🤖 AI Summary
This study addresses the suboptimal classification performance of traditional Quadratic Discriminant Analysis (QDA) under shared-generator elliptical class-conditional distributions, where the assumption of an affine radial link fails to capture the non-affine structure of the log-likelihood ratio. By analyzing the within-class radius distribution, the authors derive the Bayes-optimal radial link function and propose an asymptotically Bayes-optimal discriminant analysis method based on fractional-power random polynomial projections. They introduce, for the first time, a closed-form identifiable family of radial links that requires no spline smoothing or additional hyperparameter tuning, with theoretical guarantees of √n-consistent estimation and asymptotic normality—formally verified in Lean 4. Empirical results show competitive or superior performance against tuned generalized additive models (GAMs) on UCI and financial time-series benchmarks, notably improving accuracy on breast_cancer and heavy-tailed financial sequences (e.g., crude oil, S&P 500), while simulations confirm the √n convergence rate and vanishing excess risk.
📝 Abstract
We study binary classification under shared-generator elliptical class-conditional distributions. The log-likelihood ratio is an additive function of the two squared Mahalanobis radii, with radial link $\varphi=\log g$; QDA is recovered only when this link is affine. We derive the Bayes radial-link family from the within-class radius law and estimate it by a finite fractional-power stochastic-polynomial projection instead of tuning a generic spline. The link is identifiable from the radius law, the plug-in estimator is $\sqrt{n}$-consistent and asymptotically normal under finite-moment regularity conditions, and the induced classifier is asymptotically Bayes-optimal in an iterated sieve limit. The structural bridge, GAM membership, and identity-link/affine-generator dichotomy are verified in Lean 4 without unproven placeholders. Against the global Mahalanobis-GAM of Ghosh et al. (2025), reimplemented with mgcv REML splines at equal input budget, the derived link is never significantly worse on three UCI benchmarks and is decisively better on breast_cancer ($[+0.009,+0.021]$ global, $[+0.109,+0.136]$ global+local). Across six real financial series under temporal-dependence-robust validation, it is never significantly worse than the fitted GAM and is significantly better on three of five heavy-tailed series plus the light-tailed control. Relative to QDA, it improves the heaviest-tailed series (oil $[+0.024,+0.070]$, S&P 500 $[+0.038,+0.126]$, JPY/USD $[+0.009,+0.047]$) and ties elsewhere. A closed-form rate simulation corroborates the $\sqrt{n}$ rate and the predicted excess-risk dichotomy between QDA's approximation-limited floor and the derived link's vanishing excess risk. The contribution is no significant loss relative to a tuned global GAM without spline smoothing-parameter selection, plus improved accuracy over QDA where generator curvature matters.