A2 Copula-Driven Spatial Bayesian Neural Network For Modeling Non-Gaussian Dependence: A Simulation Study

In this paper, we introduce the A2 Copula Spatial Bayesian Neural Network (A2-SBNN), a predictive spatial model designed to map coordinates to continuous fields while capturing both typical spatial patterns and extreme dependencies. By embedding the dual-tail novel Archimedean copula viz. A2 directly into the network's weight initialization, A2-SBNN naturally models complex spatial relationships, including rare co-movements in the data. The model is trained through a calibration-driven process combining Wasserstein loss, moment matching, and correlation penalties to refine predictions and manage uncertainty. Simulation results show that A2-SBNN consistently delivers high accuracy across a wide range of dependency strengths, offering a new, effective solution for spatial data modeling beyond traditional Gaussian-based approaches.

Theoretical Foundations of the Deep Copula Classifier: A Generative Approach to Modeling Dependent Features

Traditional classifiers often assume feature independence or rely on overly simplistic relationships, leading to poor performance in settings where real-world dependencies matter. We introduce the Deep Copula Classifier (DCC), a generative model that separates the learning of each feature's marginal distribution from the modeling of their joint dependence structure via neural network-parameterized copulas. For each class, lightweight neural networks are used to flexibly and adaptively capture feature interactions, making DCC particularly effective when classification is driven by complex dependencies. We establish that DCC converges to the Bayes-optimal classifier under standard conditions and provide explicit convergence rates of O(n^{-r/(2r + d)}) for r-smooth copula densities. Beyond theoretical guarantees, we outline several practical extensions, including high-dimensional scalability through vine and factor copula architectures, semi-supervised learning via entropy regularization, and online adaptation using streaming gradient methods. By unifying statistical rigor with the representational power of neural networks, DCC offers a mathematically grounded and interpretable framework for dependency-aware classification.

Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modeling

We introduce the Symplectic Generative Network (SGN), a deep generative model that leverages Hamiltonian mechanics to construct an invertible, volume-preserving mapping between a latent space and the data space. By endowing the latent space with a symplectic structure and modeling data generation as the time evolution of a Hamiltonian system, SGN achieves exact likelihood evaluation without incurring the computational overhead of Jacobian determinant calculations. In this work, we provide a rigorous mathematical foundation for SGNs through a comprehensive theoretical framework that includes: (i) complete proofs of invertibility and volume preservation, (ii) a formal complexity analysis with theoretical comparisons to Variational Autoencoders and Normalizing Flows, (iii) strengthened universal approximation results with quantitative error bounds, (iv) an information-theoretic analysis based on the geometry of statistical manifolds, and (v) an extensive stability analysis with adaptive integration guarantees. These contributions highlight the fundamental advantages of SGNs and establish a solid foundation for future empirical investigations and applications to complex, high-dimensional data.

Can Copulas Be Used for Feature Selection? A Machine Learning Study on Diabetes Risk Prediction

Accurate diabetes risk prediction relies on identifying key features from complex health datasets, but conventional methods like mutual information (MI) filters and genetic algorithms (GAs) often overlook extreme dependencies critical for high-risk subpopulations. In this study we introduce a feature-selection framework using the upper-tail dependence coefficient ({\lambda}U) of the novel A2 copula, which quantifies how often extreme higher values of a predictor co-occur with diabetes diagnoses (target variable). Applied to the CDC Diabetes Health Indicators dataset (n=253,680), our method prioritizes five predictors (self-reported general health, high blood pressure, body mass index, mobility limitations, and high cholesterol levels) based on upper tail dependencies. These features match or outperform MI and GA selected subsets across four classifiers (Random Forest, XGBoost, Logistic Regression, Gradient Boosting), achieving accuracy up to 86.5% (XGBoost) and AUC up to 0.806 (Gradient Boosting), rivaling the full 21-feature model. Permutation importance confirms clinical relevance, with BMI and general health driving accuracy. To our knowledge, this is the first work to apply a copula's upper-tail dependence for supervised feature selection, bridging extreme-value theory and machine learning to deliver a practical toolkit for diabetes prevention.

IGNIS: A Neural Network Framework for Robust Parameter Estimation in Archimedean Copulas

Parameter estimation for Archimedean copulas remains a challenging problem, particularly for the recently developed A1 and A2 families that exhibit complex dependency structures. Traditional methods, such as the Method of Moments (MoM), Maximum Likelihood Estimation (MLE), and Maximum Pseudo-Likelihood (MPL), often struggle due to issues of non-monotonic relationship with dependency measures such as Kendall's tau (as in the case of A1) and numerical instability. In this paper, we present the IGNIS Network, a novel, unified neural framework that learns a direct mapping from observable dependency measures to copula parameters, thereby overcoming the limitations of classical approaches. Our approach is trained on simulated data spanning five Archimedean copula families including Clayton, Gumbel, Frank, A1, and A2, ensuring its general applicability across the entire family. Extensive simulation studies demonstrate that the IGNIS Network reduces estimation errors compared to MoM, while inherently enforcing parameter constraints through theory-guided post-processing. We further validate the practical utility of our method on diverse real-world datasets, including financial returns (AAPL-MSFT), healthcare metrics (CDC Diabetes indicators), and environmental measurements (PM2.5 air quality). Our results underscore the transformative potential of neural methods for robust and accurate dependence modeling in modern applications.