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# Predicting the Spectroscopic Features of Galaxies by Applying Manifold Learning on Their Broadband Colors: Proof of Concept and Potential Applications for Euclid, Roman, and Rubin LSST

Marziye Jafariyazani¹, Daniel Masters¹, Andreas L. Faisst¹, Harry I. Teplitz¹, and Olivier Ilbert²

Caltech/IPAC, 1200 E. California Boulevard, Pasadena, CA 91125, USA

²Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France

Received 2023 November 28; revised 2024 March 19; accepted 2024 March 26; published 2024 May 17

## Abstract

Entering the era of large-scale galaxy surveys, which will deliver unprecedented amounts of photometric and spectroscopic data, there is a growing need for more efficient, data-driven, and less model-dependent techniques to analyze the spectral energy distribution of galaxies. In this work, we demonstrate that by taking advantage of manifold learning approaches, we can estimate spectroscopic features of large samples of galaxies from their broadband photometry when spectroscopy is available only for a fraction of the sample. This will be done by applying the self-organizing map algorithm on broadband colors of galaxies and mapping partially available spectroscopic information into the trained maps. In this pilot study, we focus on estimating the 4000 Å break in a magnitude-limited sample of galaxies in the Cosmic Evolution Survey (COSMOS) field. We also examine this method to predict the HδA index given our available spectroscopic measurements. We use observed galaxy colors (u,g,r,i,z,Y,J,H), as well as spectroscopic measurements for a fraction of the sample from the LEGA-C and zCOSMOS spectroscopic surveys to estimate this feature for our parent photometric sample. We recover the D4000 feature for galaxies that only have broadband colors with uncertainties about twice the uncertainty of the employed spectroscopic surveys. Using these measurements, we observe a positive correlation between D4000 and the stellar mass of the galaxies in our sample with weaker D4000 features for higher-redshift galaxies at fixed stellar masses. These can be explained by the downsizing scenario for the formation of galaxies and the decrease in their specific star formation rate as well as the aging of their stellar populations over this time period.

Unified Astronomy Thesaurus concepts: Broad band photometry (184); Spectroscopy (1558); Galaxy colors (586)

## 1. Introduction

Measuring the physical properties of galaxies is the first and key step to understanding how galaxies form and evolve. One of the most common and least expensive ways to infer physical properties, such as redshift, stellar mass, and star formation rate (SFR), for statistically large numbers of galaxies over cosmic time is by measuring their energy output in multiple photometric filters. Multiband photometric measurements provide data to construct the spectral energy distribution (SED) of galaxies, and then, SED-fitting techniques can be employed to fit theoretical and empirical models to observed SEDs to infer multiple physical properties, including stellar mass, SFR, age, and reddening simultaneously, though with some limitations, and known degeneracies (Noll et al. 2009; Walcher et al. 2011; Conroy 2013; Lower et al. 2020; Pacifici et al. 2023).

On the other hand, high-resolution spectra from galaxies enable the measurement of detailed characteristics of galaxies more robustly, including details of their chemical content and their star formation history, albeit at a much higher cost both at the observing and data analysis stages. Therefore, the combination of wide multiband photometric surveys observing millions of galaxies, with more restricted spectroscopic surveys targeting thousands of galaxies, is the current scheme of galaxy observing programs aiming to measure the physical properties of galaxies.

We are now entering the era of large-scale galaxy surveys, which are going to deliver unprecedented amounts of photometric and spectroscopic data. Euclid (Laureijs et al. 2011; Amendola et al. 2013), the Nancy Grace Roman Space Telescope (Spergel et al. 2015; Akeson et al. 2019), and the Vera C. Rubin Observatory (LSST Science Collaboration et al. 2009;IVEzić et al. 2019) will each survey large areas of sky and provide multiband photometric data for billions of galaxies, which will need to be analyzed in more efficient ways. SED-fitting techniques, as classical methods of inferring the physical properties of galaxies from broadband photometry, will be computationally expensive for these large data sets. The dependence of SED-fitting results on our current models and assumptions will be another limiting factor in properly analyzing new observations. On the other hand, spectroscopic observations will still be limited to only smaller subsamples of galaxies observed photometrically. Therefore, the need for more efficient, data-driven, and less model-dependent techniques to analyze SEDs of galaxies to infer the most information from them is becoming more crucial. A recent analysis by Nersesian et al. (2024) evidently showed that relying only on SED-fitting techniques to recover spectral features of galaxies from their broadband colors is not sufficient, and incorporating techniques to inform photometric analysis with high-quality spectroscopy is necessary. Developing techniques to allow the integration of valuable but scarce spectroscopic information with broadband photometry will maximize the efficiency of both photometric and spectroscopic surveys, facilitating the estimation of more reliable properties of galaxies.

