＞6500 ?, leaving the only [O II] doublet as a feature for the identification of background sources at z ?1.2. As such,their predictive sample is wider in the parameter space than the DR16-predictive we have adopted. For a proper comparison,we have selected the T+21 candidates that fall in the GaSNets predictive space (i.e., zG=0.05-0.8, spectra SNR ＞2,zE?1.2, zE=zG+0.1) and finally obtain 778 “compatible”candidates (～50% of the original sample). We have checked the excluded 773 and found that 739 detections are, indeed,based on a single line (generally in spectra with SNR ＞2)and 29/5 are based on 2/3 lines (all with spectra SNR ＜2),according to the T+21 catalog. Hence, the majority of these“known candidates” would have been missed anyways in our HQ catalog because of the conservative selection in the number of lines to use for the classification, either in the deep learning training or visual ranking.

We have,then,matched the compatible 778 candidates with our HQ sample of 497 entries and,surprisingly,we have found a match for only 68 objects.

The positive note is that GaSNets have found ～430 new HQ candidates that have been missed by standard techniques. The negative note is that the GaSNets seem to have missed 710 candidates from T+21.

Figure 16. Sample of missing candidates in the HQ catalog from GaSNets+visual inspection. In this figure, we show the distribution of the missing candidates in the parameter space adopted for the training of the GaSNets(i.e.,spectra SNR ＞2 and zE ＜1.2). Each candidate is color-coded by the number of detected lines in their spectra (according to T+21). Most of the missed candidates are 1-line and did not qualify in our HQ sample.

Is this true? To answer this question we need to first check how many of these objects are lost by the GaSNets according to the criteria imposed on their outputs,i.e.,they do not fall in the criteria PL＞0.95 and zE-zG＞0.1. These are 327, i.e., 42%of the compatible sample.This is larger than the fraction of lost objects found in the test against the real systems in Section 4.3(i.e., 100-69=31% of “candidates” and 100-80=20%confirmed ones, having PL＜0.95). One explanation of this excess of lost objects with low PLcan be that these are mainly optimistic candidates in T+21, for which the GaSNets have given low reliability. To confirm this we have checked that 215/327 are single line detections, according to T+21, and only 87/327 have scored A+ or A in their check against lowresolution imaging.8As we will comment later, the image quality of the low-resolution DES imaging used by T+21 does not consent a firm classification, except for very clear features. Hence, we have conservatively assumed the A+ and A scores sufficient to preliminary quantify the confirmation rate.Hence, we can fairly conclude that this sample of lost candidates is overall low-valuable,having a tiny(albeit insecure) confirmation rate. This also implies that the fraction of lost SGL“real”events in our catalog is in line with the one estimated in Section 4.3, reported above (i.e., 20%).

Going to the remaining lost candidates (710-327=383),in Figure 16 we show the spectra (not line) SNR versus the estimated redshift of the background lines from T+21, colorcoded by the number of detected lines. From this figure, we observe that:

(1) The majority (286/383, i.e., 75%) of the missing candidates have 1-line detection, thus they are lost from our HQ catalog because we excluded them in our filtering (both because of〈SNRlines〉or the visual inspection,see Sections 5.1 and 5.2).According to the T+21 low-resolution grading,164/286 of the 1-line detections have A or A+ scores, which implies a rather large confirmation rate,～60%,if confirmed by higher-quality imaging. This is a sample that we can easily intercept with GaSNets, by simply releasing the conservative criterion of the 1-line.From Figure 16,we see that above z 1.05 we lose some 2-line candidates, which supports further the conclusion in Section 5.2 that we are incomplete at zE?1.2.

(2) The remaining 97 multi-line objects, in Figure 16,majorly concern us, as according to their PLand number of lines should have been picked by the GaSNets + visual inspection. First, we have found 10/97 objects classified as quasar or unknown in DR16, so these could not be in our catalog. For all the other 87 we have visually inspected the spectra and found that despite they being classified as multilines in T+21, no line, except the [O II] doublet, had an acceptable SNR.Hence,these are all candidates that have been substantially treated as 1-line from us or given a rather poor visual grade. We give some examples of these spectra in Figure 17.Since 60/97 have received A or A+scores from the low-resolution confirmation in T+21,i.e.,60%,this is a sample that is likely to be valuable and should not be missed.However,we need to point out that this sample was not lost by the GaSNets but by human selection.

