# Artificial Neural Networks for Galaxy Clustering:
Learning from the two-point correlation function of BOSS galaxies

Niccoló Veronesi<EMAIL_ADDRESS>Federico Marulli  Alfondo
Veropalumbo  Lauro Moscardini  Leiden Observatory, Leiden University, PO Box
9513, 2300 RA Leiden, The Netherlands  Dipartimento di Fisica e Astronomia
“Augusto Righi” - Alma Mater Studiorum Università di Bologna, via Piero
Gobetti 93/2, I-40129 Bologna, Italy  INAF - Osservatorio di Astrofisica e
Scienza dello Spazio di Bologna, via Piero Gobetti 93/3, I-40129 Bologna,
Italy  INFN - Sezione di Bologna, viale Berti Pichat 6/2, I-40127 Bologna,
Italy  Dipartimento di Fisica, Università degli Studi Roma Tre, via della
Vasca Navale 84, I-00146 Rome, Italy  INFN - Sezione di Roma Tre, via della
Vasca Navale 84, I-00146 Rome, Italy

###### Abstract

The increasingly large amount of cosmological data coming from ground-based
and space-borne telescopes requires highly efficient and fast enough data
analysis techniques to maximise the scientific exploitation. In this work, we
explore the capabilities of supervised machine learning algorithms to learn
the properties of the large-scale structure of the Universe, aiming at
constraining the matter density parameter, $\Omega_{m}$. We implement a new
Artificial Neural Network for a regression data analysis, and train it on a
large set of galaxy two-point correlation functions in standard cosmologies
with different values of $\Omega_{m}$. The training set is constructed from
log-normal mock catalogues which reproduce the clustering of the Baryon
Oscillation Spectroscopic Survey (BOSS) galaxies. The presented statistical
method requires no specific analytical model to construct the likelihood
function, and runs with negligible computational cost, after training. We test
this new Artificial Neural Network on real BOSS data, finding
$\Omega_{m}=0.309\pm 0.008$, which is consistent with standard analysis
results.

###### keywords:

cosmology: theory , cosmology: observations , cosmology: large-scale structure
of universe , methods: statistical

††journal: Astronomy and Computing

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##  1 Introduction

One of the biggest challenges of modern cosmology is to accurately estimate
the standard cosmological model parameters and, possibly, to discriminate
among alternative cosmological frameworks. Fast and accurate statistical
methods are required to maximise the scientific exploitation of the
cosmological probes of the large-scale structure of the Universe. During the
last decades, increasingly large surveys have been conducted both with ground-
based and space-borne telescopes, and a huge amount of data is expected from
next-generation projects, like e.g. Euclid (Laureijs et al., 2011; Blanchard
et al., 2020) and the Vera C. Rubin Observatory (LSST Dark Energy Science
Collaboration, 2012). This scenario, in which the amount of available data is
expected to keep growing with exponential rate, suggests that machine learning
techniques shall play a key role in cosmological data analysis, due to the
fact that their reliability and precision strongly depend on the quantity and
variety of inputs they are given.

According to the standard cosmological scenario, the evolution of density
perturbations started from an almost Gaussian distribution, primarily
described by its variance. In configuration space, the variance of the field
is the two-point correlation function (2PCF), which is a function of the
magnitude of the comoving distance between objects, $r$. At large enough
scales, the matter distribution can still be approximated as Gaussian, and
thus the 2PCF contains most of the information of the density field.

The 2PCF and its analogous in Fourier space, the power spectrum, have been the
focus of several cosmological analyses of observed catalogues of extra-
galactic sources (see e.g. Totsuji and Kihara, 1969; Peebles, 1974; Hawkins et
al., 2003; Parkinson et al., 2012; Bel et al., 2014; Alam et al., 2016;
Pezzotta et al., 2017; Mohammad et al., 2018; Gil-Marín et al., 2020; Marulli
et al., 2021, and references therein). The standard way to infer constraints
on cosmological parameters from the measured 2PCF is by comparison with a
physically motivated model through an analytical likelihood function. The
latter should account for all possible observational effects, including both
statistical and systematic uncertainties. In this work, we investigate an
alternative data analysis method, based on a supervised machine learning
technique, which does not require a customized likelihood function to model
the 2PCF. Specifically, the 2PCF shape will be modelled with an Artificial
Neural Network (NN), trained on a sufficiently large set of measurements from
mock data sets generated at different cosmologies.

