How Label Imbalance Shapes Geometry: A General Spectral Analysis of Multi-Label Neural Collapse

This work investigates the phenomenon of Neural Collapse (NC) in multi-label classification, extending its conceptual framework from multi-class learning to general correlated and imbalanced multi-label settings. Although recent studies have identified a ''tag-wise averaging'' structure for multi-label features, this view relies on implicit assumptions of label balance and combinatorial symmetry. Consequently, it fails to account for the geometrical distortions caused by intrinsic label correlations and data imbalance, which are common in practice. We resolve the multiplicity-one imbalance conjecture raised by Li et al. (2024), showing that higher-multiplicity prototypes obey a class-frequency-weighted synthesis rule rather than uniform averaging. To address this, we propose a rigorous spectral-control framework to analyze the terminal phase of multi-label learning under general imbalanced conditions. We introduce the label covariance spectrum $κ_m$, a scalar controlling the distribution-dependent lower-bound geometry, derived from the second-order moment matrix of the label distribution. Contrary to the averaging perspective, our analysis reveals that the centered label covariance spectrum controls the stability of terminal geometry by quantifying the weakest centered inter-class contrast directions. We prove that the classical Tag-wise Averaging emerges only as a special case under perfect orthogonality. Numerical experiments on synthetic distributions validate our theoretical bounds. This work resolves the scaled-average aspect of the imbalance conjecture and establishes a unifying theoretical framework that extends Neural Collapse to complex, imbalanced multi-label settings.