# How complex are galaxies? A non-parametric estimation of the intrinsic dimensionality of wide-band photometric data

Corentin Cadiou, Clotilde Laigle, Oscar Agertz

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Corentin Cadiou, Clotilde Laigle, Oscar Agertz. How complex are galaxies? A non-parametric estimation of the intrinsic dimensionality of wide-band photometric data. Monthly Notices of the Royal Astronomical Society, 2025, 537, pp.1869-1878. 10.1093/mnras/staf139. insu-05101694

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# How complex are galaxies? A non-parametric estimation of the intrinsic dimensionality of wide-band photometric data

Corentin Cadiou<sup>1,2*</sup> Clotilde Laigle<sup>2</sup> and Oscar Agertz<sup>1</sup>

Accepted 2025 January 22. Received 2024 December 3; in original form 2024 April 4

## ABSTRACT

Galaxies are complex objects, yet the number of independent parameters to describe them remains unknown. We present here a non-parametric method to estimate the intrinsic dimensionality of large data sets. We apply it to wide-band photometric data drawn from the COSMOS2020 catalogue and a comparable mock catalogue from the HORIZON-AGN simulation. Our galaxy catalogues are limited in signal-to-noise ratio (SNR) in all optical and near-infrared bands. Our results reveal that most of the variance in the wide-band photometry of this galaxy sample can be described with at most 4.3 ± 0.5 independent parameters for star-forming galaxies and 2.9 ± 0.2 for passive ones, both in the observed and simulated catalogues. We identify one of these parameters to be noise-driven, and recover that stellar mass and redshift are two key independent parameters driving the magnitudes. Our findings support the idea that wide-band photometry does not provide more than one additional independent parameter for star-forming galaxies. Although our sample is not mass-limited and may miss some passive galaxies due to our cut in SNR, our work suggests that dimensionality reduction techniques may be effectively used to explore and analyse wide-band photometric data, provided the used latent space is at least four-dimensional.

**Key words:** methods: data analysis – galaxies: formation – galaxies: fundamental parameters.

## 1 INTRODUCTION

In the past decades, we have collected exquisite multiwavelength photometric data for millions of galaxies in the observable Universe, e.g. on Cosmological Evolution Survey (COSMOS, Scoville et al. 2007) and Subaru/XMM-Newton Deep Survey (SXDS, Mehta et al. 2018) fields, or through the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP, Aihara et al. 2018) and Dark Energy Survey (DES, Hartley et al. 2022) surveys, among others. The pace of observations is now accelerating and current and next generations of space-based instruments, including *JWST* (Gardner et al. 2006), *Euclid* (Laureijs et al. 2011), the Vera Rubin (Ivezić et al. 2019), and the Nancy Grace Roman (Spergel et al. 2015) observatories, will provide in the next few years broad-band optical and near-infrared fluxes for almost all galaxies in the observable Universe over at least the last 10 billion years of history. These photometric data sets represent a goldmine to constrain galaxy formation, evolution, and quenching from their photometry.

Galaxy photometric fluxes are directly sensitive to the complex age and metallicity distributions of the stellar population that comprise them. Traditionally, astronomers have focused on separating star-forming and passive populations based on two-dimensional (2D) colour–colour and colour–magnitude diagrams (CMDs; e.g. de Vaucouleurs 1961; Strateva et al. 2001; Bell et al. 2003; Bell et al. 2004; Arnouts et al. 2013). Recently, more sophisticated techniques

have been explored to efficiently reduce dimensionality while preserving the topology of data, trying to unfold as much as possible the high-dimensional space of galaxy colours into a single 2D manifold.

For example, self-organized maps (SOMs; Kohonen 1982, 2001) are now widely used as a way to circumvent SED fitting (see e.g. Hemmati et al. 2019). In this approach, an SOM is first built as an unsupervised classification of galaxies based on their photometric colours. In a second step, provided a representative fraction of galaxies have a label of the quantity that we want to predict, e.g. redshift, mass, and star formation rate (SFR), this information can be efficiently propagated to the rest of the sample. This technique was tested in simulations (e.g. Davidzon et al. 2019) and applied to observed data sets to predict redshift (Wilson et al. 2020), SFR (Davidzon et al. 2022), or redshift distribution in tomographic bins for weak lensing studies, such as DES and *Euclid* (see e.g. Buchs et al. 2019; Myles et al. 2021). As an efficient way to compare the multiwavelength colour distribution of two galaxy samples, SOMs are also used to improve the completeness of spectroscopic sample, the representativity of which is pivotal to calibrate the photometric redshifts (e.g. the C3R2 spectroscopic calibration sample for *Euclid*, see Masters et al. 2015; *Euclid* Collaboration 2022b). In addition to SOMs, other unsupervised dimensionality reduction techniques have been explored to study large data sets, such as t-SNE (Steinhardt et al. 2020), locally linear embedding (Vanderplas & Connolly 2009) or auto-encoders (Portillo et al. 2020).