Supervised machine learning methods can be considered as a complementary
approach to standard data analysis techniques for the investigation of the
large-scale structure of the Universe. There are two main issues in applying
machine learning algorithms in this context. Firstly, the simulated galaxy
maps used to train the NNs have to be sufficiently accurate at all scales of
interest, including all the relevant observational effects characterizing the
surveys to be analysed. Indeed, the NN outputs rely only on the examples
provided for the training phase, and not according to any specific
instructions (Samuel, 1959; Goodfellow et al., 2016). The accuracy of the
output depends on the level of reliability of the training set, while the
precision increases when the amount of training data set increases. Moreover,
significant computational resources are required to construct the mock data
sets in a high enough number of test cosmologies.

On the other hand, both the construction of the mock data sets and the NN
training and validation have to be done only once. After that, the NN can
produce outputs with negligible computational cost, and can exploit all
physical scales on which the mock data sets are reliable, that is possibly
beyond the domain in which an analytic likelihood is available. These
represent the key advantages of this novel, complementary data analysis
technique.

Machine learning based methods should provide an effective tool in
cosmological investigations based on increasingly large amounts of
cosmological data from extra-galactic surveys (e.g. Cole et al., 2005;
Parkinson et al., 2012; Anderson et al., 2014; Kern et al., 2017; Ntampaka et
al., 2019; Ishida, 2019; Hassan et al., 2020; Villaescusa-Navarro, 2021). In
fact, these techniques have already been exploited for the analysis of the
large-scale structure of the Universe (see e.g. Aragon-Calvo, 2019; Tsizh et
al., 2020). In some cases, machine learning models have been trained and
tested on simulated mock catalogues to obtain as output the cosmological
parameters those simulations had been constructed with (e.g. Ravanbakhsh et
al., 2016; Pan et al., 2020). These works demonstrated that the machine
learning approach is powerful enough to infer tight constraints on
cosmological model parameters, when the training set is the distribution of
matter in a three-dimensional grid.

The method we are presenting in this work is different in this respect, as our
input training set consists of 2PCF measurements estimated from mock galaxy
catalogues. The rationale of this choice is to help the network to efficiently
learn the mapping between the cosmological model and the corresponding galaxy
catalogue by exploiting the information compression provided by the second-
order statistics of the density field. The goal of this work is to implement,
train, validate and test a new NN of this kind, and to investigate its
capability in providing constraints on the matter density parameter,
$\Omega_{m}$, from real galaxy clustering data sets.

The construction of proper training samples would require running N-body or
hydrodynamic simulations in a sufficiently large set of cosmological
scenarios. As already noted, this task demands substantial computational
resources and a dedicated work, which is beyond the scope of the current
analysis. A forthcoming paper will be dedicated to this fundamental task. Here
instead we rely on log-normal mock catalogues, which can reproduce the
monopole of the redshift-space 2PCF of real galaxy catalogues with reasonable
accuracy and in a minimal amount of time. This will allow us to test our NN on
training data sets with the desired format, in order to be ready for
forthcoming analyses with more reliable data sets. Moreover, future
developments of the presented method will involve estimating a higher number
of cosmological parameters at the same time. The implemented NN is provided
through a public Google Colab notebook111The notebook is available at: Colab..

The paper is organised as follows. In Section 2 we give a general overview of
the data analysis method we present in this work. In Section 3 we describe in
detail the characteristics of the catalogues used to train, validate and test
the NN. The specifics on the NN itself, together with its results on the test
set of mock catalogues are described in Section 4. The application to the BOSS
galaxy catalogue and the results this leads to are presented in Section 5. In
Section 6 we draw our conclusions and discuss about possible future
improvements. Finally, A provides details on the algorithms used to construct
the log-normal mock catalogues.

##  2 The data analysis method

The data analysis method considered in this work exploits a supervised machine
learning approach. Our general goal is to extract cosmological constraints
from extra-galactic redshift surveys with properly trained NNs. As a first
application of this method, we focus the current analysis on the 2PCF of BOSS
galaxies, which is exploited to extract constraints on $\Omega_{m}$.

The method consists of a few steps. Firstly, a fast enough process to
construct the data sets with which train, validate and test the NN is needed.
In this work we train the NN with a set of 2PCF mock measurements obtained
from log-normal catalogues. The construction of these input data sets is
described in detail in Section 3. Specifically, we create several sets of mock
BOSS-like catalogues assuming as values for the cosmological parameters the
ones inferred from the Planck Cosmic Microwave Background observations, except
for $\Omega_{m}$, that assumes different values in different mock catalogues,
and $\Omega_{\Lambda}$ that has been changed every time in order to have
$\Omega_{tot}=\Omega_{m}+\Omega_{\Lambda}+\Omega_{rad}=1$, where
$\Omega_{\Lambda}$ and $\Omega_{rad}$ are the dark energy and the radiation
density parameters, respectively. Specifically, we fix the main parameters of
the $\Lambda$-cold dark matter ($\Lambda$CDM) model to the following values:
$\Omega_{b}\,h^{2}=0.02237$, $\Omega_{c}\,h^{2}=0.1200$,
$100\,\Theta_{MC}=1.04092$, $\tau=0.0544$,
$\textrm{ln}(10^{10}\,A_{s})=3.044$, and $n_{s}=0.9649$ (Aghanim et al., 2018,
Table 2, TT,TE+lowE+lensing). Here $h$ indicates one hundredth of the Hubble
constant, $H_{0}$.

