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

2026-07-24

AI Summary

This work addresses the failure in recognizing tail classes in long-tailed class-incremental learning caused by spectral collapse of the autocorrelation matrix. The authors propose a Geometry-Spectral Rectification (GSR) framework, which for the first time reveals the spectral pathology induced by long-tailed distributions and formulates the problem as a spectral regularization task. GSR introduces an anisotropic spectral filtering mechanism through a data-dependent, structured spectral perturbation matrix that selectively amplifies collapsed feature directions of tail classes, thereby overcoming the limitations of conventional isotropic ridge regression. By integrating analytical recursive least squares learning with stable rank theory, the method significantly enhances the numerical stability of the Gram matrix and the generalization performance on tail classes while maintaining computational efficiency, achieving a new state-of-the-art on long-tailed class-incremental learning benchmarks.

Abstract

Analytic Continual Learning (ACL) offers a computationally efficient alternative to gradient-based approaches. Recent ACL methods are based on Recursive Least Squares (RLS) and have achieved the state-of-the-art results compared to other alternatives. However, they falter significantly in Class-Incremental Learning scenarios characterized by Long-Tailed distributions. While the ill-conditioning of the autocorrelation (Gram) matrix is a known limitation of RLS, we demonstrate that class imbalance exacerbates this issue into a distinct spectral pathology: "tail" classes suffer from severe spectral collapse, rendering their subspaces numerically indistinguishable from noise. Standard Ridge Regression ($L_2$) fails to address this effectively as it applies isotropic regularization - a uniform penalty that is insufficient to stabilize the tail without over-shrinking the head. To address this, we propose Geometry-Spectral Rectification (GSR), a theoretically grounded framework that treats long-tailed learning as a spectral regularization problem. Unlike standard isotropic regularization (Ridge) which uniformly penalizes all eigenvalues, GSR acts as an anisotropic spectral filter, selectively inflating the collapsed eigenvalues of tail classes. We construct a structured, data-dependent spectral perturbation matrix $Δ$ that selectively inflates collapsed tail eigen-directions of the Gram matrix. Theoretical analysis proves that GSR guarantees an improved stable rank for the Gram matrix, ensuring numerical stability. Extensive experiments show that GSR establishes a new state-of-the-art for analytic CIL, offering a superior trade-off between computational efficiency and robust generalization in long-tailed settings.

Problem

• Class-Incremental Learning
• Long-Tailed Distributions
• Spectral Collapse
• Ill-Conditioning
• Analytic Continual Learning

Innovation

• Geometry-Spectral Rectification
• Spectral Regularization
• Long-Tailed Learning
• Class-Incremental Learning
• Analytic Continual Learning