Manifold-Constrained Adversarial Training for Long-Tailed Robustness via Geometric Alignment

Abstract

Adversarial training is effective on balanced datasets, but its robustness degrades under long-tailed class distributions, where tail classes suffer high robust error and unstable decision boundaries. We propose \emph{Manifold-Constrained Adversarial Training (MCAT)}, a unified framework that enforces the semantic validity of adversarial examples by penalizing deviations from class-conditional manifolds in feature space, while promoting balanced geometric separation across classes via an ETF-inspired regularization. We provide theoretical results that link geometric separation to lower bounds on adversarially robust margins, and show that manifold-constrained adversarial risk upper-bounds robust risk on high-density semantic regions. Extensive experiments on standard long-tailed benchmarks demonstrate consistent improvements in overall, balanced, and tail-class adversarial robustness. The codes and appendix are available on https://github.com/yneversky/MCAT.