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arxiv: 2508.20266 · v2 · submitted 2025-08-27 · 🌌 astro-ph.IM · astro-ph.GA

Using Symbolic Regression to Emulate the Radial Fourier Transform of the S\'ersic profile for Fast, Accurate and Differentiable Galaxy Profile Fitting

Tim B. Miller , Imad Pasha This is my paper

Pith reviewed 2026-05-18 20:46 UTC · model grok-4.3

classification 🌌 astro-ph.IM astro-ph.GA
keywords symbolic regressionSérsic profileradial Fourier transformgalaxy profile fittingmorphological parametersemulatorFourier renderingextragalactic imaging
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The pith

Symbolic regression yields an equation that approximates the radial Fourier transform of the Sérsic profile.

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

The paper establishes that numerically computed radial Fourier transforms of the Sérsic profile vary smoothly with index and frequency, allowing symbolic regression to recover a simple approximating equation. This equation serves as a fast, differentiable emulator for rendering galaxy profiles in Fourier space during fitting. A sympathetic reader would care because profile fitting supplies the morphological parameters used in nearly all photometric analyses of galaxy surveys, and computation time becomes a bottleneck as image volumes increase. The resulting method achieves the same measurements at 2.5 times the speed of standard approaches while preserving accuracy, as verified through controlled recovery tests and real imaging.

Core claim

Symbolic regression applied to a dense numerical training set of radial Fourier transforms produces a closed-form equation that approximates the transform across the range of Sérsic indices and spatial frequencies encountered in real galaxy data. When this equation is substituted for direct numerical integration or expensive approximations inside a Fourier-space rendering pipeline, the full profile-fitting procedure runs substantially faster while returning essentially the same morphological parameters.

What carries the argument

The symbolic-regression-derived equation that emulates the radial Fourier transform of the Sérsic profile as a function of Sérsic index and spatial frequency k.

If this is right

  • Morphological parameters of galaxies can be extracted 2.5 times faster than with previous rendering methods.
  • The loss in accuracy remains small enough that scientific conclusions drawn from the fitted parameters are unchanged.
  • Differentiability of the emulator allows gradient-based optimization to proceed without extra numerical overhead.
  • The same rendering step scales directly to the data rates expected from future wide-field imaging surveys.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same regression workflow could be repeated for other analytic light profiles that lack closed-form Fourier transforms.
  • Replacing the current numerical training grid with an adaptive sampling strategy might further tighten the approximation error at high frequencies.
  • Because the emulator is built only from elementary functions, it can be ported without change to any fitting code that already supports Fourier-space operations.

Load-bearing premise

The equation recovered by symbolic regression must remain accurate for every Sérsic index and every spatial frequency that actually occurs in observed galaxy images.

What would settle it

Compute the pointwise difference between the symbolic equation and a high-resolution numerical quadrature of the true radial Fourier integral for Sérsic indices spanning 0.5 to 8 and for k values up to the Nyquist frequency of typical survey pixels; if the maximum relative error exceeds the tolerance used in the fitting tests, the speed-up claim no longer holds.

Figures

Figures reproduced from arXiv: 2508.20266 by Imad Pasha, Tim B. Miller.

Figure 1
Figure 1. Figure 1: — Showcasing the S´ersic profile and its associated radial Fourier Transform, also known as the Hankel transform. We show a set of four indices which are all normalized such that the total flux and effective radius are unity. The surface brightness profile is shown (left) along with two different views of the radial Fourier transform; log-log (middle) and linear (right). We find the transformed S´ersic pro… view at source ↗
Figure 2
Figure 2. Figure 2: — Showcasing the results of the symbolic regression fitting process comparing execution time and two measures of accuracy. The execution time is measured for rendering a model image, calculating a χ 2 log-likelihood and calculating the gradients of the input parameters. We compare this against the final loss from the symbolic regression. Each point is colored by its complexity. We find a negative correlati… view at source ↗
Figure 3
Figure 3. Figure 3: — Injection recovery tests to ensure our chosen S´ersic Fourier Emulator, Eqn. 6 accurately for profile fitting. We render a series of profiles using a heavily over-sampled pixel space renderer as the ‘truth‘ and fit each using a renderer based on the Fourier Emulator. The input values are compared to those recovered for reff and the index n. We find great agreement, with the average residual < 0.5% and th… view at source ↗
Figure 4
Figure 4. Figure 4: — The results of profile fitting for three galaxies selected from the GAMA survey at 0.4 < z < 0.5. We model the i band images from HSC using pysersic and compare two different rendering methods: The default hybrid method based on using a mixture of Gaussians and the Fourier Emulator method presented in this work. For each method we show the best fit model, residual and the median and standard deviation of… view at source ↗
Figure 5
Figure 5. Figure 5: — Comparing the recovered radius, Reff and S´ersic index,n, when using the hybrid and Fourier Emulator rendering methods for a sample of 100 galaxies with HSC imaging. The red line shows the one-to-one relation while the black line shows median calculated using the loess algorithm. (Cappellari et al. 2013). An inset is included for each parameter showcasing a histogram of the fractional differences. We fin… view at source ↗
Figure 6
Figure 6. Figure 6: — Comparing inference times in pysersic when using the default hybrid (Lang 2020) method compared to the S´ersic Fourier Emulator methods described here. For both inference techniques the rendering scheme proposed in this work performs 2.5× faster.This brings the median time to complete sampling, for example, from 64 seconds down to 23.5 seconds including all overheads. Gaussians. Similar to the injection-… view at source ↗
read the original abstract

