Symbolic Regression for Shared Expressions: Introducing Partial Parameter Sharing

Roland Herzog; Viktor Martinek

REVIEW 3 major objections 4 minor 1 cited by

The paper introduces partial parameter sharing for symbolic regression with multiple categorical variables, claiming that a middle level of sharing — parameters that vary with one category but not another — achieves comparable fit quality w

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2026-08-03 12:06 UTC pith:FLQUI5WW

load-bearing objection A clean, useful extension of symbolic regression to multiple categorical variables, but the paper overclaims generalization from in-sample fits. the 3 major comments →

arxiv 2601.04051 v3 pith:FLQUI5WW submitted 2026-01-07 cs.LG

Symbolic Regression for Shared Expressions: Introducing Partial Parameter Sharing

Viktor Martinek , Roland Herzog This is my paper

classification cs.LG

keywords symbolic regressionparameter sharingcategorical variablesgenetic programminginterpretabilitytransfer learningmulti-objective optimizationsupernova light curves

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.

The paper sets out to extend symbolic regression to problems with several categorical variables at once, not just one. It introduces a middle ground between parameters that are the same everywhere and parameters that change with every category combination: partially shared parameters that depend on one category but remain constant across others. On a synthetic example and a supernova flux dataset, the paper argues that this scheme fits data about as well as prior single-category methods while using far fewer parameter values, and that the resulting expressions separate universal, category-specific, and interaction effects. If correct, this makes symbolic regression more parameter-efficient and more interpretable for multi-category scientific datasets.

Core claim

The central claim is that intermediate levels of parameter sharing, where a parameter is tied to one categorical variable but held constant across another, give symbolic regression the flexibility to fit multi-category data with fewer individual parameters than either full sharing or full non-sharing. On a synthetic two-category problem, the paper shows that the scheme can re-identify perturbed ground-truth parameters from very few data points per combination, as long as there is enough global data to pin down shared parameters. On a supernova light-curve dataset with two categorical variables (three datasets, two photometric bands), the 'novel' variant, which uses all three sharing levels,

What carries the argument

The key object is the three-level parameter-sharing hierarchy: fully-shared parameters (one value across all category-value combinations), partially-shared parameters (one value per level of one categorical variable, constant across the other), and non-shared parameters (a distinct value for each category-value combination). This decomposition lets a single expression separate additive effects of each category from their interactions. The scheme is embedded in a multi-objective genetic-programming symbolic regression loop, where the count of individual parameters is a third objective alongside fit and complexity, and the nested least-squares fitting exploits the Jacobian's sparsity pattern t

Load-bearing premise

The load-bearing premise is that matching in-sample fit while using fewer parameters automatically guarantees better out-of-sample and zero-shot generalization, a premise the paper adopts without a held-out category-combination test.

What would settle it

Hold out all data for one category-value combination (for instance, one supernova dataset seen through one photometric band), train the predictive variant on the remaining five combinations, and check whether its predicted flux curve for the held-out combination is accurate; if it is not, the claimed zero-shot generalization fails.

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

If this is right

For scientific datasets with several categorical variables, researchers can now search for one shared functional form that distinguishes universal behavior from category-specific trends and interactions.
Data requirements per category combination can fall dramatically; in the synthetic example, some combinations need only a single data point once shared parameters are well constrained by the rest of the data.
The Pareto-frontier results suggest that parameter count, not just expression size, should be tracked as a primary complexity objective in symbolic regression.
The predictive variant offers a route to zero-shot prediction for unseen category combinations, which the paper motivates but does not empirically verify.

Where Pith is reading between the lines

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

A natural next experiment is to hold out an entire category-value combination (e.g., one supernova dataset in one photometric band) and measure whether the predictive variant's fit degrades gracefully; the paper's zero-shot claim stands or falls on this test.
The same partial-sharing principle could be applied to other structured multi-task regression settings, such as group-contribution models in chemical engineering, where parameters like group-specific contributions are already a form of categorical sharing.
Because the real-data comparison uses a fixed preprocessing from the prior study, an independent train/test split would clarify whether the reported R² values reflect in-sample fits; ideally the Pareto analysis would be repeated on held-out data.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A clean, useful extension of symbolic regression to multiple categorical variables, but the paper overclaims generalization from in-sample fits.

read the letter

Hi —

The paper is a legitimate incremental step: it generalizes symbolic regression with categorical variables from one to multiple categories, and introduces intermediate sharing levels. That is new relative to the cited work — Russeil et al. [24] flatten categories, Kronberger et al. [11] mention multiple categories but call them unfeasible. The three-way split into fully-shared, partially-shared, and non-shared parameters is a natural and clean formalization, and the nested least-squares formulation with sparsity exploitation is a sensible implementation detail.