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Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
In this context, and with the expectation of high-dimensional data soon to become available for billions of galaxies, here we aim to make use of manifold learning as a data-driven approach to recover fundamental properties of galaxies from their broadband colors more efficiently. Manifold learning is a class of unsupervised algorithms that seek to learn the high-dimensional structure of the data without predefined classifications in order to find a lower-dimensional representation of that data, which would be easier to visualize and interpret. This approach has been implemented in a handful of algorithms, each with its own capabilities and limitations, including Isometric Mapping (Tenenbaum et al. 2000), self-organizing maps (SOMs; Kohonen 1982), locally linear embedding (Roweis & Saul 2000), t-distributed stochastic neighbor embedding (van der Maaten & Hinton 2008), and Uniform Manifold Approximation and Projection (McInnes et al. 2018). The application of some of these algorithms in astronomical research has demonstrated great promise in analyzing photometric and spectroscopic data of stars, quasars, and galaxies (Daniel et al. 2011; Bu et al. 2014; Masters et al. 2015; Süveges et al. 2017; Traven et al. 2017; Anders et al. 2018; Faisst et al. 2019; Hemmati et al. 2019; Nakoneczny et al. 2019; Steinhardt et al. 2020; Zhang et al. 2020; Bódi & Hajdu 2021; Storey-Fisher et al. 2021; Sun et al. 2021; Davidzon et al. 2022; Chartab et al. 2023; Steinhardt et al. 2023).

In this paper, we focus on using the SOM algorithm, which has been previously used for the classification of light curves (Brett et al. 2004), estimating the photometric redshift of galaxies (Way & Klose 2012; Carrasco Kind & Brunner 2014; Greisel et al. 2015), and finding and classifying variable active galactic nuclei (Faisst et al. 2019). More recently, Masters et al. (2019) used these maps as a tool to develop optimal photometric redshift calibration strategies for upcoming cosmology surveys with Euclid and Roman. In another work, Hemmati et al. (2019) used this tool for both visualizing and computationally accelerating the estimation of galaxy physical properties, such as stellar mass and SFR, from photometric data by training SOMs with theoretical models, demonstrating that their method is able to estimate the full probability distribution functions for each galaxy significantly faster than traditional SED-modeling codes. Furthermore, Teimoorinia et al. (2022) used SOMs to classify the spectra of galaxies in the MaNGA survey (Bundy et al. 2015). They showed that the distribution of spectra onto a SOM for a single galaxy can reveal the presence of distinct stellar populations within the galaxy, and therefore, can be an alternative method to full spectral fitting for deriving the full probability distribution of physical properties, with the advantage of being more efficient than Bayesian methods.

In this work, we advance one step further to use SOMs to estimate specific spectroscopic features of galaxies based on their broadband colors. This will be done by informing SOMs trained with a set of broadband colors with spectroscopic measurements for a subsample of the galaxies and then estimating that feature for the rest of the training sample. In this paper, we specifically focus on estimating the 4000 Å break, which is a spectral feature of galaxies caused by the absorption of metals in stellar atmospheres, indicative of a lack of young stars and an abundance of metal absorbents (Hamilton 1985; Balogh et al. 1999). It is therefore a powerful classical probe of age and metallicity of galaxies, and its relation with other properties of galaxies can put further

constraints on galaxy formation models, including the downsizing scenario (Cowie et al. 1996). We finally test this method, given available spectroscopic data, to predict the $H\delta_A$ index, which is another age-sensitive spectral feature of galaxies (Worthey & Ottaviani 1997; Balogh et al. 1999).

This paper is organized as follows: In Section 2, we present the parent samples for our training data. Section 3 describes the algorithm we use and the training process. In Section 4, we use our method to predict the D4000 feature for our photometric sample, and we study the D4000-stellar mass relation for our sample in Section 5. We finally discuss our results in Section 6, and present $H\delta_A$ measurements in the Appendix. Throughout this work, we assume a $\Lambda$CDM cosmology with $H_0 = 70 \text{ km s}^{-1} \text{ Mpc}^{-1}$, $\Omega_\Lambda = 0.7$, and $\Omega_m = 0.3$. All magnitudes are given in the AB system (Oke & Gunn 1983).

