Exhaustive Canonical Skeletons Generation: A Formal Dataset for Decomposable Symbolic Regression

The strongest elements are the formal dataset construction and the rigorous benchmarking effort. The closure rules in Definition 3 and the normal-form operator $\mathcal{N}$ yield a deterministic enumeration pipeline whose output is provably non-redundant; as the authors note, canonicalizing existing datasets reveals a set that is substantially more compact, eliminates structural repetitions, and achieves a uniform operator distribution free of simplicity bias. The derivative-driven singularity avoidance mechanism represents a genuine algorithmic improvement over prior grid-based heuristics, and the benchmark design directly addresses the narrow-domain limitations of standard suites such as Feynman.

The term 'exhaustive' must be read in a highly restricted sense. The generating grammar is bounded by $N_U$ and $N_D$ such that the maximum operators per expression is only four and the maximum unary nesting depth is two (Table I), a drastically smaller complexity regime than NeSymRes (depth six, ten operators) or SeTGAP (depth three). This means the zero-duplication claim is achieved by pruning the search space aggressively rather than by solving redundancy in a general setting.

The multivariate utility is asserted but not empirically validated: the paper states that univariate skeletons are intended to serve the decomposable SR paradigm, yet no experiments on multivariate synthesis are presented. The benchmark evaluation also raises fairness questions—hyperparameters for thirteen diverse SR methods were tuned using a single representative equation per tier, as the authors report selecting the configuration 'yielding the average mean square error across valid (non-NaN) runs.'

The evidence supports the core claim that stochastic datasets are structurally redundant, but the comparisons deserve nuance. The authors canonicalized the baseline datasets themselves, and they acknowledge that NeSymRes multivariate expressions were re-skeletonized with respect to the primary variable $x_1$ only, a transformation that can distort structural statistics. The reported duplication rates are therefore partly a function of comparing a constrained canonical catalog against much larger unconstrained sets: ExCaUS has 71,027 unique skeletons out of 71,027 total, whereas NeSymRes yields 128,514 unique skeletons out of 10 million. On the positive side, the benchmark is refreshingly difficult, and the authors correctly observe that most symbolic regression methods struggle significantly to recover the underlying functions.

Reproducibility is mixed. The authors provide public code and data, and the grammar rules, complexity bounds, and Algorithm 1 are specified with enough detail to recreate the generation pipeline. However, critical implementation details—including the additional structural filtering rules in S.M. C-A, the full benchmark equations in S.M. E, and the hyperparameter grids in S.M. F—are deferred to supplementary material not provided here, which would block independent verification of the canonical counts and benchmark scores. Moreover, the simplification step in Algorithm 1 relies on SymPy's simplify, whose behavior is version-dependent and not formally characterized. The sampling threshold $\alpha = 0.9995$ is stated, but the adaptive-interval algorithm requires the same supplementary appendix to be fully reproduced.