5.4.First Catalog of New HQ Strong Lensing Candidates in eBOSS from Deep Learning

After having subtracted the 68 candidates already found in T+21, we obtain a final catalog of 429 new HQ candidates in eBOSS, the first fully derived using deep learning. The full catalog is reported in Appendix A. This includes information about (1) R.A./decl. coordinates; (2) plate ID; (3) MJD(Modified Julian Day),the observation date;(4)the GaSNet-L1 probability, PL; (5) the redshift of the galaxy from the eBOSS catalog; (6) the predicted redshift of the galaxy from GaSNet-L3; (7) the predicted redshift of the background source from GaSNet-L2; (8) the corrected redshift of the source from the visual inspection (see Section 6.2); the total probability,PT=PL×0.1 visual scores, i.e., combining the GaSNets and human probabilities to be a lens.

6. Discussion

In the previous section, we have presented the final list of 429 new strong galaxy lensing candidates, obtained by applying the three GaSNets to the latest eBOSS database(DR16), and further cleaning the sample via visual inspection.

Figure 17.Sample of missing candidates in the HQ catalog from GaSNets+visual inspection and found in T+21.The red vertical lines are the features identified as multi-lines in T+21, but those have been excluded by us because either too faint or embedded in noisy regions, making them poorly reliable to qualify as HQ candidates.

Strictly speaking, the GaSNets’ candidates consist of systems where, in the spectrum of a foreground galaxy, we have found emission lines that are incompatible with belonging to the same galaxy.We have assumed,so far,that all these lines come from lensing events. In reality, they can be emitted by other kinds of sources,like overlapping galaxies along the line of sight, outflows in late-type galaxies, interacting systems,etc., although we have set a redshift gap, Δz, that might have prevented the confusion with some“l(fā)ocal”phenomena.Hence,to fully assess the new catalog, we need to estimate a fiducial confirmation rate based on space observations or high-quality ground-based imaging. Such a confirmation rate is important(1) to compare with the one from standard techniques, to see whether Deep Learning can outperform them in terms of reliability of the candidates; (2) to check whether the large spectroscopically selected samples accumulated so far, are compatible with expected numbers of SGL events from theoretical predictions (see e.g., 2.1), or we might expect to find more events with more refined tools.

Besides the confirmation rate, in this section, we also discuss the possibility to use the GaSNet-L2 and GaSNet-L3 as automatic tools for redshift estimates and spectra classification. We will conclude this discussion with some perspective on the next improvements of the GaSNets.

6.1. Confirmation Rate Via Ground Based Imaging

To properly derive a fiducial confirmation rate for the 429 HQ candidates in Section 5.4, we have checked the HST archive observations to look for serendipitous matches with our newly discovered candidates but found no matches.Hence,the only remaining check we can perform is inside archive observations from the ground. There are three data sets potentially useful for the test: (1) DECaLS;9https://portal.nersc.gov/cfs/cosmo/data/legacysurvey/dr7/(2) KiDS10https://kids.strw.leidenuniv.nl/DR4/access.phpand(3)HSC.11https://hsc-release.mtk.nao.ac.jp/das_cutout/pdr3/We have found 279 matches with DECaLS,16 with KiDS, and 63 with HSC, however: (1) the quality of the DECaLS grz color images from the public data is rather poorer than other surveys and made the identification of the lensing features extremely uncertain(see Appendix B);(2)the number of KiDS matches is too small to have a fair statistics and we decided to leave the few convincing candidates for future analyses; (3) the HSC sample is the one with sufficient large statistics,image quality,and uniformity to make a fair estimate of the fraction of convincing lenses without strong biases.

Looking at this latter sample, we find that seven candidates have corrupted color images or are too close to some bright source to be used with sufficient confidence.Hence,we finally inspect 56 systems. Of these, our HQ candidates match eight known lens candidates from HSC12http://www-utap.phys.s.u-tokyo.ac.jp/～oguri/sugohi/(e.g., Sonnenfeld et al.2018, 2019), although they are all C-graded by the imaging only in their catalogs. We have visually inspected them again and, applying the ABCD scheme as in Section 5.2 and taking into account the spectroscopic evidence, we have reclassified three of them with A-grade and five with B-grade.