## 2. Parent Samples

### 2.1. Photometric Sample

We choose the training sample for this work from the latest data release of the Cosmic Evolution Survey (COSMOS; Scoville et al. 2007). COSMOS started with a Hubble Space Telescope/Advanced Camera for Survey (ACS) observation of a 1.7 deg² mosaic, reaching a depth of 27.2 AB in the F814W filter, and then has been observed with most major ground and space-based telescopes to become a field with an invaluable amount of both photometric and spectroscopic data. The new COSMOS2020 catalog (Weaver et al. 2022) includes multi-band photometry in the UV, optical, and IR for about 1.7 million sources.

### 2.2. Spectroscopic Sample

The spectroscopic measurements used in this paper come from two magnitude-limited surveys which measured specific spectroscopic features for galaxies in addition to their redshift in the COSMOS field: the Large Early Galaxy Census (LEGA-C) survey (van der Wel et al. 2016) and zCOSMOS (Lilly et al. 2007):

1. **LEGA-C:** LEGA-C is a spectroscopic survey of about 3200 K-band selected galaxies at redshift $0.6 < z < 1.0$, which is conducted with the VIMOS/Very Large Telescope (VLT; van der Wel et al. 2016). This survey has a redshift-dependent magnitude limit of $K_{AB} = 20.7-7.5 \times \log((1+z)/1.8)$, which resulted in targeting only bright galaxies to obtain high-quality spectra.

2. **zCOSMOS:** zCOSMOS is a redshift survey in the COSMOS field conducted with VIMOS/VLT (Lilly et al. 2007). For this work, we use zCOSMOS-bright data, which is an I-band limited ($I_{AB} < 22.5$) survey of about 20,000 galaxies covering the whole COSMOS ACS field. The details of the spectral measurements for this sample of galaxies at $0 < z < 1.4$ can be found in Vergani et al. (2010).

Further details on the subsample of these surveys used in this analysis will be presented in Section 4 and Table 1.

## 3. Manifold Learning on Broadband Colors of Galaxies

### 3.1. SOM Algorithm in a Nutshell

SOMs (Kohonen 1982) are a type of artificial neural network trained using unsupervised learning and are mainly used for
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Trained SOMs with seven broadband colors (u,g,r,i,z,Y,J,H) for our sample of 60,544 galaxies. The top two left panels present two sample colors (u − g and J − H) on our trained maps. The top-right panel presents the occupation map showing the number of galaxies assigned to each cell during the training step. The bottom panels present the physical properties of the trained sample measured using SED fitting, including photometric redshift, stellar mass, and sSFR. The value of each cell in these maps represents the median value of that property for the galaxies in the cell, adopted from the COSMOS2020 catalog.</figcaption></figure>

<table><caption>**Table 1**

Physical Properties of the Subsamples</caption><thead><tr><th></th><th>No. of Galaxies</th><th>Log(Stellar Mass)(M<sub>☉</sub>) [min, median, max]</th><th>Passive Galaxies (%)</th></tr></thead><tbody><tr><td>Training sample</td><td>60,544</td><td>[6, 10.4, 12.8]</td><td>32%</td></tr><tr><td>LEGA-C subsample</td><td>2241</td><td>[8.8, 10.8, 11.9]</td><td>38%</td></tr><tr><td>zCOSMOS subsample</td><td>10,622</td><td>[7.1, 10.3, 11.8]</td><td>23%</td></tr></tbody></table>

dimensionality reduction as well as data visualization and clustering. The algorithm preserves the topological relationship of the input data set, which helps reveal correlations that are not easily identifiable in a high-dimensional space. SOMs are usually used to bring down the dimension of a data set to two for visualization purposes, though the technique can be used to reduce dimension to any arbitrary number.

This neural network algorithm consists of two layers of input and output data without any hidden layers in between. The input training data is a set of vectors of *N* dimensions, and the data is a feature map; a lattice of neurons (nodes) with topological positions (e.g., X, Y), where each neuron is assigned a weight vector with the same dimensionality as the input data. In the training step, the weight vectors are initialized randomly (or by sampling from the input data), and then SOMs are trained by computing the distance (e.g., Euclidean distance) between each neuron (neuron from the output layer) and the

input data, and the neuron with the lowest distance will be the winner of the competition, known as the Best Matching Unit (BMU). After finding the BMU, the value of that neuron and its neighbors will be updated, and the steps will be repeated until the maps reach convergence and data points with similar features end up in neighboring positions on the map. More details about this algorithm and its applications can be found in Yin (2008) and Cottrell et al. (2016).

