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arxiv: 2605.15809 · v1 · pith:PCTROGYUnew · submitted 2026-05-15 · 💻 cs.NE

Diversified Residual Symbolic Regression

Koki Ikeda , Masahiro Nomura , Ryoki Hamano This is my paper

Pith reviewed 2026-05-19 18:41 UTC · model grok-4.3

classification 💻 cs.NE
keywords symbolic regressionquality diversityresidual patternsoutliersinterpretabilityastronomical datasynthetic datasetsmultiple expressions
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The pith

Symbolic regression now collects multiple expressions that differ in which observations they treat as outliers.

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

Symbolic regression seeks explicit mathematical expressions to explain data, but outliers can derail the search for the dominant relationship. Deciding which points count as outliers often requires domain expertise that is hard to encode in advance. The paper proposes diversified residual symbolic regression, which runs a quality-diversity search to retain a set of high-accuracy expressions that vary systematically in their residual patterns. Users can then inspect these candidates after the search and pick the one whose outlier treatment matches their knowledge. Experiments on synthetic mixtures and an astronomical dataset show that the approach recovers multiple plausible relationships where standard symbolic regression returns only one.

Core claim

DRSR collects multiple expressions that fit the data well but differ in how residuals are distributed, enabling post-search selection aligned with domain knowledge. On a synthetic mixture dataset, DRSR produces more diverse expressions than conventional SR while capturing multiple underlying relationships. On a real-world astronomical dataset, DRSR discovers multiple expressions consistent with known physical relationships.

What carries the argument

A Quality-Diversity archive that maintains expressions distinguished by the distribution of their residuals across the data points.

If this is right

  • Users gain the ability to examine different residual patterns and select the expression consistent with their domain expertise.
  • Symbolic regression becomes less sensitive to ambiguous outlier definitions without needing predefined thresholds.
  • A single search run can surface multiple meaningful relationships present in the same dataset.
  • Post-search selection replaces the need for upfront decisions on which observations to downweight.

Where Pith is reading between the lines

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

  • The same residual-diversity idea could be applied to other regression or modeling tasks where outlier treatment is ambiguous.
  • Interactive interfaces that let experts steer the archive during search might further improve relevance of the returned expressions.
  • Testing whether the diversity of residuals correlates with diversity of downstream predictions or decisions would strengthen the method's utility.

Load-bearing premise

Diversity in residual patterns produced by the Quality-Diversity archive corresponds to distinct, meaningful underlying relationships that domain experts can reliably distinguish and select among.

What would settle it

Domain experts reviewing the archive expressions find that the different residual patterns do not map to substantively different physical or causal interpretations of the data.

Figures

Figures reproduced from arXiv: 2605.15809 by Koki Ikeda, Masahiro Nomura, Ryoki Hamano.

Figure 1
Figure 1. Figure 1: MedAE landscapes over the coefficients for two ex [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Parameterized GP tree for 1 1+exp(𝑥 ) . in A. As a result, A can later return multiple coefficient-tuned expressions corresponding to different locally refined coefficient configurations. 5 Evaluation on Synthetic Datasets In this section, we evaluate the effectiveness of the proposed frame￾work, DRSR, on synthetic datasets. 5.1 General Setting Representation and GP Approach. DRSR follows a GP approach, it… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of trajectories of the best predictive accuracy in each generation of DRSR on Nguyen benchmarks with [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of trajectories of the best predictive accuracy in each generation of DRSR on Nguyen benchmarks with [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evaluation metric trajectories of DRSR, SR, and MOSR on the synthetic mixture dataset. We track best fitness, [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Predictive behaviors of the top 3 expressions from DRSR, SR, and MOSR with the highest component-wise accuracies [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Top 6 expressions found by DRSR for the stellar [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Log–log plots of the same expressions shown in [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of trajectories of the best predictive accuracy in each generation of SR on Nguyen benchmarks with [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of trajectories of the best predictive accuracy in each generation of SR on Nguyen benchmarks with [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of trajectories of the best predictive accuracy in each generation of MOSR on Nguyen benchmarks with [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of trajectories of the best predictive accuracy in each generation of MOSR on Nguyen benchmarks with [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
read the original abstract

