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Compact equations match black-box models in star-galaxy-QSO separation

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2026-08-02 20:03 UTC pith:QO677HJH

load-bearing objection A useful SR benchmark on SDSS DR17 with a credible MvSR result, but the headline parity claim rests on mismatched evaluation protocols and needs a matched-protocol reanalysis before it can be accepted. the 3 major comments →

arxiv 2602.24022 v2 pith:QO677HJH submitted 2026-02-27 astro-ph.IM astro-ph.GA

Comparison of symbolic regression algorithms in Star/galaxy/quasar separation

Rachit Deshpande , Shantanu Desai This is my paper
classification astro-ph.IM astro-ph.GA
keywords symbolic regressionstar-galaxy-quasar classificationSDSS DR17interpretable machine learningCohen's kapparedshift classificationmulti-view symbolic regressiondecision threshold optimization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims that simple, human-readable mathematical functions of a single quantity—spectroscopic redshift—can classify stars, galaxies, and quasars almost as well as optimized black-box machine learning models. Using four symbolic regression systems, the authors search for compact expressions (at most ten nodes) that map redshift to a score, then cut the score with two thresholds to assign the three classes. The best expression reaches a chance-corrected agreement score, Cohen's kappa, of about 0.895 and 94% accuracy, statistically indistinguishable from the random forest and support vector machine benchmarks. The paper argues this shows transparent equations can replace opaque models for this task, and that the shared ~0.90 performance ceiling reflects an actual physical overlap between galaxy and quasar spectra near redshift 1.25, not a limitation of any particular model.

Core claim

Under a unified five-fold cross-validation protocol on 100,000 spectroscopically confirmed SDSS DR17 objects, the authors find that a ten-node analytic function of redshift—s(z)=3.134−2.320/(z+e^(−2.904z))—classifies galaxies, stars, and quasars with Cohen's kappa (chance-corrected agreement) 0.8948±0.0030 and accuracy 94.11%, statistically indistinguishable from the optimized Random Forest (κ=0.8953) and Support Vector Machine (κ=0.8954) baselines. All optimized models, symbolic and black-box, converge near κ≈0.90; the authors interpret this as an information-theoretic ceiling imposed by intrinsic physical degeneracies in the redshift distribution, particularly galaxy-quasar overlap near z≈

What carries the argument

The central object is the one-dimensional decision manifold: a scalar scoring function s(z) built from spectroscopic redshift z, together with two calibrated thresholds t1<t2, such that s(z)<t1 labels a galaxy, t1≤s(z)≤t2 labels a star, and s(z)>t2 labels a quasar. The paper evaluates four symbolic regression algorithms that search for compact expressions (complexity ≤10) for s(z) under mean-squared error against an ordinal class encoding, then optimize the thresholds by grid search over Cohen's kappa. The best expression, found by the multi-view symbolic regression variant, takes the rational-exponential form s(z)=3.134−2.320/(z+e^(−2.904z)), with thresholds t1=0.776±0.002 and t2=0.830±0.00

Load-bearing premise

The parity conclusion rests on comparing symbolic regressions evaluated by five-fold cross-validation on the full 100,000-object set with black-box baselines trained on an 8,000-sample subset and tested on a 2,000-sample holdout; if those evaluation protocols are not actually matched, 'statistically indistinguishable' is not supported.

What would settle it

Retrain the random forest and support vector machine baselines under the exact five-fold cross-validation protocol on the full 100,000 objects, and also evaluate the symbolic expressions on the 8,000/2,000 holdout; if the black-box models then outperform the symbolic expressions by more than the reported error bars, the parity claim fails. A second check is to recompute the thresholds and kappa after permuting the ordinal class encoding (e.g., galaxy=0, star=2, quasar=1); if the reported threshold stability disappears, the decision manifold is an artifact of the encoding.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the parity claim holds, spectroscopic classification pipelines can replace stored black-box models with a single analytic formula and two numbers, reducing memory and compute requirements.
  • The convergence of all models at κ≈0.90 implies that adding more model capacity will not improve accuracy on this one-dimensional feature; improvement must come from additional features (e.g., colors) rather than more complex decision functions.
  • The threshold stability reported for the reinforcement-learning-discovered expression (σ<0.002) suggests that some symbolic search strategies yield decision boundaries robust to calibration noise, which could make them more reliable in automated survey pipelines.
  • The specific failure region around z≈1.25, identified across all methods, gives a concrete target for follow-up with multi-band photometry or spectral indices.