Implicitly, these works assume that the manifold occupied by galaxies in photometric magnitude space can be projected in typically

<footnote n="1"><sup>1</sup>Lund Observatory, Division of Astrophysics, Department of Physics, Lund University, Box 43, SE-221 00 Lund, Sweden</footnote>

<footnote n="2"><sup>2</sup>Sorbonne Université, CNRS, UMR 7095, Institut d'Astrophysique de Paris, 98 bis Boulevard Arago, F-75014 Paris, France</footnote>

<footnote>*E-mail: corentin.cadiou@iap.fr</footnote>
two dimensions without significant loss of information. While this implicit assumption is convenient for visualization purposes, the choice of two dimensions is not based on any rigorous argument. Indeed, the question of the complexity of galaxy formation is still open. Several studies have used a principal component analysis (PCA) to quantify the number of parameters needed to capture the variance in the observed properties of galaxies, with the main findings being that only a handful are required [in observations of 21cm emission, Disney et al. (2008), in SDSS spectra, Sharbaf, Ferreras & Lahav (2023)]. One limitation of using PCA is, however, that it is a linear method, and thus cannot capture non-linear correlations between the observables, and should thus be interpreted as an upper limit on the actual dimensionality of the data. Similarly, machine learning approaches suggest that galaxies lie on a low-dimensionality manifold (Villaescusa-Navarro et al. 2022; Chawak et al. 2023; Echeverri-Rojas et al. 2023). Taken at face value, this is at odds with theory, where models of galaxy formation involve complex physical mechanisms (see e.g. Somerville & Davé 2015). Those depend, at least, on the mass of the halo, its angular momentum distribution, the impact parameter and timing of the mergers, and the non-linear physics of gas cooling, star formation, and stellar/AGN (active galactic nucleus) feedback processes, suggesting that galaxy formation unravels in high-dimensional parameter space.

The complexity of galaxy formation has, however, been recently questioned. Indeed, simulations have revealed that, at fixed galaxy mass, the scatter in properties such as disk sizes may be driven by the misalignment between the inertia tensor and tidal tensor in the initial conditions (Cadiou, Pontzen & Peiris 2022; Moon & Lee 2024), potentially explaining the origins of scatter in well-established scaling relations such as the size-magnitude relation (Courteau et al. 2007). In this picture, galaxy formation is a self-regulated process, where only a subvolume of the parameter space is accessible by the global properties of galaxies.

In this paper, we aim to answer the question of the dimensionality of the manifold occupied by galaxies in the multidimensional space of their photometric magnitudes. We rely on observations of hundreds of thousands of galaxies from the COSMOS catalogue, and simulated ones from the HORIZON-AGN simulation; we describe these data sets in Section 2. We then present our parameter-free method, based on the approach of Granata & Carnevale (2016), to estimate their intrinsic dimensionality in Section 3. We discuss our results in Section 4 and wrap up in Section 5.

# 2 CATALOGUE DATA

## 2.1 The COSMOS2020 catalogue

We rely on the COSMOS2020 catalogue (Weaver et al. 2022) to estimate the dimensionality of galaxy photometry. COSMOS2020 is a deep multiwavelength catalogue extracted from the COSMOS field. It includes the photometry in 35 filters from ultraviolet (UV) to infrared (IR). The improvement of this catalogue with previous versions is especially driven by deeper optical, near-infrared (NIR), and IR data obtained with the HSC camera (Miyazaki et al. 2018) on the Subaru telescope, the VIRCAM instrument on the VISTA telescope (as part of the UltraVISTA survey, see McCracken et al. 2012; Euclid Collaboration 2022a), and the IRAC instrument on the Spitzer telescope (as part of the SPLASH survey, see Steinhardt et al. 2014). In the following, we use the CLASSIC version of the catalogue, that is, the one that uses the photometry obtained with SExTRACTOR (Bertin & Arnouts 1996). More specifically, galaxies were identified

from a izYJHKs chi-squared image, and magnitudes are extracted in a 2 arcsec aperture reaching a 3σ-depth of 28.1 and 25.3 in the g-band and the K-band, respectively. A full description of the catalogue preparation can be found in Weaver et al. (2022). We remove from the sample all galaxies that have been flagged as being near bright sources and areas affected by artefacts. The total area after accounting for these masks is 1.27 deg². We use galaxy properties (redshifts, masses, and rest-frame UVJ) derived through the LEPHARE code (Ilbert et al. 2006).

## 2.2 The HORIZON-AGN mock catalogue

We work with a simulated catalogue to investigate the effect of noise in the data, and to explore how different is the dimensionality of galaxy photometry when estimated either from their apparent or absolute magnitudes.

The HORIZON-AGN hydrodynamical simulation (Dubois et al. 2014), run with the adaptive mesh refinement code RAMSES (Teyssier 2002), was used to produce a mock catalogue with the same properties as the observed COSMOS2020 sample. HORIZON-AGN has a box size of 100 Mpc/h and a DM mass resolution of 8 × 10⁷ M⊙. In brief, the simulation includes gas heating, cooling, star formation, feedback from stellar winds, type Ia and type II supernovae with mass, energy, and metal release in the interstellar medium, and black hole formation. Depending on the accretion rate, AGNs release energy in a quasar or radio mode. A light cone was extracted on the fly to build realistic mocks<fnref n="1" /> (Gouin et al. 2019; Laigle et al. 2019).

