Newton-Puiseux Analysis for Interpretability and Calibration of Complex-Valued Neural Networks (2504.19176v1)

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 \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 $\mathbb{C}^2$ 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.

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

This paper presents an innovative approach to the interpretability and calibration of complex-valued neural networks (CVNNs) using a Newton-Puiseux framework. CVNNs are particularly suitable for tasks where phase information is crucial, such as radar imaging and quantum state modeling. However, their unique geometry poses challenges for standard tools used in explainable AI (XAI) and probability calibration. The authors introduce a novel method leveraging Newton-Puiseux expansions to address these challenges.

Key Contributions

1. Newton-Puiseux Framework: This methodology involves fitting a local polynomial surrogate to high-uncertainty input regions and decomposing it into fractional-power series, known as Puiseux series. This approach allows for closed-form estimates of decision robustness and network over-confidence, which are not achievable with gradient-based methods like saliency maps or perturbation techniques such as LIME and SHAP.
2. Interpretability of Decision Boundaries: By analyzing the Puiseux series of decision functions, the framework provides insights into multi-sheeted decision surfaces that CVNNs exhibit. Specifically, the coefficients obtained from the Puiseux expansions relate directly to the robustness and sensitivity of the decision boundary to phase shifts.
3. Practical Calibration Results: The application of this technique to the MIT-BIH arrhythmia corpus demonstrates significant improvements in network calibration. The authors employ phase-aware temperature scaling informed by Puiseux branches to reduce the expected calibration error from 0.087 to 0.034, a relative reduction of 61%.

Detailed Findings

• On synthetic datasets modeling complex helix structures, the surrogate achieved an RMSE of less than 0.09, effectively capturing phase-related decision complexities.
• In real-world ECG data (MIT-BIH corpus), the technique reduces calibration errors significantly via Puiseux-guided methods.
• The extracted Puiseux coefficients predict adversarial flip radii within $10^{-3}$, highlighting potential vulnerabilities in model decision-making processes.

Implications and Future Directions

The paper positions the Newton-Puiseux framework as a critical development in bridging the interpretability-calibration gap for CVNNs. Its implications are profound for domains that rely on phase information, such as biomedical signal processing and quantum machine learning. The analytic nature of Puiseux expansions offers a transparent approach to both local and global understanding of complex networks.

In terms of future research directions, expanding the applicability of Newton-Puiseux analysis to higher-dimensional CVNNs could enhance its utility across broader datasets and applications. Furthermore, integrating stochastic aspects to account for uncertainty in Euclidean decision spaces could make CVNNs more robust against real-world noise and perturbations.

Conclusion

Ultimately, this paper advances the field of complex-valued machine learning by providing both theoretical insights and practical solutions to longstanding interpretability and calibration challenges. As complex-valued networks continue to gain traction, the Newton-Puiseux analysis framework offers a promising tool to navigate the intricacies of phase-dependent data, unlocking potential improvements across multiple scientific and industrial applications.