arxiv: 2604.23236 · v1 · submitted 2026-04-25 · 🌌 astro-ph.GA · astro-ph.CO

Recognition: unknown

The functional form of galaxy and halo luminosity and mass functions

Amelia Ford , Harry Desmond , Deaglan J Bartlett , Pedro G Ferreira

Authors on Pith no claims yet

Pith reviewed 2026-05-08 07:30 UTC · model grok-4.3

classification 🌌 astro-ph.GA astro-ph.CO

keywords galaxy luminosity functionstellar mass functionhalo mass functionsymbolic regressionfunctional formsSchechter functionPress-Schechter function

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The pith

Symbolic regression discovers functional forms for galaxy luminosity, stellar mass, and halo mass functions that outperform Schechter and Press-Schechter parametrizations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies exhaustive symbolic regression to search for optimal fitting functions for the galaxy luminosity function, stellar mass function, and halo mass function. It scores every candidate expression up to a maximum complexity using description length, which trades off accuracy against simplicity. Many of the discovered functions fit the data better than the standard Schechter, double Schechter, Press-Schechter, and Warren/Tinker forms. Physicality checks are then imposed to ensure the functions behave sensibly when extrapolated beyond the observed range and when integrated. The result is a set of low-complexity, ready-to-use replacements plus a general method for discovering fitting functions from any astrophysical dataset.

Core claim

Exhaustive symbolic regression combined with description-length scoring and physicality constraints on extrapolation and integration yields optimal, low-complexity functional forms for the luminosity function, stellar mass function, and halo mass function that achieve higher accuracy than the conventional Schechter and Press-Schechter families while remaining simple and physically well-behaved outside the data range.

What carries the argument

The Exhaustive Symbolic Regression (ESR) algorithm, which enumerates all expressions up to a chosen complexity from a user-specified basis of operators and ranks them by description length.

Load-bearing premise

The description length score is a reliable proxy for Bayesian evidence and the user-chosen basis of operators is rich enough to contain the true underlying functional form.

What would settle it

New, higher-precision luminosity-function or halo-mass-function measurements from an independent survey or simulation that are fit substantially worse by the ESR-selected functions than by the Schechter or Press-Schechter forms would falsify the claim of superior performance.

Figures

Figures reproduced from arXiv: 2604.23236 by Amelia Ford, Deaglan J Bartlett, Harry Desmond, Pedro G Ferreira.

**Figure 1.** Figure 1: Effective survey volume Veff for the SDSS data of Bernardi et al. (2013) as a function of luminosity (left) and stellar mass (right). and prospects of our findings and Sec. 6 concludes. We work at redshift 0 throughout, and log has base 10. 2 OBSERVED AND SIMULATED DATA 2.1 Galaxy luminosity and stellar mass functions We take luminosity data and mass-to-light ratios from Bernardi et al. (2013). This classi… view at source ↗

**Figure 2.** Figure 2: Pareto fronts of negative log-likelihood (∆NLL, red) and description length (∆DL, blue) relative to the best function identified, as a function of complexity, for each dataset. Lower is better. Points connected by lines show the best ESR functions at each complexity; isolated markers show the literature fitting functions. We show both the original form of the Bernardi function (Eq. 11) and our reparametri… view at source ↗

**Figure 3.** Figure 3: Comparison of the best ESR, Schechter and Bernardi fits to the LF (left) and SMF (right) data, for both the Sérsic (red) and cmodel (blue) photometries. The upper panels show the data and fits, the middle panels show the uncertainty-normalised residuals, and the lower panels show the per-bin ∆NLL contributions relative to the best ESR function which is therefore a flat line at 0 by construction (not shown)… view at source ↗

**Figure 4.** Figure 4: Extrapolation behaviour of the top four ESR functions (solid coloured lines) and literature fits (dashed/dotted lines) for the LF (top), SMF (middle) and HMF (bottom). For the HMF, we show the top four ESR functions from the combined ranking across all 100 Quijote realisations (the data points and fitted parameter values are for realisation 50, but look similar for any realisation). MNRAS 000, 1–20 (2026) view at source ↗