Of the remaining 49 matches, we have classified seven candidates as A-grade and 17 as B-grade systems. Taking the A-grade as bona fide confirmed lenses and weighting the B-grade ones by a 0.5 factor to account that they may be not lenses, we conclude that the lens confirmation rate is 21/56 or 38%, which is lower than the confirmation rate estimated in Section 4.3 using space imaging.

In Figure 18 we show a gallery of the“confirmed”lens and,as a comparison,the“unconfirmed”ones(i.e.,the ones C-and D-graded). In the first row, we report some of the lenses previously found in the HSC imaging and confirmed and regraded by us, in the second and third rows some examples of new GaSNets’ confirmed lenses with A-grade, and B-grade,respectively. In the final two rows the unconfirmed C and D cases.These clearly show the variety of potential contaminants,including arc-like features of unclear nature, blue/faint background galaxies similar to other objects in the field-of-view,interacting systems,and large late-type or lenticular galaxies.In these latter examples, especially the large galaxies, if we exclude the cases where it is likely that the background emissions found in the spectra come from unlensed faint background systems as they can be seen in field-of-view, it is difficult to identify any other potential high-z emitters. This leaves the nature of these emissions unresolved.In principle we cannot exclude that, given the small area covered by the fibers in eBOSS (2″, see also Figure 18) there is some very low separation arc,embedded in the bright foreground galaxy light,remaining undetected in the seeing-confused images from HSC. In this case, we can argue that the confirmation rates estimated above (38%) might represent a lower limit.

If this conclusion is correct, we can attempt to derive a prediction of the total number of true SGL events in eBOSS,based on the current candidates from T+21 and this work.Put together they are 1551+429=1980. Assuming a pessimistic confirmation rate of 38%, they make 752 real SGL events,while for a more optimistic 46% conformation rate of SLACS+BELLS+S4TM, it makes 911 real SGL. If we add the other candidates found in BOSS from BELLS (25) and BELLS GALLERY (1713Note that more can be still found on their sample of remaining 155 candidates remaining unconfirmed. Assuming ～50% confirmation rate they can be ～70.) we reach 794 and 953 real SGL, which nicely bracket the expected number we have estimated in Section 2.1 for BOSS (～920). This suggests that we have possibly reached the full completeness of the lens population accessible by the largest spectroscopic database currently available.

6.2. Statistical Errors of GaSNet-L2 and GaSNet-L3

GaSNet-L2 and GaSNet-L3 are two CNNs that can perform the generic task to estimate the redshift of given features in 1D spectra. As such, they can be applied to spectroscopic databases regardless of the specific task of looking for strong gravitational lenses.

Certainly, the search for lenses requires a much lower accuracy in zPGand zPE,because the only condition to ring the bell for potential events is Δz=zPE-zPG＞0.1, which is rather higher than typical spectroscopic redshift errors based on the human measurements. However, this condition is physically meaningful if Δz is larger than the combination of the typical errors on zPEfrom GaSNet-L2 and zPGfrom GaSNet-L3, which also include the uncertainties that a deep learning process might introduce (activation, loss, training, etc.).

Hence, if on one hand, the assessment of the “bias” and typical “statistical errors” of the two GaSNets (L2 and L3) is needed to validate the pre-condition for the HQ candidates, on the other hand, they can also quantify the accuracy of the individual CNN as “automatic tools” for redshift measurements. In this latter case, we can possibly require the typical errors to be of the order of ＜1%, and systematics smaller than this precision.At the same time,we should expect a negligible fraction of outliers/catastrophic events.

Figure 18. Some examples of ground-based color cutouts (20″×20″) of GaSNet candidates. Top row: Match with HSC known candidates, re-graded as in the bottom-left corner.GaSNets have found them as HQ candidates independently.Bottom four rows:A,B,C and D ranked HSC counterparts of GaSNet candidates from the HQ sample in Section 5.4.

Figure 19.EyezE vs.zPE and eyezE,zPE distribution of visual inspected 429 HQ candidates.

In both cases, the scatter and accuracy are reasonably good,and so it is the outlier fraction. This result confirms that the adoption of the Δz ＞0.1 is conservative enough to account for the nominal statistical errors of the predicted redshifts.Furthermore, if we consider that the SNR is generally poor for the majority of the emission lines of the background galaxies, then we believe that both GaSNets (L2 and L3) are a very promising start and can be possibly be already used to automatically provide a first accurate guess of the redshift of galaxies in large surveys,while a more dedicate training would possibly improve the overall accuracy.We will dedicate future analyses to the GaSNets on the latter and more applications,including specialized tasks for spectra classification (e.g.,starburst galaxies, AGNs, irregular systems, etc.).