The implemented NN performs a regression analysis, that is it can assume every
value within the range the algorithm has been trained for, that in our case is
$0.24<\Omega_{m}<0.38$. In particular, the algorithm takes as input a 2PCF
monopole measurement, and provides as output a Gaussian probability
distribution on $\Omega_{m}$, from which we can extract the mean and standard
deviation (see Section 3.4 and A). Specifically, the data sets used for
training, validating and testing the NN consist of different sets of 2PCF
monopole measures, labelled by the value of $\Omega_{m}$ assumed to construct
the mock catalogues these measures have been obtained from, and assumed also
during the measure itself.

After the training and validation phases, we test the NN with a set of input
data sets constructed with random values of $\Omega_{m}$, that is, different
values from the ones considered in the training and validation sets. The
structure of the NN, the characteristics of the different sets of data it has
been fed with, and how the training process of the NN has been led are
described in Section 4.

Finally, once the NN has proven to perform correctly on a test set of 2PCF
measures from the BOSS log-normal mock catalogues, we exploit it on the real
2PCF of the BOSS catalogue. To measure the latter sample statistic, a
cosmological model has to be assumed, which leads to geometric distortions
when the assumed cosmology is different to the true one. To test the impact of
these distortions on our final outcomes, we repeat the analysis measuring the
BOSS 2PCF assuming different values of $\Omega_{m}$. We find that our NN
provides in output statistically consistent results independently of the
assumed cosmology. The results of this analysis are described in detail in
Section 5.

We provide below a summary of the main steps of the data analysis method
investigated in this work:

  1. 1.

Assume a cosmological model. In the current analysis, we assume a value of
$\Omega_{m}$, and fix all the other main parameters of the $\Lambda$CDM
cosmological model to the Planck values. A more extended analysis is planned
for a forthcoming work.

  2. 2.

Measure the 2PCF of the real catalogue to be analysed and use it to estimate
the large-scale galaxy bias. This is required to construct mock galaxy
catalogues with the same clustering amplitude of the real data. Here we
estimate the bias from the redshift-space monopole of the 2PCF at large scales
(see Section 3.3).

  3. 3.

Construct a sufficiently large set of mock catalogues. In this work we
consider log-normal mock catalogues, that can be obtained with fast enough
algorithms. The measured 2PCFs of these catalogues will be used for the
training, the validation and the test of the NN.

  4. 4.

Repeat all the above steps assuming different cosmological models. Different
cosmological models, characterised by different values of $\Omega_{m}$, are
assumed to create different classes of mock catalogues and measure the 2PCF of
BOSS galaxies.

  5. 5.

Train and validate the NN.

  6. 6.

Test the NN. This has to be done with data sets constructed considering
cosmological models not used for the training and validation, to check whether
the model can make reliable predictions also on previously unseen examples.

  7. 7.

Exploit the trained NN on real 2PCF measurements. This is done feeding the
trained machine learning model with the several measures of the 2PCF of the
real catalogue, obtained assuming different cosmological models. The reason
for this is to check whether the output of the NN is affected by geometric
distortions.

##  3 Creation of the data set

###  3.1 The BOSS data set

The mock catalogues used for the training, validation and test of our NN are
constructed to reproduce the clustering of the BOSS galaxies. BOSS is part of
the Sloan Digital Sky Survey (SDSS), which is an imaging and spectroscopic
redshift survey that used a $2.5$m modified Ritchey-Chrétien altitude-azimuth
optical telescope located at the Apache Point Observatory in New Mexico (Gunn
et al., 2006). The data we have worked on are from the Data Release 12 (DR12),
that is the final data release of the third phase of the survey (SDSS-III)
(Alam et al., 2015). The observations were performed from fall 2009 to spring
2014, with a 1000-fiber spectrograph at a resolution $R\approx 2000$. The
wavelength range goes from $360$ nm to $1000$ nm, and the coverage of the
survey is $9329$ square degrees.

The catalogue from BOSS DR12, used in this work, contains the positions in
observed coordinates (RA, Dec and redshift) of $1\,198\,004$ galaxies up to
redshift $z=0.8$.

Both the data and the random catalogues are created considering the survey
footprint, veto masks and systematics of the survey, as e.g. fiber collisions
and redshift failures. The method used to construct the random catalogue from
the BOSS spectroscopic observations is detailed in Reid et al. (2016).