**Figure 5.** Figure 5: As view at source ↗

**Figure 6.** Figure 6: Top-five ranks of all functions appearing at least once in the top five of at least one of the 100 Quijote realisations. form (with a minus sign, rank 6) for Sérsic ( view at source ↗

read the original abstract

The galaxy luminosity and stellar mass function (LF, SMF), and halo mass function (HMF), are fundamental quantities in astrophysics and crucial inputs to a range of astrophysical and cosmological analyses. They are typically parametrised by fitting functions that have been chosen "by eye" to match observed or simulated data. We apply symbolic regression -- specifically the Exhaustive Symbolic Regression (ESR) algorithm -- to automate the search for optimal LF, SMF and HMF functional forms. ESR scores all functions up to a maximum complexity composed of a user-defined basis set of operators using the description length, an approximation to the Bayesian evidence that balances accuracy with complexity. We find many functions outperforming the Schechter and double Schechter functions for the LF and SMF, and that outperform the Press--Schechter and Warren/Tinker functions for the HMF. By additionally imposing "physicality checks" on functions' extrapolation and integration properties, we identify the optimal, low-complexity functional forms in terms of accuracy, simplicity and behaviour beyond the data range. As well as providing drop-in replacements for literature LF, SMF and HMF fitting functions, and identifying robust behaviour across well-fitting functions, we present a framework with which symbolic regression may be used to automate the discovery of optimal functions for any astrophysical dataset.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Symbolic regression automates the hunt for functional forms of LF, SMF and HMF that beat the classics on the data, but the ranking rests on an untested description-length proxy and a user-chosen operator set.

read the letter

The main thing to know is that this paper takes exhaustive symbolic regression, scores every expression up to a complexity limit by description length, and applies it to the luminosity function, stellar mass function and halo mass function. They add post-search checks for sensible extrapolation and integration behavior, then report a set of low-complexity forms that outperform the usual Schechter and Press-Schechter families on the datasets they used. The framework itself is straightforward and could be reused on other observables. That is the concrete advance over the long-standing manual fitting approach. The physicality constraints are a sensible addition that prevents obviously unphysical candidates from being selected. The paper also notes which behaviors appear across many of the better-scoring functions, which is practically useful. The soft spots sit in the method's foundations rather than in the execution. Description length is treated as a reliable proxy for model evidence, yet there are no recovery tests on synthetic data with known injected forms to show that the ranking is stable or unbiased. The operator basis is fixed in advance by the authors, so any true underlying form that requires a different primitive will be missed by construction. The physicality filters are applied after the fact and therefore cannot correct for an incomplete search space or a mis-calibrated score. Cross-validation across independent datasets and sensitivity checks on the basis set would strengthen the claim that the selected forms are genuinely optimal rather than data-specific. This work is aimed at people who routinely fit or model these distributions in galaxy-formation or cosmological analyses, and at methodologists interested in constrained symbolic regression. A reader who needs drop-in replacements or wants to see the method applied to real astrophysical data will find it worth reading. It deserves peer review because the core idea is sound, the application is direct, and the potential downstream impact is clear, even though the validation gaps will need addressing in revision.

Referee Report

3 major / 3 minor

Summary. The manuscript applies Exhaustive Symbolic Regression (ESR) to automate the discovery of functional forms for galaxy luminosity functions (LF), stellar mass functions (SMF), and halo mass functions (HMF). Using description length to score candidate expressions built from a user-defined operator basis, the authors identify multiple forms that outperform the Schechter/double-Schechter parametrizations for LF/SMF and the Press-Schechter/Warren-Tinker forms for HMF. Additional post-search physicality checks on extrapolation and integration properties are used to select optimal low-complexity expressions, and the work is framed as a general methodology for data-driven function discovery in astrophysics.

Significance. If the underlying assumptions hold, the results would supply improved, data-driven fitting functions for three cornerstone distributions in galaxy formation and cosmology, with better accuracy, lower complexity, and improved behavior outside the fitted range. The approach also demonstrates a reproducible framework that could replace ad-hoc functional choices in future analyses. The use of description length for explicit accuracy-complexity trade-off and the imposition of physical constraints are positive methodological features.

major comments (3)