## 3.2. Training SOMs

We train our SOMs with seven broadband colors derived from *u*, *g*, *r*, *i*, *z*, *Y*, *J*, and *H* filters for a subsample of 60,544 galaxies in the COSMOS field. We select these galaxies from the COSMOS2020 photometric catalog such that they meet the magnitude criteria for either zCOSMOS (*I*<sub>AB</sub> < 22.5) or the LEGA-C survey (*K*<sub>AB</sub> < 21.08 at *z* ~ 0.6), with an additional condition that they should have at least four measured colors in our selected filter set. This specific filter set is analogous to what will soon become available for a statistically large sample of galaxies thanks to the Euclid and Vera Rubin telescopes. The NISP instrument on board Euclid will provide us with broadband photometry in the *Y*<sub>E</sub>, *J*<sub>E</sub>, and *H*<sub>E</sub> filters for about 15,000 deg<sup>2</sup> of the sky, where approximately 9000 deg<sup>2</sup> of the region will also be observed with the Vera Rubin telescope, providing photometry in the *u*, *g*, *r*, *i*, and *z* filters.

These trained SOMs are presented in Figure 1, where the top left panels show two sample colors (J − H and *u* − *g* and) organized on the maps. As expected, galaxies are clustered on these maps according to their colors. The top-right panel of the same figure shows an occupation map where each cell of the
<figure><img src="image_4.png" /><figcaption>**Figure 2.** This plot demonstrates the SEDs of galaxies on trained SOMs. For visualization purposes, every one out of four cells in each row of the original map is shown in the plot. The median SED of each cell (dashed line) and the number of galaxies in each cell is shown on each panel.</figcaption></figure>

map is color coded based on the number of galaxies contained in that cell, which ranges from one to 380 galaxies, with a median of 56 galaxies per cell. This classification based on colors suggests that these galaxies are also organized in terms of their physical characteristics on the map. To examine this assumption, we color coded these maps with some physical features that were not fed into the algorithm during the training step. Here, we use the physical properties derived from the SED fitting of these galaxies available in the COSMOS2020 catalog (Weaver et al. 2022). These are shown in the bottom panels of the Figure 1. The lower-left panel presents the stellar mass, where the value of each cell is the median stellar

mass of the galaxies in that cell. The data in the middle and right panels have been calculated in the same way for the specific star formation rate (sSFR) and photometric redshift of these galaxies. These three maps visually demonstrate how training SOMs with broadband colors leads to classifying galaxies based on their fundamental physical properties without any modeling assumptions.

It should also be noted that the physical properties mapped onto these SOMs are the results of SED fitting on up to 31 photometric bands available in the COSMOS2020 catalog, while we only used eight bands in training our SOMs. These maps suggest that the groups of galaxies in the cells where the
<figure><img src="image_5.png" /><figcaption>Figure 3. This plot compares the median D4000 measured from spectra taken in the LEGA-C survey and zCOSMOS for galaxies in each cell of our trained SOM. Only cells that have at least three D4000 measurements in each survey are shown here. The plot shows a general consistency between the measurements in LEGA-C and zCOSMOS.</figcaption></figure>

dispersion of their colors is small (i.e., have similar SEDs), potentially have very similar physical properties, allowing us to estimate further features that have not been explicitly measured for all the galaxies in that cell. Furthermore, measuring absolute physical properties is attainable as the color-based trained SOMs have already separated galaxies in the redshift space.

Our 30 × 30 maps categorized our ~60,000 galaxies into 900 classes. This size for our training map is determined in an iterative process such that individual cells are neither underpopulated nor overpopulated to not cause high dispersion in galaxies' colors. Figure 2 presents individual SEDs of galaxies in 64 cells of our map, corresponding to every four cells of the original 30 × 30 map. The number of galaxies and the median SED of each cell (dashed line) is also presented in each panel. It should be emphasized that while galaxies within the same cell have the highest degree of similarity, neighboring cells also have more similar properties compared to galaxies in distant regions of the map. By taking advantage of this model-free SED grouping, we then use the physical features measured for a subsample of galaxies in each cell to predict that feature for the rest of the galaxies in that cell. To investigate this approach and its potential, we estimate the 4000 Å break for our sample in Section 4, which classically requires spectroscopy or narrowband photometry for proper estimation (Stothert et al. 2018; Renard et al. 2022).