###  3.2 Two-point correlation function estimation

We estimate the 2PCF monopole of the BOSS and mock galaxy catalogues with the
Landy and Szalay (1993) estimator:

$\xi(s)=1+\frac{N_{\rm RR}}{N_{\rm DD}}\frac{DD(s)}{RR(s)}-2\frac{N_{\rm
RR}}{N_{\rm DR}}\frac{DR(s)}{RR(s)}\;,$ (1)

where $DD(s)$, $RR(s)$ and $DR(s)$ are the number of galaxy-galaxy, random-
random and galaxy-random pairs, within given comoving separation bins (i.e. in
$s\pm\Delta s$), respectively, while $N_{\rm DD}=N_{D}(N_{D}-1)/2$, $N_{\rm
RR}=N_{R}(N_{R}-1)/2$ and $N_{\rm DR}=N_{D}N_{R}$ are the corresponding total
number of galaxy-galaxy, random-random and galaxy-random pairs, being $N_{D}$
and $N_{R}$ the number of objects in the real and random catalogues. The
random catalogue is constructed with the same angular and redshift selection
functions of the BOSS catalogue, but with a number of objects that is ten
times bigger than the observed one, to minimise the impact of Poisson errors
in random pair counts.

###  3.3 Galaxy bias

As introduced in the previous Sections, we train our NN to learn the mapping
between the 2PCF monopole shape, $\xi$, and the matter density parameter,
$\Omega_{m}$. Thus, we construct log-normal mock catalogues assuming different
values of $\Omega_{m}$. A detailed description of the algorithm used to
construct these log-normal mocks is provided in the next Section 3.4.

Firstly, we need to estimate the bias of the objects in the sample, $b$. We
consider a linear bias model, whose bias value is estimated from the data.
Specifically, when a new set of mock catalogues (characterised by
$\Omega_{m}=\Omega_{m,i}$) is constructed, a new linear galaxy bias has to be
estimated. The galaxy bias is assessed by modelling the 2PCF of BOSS galaxies
in the scale range $30<s\,[h^{-1}\,\mbox{Mpc}]<50$, where $s$ is used here,
instead of $r$, to indicate separations in redshift space. We consider a
Gaussian likelihood function, assuming a Poissonian covariance matrix which is
sufficiently accurate for the purposes of this work. We model the shape of the
redshift-space 2PCF at large scale as follows (Kaiser, 1987):

$\xi_{\rm
gal}(s)=\left[(b\sigma_{8})^{2}+\frac{2}{3}f\sigma_{8}+\frac{1}{5}(f\sigma_{8})^{2}\right]\frac{1}{\sigma_{8}^{2}}\xi_{m}(r)\;,$
(2)

where the matter 2PCF, $\xi_{m}(r)$, is obtained by Fourier transforming the
matter power spectrum modelled with the Code for Anisotropies in the Microwave
Background (CAMB) (Lewis et al., 2000). The product between the linear growth
rate and the matter power spectrum normalisation parameter, $f\sigma_{8}$, is
set by the cosmological model assumed during the measuring of the 2PCF and the
construction of the theoretical model. The values of all redshift-dependent
parameters have been calculated using the mean redshift of the data catalogue,
$z=0.481$.

We assume a uniform prior for $b\sigma_{8}$ between $0$ and $2.7$. The
posterior has been sampled with a Markov Chain Monte Carlo algorithm with
$10\,000$ steps and a burn-in period of $100$ steps. The linear bias, $b$, is
then derived by dividing the posterior median by $\sigma_{8}$. The latter is
estimated from the matter power spectrum computed assuming Planck cosmology.
We note that this method implies that only the shape of the 2PCF is actually
used to constrain $\Omega_{m}$.

###  3.4 Mock galaxy catalogues

As described in Section 2, the data sets used for the training, validation and
test of the NN implemented in this work consist of 2PCF mock measurements
estimated from log-normal galaxy catalogues. Log-normal mock catalogues are
generally used to create the data set necessary for the covariance matrix
estimate, in particular in anisotropic clustering analyses (see e.g. Lippich
et al., 2019). In fact, this technique allows us to construct density fields
and, thus, galaxy catalogues with the required characteristics in an extremely
fast way, especially if compared to N-body simulations, though the latter are
more reliable at higher-order statistics and in the fully nonlinear regime.

To construct the log-normal mock catalogues, we adopted the same strategy
followed by Beutler et al. (2011). We provide here a brief explanation of the
method, while a more detailed description is given in A. As already commented
in Section 1, log-normal catalogues do not provide the optimal training sets
for this machine learning method, but are used here only to test the
algorithms with input data with the desired format.