6.3. Improvements of CNN Model

In this work we have used three independent CNN models and combined their outputs according to some physically meaningful conditions (see Figure 2), to identify strong lens candidates. In fact, because of the physics of the SGL, which involves the position of the source and lens with respect to the observer,the properties of the projected potential,etc.,the three outputs of the GaSNets are not fully independent.Rather, they must be connected via the ray-tracing equation of the SGL.For instance, one can define a more meaningful probability for spectra to have caught a lens candidate, by looking at the relative distance of the zPGand zPE, or at the absolute value of zPG(e.g.,giving a lower PLif the galaxy is a very low redshift),etc. One possible future development is to connect different individual CNN networks (just like the neurons in our brain),for example, as in Figure 20, to make a more educated probability for a spectrum to be an SGL system.

In this figure, we suggest using the prediction of zPGas conditional information for the prediction of PL,then using the prediction of zPGand PLas auxiliary information for the prediction of zE. If, on one hand, this architecture can help to improve the accuracy,the cost to pay is the model complexity,including a larger correlation among the different branches with some large back-propagation. This would make the overall model more time-consuming in terms of training and prediction, but likely more accurate and false detection free.

Figure 20. A possible scheme for increasing the interplay between the three GaSNets. Here, we foresee input GaSNet-L1 with the outputs of GaSNet-L2 and -L3, as shown in the yellow line show, to improve the PL.

7. Conclusions

In this paper,we have presented a novel deep learning tool to search for SGL events in 1D galaxy spectra. This is the first attempt to use multiple emission lines after Li+19 used Lyα only.

The new algorithm is made of different CNNs, dubbed Galaxy Spectra convolutional neural Networks (GaSNets).These are optimized to work together to provide SGL candidates, but can also perform classification and regression tasks independently. As such, they are extremely suitable for further applications in large databases of tens to hundreds of millions of spectra, like the ones expected from the next generation spectroscopic surveys (4MOST, DESI,EUCLID, CSST).

In this paper,we have started by applying these new tools to the strong lensing search in the eBOSS/DR16 database(Ahumada et al. 2020). To this aim we have introduced: (1)GaSNet-L1 giving to each eBOSS spectrum the probability to be an SGL event(PL);(2)GaSNet-L2 estimating the redshift of background sources(zPE)from a series of pre-selected emission lines (see Table 2); and (3) GaSNet-L3 estimating the redshift of the galaxy itself (zPG), using the information it learns from the continuous spectrum, including local absorption/emission features. Only working together, the three GaSNets efficiently pinpoint SGL candidates combining a high PLwith the condition that zPE＞zPG, as expected for typical strong lensing configurations.

In particular, by testing the GaSNets on a list of known spectroscopically selected gravitational lenses in SDSS/BOSS(from Bolton et al.2008;Brownstein et al.2012,and Shu et al.2017) we have found that using a PL＞0.95 we can recover about 80% of the strong lenses confirmed by HST. This very conservative probability threshold provided a reasonable tradeoff between significant completeness and a reasonably small sample to visually inspect, with low contamination from falsepositive detection.

Using this set-up,with the condition that zPE＞zPG+0.1,we have applied the GaSNets to ～1.3 million spectra from the SDSS-DR16, after having imposed some appropriate cuts to guarantee a good spectrum quality and the visibility of at least two emission lines from the putative sources (namely, [O II]and Hγ), assumed to be star-forming galaxies.

We have collected ～930 candidates that have been further cleaned by misclassified SGL events,via visual inspection.The final sample of visual HQ candidates is made of 497 spectroscopic selected objects. This catalog has been a posteriori compared to the most extended catalog of spectroscopic selected lens candidates from T+21 and found an overlap of only 68 candidates, meaning that 429 of our candidates are newly found. On the other hand, we have demonstrated that GaSNets did not recover the remaining T+21 sample because of the conservative constraints we have adopted for the number of lines to be detected(＞2).Releasing them, half of the sample from T+21 (i.e., the one for which GaSNets has PL＞0.95) remains under the GaSNets discovery reach.

For the new HQ catalog, we provide R.A., decl., the probability, PL, the redshift of the galaxy from the eBOSS catalog, the predicted redshift of the galaxy from GaSNet-L3,the predicted redshift of the background source from GaSNet-L2, the corrected redshift of the source from the visual inspection, in Appendix A.

