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ECSEL: Explainable Classification via Signomial Equation Learning

Adia Lumadjeng, Ilker Birbil, Erman Acar

TL;DR

ECSEL presents an explainable classification framework that learns closed-form signomial expressions $z_c(x)=\sum_{k=1}^K \alpha_{c,k}\prod_{j=1}^m x_j^{\beta_{c,k,j}}$ per class, enabling transparent predictions through global feature elasticities, exact counterfactuals, and local attributions via log-space derivatives. By leveraging the universal approximation properties of signomials and a gradient-based, sparsity-inducing training objective, ECSEL achieves competitive accuracy on standard benchmarks while delivering inherently interpretable models. Empirical results show ECSEL substantially outperforms symbolic regression baselines in symbolic recovery with faster runtimes and demonstrates strong performance in binary and multi-class classification across diverse domains, including e-commerce and fraud detection. The framework further provides exact analytical tools for explaining decisions, analyzing decision boundaries, and auditing dataset biases, making it practical for high-stakes applications requiring both performance and interpretability.

Abstract

We introduce ECSEL, an explainable classification method that learns formal expressions in the form of signomial equations, motivated by the observation that many symbolic regression benchmarks admit compact signomial structure. ECSEL directly constructs a structural, closed-form expression that serves as both a classifier and an explanation. On standard symbolic regression benchmarks, our method recovers a larger fraction of target equations than competing state-of-the-art approaches while requiring substantially less computation. Leveraging this efficiency, ECSEL achieves classification accuracy competitive with established machine learning models without sacrificing interpretability. Further, we show that ECSEL satisfies some desirable properties regarding global feature behavior, decision-boundary analysis, and local feature attributions. Experiments on benchmark datasets and two real-world case studies i.e., e-commerce and fraud detection, demonstrate that the learned equations expose dataset biases, support counterfactual reasoning, and yield actionable insights.

ECSEL: Explainable Classification via Signomial Equation Learning

TL;DR

ECSEL presents an explainable classification framework that learns closed-form signomial expressions per class, enabling transparent predictions through global feature elasticities, exact counterfactuals, and local attributions via log-space derivatives. By leveraging the universal approximation properties of signomials and a gradient-based, sparsity-inducing training objective, ECSEL achieves competitive accuracy on standard benchmarks while delivering inherently interpretable models. Empirical results show ECSEL substantially outperforms symbolic regression baselines in symbolic recovery with faster runtimes and demonstrates strong performance in binary and multi-class classification across diverse domains, including e-commerce and fraud detection. The framework further provides exact analytical tools for explaining decisions, analyzing decision boundaries, and auditing dataset biases, making it practical for high-stakes applications requiring both performance and interpretability.

Abstract

We introduce ECSEL, an explainable classification method that learns formal expressions in the form of signomial equations, motivated by the observation that many symbolic regression benchmarks admit compact signomial structure. ECSEL directly constructs a structural, closed-form expression that serves as both a classifier and an explanation. On standard symbolic regression benchmarks, our method recovers a larger fraction of target equations than competing state-of-the-art approaches while requiring substantially less computation. Leveraging this efficiency, ECSEL achieves classification accuracy competitive with established machine learning models without sacrificing interpretability. Further, we show that ECSEL satisfies some desirable properties regarding global feature behavior, decision-boundary analysis, and local feature attributions. Experiments on benchmark datasets and two real-world case studies i.e., e-commerce and fraud detection, demonstrate that the learned equations expose dataset biases, support counterfactual reasoning, and yield actionable insights.
Paper Structure (41 sections, 3 theorems, 66 equations, 4 figures, 13 tables)

This paper contains 41 sections, 3 theorems, 66 equations, 4 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Approach
  4. Signomial Functions
  5. Desirable Properties of ECSEL
  6. Symbolic Regression Benchmark
  7. Experimental Setup
  8. Results
  9. Classification Benchmark
  10. Experimental Setup
  11. Results
  12. Case Studies
  13. Case Study: Online Shopping Intention
  14. Case Study: PaySim Fraud Detection
  15. Limitations and Future Work
  16. ...and 26 more sections

Key Result

Theorem 3.1

Let $D \subset \mathbb{R}^m_{>0}$ be compact. For any $f \in C(D, \mathbb{R})$ and $\epsilon > 0$, there exists a signomial $z(x) = \sum_{k=1}^K \alpha_k \prod_{j=1}^m x_j^{\beta_{k,j}}$ such that $\sup_{x \in D} |f(x) -z(x)| < \epsilon$.

Figures (4)

  • Figure 1: Equation recovery rates and average computation time on 45 AI Feynman signomial equations. Comparison of ECSEL (ours), general-purpose SR methods gplearn, PySR, and DGSR (state-of-the-art), averaged over five random seeds (42--46). DGSR timeout cases ($>900$s) excluded from time average.
  • Figure 2: Overview of ECSEL's computation flow and analytical properties. Input features are transformed through class-specific signomial functions to produce score functions $z_c(x)$, which are then converted to class probabilities via softmax. The signomial structure enables three categories of desirable properties: global feature behavior (elasticities, counterfactuals, sensitivity), decision-level effects (margin analysis, probability competition), and local feature attributions (exact for $K=1$, gradient-based for $K>1$).
  • Figure 3: Interpretability properties illustrated on a toy cancer screening example with tumor size ($x_1$), biomarker level ($x_2$), and imaging irregularity ($x_3$). Four scenarios: no-cancer $(0.7, 0.7, 0.8)$, mixed $(1.4, 1.4, 1.2)$, size-driven $(3.0, 1.0, 2.0)$, biomarker-driven $(1.0, 3.0, 2.0)$. (a) Feature elasticities show context-dependent importance (Property \ref{['prop:g1']}). (b) Exact counterfactual under scaling tumor size by $q$ (Property \ref{['prop:g2']}). (c) Decision margin $M_{1,0}$ and cancer probability $p_1$ across scenarios (Properties \ref{['prop:d1']}--\ref{['prop:d2']}). (d) Gradient-based attributions $\phi_j \approx G_{c,j}(x^*)(\log x_j - \log x^*_j)$ for mixed sample at baseline $x^* = (1,1,1)$ (Property \ref{['prop:l2']}).
  • Figure 5: Visual explanations demonstrating ECSEL's desirable properties on the Online Shopping Intention dataset.

Theorems & Definitions (7)

  • Theorem 3.1: Universal Approximation for Signomials
  • Theorem 3.2: Interpretability properties of ECSEL
  • Remark 1.1
  • proof
  • Proposition 2.1
  • proof : Proof sketch
  • proof