## 4. Predicting the 4000 Å Break from Broadband Photometry

The 4000 Å break, a probe of age and metallicity of galaxies, is quantified by measuring the ratio between the continuum fluxes on the red and blue sides of this break (Bruzual 1983; Balogh et al. 1999). D4000 is one of the features that has been measured robustly in both the LEGA-C survey and zCOSMOS for a statistically large number of galaxies in the COSMOS

field, as it can be measured in lower-quality spectra compared to the higher resolutions needed to measure individual spectral lines. In this work, we use D4000 measurements for 2241 galaxies from the LEGA-C survey and 10,622 galaxies from zCOSMOS, giving us the proper statistics for this pilot study. Table 1 provides an overview of these subsamples, where the basic physical properties of these galaxies computed from SED fitting available in the COSMOS2020 catalog are presented. These include their stellar mass range and median, as well as the relative number of passive galaxies in each subsample. We employ a simple classification criteria based on the sSFR, where we categorize galaxies with an sSFR < 10⁻¹¹ yr⁻¹ as passive galaxies (Ilbert et al. 2010; Faisst et al. 2017).

In this paper, we adopt the definition of Balogh et al. (1999; D4000_n) for the strength of the 4000 Å break, as it is less sensitive to reddening effects compared to the original definition of the D4000 (D4000_w, Bruzual 1983), which uses a larger wavelength range to define the continuum around the break. We refer to D4000_n as simply D4000 throughout this paper.

$$D4000_n = \frac{F(4000 - 4100 \AA)}{F(3850 - 3950 \AA)} \quad (1)$$

In the first step, we map D4000 measurements from these two surveys on our trained SOMs described in Section 3.2. We then compare these measurements from the two surveys for the galaxies in our sample. For this purpose, we use the median D4000 measurements for the galaxies lying in each cell of the trained SOM from LEGA-C and zCOSMOS, as shown in Figure 3. There are 282 cells in our trained maps that have at least one galaxy with a measured D4000 in the LEGA-C survey (31% of the cells) and 617 cells with at least one measurement in zCOSMOS (69% of the cells). Figure 3 compares the median of these measurements for cells that have at least three measurements in each survey, showing the consistency between measurements in two surveys, with a tighter relation at lower D4000 values.

The distribution of these spectroscopic measurements on our trained SOM is presented in Figure 4. In the top row, the left panel presents the median D4000 measurements for the subsample of our galaxies observed in LEGA-C, the middle panel is from zCOSMOS, and the right panel includes D4000 measurements from both surveys. These panels reveal two distinct populations of galaxies with weak and strong D4000 features, blue and red, respectively. Comparing these maps with the sSFR map (lower-right panel of Figure 1) demonstrates that, as we expected, galaxies with lower sSFR have stronger D4000 features, and galaxies with high sSFR and abundance of younger stars have weaker D4000. The data from zCOSMOS (middle panel of Figure 4) has more coverage on the D4000 map, as it includes fainter- and lower-mass galaxies in a wider redshift range compared to LEGA-C. The right panel of Figure 4 includes all the D4000 measurements available from these two spectroscopic surveys for our training sample, which will be used to estimate this feature for the rest of our galaxies.

The bottom row of Figure 4 presents the dispersion map of D4000 measurements in each cell of the SOM. The inset plots in the bottom row present the distribution of the cells in terms of the amount of the D4000 dispersion. For the LEGA-C map, most of the cells (81%) have a dispersion of less than 0.1, and for the zCOSMOS map, most of the cells (72%) have a
<figure><img src="image_6.png" /><figcaption>**Figure 4.** The top row of this plot presents the D4000 measurements available for a fraction of our sample mapped on the trained SOMs measured from the LEGA-C survey (left), zCOSMOS (middle), and the median of the measurements from both surveys (right). The value for each cell is the median of the measured D4000 for the galaxies residing in that cell, where at least three spectroscopic measurements exist. The bottom row shows the dispersion in D4000 for every cell of the SOM. The inset histogram in each panel presents the distribution of the median dispersion among cells. zCOSMOS includes lower-mass and fainter galaxies with more uncertain measurements, and therefore, higher dispersion is seen for estimations from this survey.</figcaption></figure>

dispersion of less than 0.15. Also, cells with at least three D4000 measurements have a median sigma value of 0.12, which does not decrease significantly with an increased number of spectroscopically measured D4000 in the cell. Therefore, we evaluate the potential use of the median value of the spectroscopic D4000 measurements to estimate this feature for the rest of the galaxies in that cell with no available spectroscopy, where at least three spectroscopic measurements exist in their host cell.