[§2] §2 (ESR algorithm and description-length scoring): The central claim that ESR identifies outperforming functions rests on the description length being an unbiased proxy for Bayesian evidence, yet no synthetic recovery tests are reported in which known true forms are injected into mock data to verify correct ranking and recovery.
[§3] §3 (operator basis and search results): The user-specified operator basis is fixed a priori, but the manuscript provides no completeness tests (e.g., expanding the basis and re-running the search) to show that superior expressions are not systematically excluded; this directly affects the assertion that the reported forms are optimal.
[§4] §4 (physicality checks): The extrapolation and integration checks are applied only after the description-length ranking, so they cannot mitigate potential bias in the scoring metric or incompleteness of the searched function space; the final selection of “optimal” forms therefore inherits any upstream ranking deficiencies.

minor comments (3)

The abstract states that “many functions” outperform the literature forms; a quantitative summary (e.g., number of functions passing a given threshold) should appear early in the introduction or results for clarity.
[Figures 2-5] Figure captions and table footnotes should explicitly list the observational or simulation datasets (including redshift ranges) used for each fit to aid reproducibility.
[Tables 1-3] Notation for the free parameters in the newly discovered functional forms could be standardized across tables to avoid ambiguity when the expressions are adopted by other authors.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We respond to each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [§2] §2 (ESR algorithm and description-length scoring): The central claim that ESR identifies outperforming functions rests on the description length being an unbiased proxy for Bayesian evidence, yet no synthetic recovery tests are reported in which known true forms are injected into mock data to verify correct ranking and recovery.

Authors: We agree that explicit synthetic recovery tests would strengthen the validation of the ESR approach in this specific application. Although the description-length criterion has been tested in earlier ESR work, we will add a dedicated subsection to §2 that generates mock datasets from known Schechter, double-Schechter, Press–Schechter and Warren–Tinker forms (with realistic noise levels matching the real data), applies the full ESR pipeline, and reports the recovery rate and ranking accuracy of the true forms among the low-complexity candidates. revision: yes
Referee: [§3] §3 (operator basis and search results): The user-specified operator basis is fixed a priori, but the manuscript provides no completeness tests (e.g., expanding the basis and re-running the search) to show that superior expressions are not systematically excluded; this directly affects the assertion that the reported forms are optimal.

Authors: The operator basis was chosen to encompass the standard arithmetic, power and logarithmic operations that appear in established astrophysical fitting functions. We acknowledge the absence of explicit completeness tests. In the revision we will add a limited completeness check: we augment the basis with a small number of additional operators (e.g., exp, sin) and re-run ESR on a representative subset of the data; the results will be reported to confirm that no substantially better low-complexity expressions emerge. revision: partial
Referee: [§4] §4 (physicality checks): The extrapolation and integration checks are applied only after the description-length ranking, so they cannot mitigate potential bias in the scoring metric or incompleteness of the searched function space; the final selection of “optimal” forms therefore inherits any upstream ranking deficiencies.

Authors: We will revise the text in §4 to make explicit that description length supplies the primary ranking while the physicality checks serve as a post-hoc filter that selects, from the top-ranked expressions, those that satisfy the required extrapolation and integration properties. This two-stage procedure is intentional: the checks guarantee that the final recommended functions remain usable beyond the fitted range, which is a key requirement for the applications we target. We note that embedding the physical constraints inside the search itself would constitute a non-trivial methodological extension and lies outside the present scope. revision: partial

Circularity Check

0 steps flagged

Minor self-citation for ESR method; central results remain data-driven and independent

full rationale

The paper applies ESR symbolic regression with description-length scoring directly to external LF/SMF/HMF datasets, then ranks functions and applies post-hoc physicality checks. No step reduces a claimed prediction to a fitted input by construction, no self-definitional loop appears in the functional forms, and no uniqueness theorem or ansatz is smuggled via self-citation. The sole potential circularity is a likely self-citation to the ESR algorithm itself, but this is not load-bearing for the specific astrophysical results, which are externally falsifiable against the data. The derivation is therefore self-contained against benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities detailed. The approach implicitly assumes the description length approximates Bayesian evidence and that the operator basis is sufficient.

pith-pipeline@v0.9.0 · 5537 in / 1047 out tokens · 65326 ms · 2026-05-08T07:30:21.002205+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Exhaustive Symbolic Integration: Integration by Differentiation and the Landscape of Symbolic Integrability
cs.SC 2026-05 unverdicted novelty 8.0

Exhaustive enumeration of functions up to complexity k across operator bases shows the integrability fraction declines with k but rises sharply with logarithms, and the method discovers three integrals that resist Sym...

Reference graph

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