Galaxies were extracted based on the position of stellar particles using the ADAPTAHOP structure finder (Aubert, Pichon & Colombi 2004; Tweed et al. 2009). Mock spectra were produced for each galaxy, assuming Bruzual & Charlot (2003) synthetic stellar population models and a Chabrier initial mass function (Chabrier 2003). Dust attenuation was implemented for each star particle using the gas-metal mass distribution along the line of sight as a proxy for the dust distribution. All details about the spectra computation can be found in Laigle et al. (2019). In addition to spectra, both absolute and apparent magnitudes were extracted in the same photometric filters as available in the COSMOS2020 catalogue. Mock photometric fluxes were perturbed by assuming a combination of Poisson noise and Gaussian noise, considering in each filter the same depth as the COSMOS2020 catalogue, so that galaxies of similar luminosity have a similar signal-to-noise ratio (SNR) in the mock and in the observed catalogues.

Overall, the galaxy populations agree well with the observations, but some discrepancies must be mentioned (Kaviraj et al. 2017). HORIZON-AGN produces galaxies that are too massive for their halo, in particular at low masses and low redshift. As a result, the stellar-to-halo mass ratio is up to 10 times larger than expected observationally in haloes of mass 10¹<fnref n="1" /> M⊙ (Shuntov et al. 2022). Furthermore, massive passive galaxies at low redshift are not as numerous as in observations, due to residual star formation in massive galaxies, and low-mass galaxies are not star-forming enough (see e.g. Dubois et al. 2016).

<footnote n="1">¹A version of which being publicly available at Horizon-AGN Virtual Observatory.</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Colour–colour diagram used to distinguish passive from star-forming galaxies in the COSMOS2020 (**left**, NUV − *r* versus *r* − *J* rest-frame plane) and HORIZON-AGN (**right**, NUV − *r* versus *r* − *K*<sub>s</sub> rest-frame plane) catalogues. Here, we display galaxies in the redshift range 0 < *z* < 1 and with mass larger than 10<sup>9.5</sup> M<sub>☉</sub>. Passive galaxies are located in the top left corner.</figcaption></figure>

## 2.3 Data set selection

### 2.3.1 Filter set

We estimate the dimensionality of the galaxy distribu-
tion based on the following 14 fiducial broad-band filters: $u^*$, $g$, $r$, $i$, $z$, $y$, $Y$, $J$, $H$, $K_s$, $B$, $V$ and IRAC's CH1 and CH2. The full definition of these filters is provided in Weaver et al. (2022). Unless stated otherwise, this set of 14 filters is the one used throughout the analysis. However, because of their width, pho- tometry in broad-band filters cannot easily isolate the emission by the continuum from the nebular emission lines. The latter, which could carry additional information with respect to the con- tinuum, can be captured when adding narrow-band filters to the filter set. To that end, we also consider the fourteen narrow- band filters are also available in the COSMOS2020 catalogue in the range [0.5, 0.8] µm. We will also quantify in the following the change in dimensionality due to their addition to our fiducial filter set.

### 2.3.2 Passive galaxy identification

Passive galaxies in HORIZON-AGN were identified from their
position in the NUV − *r* versus *r* − *K*<sub>s</sub> rest-frame plane.
More specifically, we define passive galaxies as those with
(NUV − *r* > 2.8(*r* − *K*<sub>s</sub>) + 2.2) & (NUV − *r* > 3.6). This empiri-
cal definition provides the best segregation between passive and
star-forming galaxies in the HORIZON-AGN simulated light cone. In
observations, we adopt the usual definition based on the NUV − *r*
versus *r* − *J* rest-frame plane, to match previous observational
works (Ilbert et al. 2013; Laigle et al. 2016; Weaver et al. 2023),
with (NUV − *r* > 3(*r* − *J*) + 1.2) & (NUV − *r* > 3.1). We show
the colour–colour diagrams on Fig. 1 together with our selection
for the COSMOS and HORIZON-AGN data sets. We also note that
the definition of passive galaxies is slightly different in the simulated
and observed data sets, to account for the difference outlined in
Section 2.2 in terms of star formation between the observed and
simulated populations. However, the method we propose in the
following provides an ensemble average of the dimensionality and
is thus insensitive to where exactly the cuts are done, as long as they
are sufficiently far from the bulk of the distribution.

### 2.3.3 Signal-to-noise ratio

In order to limit the impact of noise on our results, we keep only
 galaxies for which SNR > 2 in each band in each catalogue. We
 note that this cut will introduce a bias in the sample, since it imposes
 that galaxies emit flux even in the bluest bands. It therefore excludes
 those very passive and faint galaxies which remain undetected in the
 bluest bands, and limits the maximum redshift that we can probe. As
 an illustration, we show in Fig. 2 the distribution of passive galaxies
 as a function of mass and redshift with and without the SNR cut.
 Passive galaxies at high redshift and in the relatively low-mass range
 are not present with the SNR cut. Similarly, very dusty star-forming
 galaxies are likely to be excluded from the sample with this SNR cut.