Where Pith is reading between the lines

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

  • A natural extension is to feed the discovered symbolic scores as a feature into a higher-dimensional classifier; because the equation saturates the one-dimensional ceiling, any gain would isolate information carried by color features.
  • The claim of physical degeneracy is inferred from the plateau, but the paper does not compute a Bayes error rate from the full multi-dimensional data; estimating that rate independently would test whether the plateau truly reflects intrinsic class overlap.
  • The ordinal encoding (galaxy=0, star=1, quasar=2) is one of many possible scalings; re-checking threshold stability under different monotonic encodings would show whether the fitted constants depend on the chosen scale.
  • The authors caution that the expressions are empirical and carry no physical significance; a patient reader might still probe whether the e-fold scale in the denominator corresponds to any characteristic quasar spectral feature, but the paper itself disavows that interpretation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper compares four symbolic regression (SR) algorithms — PySR, ESR, PhySO, and MvSR — for the three-class star/galaxy/quasar classification of 100,000 SDSS DR17 objects using only the spectroscopic redshift z. For each method, a compact expression of complexity ≤10 is discovered on a 10,000-object subset, then two thresholds (t1, t2) are calibrated per fold by maximizing Cohen's κ under a 5-fold stratified cross-validation protocol on the full sample. MvSR is reported as the best SR method (κ = 0.8948 ± 0.0030, accuracy 94.11%), with Random Forest, SVM, and MLP baselines reaching κ ≈ 0.895. The paper concludes that low-complexity symbolic expressions can match black-box models and that the residual error reflects an intrinsic physical degeneracy / Bayes error in the redshift distribution.

Significance. If the parity claim survives a matched evaluation, the paper provides a useful, reproducible demonstration that interpretable low-complexity formulae can rival tuned classifiers on a standard astronomical benchmark. The explicit caveat that the equations are empirical, the unified threshold-optimization protocol for the SR methods, and the reporting of fold-to-fold standard deviations are strengths. The comparison against three ML families is a reasonable control. However, the scientific significance depends on two points that the manuscript does not yet establish: the ML baselines must be evaluated under the same 100k 5-fold CV protocol as the SR models, and the interpretation of the κ≈0.90 ceiling as a fundamental Bayes error must be justified rather than asserted.

major comments (3)
  1. [§3.4.4 vs §3.2, Table 4] The ML baselines are evaluated on a single stratified holdout from a 10,000-object subset (8,000 train / 2,000 test, Section 3.4.4), whereas all SR results are 5-fold CV on 100,000 with thresholds tuned per fold. Table 4 reports 'CVκ(Discovery Phase)' and 'CVκ(Global Test Set)' without explaining how a holdout yields CV quantities or what the global test set is. The Section 5.1 claim that MvSR's κ=0.8948±0.0030 is 'statistically indistinguishable' from RF/SVM is therefore not supported by a matched comparison. Please rerun baselines under the identical 100k 5-fold CV protocol, or report a matched holdout for the SR models, and adjust the parity claim accordingly.
  2. [Abstract vs §4.1.4, Table 3] The abstract describes 'an 80,000-sample 5-fold cross-validation threshold optimization phase and a subsequent 10,000-sample unseen hold-out test set' and reports MvSR κ=0.8956 (0.8876 on holdout). Section 4.1.4 and Table 3 report κ=0.8948±0.0030 via 5-fold CV on 100,000. These are different protocols and different numbers; the holdout value 0.8876 does not appear in Table 3 or Table 4. The headline result is thus ambiguous. Please align the abstract with the actual analysis and tabulate protocol-specific metrics explicitly.
  3. [§5.2, §4.2.1] The claim that all models converge at κ≈0.90 and that 'the remaining classification error is rooted in intrinsic physical degeneracies ... fundamental Bayes error rate' is an unsupported interpretation. Convergence of models trained on the same single feature z only shows that these model classes saturate the information contained in z under their training criteria; it does not establish a Bayes error of the redshift distribution. No estimate of the Bayes error, no information-theoretic bound, and no experiment with additional features are provided. Please qualify or remove this claim to avoid overstating the astrophysical conclusion.
minor comments (5)
  1. [§4.1.2, Fig. 16 caption] Caption reads 'The curve for recoil as a function of redshift' — should be 'recall'.
  2. [§3.5.1] The ordinal target mapping is never specified (e.g., STAR=0, GALAXY=1, QSO=2). Please state the exact encoding used for the regression targets; otherwise the numeric constants in Eqs. 34–40 cannot be reproduced or interpreted.
  3. [Table 3] Accuracy and balanced accuracy are reported in inconsistent units: text says '0.9411±0.0017' while the column header says 'Accuracy' without a % sign; the text later says '94.11±0.0017%'. Clarify whether values are fractions or percentages and use consistent notation.
  4. [Eq. (7)] The selection score in Eq. (7) uses Accuracy and κ, but it is not stated whether these are computed on the validation fold after threshold optimization or on the training fold. Please specify the exact evaluation set for model selection.
  5. [Fig. 7 caption] The caption refers to 'Φ-SO' while the method is called PhySO in the text and Table 1. Align the notation.