Galaxy profile fitting is a ubiquitous technique that provides the backbone for photometric and morphological measurements in modern extragalactic surveys. A recent innovation in profile fitting algorithms is to render, or create, the model profile in Fourier space, which aims to provide faster and more accurate results. However, the most common parameterization, the S\'ersic profile, has no closed form Fourier transform, requiring the use of computationally expensive approximations. In this paper our goal is to develop an emulator to mimic the radial Fourier transform of the S\'ersic profile, for use in profile fitting. We first numerically compute the radial Fourier transform and demonstrate that it varies smoothly as a function of the S\'ersic index and $k$, the spatial frequency coordinate. Using this set of numerically calculated transforms as a training set, we use symbolic regression to discover an equation which approximates its behavior. This ensures the emulator will be based on computationally efficient and differentiable building blocks. We implement this novel rendering method in the pysersic profile fitter, and ensure it is accurate by conducting both injection-recovery tests using model galaxy profiles and applying multiple rendering methods to a real sample of galaxies in HSC-SSP imaging. Crucially, the Fourier emulator rendering technique enables measurements of morphological parameters of galaxies 2.5 times faster than standard methods with minimal loss in accuracy. This increased performance while maintaining accuracy is a step that ensures these tools can continue to scale with the ever-increasing flow of incoming data.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript develops an emulator for the radial Fourier transform of the Sérsic profile by first computing it numerically over a grid of Sérsic indices and spatial frequencies k, then applying symbolic regression to discover a closed-form approximating equation built from efficient, differentiable operations. This emulator is integrated into the pysersic profile-fitting code to enable Fourier-space rendering of galaxy models. The authors validate the approach via injection-recovery tests on simulated galaxies and by comparing fits to real HSC-SSP imaging using both the emulator and standard methods, reporting a 2.5× speed-up with minimal loss in accuracy.

Significance. If the emulator maintains sufficient accuracy across the parameter ranges encountered in real data, the work provides a practical route to scaling Fourier-space profile fitting to the data volumes expected from next-generation surveys. The use of symbolic regression to obtain an interpretable, differentiable approximation is a constructive contribution that could be adopted in other rendering contexts.

major comments (1)
  1. [Validation section] Validation section (injection-recovery and HSC-SSP tests): the reported performance metrics are averages; no maximum relative error, bias maps, or error distributions are shown specifically for n > 4 and intermediate k values (roughly 0.5–5) that dominate the likelihood surface for typical galaxy fits. Without these, it is not possible to confirm that downstream morphological parameters remain unbiased at the level implied by the “minimal loss in accuracy” claim.
minor comments (2)
  1. [Abstract] The abstract states that the transform “varies smoothly”; a brief quantitative statement of the smoothness (e.g., maximum second derivative or grid spacing used) would help readers assess the adequacy of the training-set density.
  2. [Methods] Notation for the spatial-frequency coordinate is introduced as k but later appears in figures without explicit definition of its normalization; a short clarifying sentence in the methods would remove ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. We have considered the major comment on the validation section and provide a point-by-point response below. We agree that additional detail will strengthen the paper and will incorporate the requested analyses in the revised manuscript.

read point-by-point responses

  1. Referee: [Validation section] Validation section (injection-recovery and HSC-SSP tests): the reported performance metrics are averages; no maximum relative error, bias maps, or error distributions are shown specifically for n > 4 and intermediate k values (roughly 0.5–5) that dominate the likelihood surface for typical galaxy fits. Without these, it is not possible to confirm that downstream morphological parameters remain unbiased at the level implied by the “minimal loss in accuracy” claim.

    Authors: We agree that reporting only average metrics leaves open the possibility of localized biases in the regimes most relevant to typical galaxy fits. In the revised manuscript we will add (i) maximum relative errors, (ii) two-dimensional bias maps in the (n, k) plane, and (iii) histograms of residuals, all restricted to n > 4 and 0.5 < k < 5. These quantities will be computed from the existing injection-recovery suite and from the HSC-SSP comparison sample, allowing direct assessment of whether morphological parameters remain unbiased at the level stated in the abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity; emulator derived from independent numerical training data with separate validation

full rationale

The paper first numerically computes the radial Fourier transform of the Sérsic profile as a function of index and frequency, assembles these values into a training set, and applies symbolic regression to discover an approximating functional form. This discovered equation is then implemented as an emulator inside the pysersic fitter and tested via injection-recovery experiments on model profiles plus direct comparison against other rendering methods on real HSC-SSP galaxies. None of the load-bearing steps reduce by construction to the target morphological parameters or to a self-citation chain; the numerical training data and downstream validation samples are generated independently of the final fitting results. The 2.5× speed-up claim therefore rests on external numerical and observational checks rather than tautological re-use of fitted quantities.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central contribution rests on numerical pre-computation of the transform and the assumption of smoothness; the symbolic regression step introduces no additional free parameters beyond the training grid choice.

free parameters (1)
  • Sampling grid for Sérsic index and k in numerical training set
    The density and range of the numerically computed training points determine the quality of the symbolic regression fit and are chosen by the authors.
axioms (1)
  • domain assumption The radial Fourier transform of the Sérsic profile varies smoothly as a function of Sérsic index and spatial frequency k
    Explicitly stated as a demonstration prior to applying symbolic regression.

pith-pipeline@v0.9.0 · 5806 in / 1242 out tokens · 36004 ms · 2026-05-18T20:46:11.395001+00:00 · methodology

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Reference graph

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