Differential Spectral Damping Gap Adaptive Regularization for Ill-Conditioned Kernel Methods

arXiv:2606.23407v1 Announce Type: new Abstract: Kernel methods requiring matrix inversion -- particularly Least-Squares Twin Support Vector Machines (LSTSVM) -- suffer from exponential eigenvalue decay in their system matrices, producing severely ill-conditioned problems where standard Tikhonov regularization applies uniform damping regardless of eigenvector reliability. We propose Differential Spectral Damping (DSD), a regularization formula that adapts its penalty to localized eigengap structure: preserving eigenvectors with large spectral gaps (reliable per Davis-Kahan perturbation theory) while aggressively suppressing those with small gaps (directionally corrupted beyond recovery). We motivate DSD through a principled design procedure grounded in the Davis-Kahan sin(Theta) theorem, systematically deriving the requirements for a reliability-aware damping function and selecting the exponential form for its smoothness, differentiability, and natural saturation properties. Through rigorous paired testing with fairly optimized baselines (including gradient-optimized Tikhonov receiving equal optimization opportunity), we demonstrate that DSD improves LSTSVM classification accuracy by +4.8 percentage points on real-world GINA (d=970, Cohen's d = 4.49, p 10^3) and document where simpler methods suffice, providing practitioners with clear deployment guidance.

Source: arXiv cs.LG | 2026-06-23