Circularity Check

1 steps flagged

Main SR-vs-baseline benchmark is held-out and non-circular; minor circularity in the 'information-theoretic ceiling' interpretation.

specific steps
  1. self definitional [Section 5.2 (and Section 4.2.1, Table 4)]
    "The convergence of all optimized models at a performance ceiling of κ≈ 0.90 indicates that the remaining classification error is rooted in intrinsic physical degeneracies of the SDSS DR17 redshift distribution. Specifically, the overlap in spectroscopic signatures between galaxies and quasars near z≈ 1.25 represents a fundamental Bayes error rate within a one-dimensional feature space."

    The 'ceiling' is not independently derived; it is the empirical CV κ of the same baselines (Table 4: 0.8953–0.8954). The paper then labels this measured value an 'information-theoretic ceiling' and a 'fundamental Bayes error rate', and uses it to conclude that the residual error is 'intrinsic'. No external or first-principles estimate of the Bayes error is given, so the explanation reduces to the observed convergence: the models' own performance defines the ceiling and is then said to reach it. This is a self-referential interpretation, though it is secondary to the main held-out benchmark.

full rationale

The central derivation is not circular. Symbolic expressions are fit on a 10,000-object discovery subset, decision thresholds are grid-searched per fold on 80,000 training samples, and κ/accuracy are computed on held-out 20,000-sample folds. The ML baselines are independent black-box models, so the reported parity is an empirical comparison rather than a fit renamed as a prediction; the mismatched evaluation protocol (8k/2k holdout vs full-data 5-fold CV) is a validity concern, not a circularity. The paper explicitly states the equations are empirical and no physical significance should be ascribed to them. Self-citations (Bethapudi & Desai 2018; Krishak & Desai 2020; Desai et al. 2012) are standard references and not load-bearing. The only circular-adjacent step is the Section 5.2 claim of an 'intrinsic' information-theoretic ceiling, which is defined by the models' own converged performance and then used to explain that same performance. Because this interpretive step is secondary and does not drive the main predictive-parity claim, the overall circularity score is 2.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The central result rests on the SDSS DR17 labels, an assumed ordinal encoding of the three classes, the representativeness of the 10k subset, BIC/MDL for model selection, and the fitted constants/thresholds. No new physical entities are introduced; the equations are explicitly empirical.

