Papers › Newton-Puiseux Analysis for Interpretability and Calibration of Complex-Valued Neural Networks
Newton-Puiseux Analysis for Interpretability and Calibration of Complex-Valued Neural Networks
Piotr Migus
Complex-valued neural networks (CVNNs) excel where phase matters, yet their multi-sheeted decision surfaces defy standard explainability and calibration tools. We propose a \emph{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 ℂ² 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.
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