We list in Table 1 all the data sets together with the selection
operated, along with the number of galaxies in each sample. We
note that, for the sake of consistency, we use the same subset
of simulated galaxies that have a SNR > 2, including in the data
sets with no simulated noise. We checked that including galaxies
with SNR < 2 does not change the results obtained for the mock
data set with no simulated noise, and yields consistent results
for the COSMOS catalogues, albeit with larger error bars. We
however note that those faint galaxies may have an intrinsically
different dimensionality. However, we find that, should that be
case, the signal would be hidden beyond the relatively high noise
level.

## 3 DIMENSIONALITY ESTIMATION

In this work, we aim to measure the intrinsic dimensionality *D* of wide-band photometric data of galaxies. Our data are comprised of *N* bands with a sample size of *N*<sub>*g*</sub> galaxies. To that end, we rely on intrinsic dimensionality estimation methods. We note that, although intrinsic dimensionality estimation bears similarities with dimensionality reduction techniques (variational auto-encoders, PCA, t-SNE, etc.), their goal differ fundamentally: while dimensionality reduction techniques aim at finding a low-dimensional representation of the data that preserves the topology of the data, intrinsic dimensionality estimation aims at finding the number of independent parameters that are needed to describe the data. The latter can then be used to inform the former,
<figure><img src="image_3.png" /><figcaption>**Figure 2.** The mass and redshift distribution of passive galaxies in the whole sample (left) and after having applied the SNR > 2 cut (right). Our sample is biased towards high-mass and low-redshift passive galaxies, for which significant flux may be detected in all bands.</figcaption></figure>

<table><caption>**Table 1.** Measure of intrinsic dimensionality (ID). Here, we only use high SNR data. Contributions to ID: (1) measurement noise, (2) stellar mass, (3) redshift. For star-forming (SF) galaxies, an extra parameter is (4) the sSFR.</caption><thead><tr><th>Data set</th><th>Comment</th><th>Selection</th><th>Size</th><th>Dimensionality</th><th>Figure</th></tr></thead><tbody><tr><td rowspan="10">HAGN</td><td rowspan="3">No noise<sup>a</sup></td><td>–</td><td>715 006</td><td>4.3 ± 0.2</td><td rowspan="3">5</td></tr><tr><td>Star forming</td><td>706 720</td><td>4.5 ± 0.3</td></tr><tr><td>Passive</td><td>8286</td><td>2.9 ± 0.2</td></tr><tr><td rowspan="2">No noise<sup>a</sup></td><td>–</td><td>715 006</td><td>3.3 ± 0.1</td><td rowspan="2">A3</td></tr><tr><td>Star forming</td><td>706 720</td><td>3.3 ± 0.1</td></tr><tr><td>No noise<sup>a</sup></td><td>Passive</td><td>8286</td><td>2.2 ± 0.3</td><td rowspan="2">A3</td></tr><tr><td>No noise<sup>a</sup></td><td>10.0 ≤ log<sub>10</sub>(M<sub>*</sub>/M<sub>☉</sub>) ≤ 10.5</td><td>105 296</td><td>2.3 ± 0.1</td></tr><tr><td>No noise<sup>a</sup></td><td>10.5 ≤ log<sub>10</sub>(M<sub>*</sub>/M<sub>☉</sub>) ≤ 11.0</td><td>28 300</td><td>2.4 ± 0.1</td><td rowspan="2">A3</td></tr><tr><td>No noise, absolute magnitudes<sup>a</sup></td><td>–</td><td>715 006</td><td>2.2 ± 0.1</td></tr><tr><td rowspan="5">COSMOS</td><td rowspan="5"></td><td>–</td><td>393 118</td><td>4.3 ± 0.2</td><td rowspan="5">5</td></tr><tr><td>Star forming</td><td>359 159</td><td>4.6 ± 0.2</td></tr><tr><td>Passive</td><td>30 260</td><td>2.8 ± 0.1</td></tr><tr><td>10.0 ≤ log<sub>10</sub>(M<sub>*</sub>/M<sub>☉</sub>) ≤ 10.5</td><td>32 190</td><td>3.0 ± 0.2</td></tr><tr><td>10.5 ≤ log<sub>10</sub>(M<sub>*</sub>/M<sub>☉</sub>) ≤ 11.0</td><td>15 581</td><td>3.2 ± 0.2</td></tr></tbody></table>

by providing a lower bound on the dimensionality of the latent space.

Intrinsic dimensionality estimation is a well-studied problem in machine learning and statistics, and many methods have been proposed to estimate it (see e.g. Pestov 2007; Gomtsyan et al. 2019; Cloninger & Klock 2021; Glielmo et al. 2022). However, most of the proposed algorithms are local: the dimensionality is estimated from the statistics of points found within some neighbourhood (e.g. using the distribution of the second-nearest neighbour; Glielmo et al. 2022) before being averaged out over the whole data set. As such, they are not optimal for astronomical data sets where the noise in the data is large, and hence local estimations equally noisy. Alternatively, Granata & Carnevale (2016) proposed an algorithm that allows to estimate the global – rather than local – intrinsic dimensionality of a data set that is robust to local sampling noise and noise in the data (we will discuss this later). In addition to its robustness, this method has the advantage of being rather simple and of having a geometrical interpretation which, we hope, will prove useful to the reader. We delay the application of more involved methods (e.g. Stubbemann, Hanika & Schneider 2023) to future work.