The algorithm used in this work to generate these mock catalogues for the
machine learning training process takes as input one data catalogue and one
corresponding random catalogue. These are used to define a grid with the same
geometric structure of the data catalogue and a visibility mask function which
is constructed from the pixel density of the random catalogue. A cosmological
model has to be assumed to compute the object comoving distances from the
observed redshifts and to model the matter power spectrum. The latter is used
to estimate the galaxy power spectrum, which is the logarithm of the variance
of the log-normal random field.

The density field is then sampled from this random field. Specifically, once
the algorithm has associated to all the grid cells their density value, we can
extract from each of them a certain number of points that depend on the
density of the catalogue and on the visibility function. These points
represent the galaxies of the output mock catalogue.

Once the mocks are created, the same cosmological model is assumed also to
measure the 2PCF. We repeated the same process for all the test values of
$\Omega_{m}$. That is, for each $\Omega_{m}$ we created a set of log-normal
mock catalogues and measured the 2PCF. This is different with respect to
standard analyses, where a cosmology is assumed only once for the clustering
measurements, and geometric distortions are included in the likelihood.

As an illustrative example, Figure 1 shows different measures of the 2PCF
obtained in the scale range $8<s\,[h^{-1}\,\mbox{Mpc}]<50$, in $30$
logarithmic bins of $s$, and assuming the lowest and the highest values of
$\Omega_{m}$ that have been considered in this work, that are
$\Omega_{m}=0.24$ and $\Omega_{m}=0.38$. The black and red dots show the
measures obtained from the corresponding two sets of $50$ log-normal mock
galaxy catalogues, while the dashed lines are the theoretical 2PCF models
assumed to construct them. As expected, the average 2PCFs of the two sets of
log-normal catalogues are fully consistent with the corresponding theoretical
predictions. Indeed, a mock log-normal catalogue characterised by
$\Omega_{m}=\Omega_{m,i}$ provides a training example for the NN describing
how galaxies would be distributed if $\Omega_{m,i}$ was the true value.
Finally, the blue and green squares show the 2PCFs of the real BOSS catalogue
obtained by assuming $\Omega_{m}=0.24$ and $\Omega_{m}=0.38$, respectively,
when converting galaxy redshifts into comoving coordinates. The differences in
the latter two measures are caused by geometric distortions. As can be seen,
neither of these two data sets are consistent with the corresponding 2PCF
theoretical models, that is both $\Omega_{m}=0.24$ and $\Omega_{m}=0.38$
appear to be bad guesses for the real value of $\Omega_{m}$. As we will show
in Section 5, the NN presented in this work is not significantly affected by
geometric distortions.

Figure 1: Different measures of the 2PCF monopole. The dots represent the
measures obtained from $50$ log-normal mock catalogues constructed with
$\Omega_{m}=0.24$ (black dots) and $\Omega_{m}=0.38$ (red dots). The black and
red dashed lines show the corresponding theoretical 2PCF models. The blue and
green squares show the real BOSS 2PCFs measured assuming the same cosmologies
of the mock catalogues, that is $\Omega_{m}=0.24$ and $\Omega_{m}=0.38$,
respectively, when converting redshifts into comoving coordinates.

##  4 The Artificial Neural Network

###  4.1 Architecture

Regression machine learning models take as input a set of variables that can
assume every value and provide as output a continuous value. In our case, the
input data set is a 2PCF measure, while the output is the predicted Gaussian
probability distribution of $\Omega_{m}$. Specifically, the regression model
we are about to describe has been trained with 2PCF measures in the scale
range $8-50h^{-1}\,\mbox{Mpc}$, in $30$ logarithmic scale bins.

The architecture of the implemented NN is schematically represented in Figure
2. It consists of the following layers:

  * 1.

Input layer that feeds the $30$ values of the measured 2PCF to the model;

  * 2.

Dense layer222A layer is called dense if all its units are connected to all
the units of the previous layer. with 2 units. Every unit of this layer
performs the following linear operation on the input elements:

$y=\sum_{i}w_{i}\cdot x_{i}+b\ \ ,$ (3)

where $y$ is the output of the unit, $x_{i}$ is the $i$-th element of the
input, $w_{i}$ is its corresponding weight, and $b$ is the intercept of the
linear operation. No nonlinear activation function has been used in this
layer. This choice has been proven to be convenient a posteriori, providing
excellent performance of the NN during the test phase (see Section 4.3). We
thus decided not to introduce any activation function to keep the architecture
of the regression model as simple as possible;

  * 3.

Distribution layer that uses the two outputs of the dense layer as mean and
standard deviation to parameterise a Gaussian distribution, which is given as
output.

Figure 2: Schematic representation of the regression model considered in this
work. The input layer is represented by blue dots, the hidden dense layer by
green dots, and the output layer by the orange dot.

This architecture has been chosen because it is the simplest one we tested
which is able to provide accurate cosmological constraints. Deeper models have
been tried out, but no significant differences were spotted in the output
predictions.