What it does well: the synthetic data-sharing experiment is honest and informative — it shows minimum data requirements track the number of individual parameters, not the number of category combinations, and it explicitly shows when re-identification fails. The real-data comparison on the supernovae dataset is run inside the same framework against three baselines, and the Pareto frontier plots are a fair way to compare. The reported expressions do achieve comparable R² with fewer individual parameters than the published [24] results. That part of the claim is credible.

The soft spot is real: none of the R² numbers are out-of-sample. There is no described train/test split for the supernovae experiment, so Figure 3.1 and Table 3.3 are fitting metrics. The conclusion then says the approach 'facilitates zero-shot generalization to unseen category combinations' — but no experiment holds out a category combination. The predictive variant is described as designed for zero-shot, but that's a design intent, not a result. The synthetic transfer test only re-fits perturbed ground-truth parameters; it doesn't discover expressions for new category values. So the generalization claims are untested. Also, five runs per variant with no error bars — the niching step in the plot may hide run-to-run variance. No code or data release is provided, which would help.

That said, the core idea holds up; the flaw is in the evaluation, not the math. The parameter-reduction benefit is visible even in-sample, and the synthetic experiment is internally consistent. This is the kind of paper that should get a serious referee, not a desk reject.

Who it's for: people working on symbolic regression with factorial structure, and applied scientists who need interpretable models across multiple conditions. I'd bring it to a reading group if you're in that area, and I'd cite it once the evaluation gap is closed — the idea is worth knowing now.

Recommendation: send to peer review, with a request for a proper train/test split, a held-out category-combination experiment, error bars, and code release. If those are added, the paper would be solid.

Referee Report

3 major / 4 minor

Summary. The paper extends symbolic regression with categorical variables by introducing partially shared parameters that are shared across one categorical variable but depend on another, alongside fully shared and non-shared parameters. After defining the sharing taxonomy and describing its implementation in the TiSR framework, the paper reports a synthetic parameter-recovery/data-reduction experiment (Section 3, Eq. 3.1) and a five-variant comparison on a supernovae dataset previously used in [24]. The central claim is that the proposed variants achieve comparable fit quality with significantly fewer parameters while revealing category structure. The conclusion further asserts superior Pareto trade-offs, transfer learning, and zero-shot generalization to unseen category combinations.

Significance. If the claims are validated, the proposed partial-sharing taxonomy is a useful extension of single-category symbolic regression, particularly for separating category-agnostic, category-specific, and interaction effects. The synthetic identifiability experiment is internally consistent, and the comparison of all five variants inside the same algorithmic framework is a strength that controls for implementation confounds. However, the most distinctive claims — out-of-sample trade-off superiority, transfer learning, and zero-shot generalization — are not supported by the experiments as reported. The central derivation is sound, but the evidence is incomplete for the paper's strongest conclusions.

major comments (3)

[Section 3, Figure 3.1 and Tables 3.2–3.3] No train/test split is described for the supernovae experiment; all reported R² values are therefore in-sample fits. Since the search simultaneously optimizes R² and parameter count, the Pareto comparison shows only that the proposed variants can fit the training data with fewer parameters. It does not establish 'superior trade-offs between fit quality and parameter count' as a predictive statement. Please add a hold-out or cross-validated evaluation, or explicitly revise the claim to 'in-sample fit quality'.
[Section 4 (Conclusion)] The conclusion claims that 'the proposed approach facilitates zero-shot generalization to unseen category combinations,' but no experiment holds out a category combination. The predictive variant is described as designed for this purpose by omitting non-shared parameters, but design intent is not evidence. Either report an experiment that evaluates held-out category combinations, or remove/qualify the zero-shot generalization claim.
[Section 3, Eq. (3.1) and Table 3.1] The synthetic 'transfer learning' test only re-fits perturbed ground-truth parameters of a known expression under data subsampling. This tests parameter recovery and data requirements, not the transfer of discovered expressions to new category values or new physical settings. The abstract's reference to testing 'transfer learning' therefore overstates what is measured. Please clarify that this is an identifiability/data-requirement experiment and temper the transfer-learning language accordingly.