To estimate a tentative confirmation rate of these candidates,we have matched the coordinates with archive HST observations and found no matches. Instead, we have found optical counterparts in DECaLS, KiDS, and HST observations, but only HSC has provided sufficient statistics and image quality to confidently confirm the first sample of GaSNets’ candidates.Among these, we have independently confirmed eight SGL candidates from previous HSC lens imaging searches, thus providing spectroscopic evidence of lensing events, even though for only three of them we have found convincing features in the imaging to be “sure lens.” Besides these“known” lenses, we have found a preliminary optical confirmation of a further 24 GaSNet HQ candidates, although,also in this case,for 17 of them the HSC images allowed only a“maybe lens” B-grade, and only seven have a “sure lens”A-grade. Taking the A-grade as bona fide lenses and giving a 0.5 weight to the B-grade candidates, we have estimated a confirmation rate of 38% for our HQ catalog.

Some examples of the HSC matched are shown in Figure 18,where we also show low-graded imaging of GaSNet candidates. The possible contaminants are higher redshift galaxies,overlapping in the fiber spectra, or maybe local phenomena mimicking an SGL event. For example, local gas outflows,with typical velocities of 3×103km s-1, will introduce asymmetric velocity distribution along the ejection direction(Veilleux et al. 2020), which would shift the wavelength of some characteristic emission lines.Among these the[O III]line could deviate from the zGby ～50 ? and produce a false positive.

In this paper, we have demonstrated that Deep Learning represents a very efficient method to search for strong lenses in galaxy spectra. This can be applied to next generation spectroscopic surveys in a fast and automated way. This first application to the eBOSS database has confirmed that the spectroscopic selection of SGL candidates is complementary to the imaging-based SGL searches. For instance, of the 32 A/B grade candidates from the GaSNets matching with HSC imaging, only eight were found previously on HSC images.This over-performance of the spectroscopic searches with respect to imaging is particularly evident for ground-based observations, where the typical seeing has no impact on emission lines of background sources in spectra but makes it hard to resolve low-separation gravitational arcs of the same sources.

For this first application,we have made conservative choices regarding (1) the number of features to use for the training of the GaSNets;(2)the overall Network architecture,e.g.,limiting the interconnections between the three GaSNets (3) the probability threshold to optimize the sample to visual inspect and keep the false positive under control. These are all directions to consider for future improvements. As a final positive note, we have discussed that the GaSNet-L3, in particular, has reached an accuracy and scatter of its predictions, sufficient to be used to automatically measure galaxy redshifts in large spectroscopic surveys.

Acknowledgments

We thank Dr. C. Tortora and Dr. Y. Shu for useful comments on the manuscripts. We acknowledge the science research grants from the China Manned Space Project (CMSCSST-2021-A01). R.L. acknowledges the support from K.C.Wong Education Foundation. N.R.N. acknowledges financial support from the“One hundred top talent program of Sun Yatsen University”grant No.71000-18841229,from the Research Fund for International Scholars of the National Science Foundation of China, grant No. 12150710511.

Data Availability

The data that support the findings of this study are available at the URLs provided in the text and the Table in Appendix A.All other data that are not provided in the paper can be requested from the authors.

Appendix A HQ Catalog from GaSNets

In this appendix we list the HQ candidates obtained with the GaSNets and described in Section 5.The table content is listed in Section 5.4.

