Newton-Puiseux Analysis for Interpretability and Calibration of Complex-Valued Neural Networks

Code

• github.com/piotrmgs/puiseux-cvnn
  OfficialIn paperpytorch★ 1

Abstract

Complex-valued neural networks (CVNNs) excel where phase matters, yet their multi-sheeted decision surfaces defy standard explainability and calibration tools. We propose a Newton-Puiseux framework that fits a local polynomial surrogate to a high-uncertainty input and analytically decomposes this surrogate into fractional-power series. The resulting Puiseux expansions, dominant Puiseux coefficients, and phase-aligned curvature descriptors deliver closed-form estimates of robustness and over-confidence that gradient - or perturbation-based methods (saliency, LIME, SHAP) cannot provide. On a controlled C^2 helix the surrogate attains RMSE < 0.09 while recovering the number of decision sheets; quartic coefficients predict adversarial flip radii within 10^-3. On the real-world MIT-BIH arrhythmia corpus, Puiseux-guided, phase-aware temperature scaling lowers expected calibration error from 0.087 to 0.034, contributing to the advancement of CVNNs. Full code, pre-trained weights, and scripts are at https://github.com/piotrmgs/puiseux-cvnn.

Reproductions