To evaluate this approach, we use 80% of our spectroscopic data to estimate the D4000 features for cells with at least three measurements. We then compare our estimated values and their uncertainty with the actual D4000 measurement from spectroscopy for the remaining 20% of the spectroscopic sample. The top panels of Figure 5 show this comparison, where the predicted D4000 for 20% of the sample is based on the trained SOM, with data for 80% of the sample presented as a function of D4000 measured in spectroscopic surveys for the same galaxies. From left to right, each panel shows the estimation based on LEGA-C data, zCOSMOS data, and the combination of both surveys. The bottom panels of Figure 5 present the distribution of uncertainties of the predicted D4000 values for each case. As expected from different depth and measurement errors of the LEGA-C survey and zCOSMOS, which significantly affect the total uncertainty, estimations based on LEGA-C, which targeted brighter galaxies with lower-measurement errors have lower uncertainties with the peak of uncertainty distribution at ~0.08, whereas estimations based on zCOSMOS, which includes fainter galaxies with higher-measurement errors are more uncertain with the peak of distribution at ~0.16. Comparing these uncertainties with the median spectroscopic measurement errors for this sample in the

two surveys (~0.05 for LEGA-C and ~0.08 for zCOSMOS) suggests that our method, with the number of spectroscopic measurements we have, can recover the D4000 feature based on broadband SEDs of galaxies without any modeling assumption, with an uncertainty on the order of twice the uncertainty of the measurement error of the employed spectroscopic data. It should be noted that for areas of the map where we do not have enough D4000 measurements from spectroscopy, specifically for cells without any measurements of their nearby neighbors as well, we do not constrain the D4000 feature, and galaxies in these cells can be targets of great interest for future spectroscopic follow-ups.

To compare the distribution of the galaxies with spectroscopic measurements with galaxies that we are able to predict their D4000 feature and our parent photometric sample, we then look at the distributions of the redshift and stellar mass of these samples. The top left panels of Figure 6 present the redshift distributions of our parent photometric sample, LEGA-C galaxies with measured D4000 and zCOSMOS galaxies with measured D4000, and the top-right panel shows the stellar mass distributions of these three samples. This is to demonstrate that our magnitude-limited photometric sample has an extended range of redshift and stellar mass compared to the employed spectroscopic samples. The bottom two panels of Figure 6 present the distributions for the photometric sample and the predicted sample, that is, the galaxies for which we are able to predict their D4000, which is ~79% of our photometric sample. The bottom panels show that our predicted sample has an approximately similar redshift and stellar mass distribution compared to the parent sample, extending our estimations beyond the redshift and stellar mass limits of our employed spectroscopic samples.
<figure><img src="image_7.png" /><figcaption>**Figure 5.** The top panels present the predicted D4000 for 20% of the sample based on data for the remaining 80% of the sample as a function of the true value of D4000 measured in spectroscopy. The left panel presents predictions based on LEGA-C only, the middle panel is for zCOSMOS only, and the right panel is based on both surveys combined. The median error bar of all the measurements is shown in the lower-right corner of the panels. The D4000 for blue galaxies that dominate the sample are best estimates, as can be seen in these plots. The bottom panels show a histogram of galaxies based on the uncertainty of their D4000 predictions (a combination of spectroscopic measurement errors and dispersion of D4000 in each cell of the SOM) based on each survey. Estimates based on the LEGA-C survey are more accurate as it has a lower-measurement error and includes a less diverse population of galaxies.</figcaption></figure>

**5. D4000–Stellar Mass Relation**

In this section, we use our predicted D4000 features based on SOMs to study the D4000–stellar mass relation for the galaxies in our sample. This relation has been studied in the literature based on D4000 features from spectroscopy and narrowband photometry (Haines et al. 2017; Siudek et al. 2017; Kim et al. 2018; Renard et al. 2022). These studies observed a bimodal distribution of galaxies on the D4000–stellar mass plane, where they found blue cloud and red-sequence galaxies differentiated from each other. Here, we use the estimated stellar mass of each galaxy based on the SED fitting available in the COSMOS2020 catalog. Figure 7 presents the D4000–stellar mass relation for the galaxies in our sample, where each data point represents an individual cell of SOM, with median properties of galaxies lying on that cell. We divide our galaxies into three redshift bins of $z<0.5$, $0.5<z<1$, and $z>1$. Overall, we observe a positive correlation between the strength of the 4000 Å break and the stellar mass of galaxies. This behavior can be explained by the downsizing scenario for the formation history of galaxies where stars formed earlier in more massive galaxies during intense star formation episodes and therefore produced a stronger 4000 Å break (Cowie et al. 1996; De Lucia et al. 2006; Pérez-González

et al. 2008; Pacifici et al. 2016). Also, galaxies at the lowest-redshift bin tend to have stronger breaks compared to the higher-redshift galaxies at fixed stellar masses. The increase in D4000 over time could be explained by the overall aging of stellar populations of galaxies as well as the decrease in the sSFR of the main sequence of galaxies over the past $\sim 10^9$ yr (Noeske et al. 2007; Ilbert et al. 2015).

