Agnideep Aich

Agnideep Aich, Ashit Baran Aich

We present the Deep Copula Classifier (DCC), a class-conditional generative model that separates marginal estimation from dependence modeling using neural copula densities. DCC is interpretable, Bayes-consistent, and achieves excess-risk $O(n^{-r/(2r+d)})$ for $r$-smooth copulas. In a controlled two-class study with strong dependence ($|ρ|=0.995$), DCC learns Bayes-aligned decision regions. With oracle or pooled marginals, it nearly reaches the best possible performance (accuracy $\approx 0.971$; ROC-AUC $\approx 0.998$). As expected, per-class KDE marginals perform less well (accuracy $0.873$; ROC-AUC $0.957$; PR-AUC $0.966$). On the Pima Indians Diabetes dataset, calibrated DCC ($τ=1$) achieves accuracy $0.879$, ROC-AUC $0.936$, and PR-AUC $0.870$, outperforming Logistic Regression, SVM (RBF), and Naive Bayes, and matching Logistic Regression on the lowest Expected Calibration Error (ECE). Random Forest is also competitive (accuracy $0.892$; ROC-AUC $0.933$; PR-AUC $0.880$). Directly modeling feature dependence yields strong, well-calibrated performance with a clear probabilistic interpretation, making DCC a practical, theoretically grounded alternative to independence-based classifiers.

Agnideep Aich, Ashit Baran Aich, Bruce Wade

The convergence of gradient descent (GD) on the non-convex loss landscapes of deep neural networks (DNNs) presents a fundamental theoretical challenge. While recent work has established that GD converges to a stationary point at a sublinear rate within locally quasi-convex regions (LQCRs), this fails to explain the exponential convergence rates consistently observed in practice. In this paper, we resolve this discrepancy by proving that under a mild assumption on Neural Tangent Kernel (NTK) stability, these same regions satisfy a local Polyak-Lojasiewicz (PL) condition. We introduce the concept of a Locally Polyak-Lojasiewicz Region (LPLR), where the squared gradient norm lower-bounds the suboptimality gap, prove that properly initialized finite-width networks admit such regions around initialization, and establish that GD achieves linear convergence within an LPLR, providing the first finite-width guarantee that matches empirically observed rates. We validate our theory across diverse settings, from controlled experiments on fully-connected networks to modern ResNet architectures trained with stochastic methods, demonstrating that LPLR structure emerges robustly in practical deep learning scenarios. By rigorously connecting local landscape geometry to fast optimization through the NTK framework, our work provides a definitive theoretical explanation for the remarkable efficiency of gradient-based optimization in deep learning.

Agnideep Aich, Sameera Hewage, Md Monzur Murshed et al.

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.

Agnideep Aich, Md Monzur Murshed, Sameera Hewage et al.

Effective feature selection is vital for robust and interpretable medical prediction, especially for identifying risk factors concentrated in extreme patient strata. Standard methods emphasize average associations and may miss predictors whose importance lies in the tails of the distribution. We propose a computationally efficient supervised filter that ranks features using the Gumbel copula upper tail dependence coefficient ($λ_U$), prioritizing variables that are simultaneously extreme with the positive class. We benchmarked against Mutual Information, mRMR, ReliefF, and $L_1$ Elastic Net across four classifiers on two diabetes datasets: a large public health survey (CDC, N=253,680) and a clinical benchmark (PIMA, N=768). Evaluation included paired statistical tests, permutation importance, and robustness checks with label flips, feature noise, and missingness. On CDC, our method was the fastest selector and reduced the feature space by about 52% while retaining strong discrimination. Although using all 21 features yielded the highest AUC, our filter significantly outperformed Mutual Information and mRMR and was statistically indistinguishable from ReliefF. On PIMA, with only eight predictors, our ranking produced the numerically highest ROC AUC, and no significant differences were found versus strong baselines. Across both datasets, the upper tail criterion consistently identified clinically coherent, impactful predictors. We conclude that copula based feature selection via upper tail dependence is a powerful, efficient, and interpretable approach for building risk models in public health and clinical medicine.