Let us present this method qualitatively. If the distance between points in the feature space is given by the function *d*, let us define

the probability of finding a point at separation *r* as

$$p(r) = \frac{2}{N(N-1)} \sum_{i<j} \delta_D(d(\mathbf{x}_i, \mathbf{x}_j) - r), \quad (1)$$

where $\delta_D$ is the Dirac delta function and $\mathbf{x}_i$ is the data-vector of the $i^{th}$ data point. Note that, here, one could approximate the distribution another kernel (e.g. a Gaussian kernel) instead of a Dirac delta function. While this would allow to accommodate for noise, we found it to have a negligible impact on the results.<fnref n="2" /> If the manifold is compact, as is the case for photometric data, this probability eventually falls to zero as there are no points infinitely far from one another. Similarly, the probability of finding two points vanishes at small separations, unless the intrinsic dimension is null as well. Indeed, at small separations, the number of points within a distance *r* from a point at position *x* goes as $r^D$, so $p(r)$ scales as $r^{D-1}$. If the distance function *d* is the Euclidean distance in the embedding space, Eckmann & Ruelle (1992) showed that the number of points required for accurate measurement of the intrinsic dimensionality scales as $N \sim 10^{D/2}$. This is a direct consequence of the well-known

<footnote><sup>a</sup>For consistency, we select the simulated galaxies that have a large signal to noise in the catalogue with simulated noise.</footnote>

<footnote n="2">²As will be shown on Fig. 5, changing the kernel would affect the left side of $p(r)$ which we discard anyway.</footnote>
curse of dimensionality and the fact that limiting the analysis to the low-separation limit of $p(r)$ discards a large fraction of the data.

The procedure can, however, be improved by addressing these two issues, as was shown in Granata & Carnevale (2016). First, if the manifold is folded onto itself, as illustrated Fig. 3, $p(r)$ will display peaks at separations corresponding to the separation between the folds. This can be addressed by substituting the Euclidean distance between points with a graph distance, which is the length of the shortest path that connects them through the graph of $k$-nearest neighbours; we illustrate it in Fig. 3. This essentially allows to ‘unroll’ the manifold and regularises $p(r)$, which Granata & Carnevale (2016) showed to then be well approximated by the distribution of a $D$-dimensional hyper-sphere up to the maximum of $p(r)$. The latter has a simple closed-form expression, $p_{sphere,D}(r) \propto \sin^{D-1}(r\pi/2)$, which we can fit to the data. This estimator was shown to yield robust estimates of the dimensionality of a 20-dimensional data set from a few thousand points.

In our case, we seek to measure the intrinsic dimensionality of data with shape $(N_g, N)$, where $N_g$ is of the order of 10 000 to 100 000 and $N = 14$ wide photometric bands, which is well within the reach of the method. To that end, we first rescale the magnitudes into the range [0, 1] using the same mapping for all bands, and we build a KD-Tree using the Euclidean distance in the embedded space. We then build the connectivity graph, where edges connect each data point to its $k$-nearest neighbours, increasing $k$ from one until the graph is fully connected. Ideally, one would then compute the graph distance between all pairs of points (e.g. with the Floyd–Warshall algorithm, with complexity $O(N_g^3)$). However, this proved to be prohibitively expensive and we instead resort to using a small subset of points. To that end, we pick $N_{sub} = 200$ points at random<fnref n="3" /> and compute the graph distance from each of those to all others using Dijkstra's algorithm (with complexity $O(N_{sub} \times N_g \log N_g)$; Dijkstra 1959). Plugging these distances back into equation (1) yields estimates of the pairwise distance probability $p(r)$. We split this sample of 200 estimates of $p(r)$ into 10 subsamples, and we estimate the intrinsic dimensionality of each subsample as follows. For each of those, we normalise $p(r)$ by its maximum value and $r$ by the corresponding radius. We then perform a least-square regression with the analytical expression of a $D$-dimensional hyper-sphere, $p_{sphere,D}(r/r_{max}) = C \sin^{D-1}(r\pi/2r_{max})$, with free parameters $D$ and $C$. On the low-separation end, we remove radii that contain less than a thousand pairs to limit shot-noise. On the high-separation end, we truncate at $r = 0.8r_{max}$ where the fit to the hyper-sphere started diverging significantly. We note that our results are robust to changes to these choices. We obtain our estimated dimensionality and its uncertainty by taking the mean and standard deviation of the ten estimates.