We then divide our sample into two populations based on their D4000 as late-type (blue, D4000 $< 1.55$) and early-type (red, D4000 $> 1.55$) galaxies following Kauffmann et al. 2003. Figure 7 demonstrates that the dependence of the strength of D4000 on the stellar mass of galaxies is slightly shallower for red galaxies with a slope of 0.07, compared to the slope of 0.12 for blue galaxies. The D4000–stellar mass relation measured for our blue galaxies is comparable to the relations reported in Haines et al. (2017), who measured a slope of 0.12–0.14 for the D4000–stellar mass relation for their blue galaxies with little to no redshift dependence at $0.5 < z < 1.1$, based on data for $\sim 65,000$ galaxies from the VIMOS Public Extragalactic Redshift Survey (Guzzo et al. 2014). Renard et al. (2022) also measured a slope of $\sim 0.09–0.12$ for blue clouds based on D4000 measured from narrowband photometry for $\sim 17,000$
<figure><img src="image_8.png" /><figcaption>**Figure 6.** The top left panel presents the redshift distributions of our magnitude-limited parent photometric sample compared to our spectroscopic samples from the LEGA-C survey and zCOSMOS. The top-right panel presents the stellar mass distributions of these three samples. The bottom panels present the same distributions for the photometric sample compared to the predicted sample, which is ~79% of galaxies in our photometric sample for which we are able to predict their D4000.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 7.** D4000 as a function of stellar mass based on an estimated 4000 Å break from SOMs where each data point represents the median D4000 and stellar mass of a cell. Galaxies are divided into two populations of blue (D4000 < 1.55) and red sources (D4000 > 1.55) in three redshift bins of z < 0.5, 0.5 < z < 1, and z > 1. A linear fit to the blue and red populations at all redshifts is also presented in the plot.</figcaption></figure>

galaxies at 0.5 < z < 1 in the Physics of the Accelerating Universe Survey (Benítez et al. 2009). However, both of these studies measured a steeper slope for their red-sequence galaxies, ~0.23 in Haines et al. (2017) and 0.25–0.32 in Renard et al. (2022). One potential source of larger discrepancy in the measured relation for the red population can arise from the wider redshift range of our sample as well as the fact that these samples are magnitude limited and therefore are not mass complete, leaving us with the most massive and bluer galaxies

at higher-redshift bins and a low fraction of red galaxies in general. For instance, only ~20% of our galaxies and 35% of the sample in Renard et al. (2022) meet the criteria for a red sequence in each study.

## 6. Discussion

This work demonstrates the power of manifold learning to predict spectroscopic features of statistically large samples of galaxies when we have their broadband colors, as well as spectroscopy only for a fraction of the sample. Throughout this paper, we mainly focused on the D4000 feature, which was measured robustly for a statistically large number of galaxies over a wide redshift range in spectroscopic surveys, giving us the proper statistic we needed for this analysis. However, this method can be extended to predict any spectroscopic features, such as emission and absorption lines, from broadband photometric surveys when we have a certain number of robust spectroscopic measurements to fill in any trained SOM.

We present another example of employing this method to predict the $H\delta_A$ feature for the galaxies in our sample in the Appendix. We include this analysis in the Appendix because of the large uncertainty of the $H\delta_A$ measurements from zCOSMOS, which resulted in predictions with lower precision based on this survey. The $H\delta_A$ measurements from the LEGA-C survey are robust enough and resulted in reliable predictions for further potential science analysis. However, the relatively smaller sample size and the lower magnitude limit of the LEGA-C survey make it less representative of our training sample, constraining our ability to perform a comprehensive scientific analysis of the entire sample solely based on the predictions using its data.