Symbolic regression (SR) aims to discover explicit mathematical expressions that explain observed data and is widely used in domains where interpretability is essential. Because interpretability requires expressions to reflect meaningful regularities, SR is sensitive to observations that deviate from the dominant relationship. Such irregular observations, or outliers, are common in real-world data and can hinder SR from identifying underlying regularities. Robust regression mitigates this by downweighting observations with large residuals. However, deciding which observations should be treated as outliers is often ambiguous and depends on user interpretation and domain knowledge, a perspective largely overlooked in existing SR studies. This motivates approaches that present multiple candidate expressions, allowing users to examine different residual patterns and choose expressions consistent with their expertise. We propose diversified residual symbolic regression (DRSR), which achieves high predictive accuracy while promoting diversity with respect to residual patterns based on the Quality-Diversity paradigm. DRSR collects multiple expressions that fit the data well but differ in how residuals are distributed, enabling post-search selection aligned with domain knowledge. On a synthetic mixture dataset, DRSR produces more diverse expressions than conventional SR while capturing multiple underlying relationships. On a real-world astronomical dataset, DRSR discovers multiple expressions consistent with known physical relationships.

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

3 major / 3 minor

Summary. The paper proposes Diversified Residual Symbolic Regression (DRSR), an algorithm that applies the Quality-Diversity (QD) paradigm to symbolic regression. It maintains an archive of expressions that achieve high predictive accuracy while differing in their residual patterns (via a residual descriptor and niching mechanism). The central claims are that DRSR yields more diverse expressions than standard SR on a synthetic mixture dataset, capturing multiple underlying relationships, and that it recovers multiple expressions consistent with known physical relationships on a real-world astronomical dataset.

Significance. If the central empirical claims hold with proper validation, the work would be moderately significant for the symbolic regression community. It directly addresses the practical problem of outlier ambiguity and multi-modal data by shifting from a single best-fit expression to a curated set of alternatives, which aligns with domain-expert selection. The use of QD for residual-pattern diversity is a novel algorithmic angle, though its impact depends on demonstrating that the produced diversity is semantically meaningful rather than artifactual.

major comments (3)
  1. [§4] §4 (Experiments on synthetic data): The claim that DRSR 'captures multiple underlying relationships' on the synthetic mixture dataset is load-bearing for the central contribution, yet the manuscript provides no quantitative recovery metrics (e.g., per-component R² on held-out subsets, symbolic equivalence checks against ground-truth components, or alignment between archive niches and generative processes). Without these, residual-pattern diversity cannot be confirmed to correspond to distinct relationships rather than fitting noise or equivalent rewrites.
  2. [§3.2] §3.2 (Residual descriptor and archive grid): The definition of the residual descriptor used for niching is central to the diversity claim, but the paper does not report sensitivity analysis or ablation on the choice of descriptor features and grid resolution. If the descriptor primarily captures magnitude rather than pattern shape, the QD archive may simply rediscover scaled variants of the same expression.
  3. [Results tables] Table 2 or equivalent results table: The reported diversity advantage over conventional SR lacks statistical tests (e.g., paired t-tests or Wilcoxon tests across multiple runs) and baseline comparisons against other multi-expression SR methods such as Pareto-front or ensemble SR approaches. This weakens the assertion that the QD mechanism is the source of improved diversity.
minor comments (3)
  1. [Abstract] The abstract states empirical outcomes but supplies no quantitative metrics, baseline comparisons, or implementation details; these should be summarized with effect sizes even in the abstract.
  2. [§3] Notation for the QD archive parameters (e.g., niche count, diversity metric weights) should be introduced with explicit symbols in §3 and used consistently in the experimental section.
  3. [Figures] Figure captions for the astronomical dataset results should explicitly state which physical relationships each discovered expression is claimed to recover, with reference to the relevant literature.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. The comments highlight important areas for strengthening the empirical validation and presentation of results. We have revised the manuscript accordingly by adding quantitative recovery metrics, sensitivity analyses, statistical tests, and additional baselines. Point-by-point responses follow.