As an illustrative example, we show in Fig. 3 (right panel) our estimate of $p(r)$ for the S-curve data set presented in Fig. 3 (left panel) sampled here with 100 000 points and embedded in three-dimensional (3D). We recover here an excellent agreement between the measured dimensionality and the truth value, 2. To assess the accuracy of our method, we then draw 100 000 points from a $D$-dimensional ball embedded in a $N$-dimensional space. We chose here $N = 14$ and vary $D$ from 1 to 8. In practice, we uniformly draw points within a $D$-dimensional unit ball and fill the remaining dimensions with a small $(10^{-2})$ uniformly distributed value to mimic

the effect of a small signal to noise. Fig. 4 shows the estimated dimensionality together with the truth value as a dashed line. We see that the accuracy of the measurement depends on the intrinsic dimensionality, with the most accurate measurements obtained for low-dimensional data sets, and on the number of points used to estimate it. We also note that our estimator is biased towards an overestimation of the actual intrinsic dimensionality by up to 0.5. While the bias may depend on the exact geometry of the manifold, we however note that, for a fixed geometry, our estimator can correctly distinguish data sets with different dimensionality. In practical terms, with a sample size of 100 000 points, we can distinguish data sets with up to seven intrinsic dimensions within one standard deviation. With a sample size of 10 000, we can distinguish data sets with up to six intrinsic dimensions.

In the following, we will mostly focus on the variation of the measured dimensionality, so that we limit the extent to which the bias is the estimator contaminates our conclusions.

## 4 RESULTS

First, let us compare the observed and simulated catalogues. We measure their dimensionality and present the results on Fig. 5, with data including all galaxies from COSMOS (top) and the HORIZON-AGN mock catalogue (bottom). In both cases, we find that the dimensionality is around 4.2. Despite data from the simulation missing passive galaxies at the high-mass end and star-forming ones at the low-mass end, we, however, find that the simulated and the observed catalogues have overall similar dimensionality. To confirm that this dimensionality is driven by actual correlations in the data set rather than an effect of the limited size of our sample, we randomly reshuffle the entries of the COSMOS data sets independently for each column to erase possible correlations. Doing so, we find a much larger dimensionality of $9.2 \pm 1.2$. We however recall that such, for such high dimensionalities, our method starts saturating and this value should thus be interpreted as a lower bound on the actual intrinsic dimensionality of the data. However, this confirms that the measured dimensionality in the fiducial, unshuffled, data set reflects internal structures rather than be an artefact of the limited size of our sample.

With this in mind, we then use the noise-free magnitudes to estimate the impact noise has on the apparent complexity of the observations. Our finding is that noise masquerades an intrinsic dimension of the data set, so removing noise decreases the overall dimensionality by one (to $3.2 \pm 0.1$), and this despite having limited ourselves to galaxies with an SNR larger than 2. The simulated data set also allows us to employ absolute, rather than apparent, magnitudes. This removes the effect of redshift from the data. Working with noise-free apparent magnitudes, we observe that the dimensionality further decreases by one (to $2.2 \pm 0.1$). We interpret this as the (unsurprising) fact that one key variable driving the magnitudes in different bands is the redshift of the galaxy.

To gain further insight into the data set, let us now split the galaxies between star-forming and passive ones, as described in Section 2.3. Doing so, we find that passive galaxies have a dimensionality of $2.8 \pm 0.1$ ($2.9 \pm 0.2$ in the simulated data set), while star-forming galaxies have $4.6 \pm 0.2$ ($4.5 \pm 0.3$ in the simulated data set).

We then include in our data sets the 14 narrow-band filters, with the expectation those would encode independent information about the star formation activity of the galaxies. However, we find that the addition of those filters does not change significantly the estimate

<footnote n="3">³We found 200 points to be sufficient to have a converged estimate of the dimensionality while remaining computationally affordable.</footnote>
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 3.** The dimensionality of a data set can be estimated from the distribution of distances between points. On the left (a) the distribution of 100 000 points drawn from a 2D S-curve and embedded in 3D. The distance between two points (large black dots) can be computed by finding the shortest path between them (black dotted line, we represent one point every ten for clarity) from nearest neighbour to nearest neighbour. We show all the points within a distance *r* from the starting point in gray. The number of such points grows here as *r*²; in general, it grows as *r*⁵. By measuring the probability *p* of finding a point within a distance *r*, we can thus estimate the intrinsic dimensionality of the data set, as is shown on the right (b). We detail the procedure in the text.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 4.** We use mock data sets that consist of points drawn from *D*-dimensional balls and measure their dimensionality as a function of the number of sample points. The ground truth values are shown as dashed lines with our estimates of the dimensionality as points with matching colour; the error bars represent the 1σ dispersion. The accuracy of the dimensionality estimation increases with the increasing number of points. For this particular sample, our method is biased towards overestimating the actual dimensionality, but it is able to correctly distinguish data sets with an intrinsic dimensionality of up to eight.</figcaption></figure>

of the intrinsic dimensionality which goes from 4.2 ± 0.4 to 4.5 ± 0.3 when including the narrow-band filters. We reach a similar conclusion for star-forming galaxies (from 4.6 ± 0.2 to 4.6 ± 0.3) or passive ones (from 2.8 ± 0.1 to 2.8 ± 0.2).

Finally, we also apply our estimator to spectra in the mock catalogue, finding a dimensionality of 5.0 ± 0.3. We warn that a careful analysis of the performance of our method should be carried out in the limit of large embedding dimensions (here, 2000) to be able to reach definitive conclusions. Notwithstanding, the measured dimensionality here is larger than the corresponding dimensionality of the wide-band photometric data set (which is 3.2 ± 0.1). We interpret this as evidence that spectra are more information-rich and that at least two additional parameters may be extracted from them. We delay further exploration of spectra to the future.