minor comments (4)

[Section 3, text before conclusion] Typo and grammar: 'An additional benefitofthe proposedapproachisthatmore informationiscan be extracted' should be corrected. Also, 'which act like 48 parameters' is unclear; the intended meaning seems to be that 20 individual parameters replace 48 fully non-shared parameters, but the phrasing should be made explicit.
[Section 3, Figure 3.1] The niching process and horizontal jitter are described, but it is unclear how many unique solutions remain per variant and whether the plotted points are the result of a single best run or aggregated across the five runs. Please clarify the aggregation and consider showing run-level variation or confidence information.
[Section 3, synthetic experiment] The 100 processions are described but no random seed or variance summary is given for the perturbation scale in Eq. (3.1). Reporting the seed would improve reproducibility.
[References] In the introduction, 'Virgolin, Pissis [29]' lacks a year; please add it. Also check the reference list for consistent formatting of arXiv identifiers.

Circularity Check

0 steps flagged

No circularity: the proposed method is a constrained fitting/search procedure evaluated against an external benchmark, and the synthetic test is explicitly fitting-only.

full rationale

The derivation chain is self-contained. The method section defines a least-squares objective (Eq. 2.2) over explicit shared, partially-shared, and non-shared parameters, and the GP search is a standard optimization procedure. The synthetic experiment uses a known ground-truth equation (Eq. 2.3) and measures parameter re-identification under data subsampling and perturbation (Eq. 3.1); the paper itself labels this “synthetic, fitting-only,” so it is not a disguised prediction of new symbolic structure. The real-data comparison adopts the external benchmark and preprocessing of [24], re-implements all variants inside the same TiSR framework, and reports fitted R² and parameter counts; no parameter or expression from [24] is used as an input to derive the proposed variants' results. The self-citation to TiSR [16, 18] is a citation to open-source code used as the implementation base, and the other self-citations (e.g., [17]) are contextual examples, not load-bearing assumptions. Some stronger claims in the conclusion—zero-shot generalization to unseen category combinations and out-of-sample superiority—are asserted without a corresponding holdout experiment, but that is a validation gap, not circularity. No equation is defined in terms of the quantity it is claimed to predict, and no fitted parameter is renamed as a prediction.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 1 invented entities

The central claims rest on several experimental design choices (complexity cap, run duration, initialization probability, perturbation scale, number of seeds) and on unproven domain assumptions. The main uncharged assumption is that in-sample R² and parameter counts imply generalization/zero-shot behavior, which appears in Sec. 3 and the conclusion. No physical entities are introduced.

free parameters (5)

Maximum expression complexity = 12
Chosen to allow direct comparison with [24]; caps the search space and shapes the Pareto front in Figure 3.1.
Run duration per variant = 15 minutes
Arbitrary compute budget; different budgets could change which expressions are found and the reported trade-offs.
Probability of introducing partially/non-shared parameters in new expressions = 0
Editorial note in Sec. 2.3: 'We set the probability of including any but fully-shared parameters into new expression to zero.' This hand-chosen initialization affects the search.
Perturbation scale in synthetic transfer experiment = 0.1
Eq. (3.1) uses p + 0.1·p·r; this choice sets how hard re-identification is and therefore the measured minimum data requirement.
Number of runs and random seeds = 5 runs, seeds not reported
Five 15-minute runs per variant; no seed list is given, so variance across runs cannot be assessed.