New HQ candidates from GaSNets R.A.Decl. PLATE MJD FIBER PLzzPGzPE meaneyeZEPT 1 27.6755 -7.1667 7164 56597 275 1.0 0.126 0.125 0.322 0.297 1.0 2 356.6865 34.8954 7143 56572 678 1.0 0.095 0.096 0.291 0.276 1.0 3 124.173 18.6976 4486 55588 218 1.0 0.221 0.221 0.636 0.641 1.0 4 139.3617 31.157 5808 56325 434 1.0 0.239 0.238 0.823 0.82 1.0 5 205.2421 18.6312 5862 56045 110 1.0 0.244 0.244 0.638 0.645 1.0 6 233.5735 3.4663 4805 55715 326 1.0 0.291 0.288 0.707 0.708 1.0 7 22.0923 22.3834 5107 55940 409 1.0 0.277 0.275 0.70.695 1.0 8 356.4175 -2.1613 4356 55829 729 1.0 0.294 0.294 0.659 0.658 1.0 99.4789 17.8124 6193 56237 298 1.0 0.298 0.297 0.695 0.695 1.0 10 4.8287 26.2703 6276 56269 26 1.0 0.262 0.262 0.60.604 1.0 11 244.2444 59.9424 6976 56448 830 1.0 0.257 0.258 0.509 0.507 1.0 12 221.3814 60.9204 6982 56444 516 1.0 0.258 0.257 0.501 0.511 1.0 13 228.0721 14.4225 5486 56030 270 1.0 0.327 0.327 0.706 0.708 1.0 14 143.3092 22.088 5770 56014 842 1.0 0.35 0.35 0.694 0.694 1.0 15 352.9407 29.3635 6581 56540 976 1.0 0.313 0.312 0.68 0.678 1.0 16 202.7383 47.6651 6743 56385 86 1.0 0.349 0.35 0.662 0.662 1.0 17 145.3751 1.865 4736 55631 264 1.0 0.353 0.354 0.719 0.722 1.0 18 212.4583 58.3783 6804 56447 957 1.0 0.36 0.36 0.744 0.742 1.0 19 189.6944 51.0158 6674 56416 262 1.0 0.388 0.387 0.762 0.766 1.0 20 358.2568 32.3241 6498 56565 550 1.0 0.367 0.366 0.729 0.731 1.0 21 215.0974 30.7816 3866 55623 45 1.0 0.466 0.47 0.813 0.814 1.0 22 127.2466 6.7193 4866 55895 335 1.0 0.47 0.471 0.878 0.877 1.0 23 176.7038 31.9488 4614 55604 299 1.0 0.478 0.479 0.845 0.846 1.0 24 322.7388 -1.587 4384 56105 873 1.0 0.506 0.509 0.81 0.814 1.0 25 338.7334 15.1238 5038 56235 231 1.0 0.514 0.513 0.822 0.825 1.0 26 118.3424 11.0724 4511 55602 755 1.0 0.565 0.558 0.82 0.824 1.0 27 189.7042 24.9656 5984 56337 333 1.0 0.559 0.558 0.826 0.828 1.0 28 153.3944 25.7594 6465 56279 537 1.0 0.616 0.619 0.825 0.828 1.0 29 321.1025 1.1421 4193 55476 665 1.0 0.70.7 0.875 0.878 1.0 30 358.3876 -8.6495 7166 56602 784 1.0 0.167 0.167 0.653 0.657 1.0 31 253.2719 50.6749 6311 56447 56 1.0 0.095 0.093 0.57 0.575 1.0 32 321.5421 1.9777 5143 55828 640 1.0 0.152 0.151 0.772 0.78 1.0 33 150.7402 6.319 4874 55673 82 1.0 0.225 0.225 0.781 0.781 1.0 34 17.3453 18.7404 5124 55894 36 1.0 0.226 0.226 0.645 0.644 1.0 35 207.8211 -1.7923 4041 55361 522 1.0 0.208 0.208 0.77 0.774 1.0 36 31.4846 -2.3339 4347 55830 156 1.0 0.283 0.281 0.462 0.462 1.0 37 181.6564 37.855 4700 55709 242 1.0 0.263 0.262 0.607 0.609 1.0 38 26.7007 22.0027 5108 55888 811 1.0 0.267 0.266 0.745 0.745 1.0 39 128.9045 8.3751 5285 55946 510 1.0 0.259 0.258 0.555 0.558 1.0 40 42.3078 -2.8227 4342 55531 112 1.0 0.292 0.29 0.459 0.459 1.0 41 18.952 5.1636 4425 55864 376 1.0 0.337 0.336 0.607 0.606 1.0 42 173.7037 54.187 6697 56419 386 1.0 0.302 0.301 0.614 0.613 1.0 43 134.3035 19.7311 5175 55955 804 1.0 0.377 0.379 0.632 0.629 1.0

(Continued)

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(Continued)