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Wide-band photometric data from the COSMOS catalogue have a low dimensionality and are intrinsically four-dimensional. We measure the pairwise distance probability *p* as a function of separation *r* and compare it to the analytical formula of a *D*-dimensional hypersphere. We show in dashed red the result of the best fit. Both simulated and observed data have an intrinsic dimensionality between four and five.</figcaption></figure>
## 5 CONCLUSION AND DISCUSSION

The goal of this paper was to estimate the dimensionality of broad-band photometric data. To that end, we have adapted the dimensionality-measurement technique of Granata & Carnevale (2016). Our main results are the following:

(i) The intrinsic large-scale dimensionality of observed broad-band data is between four and five. Mocks are consistent with observations when similar level of noise and cut are applied.

(ii) Realistic estimates of the noise in the mock catalogue account for one of these dimensions.

(iii) Star-forming galaxies are intrinsically more complex ($D \sim 3-4$) than passive galaxies ($D \sim 2-3$).

Our results highlight that most of the variance in global, large-scale properties of galaxies can be described with a small number of independent parameters. This is in line with similar findings of Villaescusa-Navarro et al. (2022), Chawak et al. (2023). These previous studies relied on the use of machine-learning and focused on physical parameters (masses, radii, velocities) rather than observational ones. Our method aims at providing the number of ‘main’ parameters driving scaling relations, which we showed here to be consistent with mass, redshift, and SFR. In the future, we suggest extending the mock data set to mix observational information with physical quantities, such as mass, radius, or spin. Any increase in the measured dimensionality would then suggest that the physical quantities cannot be derived from the observational ones alone. We delay such analysis to future work, as this would require carefully rescaling physical and observational parameters to allow building a similarity metric between them.

We emphasize that relatively small deviations from scaling relations are only revealed once the driving effect of these main parameters is removed. For example, when we bin by stellar mass in the noise-free absolute-magnitude simulated data set, we find that this does not result in a decrease in the number of dimensions. Indeed, once the main parameters driving the absolute magnitudes were fixed, we started probing secondary parameters that drive the scatter around scaling relations, such that the dimensionality measurement is now probing smaller variations in magnitudes.

Our approach stands out compared to more traditional PCA-based approaches (e.g. Yip et al. 2004; Ferreras et al. 2006; Portillo et al. 2020; Sharbaf et al. 2023) or singular-value decomposition (SVD; Cooray et al. 2023). For example, Sharbaf et al. (2023) employed a PCA to show that most of the variance in SDSS spectra can be explained by a small number of components ($\lesssim 3$), yielding a similar result as ours. PCA-based approaches have the advantage of being relatively simple and efficient, yet they suffer from two main limitations. First, the analysis on the number of components depends on the fraction of the total variance one wants to reproduce. Secondly, PCAs cannot reproduce non-linear relations between variables, as we showed on Fig. 3 (PCA would yield three components, while the true dimensionality is two). In comparison, our non-parametric approach is able to capture non-linear relations between variables and therefore relaxes the assumption that the data is linearly separable and well-represented by a multivariate Gaussian distribution.

Ferreras et al. (2023) use information theory to recover information about the stellar component. One of the authors’ main findings is that the information content of spectra is rather low, with most of the information being contained in a few components (4000 Å break and Balmer absorption lines for stellar components). In our paper, we focus on measuring the intrinsic dimensionality of the data, which has strong connection with information content:

larger numbers of intrinsic dimensions require more information to fully encode the data. Our findings are consistent with theirs in that the intrinsic dimensionality of the data is low, although a direct comparison is not possible due to the difference in the input data (wide-band photometry versus spectra) and targeted inferred properties (global galaxy properties versus stellar properties).

The exact dimensionality revealed here provides guiding numbers to employ dimensionality-reducing techniques, such as self-organized maps (Davidzon et al. 2019, 2022; Teimoorinia et al. 2022; Sanjaripour et al. 2024) locally-linear embedding (Vanderplas & Connolly 2009) or variational auto-encoders (Portillo et al. 2020; Piras & Lombriser 2024). Indeed, the latent space should have at least four to five dimensions to ensure the projection does not lose information.

Our results support an emerging scenario in which the main global galaxy properties evolve towards a low-dimensional attractor. This was first suggested by Disney et al. (2008), although our findings support more complexity than the one-dimensional attractor they reported. We however note that our data sets rely on photometry – and thus are affected by redshift–while theirs did not. In addition, they excluded colour information, and thus information about star formation rate, from their analysis, while we found this information to be relevant in driving the dimensionality up. The closest comparison is the work of Cooray et al. (2023) that utilize data from Galaxy Evolution Explorer (GALEX), Sloan Digital Sky Survey (SDSS), and United Kingdom Infra-Red Telescope (UKIRT) Infrared Deep Sky Survey (UKIDSS) in 11 wide-band photometric bands. Using SVD, they find that more than 90 % (but not all) of the variance in the data can be explained with only two components. Although we concur in that the data can be well-represented by a low-dimensional manifold, we emphasize that our reported intrinsic dimensionality of three to four cannot be compared directly to their two components.