axioms (6)

domain assumption The search heuristics (NSGA-II/GP) can find expressions close to the true Pareto frontier within 15-minute runs.
The comparisons in Fig. 3.1 assume the runs are long enough and the random search effective enough; no convergence analysis is given.
ad hoc to paper Fitted least-squares parameters and R² on the benchmark dataset are meaningful evidence of model quality without a reported hold-out split.
Section 3 reports R² values for the supernovae dataset and draws generalization/zero-shot conclusions, but no train/test split or unseen-category evaluation is described.
ad hoc to paper For nonlinear expressions, satisfying the stated per-parameter data-count conditions guarantees recoverability and minimum data requirements.
Sec. 3 states conditions for each parameter type; these are plausible for identifiable linear-in-parameter terms but not proven for the nonlinear expressions actually searched.
domain assumption Perturbing ground-truth parameters and re-identifying them measures transfer learning.
The synthetic experiment re-fits a known equation (2.3) with perturbed parameters; success is equated with data sharing/transfer, a stronger conclusion than parameter recovery.
domain assumption The supernovae dataset preprocessing from [24] is appropriate and was not selectively altered for the five variants.
The paper adopts the dataset and preprocessing of [24] without describing it; comparability depends on this carried-over pipeline.
standard math The nonlinear least-squares parameter identification (Sec. 2.2) converges to a good minimum for the expressions examined.
The method relies on numerical optimization for parameters; the paper notes the landscape is multi-modal (Sec. 2.3) but does not provide convergence guarantees.

invented entities (1)

Partially-shared parameters (ppartial,U, ppartial,L / C category) no independent evidence
purpose: Parameters shared across one categorical variable but varying with another, capturing category-specific trends without full interactions.
The paper demonstrates the construct on a synthetic equation and a supernovae dataset, but provides no external prediction (e.g., a held-out category combination) that only the partial-sharing model forecasts; the zero-shot claim is untested. It is a modeling construct rather than a physical entity.

pith-pipeline@v1.3.0-alltime-deepseek · 11183 in / 14097 out tokens · 136533 ms · 2026-08-03T12:06:50.476742+00:00 · methodology

0 comments

read the original abstract

Symbolic regression aims to find symbolic expressions that describe datasets. Due to its inherent interpretability, symbolic regression (SR) is a powerful paradigm for scientific discovery. Recent advances have expanded SR to describe related phenomena using a single expression with varying sets of parameters, thereby introducing a single categorical variable. To illustrate, this enables the search for a single expression describing temperaturedependent viscosity across multiple fluids, while simultaneously identifying a distinct set of fluid-specific parameters. We expand upon prior efforts by considering multiple categorical variables and introducing intermediate levels of parameter sharing. Rather than parameters being either entirely universal or entirely unique, some parameters can also be shared across specific categories while remaining distinct for others. This allows for separating universal effects (shared parameters), category-specific trends (partially-shared parameters), and category interactions (non-shared parameters). We test the limits of this setup in terms of reducing data requirements and transfer learning using a synthetic, fitting-only example. Furthermore, we apply the method to an astrophysics dataset also used in a previous single-category study. In comparison, we achieve similar fit quality with significantly fewer parameters while extracting additional information about the problem.

Figures

Figures reproduced from arXiv: 2601.04051 by Roland Herzog, Viktor Martinek.

**Figure 2.1.** Figure 2.1: Illustrative example of a two-category problem with all possible category-value combinations. Non-shared parameters are illustrated in the top-left, fully-shared ones in the top-right, and partially-shared ones in the bottom two illustrations. As an intuitive example, if U were patient groups and L were medications, a parameter partially shared across the patient types may account for the effects of a pa… view at source ↗

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 2.2.** Figure 2.2: Sparsity pattern of the Jacobian required for identifying the parameters of Equation (2.3) using the example of Table 2.1. The circular points correspond to shared parameters, squares correspond to partially-shared parameters, and triangles pertain to non-shared parameters. 2We refer to the real-valued parameters contained in each of the “high-level” parameters as individual parameters [PITH_FULL_IMAG… view at source ↗

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 3.1.** Figure 3.1: Performance comparison of five categorical variable handling variants, plotted by 1−R2 versus the number of individual parameters. Data points represent expressions from five 15-minute runs per variant. To improve clarity, a “niching” process was applied to each variant independently: 1 − R2 values are rounded to three significant digits, and only unique solutions are retained. A horizontal jitter is ap… view at source ↗

discussion (0)

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doi:10.5281/zenodo.4041459

doi:10.5281/zenodo.4041459