New HQ candidates from GaSNets R.A.Decl. PLATE MJD FIBER PLzzPG zPE meaneyeZE PT 201 233.7202 30.6619 4722 55735 603 0.992 0.546 0.544 0.669 0.67 0.89 202 23.0487 25.9812 5694 56213 919 0.992 0.494 0.495 0.704 0.678 0.89 203 163.2455 40.0812 4629 55630 852 0.991 0.302 0.303 0.638 0.583 0.89 204 191.086 -2.7746 3793 55214 203 0.991 0.55 0.55 0.713 0.716 0.89 205 238.0017 12.4111 4882 55721 634 0.99 0.107 0.108 0.486 0.483 0.89 206 163.8696 31.3152 6445 56366 883 0.99 0.453 0.451 0.802 0.816 0.89 207 178.0232 21.705 6407 56311 543 0.99 0.618 0.619 0.772 0.775 0.89 208 339.1266 9.5256 5054 56191 495 0.989 0.424 0.425 0.74 0.74 0.89 209 29.6199 1.0778 4234 55478 548 0.988 0.325 0.324 0.781 0.781 0.89 210 176.0852 30.9175 6433 56339 930 0.987 0.076 0.076 0.623 0.695 0.89 211 228.2038 53.9555 6713 56402 396 0.985 0.441 0.44 0.621 0.624 0.89 212 336.7086 5.148 4428 56189 98 0.984 0.09 0.09 0.401 0.406 0.89 213 174.3192 32.0515 4616 55617 283 0.984 0.456 0.461 0.641 0.641 0.88 214 157.7968 28.1283 6456 56339 122 0.983 0.312 0.311 0.549 0.555 0.88 215 233.2035 26.8323 3959 55679 69 0.982 0.523 0.524 0.668 0.664 0.88 216 186.9392 5.4132 4833 55679 266 0.979 0.47 0.47 0.634 0.625 0.88 217 336.3138 6.0182 4428 56189 199 0.979 0.498 0.5 0.736 0.734 0.88 218 174.0321 12.571 5376 55987 258 0.977 0.486 0.487 0.727 0.731 0.88 219 21.6396 14.2556 5140 55836 213 0.976 0.423 0.418 0.648 0.648 0.88 220 252.8827 36.6409 5198 55823 396 0.973 0.603 0.605 0.712 0.718 0.88 221 252.9652 35.5515 5198 55823 372 0.964 0.412 0.414 0.71 0.708 0.87 222 25.5021 31.3093 6601 56335 413 0.961 0.5 0.502 1.068 0.572 0.86 223 118.0482 15.4782 4495 55566 310 0.96 0.282 0.282 0.622 0.623 0.86 224 345.3411-1.4885 4362 55828 827 0.959 0.434 0.435 0.73 0.734 0.86 225 221.6961 7.9757 4858 55686 786 0.955 0.269 0.268 0.711 0.568 0.86 226 232.7992 10.7432 5493 56009 967 0.954 0.563 0.56 0.823 0.825 0.86 227 23.8523 14.5389 5137 55833 167 0.954 0.449 0.449 0.719 0.72 0.86 228 149.5008 10.1101 5324 55947 843 0.951 0.49 0.491 0.803 0.807 0.86 229 8.8946 9.0665 4540 55863 691.0 0.214 0.213 0.799 0.805 0.85 230 204.6975 5.1005 4786 55651 6781.0 0.271 0.271 0.561 0.559 0.85 231 27.2466 21.0868 5108 55888 991.0 0.277 0.276 0.753 0.752 0.85 232 11.5709 25.3449 6286 56301 2651.0 0.428 0.429 0.822 0.828 0.85 233 190.6792-0.3908 3793 55214 6631.0 0.454 0.452 0.815 0.818 0.85 234 246.8241 19.5842 4061 55362 4991.0 0.462 0.463 0.848 0.848 0.85 235 200.1508 12.4425 5427 56001 7001.0 0.528 0.529 0.818 0.821 0.85 236 28.1217 9.5832 4530 55564 3081.0 0.557 0.555 0.841 0.845 0.85 237 11.9627 31.0756 6872 56540 4431.0 0.643 0.643 0.844 0.847 0.85 238 170.9226 13.9839 5370 56003 6831.0 0.521 0.52 0.888 0.886 0.85 239 207.1491 9.5015 5442 55978 7111.0 0.578 0.577 0.812 0.812 0.85 240 159.9814 8.136 5349 55929 3791.0 0.617 0.619 0.873 0.875 0.85 241 169.8367 65.4877 7110 56746 8331.0 0.529 0.527 0.826 0.826 0.85 242 182.5102 3.7887 4749 55633 3811.0 0.596 0.597 0.885 0.885 0.85 243 150.4387 37.4688 4637 55616 7051.0 0.677 0.677 0.847 0.848 0.85 244 248.9092 49.4879 8056 57186 7691.0 0.671 0.672 0.89 0.891 0.85 245 338.5431 32.4454 6509 56486 7881.0 0.422 0.422 0.84 0.842 0.85 246 195.6242 5.7652 4838 55686 4671.0 0.492 0.493 0.879 0.883 0.85 247 154.1187 9.2856 5336 55957 3211.0 0.545 0.542 0.86 0.86 0.85 248 160.0284 63.5812 7097 56667 8651.0 0.528 0.528 0.836 0.836 0.85 249 143.962 15.6466 5315 55978 6231.0 0.605 0.608 0.86 0.86 0.85 250 204.0295 33.6021 3985 55320 773 0.999 0.484 0.486 0.917 0.92 0.85 New HQ candidates from GaSNets R.A.Decl. PLATE MJD FIBER PLzzPG zPE meaneyeZE PT 251 226.5216 40.6238 6054 56089 256 0.999 0.483 0.481 0.628 0.628 0.85 252 171.3643 41.4413 4699 55684 447 0.999 0.487 0.485 0.914 0.912 0.85 253 24.3379 16.7282 5130 55835 545 0.999 0.471 0.472 0.651 0.65 0.85 254 154.1552 54.3047 6696 56398 487 0.999 0.487 0.488 0.859 0.862 0.85 255 159.2973 10.5324 5346 55955 829 0.998 0.199 0.618 1.109 1.111 0.85 256 38.4525 -7.1342 4388 55536 88 0.998 0.335 0.335 0.638 0.642 0.85