This idea that galaxies are governed by a small number of parameters has found renewed interest with recent research suggesting that one previously unknown key parameter driving galaxy properties is the misalignment between the inertia tensor and the tidal tensor in the initial conditions, $\tau$. This parameter correlates well with properties in the evolved Universe such as spin and formation time (Moon & Lee 2024) or the morphology and angular momentum support of galaxies at high-redshift (Park et al. 2022). In Cadiou et al. (2022), it was additionally shown to be a cause-and-effect relationship, with higher values of the misalignment yielding less bulgy and more discy, more extended galaxies at fixed halo and stellar masses. This provides four independent parameters – (halo) mass, local density, gravitational shear ($q$, see e.g. Paranjape, Hahn & Sheth 2018) and tidal misalignment ($\tau$) – that have been shown to correlate with galaxy properties. Remarkably, this number of independent parameters is consistent with the number of dimensions we obtained from photometric surveys.

This raises the wider question of whether this set of four parameters is enough to build a complete theory capable of fully predicting the wide-band photometric survey. While this goes beyond the scope of this paper, we note that large cosmological volume simulations are finally ripe to tackle this problem from a statistical standpoint, as we did in this paper. In parallel, genetic modifications of the initial conditions (Rey & Pontzen 2018; Stopyra et al. 2021; Cadiou et al. 2021a; Cadiou et al. 2021b; Rey et al. 2023) offer avenues to confirm the causal relation between those parameters and observables.
## 5.1 Comparison to other works

Compared to previous work (Davidzon et al. 2022; Teimoorinia et al. 2022; Sanjaripour et al. 2024) that utilize SOMs with the latent-space dimension as an input parameter, we provide here a measurement of what this latent space should be. Zeraatgari et al. (2024) compared different machine learning-based approaches to classify galaxies, quasars, emission-line galaxies and stars from photometric data. In particular, they use *k*-nearest neighbours (KNN) to classify galaxies that are nearby in feature space as belonging to the same class (see also Ascasibar & Sánchez Almeida 2011 for a similar work using spectra). Our approach relies on a generalization of this, where we replace the Euclidean distance with an approximation of the geodesic distance on the manifold, similar to the UMAP approach (McInnes, Healy & Melville 2020).

One limitation of our proposed approach is to rely on wide-band integrated photometry rather than resolved observations or spectra. As such, it does not include spatial information (yet) as was done with convolutional neural networks (Buck & Wolf 2021; Liew-Cain et al. 2021). This prevents us from probing the intrinsic dimensionality of resolved galaxies. The main limitation is that

## ACKNOWLEDGEMENTS

CC and OA acknowledge support from the Knut and Alice Wallenberg Foundation, the Swedish Research Council (grant 2019-04659), and the Swedish National Space Agency (SNSA Dnr 2023-00164). CL acknowledges funding from the PNCG and the French Agence Nationale de la Recherche for the project iMAGE (grant ANR-22-CE31-0007). This work has made use of the CANDIDE computer system at the IAP supported by grants from the PNCG, CNES, and the DIM-ACAV, as well as the Infinity Cluster, both hosted by Institut d'Astrophysique de Paris. We thank Stéphane Rouberol for running them smoothly for us. The authors thank C. Pichon, M. Rey, A. Glielmo, and O. Ilbert for fruitful discussions. This research was supported in part by grant NSFPHY-1748958 to the Kavli Institute for Theoretical Physics (KITP).

This project has used GNU PARALLEL (Tange 2018), JUPYTER notebooks (Kluyver et al. 2016), MATPLOTLIB (Hunter 2007), NETWORKX (Hagberg, Schult & Swart 2008), NUMPY (Harris et al. 2020), SCIPY (Virtanen et al. 2020), and ASTROPY<fnref n="4" /> a community-developed core PYTHON package and an ecosystem of tools and resources for astronomy (Astropy Collaboration 2013, 2018, 2022).

## DATA AVAILABILITY

The data underlying this article will be shared on reasonable request to the corresponding author.

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## APPENDIX A: DIFFERENT CUTS

In this section, we provide the plots corresponding to the data in Table 1.

We show on Fig. A3 the dimensionality of the mock data sets including only passive galaxies (top) and only star-forming galaxies (bottom). We find that star-forming galaxies cover a region of the parameter space with higher dimensionality.

<figure><img src="image_9.png" /></figure>

**Figure A1.** Same as Fig. 5, but comparing the COSMOS data with only passive (top) and only star-forming (bottom) galaxies. Star-forming galaxies have a higher intrinsic dimensionality than passive galaxies.
<figure><img src="image_10.png" /><figcaption>**Figure A2.** Same as Fig. 5, but comparing the COSMOS data with only star-forming galaxies in two mass bins. Selecting galaxies within a narrow mass range decreases the intrinsic dimensionality by one.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure A3.** Same as Fig. 5, but comparing the mock data with only passive (top) and only star-forming (bottom) galaxies. Star-forming galaxies have a higher intrinsic dimensionality than passive galaxies.</figcaption></figure>

This paper has been typeset from a TeX/LaTeX file prepared by the author.