###  4.2 Training and validation

The training and validation sets are constructed separately, and consist of
$2\,000$ and $800$ examples, respectively. Regression machine learning models
work better if the range of possible outputs is well represented in the
training and validation sets. We construct mock catalogues with $40$ different
values of $\Omega_{m}$: $29$ from $\Omega_{m}=0.24$ to $\Omega_{m}=0.38$ with
$\Delta\Omega_{m}=0.005$ and $11$ from $\Omega_{m}=0.2825$ to
$\Omega_{m}=0.3325$, also separated by $\Delta\Omega_{m}=0.005$. The latter
are added to improve the density of inputs in the region that had proven to be
the one where the predictions were more likely, during the first attempts of
the NN training. The training and validation sets consist of $50$ and $20$
2PCF measures for each value of $\Omega_{m}$, respectively. All the mock
catalogues used for the training and validation have the same dimension of the
BOSS 2PCF measures.

The loss function used for the training process is the following:

$J=-\sum_{i=1}^{N}\log\left[\frac{1}{\sqrt{2\pi}\sigma_{i}}\exp{-\frac{(l_{i}-\mu_{i})^{2}}{2\sigma_{i}^{2}}}\right]\;,$
(4)

where $N$ is the number of examples, $l_{i}$ is the label correspondent to the
$i$-th 2PCF used for the training or the validation (i.e. the true value of
$\Omega_{m}$ that has been used to create the $i$-th mock catalogue and to
measure its 2PCF), while $\sigma_{i}$ and $\mu_{i}$ are the standard deviation
and the mean of the Normal probability distribution function that is given as
output for the same $i$-th 2PCF example.

During the training process, we apply the Adam optimisation (see Kingma and
Ba, 2014, for a detailed description of its functioning and parameters) with
the following three different steps:

  * 1.

750 epochs with $\eta=0.002$ ,

  * 2.

150 epochs with $\eta=0.001$ ,

  * 3.

100 epochs with $\eta=0.0005$ ,

where $\eta$ indicates the learning rate. During the three steps, the values
of the other two parameters of this optimisation algorithm are kept fixed to
$\beta_{1}=0.9$ and $\beta_{2}=0.999$, and the training set was not divided
into batches. Variations in the parameters of the optimization have been
performed and did not lead to significantly different outputs.

Gradually reducing the learning rate during the training helps the model to
find the correct global minimum of the loss function (Ntampaka et al., 2019).
The first epochs, having a higher learning rate, lead the model towards the
area surrounding the global minimum, while the last ones, having a lower
learning rate, and therefore being able to take smaller and more precise steps
in the parameter space, have the task to lead the model towards the bottom of
that minimum. In this model, minimizing the loss function means both to drive
the mean values of the output Gaussian distributions towards the values of the
labels (i.e. the values of $\Omega_{m}$ corresponding to the inputs), and to
reduce as much as possible the standard deviation of those distributions.

During the training process, the interval of the labels, that is
$0.24\leq\Omega_{m}\leq 0.38$, has been mapped into $[0,1]$ through the
following linear operation:

$L=\frac{l-0.24}{0.14}\;,$ (5)

where $l$ is the original label and $L$ is the one that belongs to the $[0,1]$
interval. Once the mean, $\mu_{L}$, and the standard deviation, $\sigma_{L}$,
of the predictions are obtained in output, we convert them back to match the
original interval of the labels. The standard deviation is then computed as
follows:

$\frac{d\mu_{L}}{d\sigma_{L}}=\frac{d\mu_{l}}{d\sigma_{l}}\;,$ (6)

where $\sigma$ is the standard deviation of $\mu$, so that the uncertainty on
$\mu$ is given by $0.14\,\sigma_{L}$.

###  4.3 Test

The test set consists of $5$ different 2PCF measures obtained from log-normal
mock catalogues with $16$ different randomly generated values of $\Omega_{m}$.
Therefore, we have a total of $80$ measures. The $\Omega_{m}$ predictions and
uncertainties are estimated as the mean and standard deviation of the Gaussian
distributions obtained in output (Matthies, 2007; Der Kiureghian and
Ditlevsen, 2009; Kendall and Gal, 2017; Russell and Reale, 2019). Our model is
thus able to associate to every point of the input space an uncertainty on the
output that depends on the intrinsic scatter of the 2PCF measures.