free parameters (6)
  • PySR expression constants = 0.8395, 78.28, 0.1141
    Fitted to minimize MSE on the 10k-sample discovery subset (Eq. 34).
  • ESR expression constants = 0.7, 11.88
    Fitted during the exhaustive search and L-BFGS refinement (Eq. 36).
  • PhySO expression constants = 2.4027, 0.5786
    Fitted by the reinforcement-learning agent and L-BFGS refinement (Eq. 38).
  • MvSR expression constants = 3.134, 2.320, 2.904
    Fitted per-view and globally in the multi-view regression (Eq. 40).
  • Decision thresholds t1, t2 per method = e.g., PySR: 0.796±0.017, 0.981±0.011; MvSR: 0.776±0.002, 0.830±0.003
    Grid-searched on each validation fold to maximize Cohen's Kappa (Section 3.6, Table 3).
  • Selection weights in Eq. 7 = 0.4, 0.4, 0.2
    Chosen by hand to balance accuracy, Kappa, and complexity in PySR model selection.
axioms (6)
  • domain assumption SDSS DR17 spectroscopic labels (GALAXY/STAR/QSO) are correct and complete.
    The classification ground truth is taken from the survey; any systematic label error propagates into the fitted scores and thresholds (Section 2).
  • domain assumption An ordinal scalar encoding of the three classes is a valid regression target for the symbolic engines.
    The paper states the target is an 'ordinal encoding' (Section 3.5.1) but never specifies the exact mapping; the fitted expressions and thresholds are meaningless if the mapping is arbitrary.
  • domain assumption A single scalar function s(z) with two thresholds can represent the optimal decision boundary along the redshift axis.
    The decision manifold in Eq. 1 assumes a total order (STAR < GALAXY < QSO in score space); this is suggested by Fig. 2 but is not derived.
  • domain assumption The 10,000-sample subset used for symbolic discovery is representative of the full 100,000-object dataset.
    The discovery phase uses a static 10% sample (Section 3.2), and the authors rely on it to find expressions that then generalize in the full CV.
  • standard math BIC/MDL as implemented in Eq. 12 provides valid model selection for this symbolic regression task.
    The paper uses the BIC formulation of MDL, a standard model-selection criterion, but its validity for the non-nested model space of symbolic regression is not established.
  • ad hoc to paper Convergence of all tested models near κ≈0.90 implies a fundamental Bayes error rate for the SDSS DR17 dataset.
    Section 5.2 concludes the remaining error is 'intrinsic physical degeneracy', but the convergence is only demonstrated within the one-dimensional redshift feature space; no higher-dimensional feature set is explored.

pith-pipeline@v1.3.0-alltime-deepseek · 17054 in / 12954 out tokens · 107641 ms · 2026-08-02T20:03:21.648525+00:00 · methodology

0 comments
read the original abstract

This work investigates symbolic regression (SR) as an interpretable alternative to black-box machine learning for the classification of stars, galaxies, and quasars in the Sloan Digital Sky Survey Data Release 17 (SDSS DR17). We conduct a systematic comparative study of four state-of-the-art SR frameworks: {\tt PySR}, Exhaustive Symbolic Regression ({\tt ESR}) with MDL-based selection, Physical Symbolic Optimization ({\tt PhySO}) using deep reinforcement learning, and Multi-View Symbolic Regression ({\tt MvSR}). By deriving compact analytic functions (complexity $\leq$ 10) on a representative training subset and subsequently evaluating them via an 80,000-sample 5-fold cross-validation threshold optimization phase and a subsequent 10,000-sample unseen hold-out test set, we map spectroscopic redshift ($z$) to continuous classification scores. Our results demonstrate that these low-complexity expressions achieve high predictive reliability, with {\tt MvSR} reaching a cross-validation Cohen's Kappa of 0.8956 (0.8876 on the hold-out set) and {\tt PhySO} achieving exceptional parametric stability ($\sigma < 0.002$). We note however that the resulting equations returned by Symbolic regression are purely empirical and no physical significance should be ascribed to these equations.

Figures

Figures reproduced from arXiv: 2602.24022 by Rachit Deshpande, Shantanu Desai.