read point-by-point responses

  1. Referee: §4 (Experiments on synthetic data): The claim that DRSR 'captures multiple underlying relationships' on the synthetic mixture dataset is load-bearing for the central contribution, yet the manuscript provides no quantitative recovery metrics (e.g., per-component R² on held-out subsets, symbolic equivalence checks against ground-truth components, or alignment between archive niches and generative processes). Without these, residual-pattern diversity cannot be confirmed to correspond to distinct relationships rather than fitting noise or equivalent rewrites.

    Authors: We agree that quantitative recovery metrics are necessary to substantiate the claim. In the revised manuscript, we have added per-component R² scores on held-out subsets for each generative component of the mixture, symbolic equivalence checks (via expression simplification and tree-edit distance) against the ground-truth expressions, and an explicit alignment between the QD archive niches and the underlying generative processes. These new results confirm that the observed residual diversity corresponds to distinct relationships. revision: yes

  2. Referee: §3.2 (Residual descriptor and archive grid): The definition of the residual descriptor used for niching is central to the diversity claim, but the paper does not report sensitivity analysis or ablation on the choice of descriptor features and grid resolution. If the descriptor primarily captures magnitude rather than pattern shape, the QD archive may simply rediscover scaled variants of the same expression.

    Authors: We acknowledge the value of validating the descriptor design. We have performed and now report sensitivity analyses on descriptor features (comparing raw moments, binned histograms, and normalized patterns) and multiple grid resolutions. The updated Section 3.2 and supplementary material show that the niching mechanism preserves diversity in residual shape even when magnitude is controlled, with the archive consistently separating expressions that differ in residual distribution rather than producing scaled variants of the same model. revision: yes

  3. Referee: Table 2 or equivalent results table: The reported diversity advantage over conventional SR lacks statistical tests (e.g., paired t-tests or Wilcoxon tests across multiple runs) and baseline comparisons against other multi-expression SR methods such as Pareto-front or ensemble SR approaches. This weakens the assertion that the QD mechanism is the source of improved diversity.

    Authors: We agree that statistical tests and broader baselines strengthen the comparison. The revised results table now includes Wilcoxon signed-rank tests across ten independent runs, confirming statistical significance of the diversity gains. We have also added direct comparisons to a Pareto-front multi-objective SR baseline and an ensemble SR method; these show that the residual-pattern QD approach yields distinct forms of diversity not captured by complexity-accuracy trade-offs or averaging ensembles. revision: yes

Circularity Check

0 steps flagged

No circularity; algorithmic proposal validated on external datasets

full rationale

The paper proposes DRSR as an algorithmic method that applies the Quality-Diversity paradigm to symbolic regression in order to collect expressions differing in residual patterns. Its central claims of greater diversity on synthetic mixtures and consistency with known physical relationships on astronomical data are presented as outcomes of empirical evaluation on those datasets, without any derivation, equation, or self-citation that reduces the reported diversity or predictive results to quantities defined by the same fitted parameters or archive metrics by construction. The approach is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The proposal rests on the assumption that Quality-Diversity can be directly applied to residual patterns in symbolic regression and that the resulting archive will contain expressions reflecting distinct regularities; no new physical entities or ad-hoc constants are introduced.

free parameters (1)
  • Quality-Diversity archive parameters and diversity metric weights
    Hyperparameters controlling the diversity objective and archive maintenance are required for the algorithm and are expected to be tuned on the target datasets.
axioms (1)
  • domain assumption Quality-Diversity optimization can be applied to symbolic regression objectives to promote diversity in residual patterns
    Invoked when the paper defines DRSR as achieving diversity with respect to residual patterns based on the Quality-Diversity paradigm.

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