Figure 3: Values of $\Omega_{m}$ and related uncertainties predicted by our
NN, compared with the best-fit linear model (dashed line). The best-fit slope
is $0.995\pm 0.014$, while the best-fit normalisation is $0.001\pm 0.005$. The
error bars in this plot represent the standard deviations of the distributions
for $\Omega_{m}$ obtained as output of the NN

Figure 3 shows the predictions on the test set, compared to the true values of
$\Omega_{m}$ the mocks in the test were constructed with. As can be seen, the
implemented NN is able to provide reliable predictions when fed with measures
from mock catalogues characterised by $\Omega_{m}$ values that were not used
during the training and validation phases. In fact, fitting this data set with
a linear model:

$\Omega_{m}^{\rm pred}=\alpha\cdot\Omega_{m}^{\rm true}+\beta\;,$ (7)

we get $\alpha=0.995\pm 0.014$ and $\beta=0.001\pm 0.005$, which are
consistent with the slope and the intercept of the bisector of the quadrant.
Figure 3 also shows that the estimates near the limits of the training range
at $\Omega_{m}=0.24$ and $0.38$ are not biased, as the predictions on the test
set, also for the most external examples, are consistent with the bisector of
the quadrant.

##  5 Application to BOSS data

After training, validating and testing the NN, we can finally apply it to the
real 2PCF of the BOSS galaxy catalogue. To test the impact of geometric
distortions caused by the assumption of a particular cosmology during the
measure, we apply the NN on $40$ measures of the 2PCF of BOSS galaxies
obtained with different assumptions on the value of $\Omega_{m}$ when
converting from observed to comoving coordinates.

Figure 4 shows the results of this analysis, compared with the $\Omega_{m}$
constraints provided by Alam et al. (2017).

Figure 4: Machine learning predictions (black dots and bars) on $\Omega_{m}$
from the measured 2PCF of BOSS galaxies, as a function of the $\Omega_{m}$
assumed in the measurements. The predictions are compared to the $\Omega_{m}$
constraints provided by Alam et al. (2017), here represented by the green
stripe.

The predictions of the NN have been fitted with a linear model:

$\Omega_{m}^{\rm pred}=\alpha\cdot\Omega_{m}^{\rm ass}+\beta\;,$ (8)

where $\Omega_{m}^{\rm pred}$ is the prediction of the regression model, while
$\Omega_{m}^{\rm ass}$ is the value assumed to measure the 2PCF. The best fit-
values of the parameters we get are $\alpha=-0.001\pm 0.025$ and
$\beta=0.309\pm 0.008$. In particular, the slope is consistent with zero, that
is all the $\Omega_{m}$ predictions are consistent, within the uncertainties,
independently of the value assumed in the measurement. This demonstrates that
the NN is indeed able to make robust predictions over observed data, without
being biased by geometric distortions.

Our final $\Omega_{m}$ constraint is thus estimated from the best-fit
normalization, $\beta$, that is

$\Omega_{m}=0.309\pm 0.008\;.$ (9)

This result is consistent with the one obtained by Alam et al. (2017), which
is $\Omega_{m}=0.311\pm 0.006$.

##  6 Conclusions

In this work we investigated a supervised machine learning data analysis
method aimed at constraining $\Omega_{m}$ from observed galaxy catalogues.
Specifically, we implemented a regression NN that has been trained and
validated with mock measurements of the 2PCFs of BOSS galaxies. Such measures
are used as a convenient summary statistics of the large-scale structure of
the Universe. The goal of this work was to infer cosmological constraints
without relying on any analytic 2PCF model to construct the likelihood
function.

To train and validate our NN, we use $2\,800$ 2PCF examples, constructed with
$40$ different values of $\Omega_{m}$, in $0.24\leq\Omega_{m}\leq 0.38$. The
trained NN has been finally applied to the real 2PCF monopole of the BOSS
galaxy catalogue. We get $\Omega_{m}=0.309\pm 0.008$, which is in good
agreement with the value found by Alam et al. (2017).

This work confirms that NNs can be powerful tools also for cosmological
inference analyses, complementing the more standard analyses that make use of
analytical likelihoods. One obvious improvement of the presented work would be
to consider more accurate mock catalogues than the log-normal ones, for the
training, validation and test phases. In particular, N-body or hydrodynamic
simulations would be required to exploit higher-order statistics, as in
particular the three-point correlation function, as features to feed the model
with.

The higher the number of reliable features is, the more accurate the
predictions of the NN will be. Furthermore, multi-labelled regression models
can be used to make predictions on multiple cosmological parameters at the
same time. To do that, however, bigger data sets are required, in order to
have a proper mapping of the input space, characterised by the different
values each label can have. The analysis presented in this work should also be
extended to larger scales, though in this case more reliable mock catalogues
are required not to introduce biases in the training, in particular at the
Baryon Acoustic Oscillations scales. Finally, a similar analysis as the one
performed in this work could be done using the density map of the catalogue,
or directly the observed coordinates of the galaxies. This approach would be
less affected by the adopted data compression methods considered. On the other
hand, it would have a significantly larger computational cost for the
training, validation and test phases, which should be estimated with a
dedicated feasibility study. All the possible improvements described above
will be investigated in forthcoming papers.