Figure 1
Figure 1. Figure 1: Class distribution of the SDSS DR17 dataset across GALAXY, STAR, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the u − g color–redshift relation, revealing systematic population differences. Stars follow the well-known stellar locus with minimal redshift evolution, while galaxies occupy a relatively stable intermediate-color region across the observed range (Desai et al., 2012). Quasars, however, exhibit [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Unified experimental pipeline for SDSS DR17 classification. [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Pareto-optimal frontier for the PySR discovery phase. The plot illustrates the trade-off between functional complexity and L2 loss. A significant performance gain is observed at C = 6, corresponding to the introduction of the exponential (exp) operator, which captures the non-linear population transitions. The final scoring function s(z) (indicated by the star) was selected at the complexity limit of C = 1… view at source ↗
Figure 6
Figure 6. Figure 6: Model selection in ESR via the Minimum Description Length (MDL) principle. The plot illustrates the information-theoretic competition between the negative log-likelihood (goodness of fit, Eq. 13) and the complexity prior (parsimony penalty, Eq. 12). The final model is selected at the global minimum of the total description length (indicated by the green dot), ensuring the discovery of a non-overfitted, phy… view at source ↗
Figure 7
Figure 7. Figure 7: Unit-constrained sampling in Φ-SO. The reinforcement learning agent utilizes a masking function to eliminate candidate tokens that would violate mathematical or structural consistency (red crosses). By restricting sampling to a valid functional manifold (shaded region), the framework efficiently identifies stable and physically plausible scoring motifs (green circles), as described in Tenachi et al. (2023)… view at source ↗
Figure 8
Figure 8. Figure 8: Specialist view partitioning in MvSR. The redshift domain is de￾composed into three contiguous, non-overlapping regimes: V1 (Stars), V2 (Galaxies), and V3 (Quasars). This partitioning is implemented by dividing the dataset into three equal-sized segments, ensuring each specialist popula￾tion has sufficient training density. This multi-view decomposition enables the framework to adapt a single global symbol… view at source ↗
Figure 9
Figure 9. Figure 9: Average Row-Normalized Confusion Matrix — [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: PySR Error Analysis. Left: Error density distribution showing correctly classified objects (gray) versus misclassifications (red) across redshift. Right: The Error rate vs. redshift profile, revealing a sharp failure spike peaking near z ≈ 1.25. Misclassifications are nearly absent at z ≈ 0 (stellar locus) but become prominent between z ≈ 0.8 and z ≈ 1.5, where the analytic form struggles to separate over… view at source ↗
Figure 14
Figure 14. Figure 14: ESR Error Analysis. Left: Error density distribution showing cor￾rectly classified objects (gray) versus misclassifications (red) across redshift. Right: Error rate vs. redshift profile indicating a peak failure rate of approxi￾mately 0.38 near z ≈ 1.25. The Error Rate vs. Redshift profile ( [PITH_FULL_IMAGE:figures/full_fig_p011_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Kappa Sensitivity Heatmap — ESR. A 2D grid visualizing Cohen’s κ as a function of decision thresholds t1 and t2, highlighting an expansive optimal plateau. The extensive high-value plateau indicates that the analytic form is less sensitive to minor threshold shifts, suggesting that the MDL-based selection successfully identifies a physically robust scoring manifold that generalizes well across different d… view at source ↗
Figure 16
Figure 16. Figure 16: Recall vs. Redshift — Physical Failure Analysis ( [PITH_FULL_IMAGE:figures/full_fig_p012_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: PhySO Error Analysis. Left: Error density distribution comparing correctly classified objects (gray) versus misclassifications (red) across redshift. Right: Error rate vs. redshift profile showing a peak error rate of approximately 0.34 near z ≈ 1.0. Recall Profiles and Physical Transition Dynamics. The Recall vs. Redshift profile ( [PITH_FULL_IMAGE:figures/full_fig_p012_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Kappa Sensitivity Heatmap — PhySO. A 2D grid visualizing Cohen’s κ as a function of decision thresholds t1 and t2, highlighting a stable and clearly defined optimal region [PITH_FULL_IMAGE:figures/full_fig_p013_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Recall vs. Redshift — Physical Failure Analysis ( [PITH_FULL_IMAGE:figures/full_fig_p013_20.png] view at source ↗
Figure 24
Figure 24. Figure 24: Recall vs. Redshift — Physical Failure Analysis ( [PITH_FULL_IMAGE:figures/full_fig_p014_24.png] view at source ↗

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