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(Continued)

Appendix B Low-resolution Image Grading from HSC versus DECaLS

In Section 6.1, we have motivated the choice to base the estimation of the confirmation rate of the HQ catalog,presented in Section 5.4, on the 56 candidates matching the HSC public image database, instead of the 279 matches we have found for the DECaLS image database, with the better image quality of the former sample. In this Appendix, we want to visually quantify this and possibly drive some conclusions about the impact that the choice of poorer quality imaging could have on our results.We start by clarifying that our arguments cannot be extended to other analyses that might have made a different choice (e.g., T+21), because the grading of the imaging data has a large level of subjectivity,hence we cannot generalize the conclusions we can make on the basis of the criteria adopted in this paper.

Given this necessary preamble,in Figure 21,we show the gri color combined images of six of the eight HSC lens candidates confirmed with the GaSNets (see Section 6.1), which we consider rather solid having received high grades both from HSC imaging and eBOSS spectroscopy from the GaSNets.Compared to face-to-face, we also show the grz combined images obtained by the DECaLS database, for which we have given a visual grading according to the standard adopted in 6.1.It looks fairly clear that HSC images have overall higher quality and that this impacts the grading of the lensing features one is expected to judge from the images. The leverage of our grading has been even slightly lowered since we have started from the evidence of some background sources which might help convince our brain that the feature in the images can be a lens, but not too much because we had to consider the possibility of occasional overlap with standard galaxies along the line-of-sight. As a consequence of that, the grading of DECaLS images might be slightly higher than we typically give in the visual inspection of imaging candidates (e.g., Li et al. 2021b). The net result is that for this sample of “secure”gravitational lenses if we used the DECaLS images we would have possibly excluded the 50%.If we take the test made over this small sample of galaxies at face value,we can predict that DECaLS imaging would lead us to an estimate of the success rate of the order of 20%–25%.To confirm this we have visually inspected the 274 matched to DECaLS and indeed find that maybe 1/5 of them had some convincing lensing features.T+21 used the DECaLS color images to find “l(fā)ensing evidence” for 447/1551 of their spectroscopical sample,corresponding to a 29%, which is not far from what we have predicted above, meaning that their confirmation rate is also rather underestimated, and possibly closer to the upper limit found by us (57%).

Figure 21.A sample of color images of the HSC lens candidates confirmed with the GaSNets vs.the corresponding DECaLS images.For each strong lens,we show the gri combined images from the HSC database left panel,against the grz color image of the same pointing from DECaLS(right panel).Color combined images are used with the same set-up on the individual band.Part of the lower quality of DECaLS can be due to the z-band imaging which was the only redder band available.In each DECaLS panel, we show the visual grading given by us. In the top left pair of figures, we also show the size of the fiber from eBOSS (top-right corner).