## Acknowledgements

We thank the anonymous referee for her/his comments that helped to improve the
presentation of our results. We acknowledge the use of computational resources
from the parallel computing cluster of the Open Physics Hub
(https://site.unibo.it/openphysicshub/en) at the Physics and Astronomy
Department in Bologna. FM and LM acknowledge the grants ASI n.I/023/12/0 and
ASI n.2018-23-HH.0. LM acknowledges support from the grant PRIN-MIUR 2017
WSCC32.

Software: Numpy (Harris et al., 2020); Matplotlib (Hunter, 2007); SciPy
(Virtanen et al., 2020); CosmoBolognaLib (Marulli et al., 2016); CAMB (Lewis
et al., 2000); FFTLog (Hamilton, 2000); TensorFlow (Abadi et al., 2015); Keras
(Chollet et al., 2015).

##  Appendix A Log-normal mock catalogues

In the following, we describe the algorithm used in this work to construct
log-normal mock catalogues. We followed the same strategy as in Beutler et al.
(2011, see their Appendix B). Let us define a random field in a given volume
as a field whose value at the position $\vec{r}$ is a random variable
(Peebles, 1993; Xavier et al., 2016). One example is the Gaussian random
field, $N(\vec{r})$. In this case the one-point probability density function
(PDF) is a Gaussian distribution, fully characterised by the mean, $\mu$, and
the variance, $\sigma^{2}$. If $n$ positions are considered, instead of just
one, the PDF is the multivariate Gaussian distribution:

$f_{n}(\vec{N})=\frac{1}{(2\pi)^{n/2}|\boldsymbol{M}|^{1/2}}\exp\left[-\frac{1}{2}\sum_{i,j}\boldsymbol{M}_{ij}^{-1}N_{i}N_{j}\right]\;,$
(10)

where $N_{i}=N(\vec{r}_{i})$, and $\boldsymbol{M}$ is the covariance matrix,
the elements of which are defined as follows:

$\boldsymbol{M}_{ij}=\langle(N_{i}-\mu)(N_{j}-\mu)\rangle\;.$ (11)

The PDF of the primordial matter density contrast at a specific position,
$\delta(\mathbf{x})$, can be approximated as a Gaussian distribution, with
null mean and the correlation function as variance. The same definitions can
be used in Fourier space as well. In this case the variance is the Fourier
transform of the 2PCF, that is the power spectrum, $P(k)$. In more general
cases, the Gaussian random field is only an approximation of the real random
field that may present features, such as significant skewness and heavy tails
(Xavier et al., 2016).

Coles and Barrow (1987) showed how to construct non-Gaussian fields through
nonlinear transformations of a Gaussian field. One example is the log-normal
random field (Coles and Jones, 1991), which can be obtained through the
following transformation:

$L(\vec{r})=\exp[N(\vec{r})]\;.$ (12)

The log-normal transformation results in the following one-point PDF:

$f_{1}(L)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{(\log(L)-\mu)^{2}}{2\sigma^{2}}\right]\frac{dL}{L}\;,$
(13)

where $\mu$ and $\sigma^{2}$ are the mean and the variance of the underlying
Gaussian field $N$, respectively.

The multivariate version for the log-normal random field is defined as
follows:

$f_{n}(\vec{L})=\frac{1}{(2\pi)^{n/2}|\boldsymbol{M}|^{1/2}}\exp\left[-\frac{1}{2}\sum_{i,j}\boldsymbol{M}_{ij}^{-1}\log(L_{i})\log(L_{j})\right]\prod_{i=1}^{n}\frac{1}{L_{i}}\;,$
(14)

where $\boldsymbol{M}$ is the covariance matrix of the N-values.

To construct the log-normal mock catalogues used to train, validate and test
our NN, we start from a power spectrum template, assuming a value for the
bias. After creating a grid according to this power spectrum, a Fourier
transform is performed to obtain the 2PCF, which is used to calculate the
function $\log[1+\xi(r)]$. We then revert to Fourier-space obtaining a
modified power spectrum, $P_{ln}(k)$, and assign to each point of the grid a
value of the Fourier amplitude, $\delta(k)$, sampled from a Normal
distribution that has $P_{ln}(k)$ as standard deviation. A new Fourier
transform is performed to obtain a density field $\delta(x)$ from which we
sample the galaxy distribution. In our calculations we used a grid size of
$10\,\mbox{Mpc}\,h^{-1}$, while the minimum and the maximum values of $k$ are
$0.0001\,h\,\mbox{Mpc}^{-1}$ and $100\,h\,\mbox{Mpc}^{-1}$, respectively.

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