Introducing our novel approach, one of the key benefits of employing this technique is that it is purely data driven and is completely independent of any modeling assumption, unlike traditional methods such as SED fitting. Another benefit is that when we acquire a minimum number of spectra to fill in the cells of our trained SOM, we are then able to predict specific spectroscopic features for an unlimited number of galaxies with observed colors, with the only requirement that they meet the criteria of the magnitude limit of the spectroscopic data. The minimum number of required spectra will depend on the diversity of galaxies under investigation as well as the required precision in the estimation of any specific feature. For example, we showed that in this pilot study, we are able to estimate D4000 features for galaxies in cells hosting an average of ~67 galaxies when we have at least three spectroscopic measurements in that cell, and the uncertainty in our estimations does not change significantly when having more spectroscopic measurements in each cell. Also, using this method, we can potentially identify cells filled with galaxies with little or no spectroscopic measurements, which will be high-priority candidates for follow-up spectroscopy to build unbiased and complete galaxy samples for any science investigation.

Another promising potential of this method is to identify outlier galaxies. After the training step, distinct galaxies in individual SOM cells, cells with very distinct properties compared to their neighboring cells, as well as cells that host very low numbers of galaxies, can be flagged to be further analyzed as potential outlier sources. These outliers could be the result of data-processing issues, observational constraints, such as blended sources, or can lead to the discovery of intrinsically unique and rare sources. In each case, correctly identifying these sources has huge benefits both for improving
data analysis pipelines and for addressing specific science questions that can transform our view of a specific subject. The routine to identify outlier sources with this method is now being developed within our team and will be published in a subsequent paper.

## Appendix

Here, we showcase the outcomes of applying the same procedure used for predicting D4000 to the prediction of $H\delta_A$. $H\delta_A$ is an absorption feature observed in the spectra of galaxies, which is caused by A-type stars (Worthey & Ottaviani 1997; Balogh et al. 1999; Paulino-Afonso et al. 2020). Therefore, it is commonly used as a proxy for the age of galaxies; however, we should caution that there is also a degeneracy between age and

metallicity since at a fixed stellar age, metal-poor galaxies may tend to have weaker $H\delta_A$. We show the available $H\delta_A$ measurements for our sample from the LEGA-C survey and zCOSMOS in the top panels of Figure 8. We also included the same maps for D4000 measurements in the middle panel of this plot to enable visual comparison between the $H\delta_A$ and D4000 features on the maps.

We should again note that these maps are trained only with observed broadband colors without any spectroscopic information. In the next step, color coding them according to D4000 measurements available for a fraction of the sample revealed two distinctive galaxy populations. Here, we also observe color coding the trained map with $H\delta_A$ measurements, which also differentiates these two populations where blue cells in D4000

<figure><img src="image_10.png" /><figcaption>**Figure 8.** The top row presents available $H\delta_A$ measurements for our sample mapped on the trained SOMs measured from the LEGA-C survey (left), zCOSMOS (middle), and the median of the measurements from both surveys (right), similar to the top row of Figure 4 for D4000. The bottom row shows the predicted $H\delta_A$ again for 20% of the sample based on data of the remaining 80% of the sample as a function measured $H\delta_A$, akin to the top row of Figure 5. The left panel presents predictions based on LEGA-C only, the middle panel is for zCOSMOS only, and the right panel is based on both surveys combined. Large uncertainties in the measured $H\delta_A$ from zCOSMOS spectra make predictions solely based on this survey very uncertain.</figcaption></figure>
maps have $H\delta_A \gtrsim 5$ Å, representative of young star-forming galaxies. Whereas red regions of the D4000 map (D4000 $\gtrsim 1.55$), indicative of early-type galaxies, have $H\delta_A \lesssim 2$ Å, another indication for old stellar populations of galaxies lying in these cells.

We then employed the same procedure to predict D4000 in Section 4 for the prediction of $H\delta_A$. The bottom panels of Figure 8 present the $H\delta_A$ predictions for 20% of the sample based on the data for the remaining 80% of the sample from LEGA-C (left panel), zCOSMOS (middle panel), and the combination of both surveys (right panel). As previously mentioned, zCOSMOS measurements lacked the precision needed for robust predictions, and therefore, have not been used for further science investigations. Predictions based on the LEGA-C sample are reliable for science use; however, because of the smaller sample size of the LEGA-C survey, limited regions of the map have predicted $H\delta_A$, constraining our ability to conduct a comprehensive science analysis of the entire sample. This highlights the need for future inclusive, diverse, and deep enough spectroscopic surveys to represent galaxies from all of the different galaxy populations being observed in photometric surveys.

## ORCID iDs

Marziye Jafariyazani https://orcid.org/0000-0001-8019-6661

Daniel Masters https://orcid.org/0000-0001-5382-6138

Andreas L. Faisst https://orcid.org/0000-0002-9382-9832

Harry I. Teplitz https://orcid.org/0000-0002-7064-5424

Olivier Ilbert https://orcid.org/0000-0002-7303-4397

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