On the Limits of Interpretable Machine Learning in Quintic Root Classification

TL;DR

This study evaluates various machine learning models on classifying polynomial root configurations, finding neural networks perform well but do not recover explicit symbolic rules, highlighting the need for structural biases for interpretability.

Contribution

The paper systematically compares ML models on polynomial root classification, revealing that current models learn geometric approximations rather than symbolic invariants, emphasizing the importance of explicit structural biases.

Findings

Neural networks achieve 84.3% accuracy on in-distribution data.

Decision trees match neural performance with explicit features.

Models do not recover human-interpretable symbolic rules.

Abstract

Can Machine Learning (ML) autonomously recover interpretable mathematical structure from raw numerical data? We aim to answer this question using the classification of real-root configurations of polynomials up to degree five as a structured benchmark. We tested an extensive set of ML models, including decision trees, logistic regression, support vector machines, random forest, gradient boosting, XGBoost, symbolic regression, and neural networks. Neural networks achieved strong in-distribution performance on quintic classification using raw coefficients alone (84.3% + or - 0.9% balanced accuracy), whereas decision trees perform substantially worse (59.9% + or - 0.9\%). However, when provided with an explicit feature capturing sign changes at critical points, decision trees match neural performance (84.2% + or - 1.2%) and yield explicit classification rules. Knowledge distillation reveals that this single invariant accounts for 97.5% of the extracted decision structure. Out-of-distribution, data-efficiency, and noise robustness analyses indicate that neural networks learn continuous, data-dependent geometric approximations of the decision boundary rather than recovering scale-invariant symbolic rules. This distinction between geometric approximation and symbolic invariance explains the gap between predictive performance and interpretability observed across models. Although high predictive accuracy is attainable, we find no evidence that the evaluated ML models autonomously recover discrete, human-interpretable mathematical rules from raw coefficients. These results suggest that, in structured mathematical domains, interpretability may require explicit structural inductive bias rather than purely data-driven approximation.