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.

翻译：复值神经网络（CVNNs）特别适用于处理对相位敏感的信号，包括心电图（ECG）、雷达/声纳以及无线同相/正交（I/Q）数据流。然而，其**可解释性**与**概率校准**仍未得到充分研究。在本工作中，我们提出一个牛顿-普适框架，通过以下方式考察训练后CVNN的**局部决策几何**：（i）在不确定输入点附近，用一个小的、能感知拐点的多项式代理模型来拟合**对数几率差**；（ii）利用牛顿-普适展开对该代理模型进行因式分解，从而导出解析分支描述符，包括指数、重数和方向。这些描述符提供了相位对齐的方向，能在原始网络中引发类别翻转，并允许进行简单、**重数引导的**温度调整以改进校准。我们概述了代理模型具有信息性的假设条件和诊断措施，并刻画了由分段全纯激活函数（如modReLU）引起的潜在失效模式。我们的相位感知分析识别了敏感方向，并在两个案例研究——即MIT-BIH心律失常（ECG）数据集和RadioML 2016.10a（无线调制）——中，相较于未经校准的softmax和标准事后基线方法，提升了预期校准误差（超出受控的$\C^2$合成基准测试）。我们还提供了置信区间、非参数检验，并量化了对分支重数估计误差的敏感性。关键在于，该方法无需修改网络架构，适用于任何具有转换为实数模的复数对数几率的CVNN。