Spectral-Aware Analytic Class-Incremental Learning for Long-Tailed Distributions

Analytic Continual Learning (ACL) offers a computationallyefficient alternative to gradient-based approaches. Recent ACL methodsare based on Recursive Least Squares (RLS) and have achieved the state-of-the-art results compared to other alternatives. However, they faltersignificantly in Class-Incremental Learning scenarios characterized byLong-Tailed distributions. While the ill-conditioning of the autocorrela-tion (Gram) matrix is a known limitation of RLS, we demonstrate thatclass imbalance exacerbates this issue into a distinct spectral pathology:"tail" classes suffer from severe spectral collapse, rendering their sub-spaces numerically indistinguishable from noise. Standard Ridge Regres-sion (L2 ) fails to address this effectively as it applies isotropic regular-ization - a uniform penalty that is insufficient to stabilize the tail with-out over-shrinking the head. To address this, we propose Geometry-Spectral Rectification (GSR), a theoretically grounded frameworkthat treats long-tailed learning as a spectral regularization problem. Un-like standard isotropic regularization (Ridge) which uniformly penalizesall eigenvalues, GSR acts as an anisotropic spectral filter, selectively in-flating the collapsed eigenvalues of tail classes. We construct a structured,data-dependent spectral perturbation matrix ∆ that selectively inflatescollapsed tail eigen-directions of the Gram matrix. Theoretical analysisproves that GSR guarantees an improved stable rank for the Gram ma-trix, ensuring numerical stability. Extensive experiments show that GSRestablishes a new state-of-the-art for analytic CIL, offering a superiortrade-off between computational efficiency and robust generalization inlong-tailed settings.