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

TL;DR

This work tackles the interpretability and probability calibration challenges of complex-valued neural networks (CVNNs) by introducing a Newton--Puiseux framework that analyzes local decision geometry around uncertain inputs. It fits a kink-aware, low-degree polynomial surrogate to the logit difference in a small neighborhood and factorizes it via Newton--Puiseux expansions to produce analytic descriptors: branch multiplicities, exponents, and orientations, which yield phase-aligned robustness directions and a multiplicity-guided calibration signal. The authors implement a phase-aware temperature scaling that adjusts calibration based on estimated branch multiplicity and validate the method on a synthetic C^2 benchmark plus two real-world CVNN tasks: MIT--BIH ECG and RadioML wireless I/Q modulation. Across datasets, the approach improves calibration relative to uncalibrated softmax and standard baselines, while remaining architecture-agnostic and applicable to CVNNs with complex logits. The contribution provides a practical, local, algebraic-geometry-based lens on CVNN interpretability and calibration with potential applicability across radar, ECG, wireless, and beyond.

Abstract

Complex-valued neural networks (CVNNs) are particularly suitable for handling phase-sensitive signals, including electrocardiography (ECG), radar/sonar, and wireless in-phase/quadrature (I/Q) streams. Nevertheless, their \emph{interpretability} and \emph{probability calibration} remain insufficiently investigated. In this work, we present a Newton--Puiseux framework that examines the \emph{local decision geometry} of a trained CVNN by (i) fitting a small, kink-aware polynomial surrogate to the \emph{logit difference} in the vicinity of uncertain inputs, and (ii) factorizing this surrogate using Newton--Puiseux expansions to derive analytic branch descriptors, including exponents, multiplicities, and orientations. These descriptors provide phase-aligned directions that induce class flips in the original network and allow for a straightforward, \emph{multiplicity-guided} temperature adjustment for improved calibration. We outline assumptions and diagnostic measures under which the surrogate proves informative and characterize potential failure modes arising from piecewise-holomorphic activations (e.g., modReLU). Our phase-aware analysis identifies sensitive directions and enhances Expected Calibration Error in two case studies beyond a controlled $\C^2$ synthetic benchmark -- namely, the MIT--BIH arrhythmia (ECG) dataset and RadioML 2016.10a (wireless modulation) -- when compared to uncalibrated softmax and standard post-hoc baselines. We also present confidence intervals, non-parametric tests, and quantify sensitivity to inaccuracies in estimating branch multiplicity. Crucially, this method requires no modifications to the architecture and applies to any CVNN with complex logits transformed to real moduli.

Figures (26)

• Figure 1: Reliability diagrams (quantile binning, $10$ bins).
• Figure 2: Sensitivity heatmaps used to select $(\tau^\star,\delta^\star)$.
• Figure 3: Uncertainty distributions at the designated operating point.
• Figure 4: Case #11 (fragile).
• Figure 5: Case #14 (benign). No class flip within the probed radius.