Agnideep Aich, Ashit Baran Aich

Kernel Stein discrepancies (KSDs) have become a principal tool for goodness-of-fit testing, but standard KSDs are often insensitive to higher-order dependency structures, such as tail dependence, which are critical in many scientific and financial domains. We address this gap by introducing the Copula-Stein Discrepancy (CSD), a novel class of discrepancies tailored to the geometry of statistical dependence. By defining a Stein operator directly on the copula density, CSD leverages the generative structure of dependence, rather than relying on the joint density's score function. For the broad class of Archimedean copulas, this approach yields a closed-form Stein kernel derived from the scalar generator function. We provide a comprehensive theoretical analysis, proving that CSD (i) metrizes weak convergence of copula distributions, ensuring it detects any mismatch in dependence; (ii) has an empirical estimator that converges at the minimax optimal rate of $O_P(n^{-1/2})$; and (iii) is provably sensitive to differences in tail dependence coefficients. The framework is extended to general non-Archimedean copulas, including elliptical and vine copulas. Computationally, the exact CSD kernel evaluation scales linearly in dimension, while a novel random feature approximation reduces the $n$-dependence from quadratic $O(n^2)$ to near-linear $\tilde{O}(n)$, making CSD a practical and theoretically principled tool for dependence-aware inference.

Jose Cribeiro-Ramallo, Agnideep Aich, Florian Kalinke et al.

Kernel Stein discrepancies (KSDs) have emerged as a powerful tool for quantifying goodness-of-fit over the last decade, featuring numerous successful applications. To the best of our knowledge, all existing KSD estimators with known rate achieve $\sqrt n$-convergence. In this work, we present two complementary results (with different proof strategies), establishing that the minimax lower bound of KSD estimation is $n^{-1/2}$ and settling the optimality of these estimators. Our first result focuses on KSD estimation on $\mathbb R^d$ with the Langevin-Stein operator; our explicit constant for the Gaussian kernel indicates that the difficulty of KSD estimation may increase exponentially with the dimensionality $d$. Our second result settles the minimax lower bound for KSD estimation on general domains.

Agnideep Aich, Ashit Baran Aich, Bruce Wade

The scalable Markov chain Monte Carlo (MCMC) algorithms that underpin modern Bayesian machine learning, such as Stochastic Gradient Langevin Dynamics (SGLD), sacrifice asymptotic exactness for computational speed, creating a critical diagnostic gap: traditional sample quality measures fail catastrophically when applied to biased samplers. While powerful Stein-based diagnostics can detect distributional mismatches, they provide no direct assessment of dependence structure, often the primary inferential target in multivariate problems. We introduce the Copula Discrepancy (CD), a principled and computationally efficient diagnostic that leverages Sklar's theorem to isolate and quantify the fidelity of a sample's dependence structure independent of its marginals. Our theoretical framework provides the first structure-aware diagnostic specifically designed for the era of approximate inference. Empirically, we demonstrate that a moment-based CD dramatically outperforms standard diagnostics like effective sample size for hyperparameter selection in biased MCMC, correctly identifying optimal configurations where traditional methods fail. Furthermore, our robust MLE-based variant can detect subtle but critical mismatches in tail dependence that remain invisible to rank correlation-based approaches, distinguishing between samples with identical Kendall's tau but fundamentally different extreme-event behavior. With computational overhead orders of magnitude lower than existing Stein discrepancies, the CD provides both immediate practical value for MCMC practitioners and a theoretical foundation for the next generation of structure-aware sample quality assessment.

Agnideep Aich, Ashit Baran Aich, Md Monzur Murshed et al.

Modern neural networks, despite their high accuracy, often produce poorly calibrated confidence scores, limiting their reliability in high-stakes applications. Existing calibration methods typically post-process model outputs without interrogating the internal consistency of the predictions themselves. In this work, we introduce a novel, non-parametric statistical probe, the Bag-of-Coins (BoC) test, that examines the internal consistency of a classifier's logits. The BoC test reframes confidence estimation as a frequentist hypothesis test: does the model's top-ranked class win 1-v-1 contests against random competitors at a rate consistent with its own stated softmax probability? When applied to modern deep learning architectures, this simple probe reveals a fundamental dichotomy. On Vision Transformers (ViTs), the BoC output serves as a state-of-the-art confidence score, achieving near-perfect calibration with an ECE of 0.0212, an 88% improvement over a temperature-scaled baseline. Conversely, on Convolutional Neural Networks (CNNs) like ResNet, the probe reveals a deep inconsistency between the model's predictions and its internal logit structure, a property missed by traditional metrics. We posit that BoC is not merely a calibration method, but a new diagnostic tool for understanding and exposing the differing ways that popular architectures represent uncertainty.

Agnideep Aich, Ashit Baran Aich, Dipak C. Jain

We propose Temporal Conformal Prediction (TCP), a distribution-free framework for constructing well-calibrated prediction intervals in nonstationary time series. TCP couples a modern quantile forecaster with a split-conformal calibration layer on a rolling window and, in its TCP-RM variant, augments the conformal threshold with a single online Robbins-Monro (RM) offset to steer coverage toward a target level in real time. We benchmark TCP against GARCH, Historical Simulation, and a rolling Quantile Regression (QR) baseline across equities (S&P 500), cryptocurrency (Bitcoin), and commodities (Gold). Three results are consistent across assets. First, rolling QR yields the sharpest intervals but is materially under-calibrated (e.g., S&P 500: 83.2% vs. 95% target). Second, TCP (and TCP-RM) achieves near-nominal coverage across assets, with intervals that are wider than Historical Simulation in this evaluation (e.g., S&P 500: 5.21 vs. 5.06). Third, the RM update changes calibration and width only marginally at our default hyperparameters. Crisis-window visualizations around March 2020 show TCP/TCP-RM expanding and then contracting their interval bands promptly as volatility spikes and recedes, with red dots marking days where realized returns fall outside the reported 95% interval (miscoverage). A sensitivity study confirms robustness to window size and step-size choices. Overall, TCP provides a practical, theoretically grounded solution to calibrated uncertainty quantification under distribution shift, bridging statistical inference and machine learning for risk forecasting.

Agnideep Aich, Md Monzur Murshed, Sameera Hewage et al.

Diabetes mellitus poses a significant health risk, as nearly 1 in 9 people are affected by it. Early detection can significantly lower this risk. Despite significant advancements in machine learning for identifying diabetic cases, results can still be influenced by the imbalanced nature of the data. To address this challenge, our study considered copula-based data augmentation, which preserves the dependency structure when generating data for the minority class and integrates it with machine learning (ML) techniques. We selected the Pima Indian dataset and generated data using A2 copula, then applied five machine learning algorithms: logistic regression, random forest, gradient boosting, extreme gradient boosting, and Multilayer Perceptron. Overall, our findings show that Random Forest with A2 copula oversampling (theta = 10) achieved the best performance, with improvements of 5.3% in accuracy, 9.5% in precision, 5.7% in recall, 7.6% in F1-score, and 1.1% in AUC compared to the standard SMOTE method. Furthermore, we statistically validated our results using the McNemar's test. This research represents the first known use of A2 copulas for data augmentation and serves as an alternative to the SMOTE technique, highlighting the efficacy of copulas as a statistical method in machine learning applications.

Agnideep Aich

Classical estimators, the cornerstones of statistical inference, face insurmountable challenges when applied to important emerging classes of Archimedean copulas. These models exhibit pathological properties, including numerically unstable densities, a restrictive lower bound on Kendall's tau, and vanishingly small likelihood gradients, making MLE brittle and limiting MoM's applicability to datasets with sufficiently strong dependence (i.e., only when the empirical Kendall's $τ$ exceeds the family's lower bound $\approx 0.545$). We introduce \textbf{IGNIS}, a unified neural estimation framework that sidesteps these barriers by learning a direct, robust mapping from data-driven dependency measures to the underlying copula parameter $θ$. IGNIS utilizes a multi-input architecture and a theory-guided output layer ($\mathrm{softplus}(z) + 1$) to automatically enforce the domain constraint $\hatθ \geq 1$. Trained and validated on four families (Gumbel, Joe, and the numerically challenging A1/A2), IGNIS delivers accurate and stable estimates for real-world financial and health datasets, demonstrating its necessity for reliable inference in modern, complex dependence models where traditional methods fail. To our knowledge, IGNIS is the first \emph{standalone, general-purpose} neural estimator for Archimedean copulas (not a generative model or likelihood optimizer), delivering direct, constraint-aware $\hatθ$ and readily extensible to additional families via retraining or minor output-layer adaptations.

Agnideep Aich, Ashit Aich

We introduce the \emph{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.