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

arXiv: https://arxiv.org/abs/2605.01897 · HTML: https://mlsanity.com/paper/2605.01897/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 44 references that are in the map.
- First posted on arXiv 3 May 2026.

## Builds on (most cited first, 30 of 32)

- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Learning Imbalanced Datasets with Label-Distribution-Aware Margin Loss](https://mlsanity.com/paper/1906.07413/) (2019)
- [Decoupling Representation and Classifier for Long-Tailed Recognition](https://mlsanity.com/paper/1910.09217/) (2019)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [A Geometric Analysis of Neural Collapse with Unconstrained Features](https://mlsanity.com/paper/2105.02375/) (2021)
- [Exploring Deep Neural Networks via Layer-Peeled Model: Minority Collapse in Imbalanced Training](https://mlsanity.com/paper/2101.12699/) (2021)
- [Neural Collapse Under MSE Loss: Proximity to and Dynamics on the Central Path](https://mlsanity.com/paper/2106.02073/) (2021)
- [Learning Diverse and Discriminative Representations via the Principle of Maximal Coding Rate Reduction](https://mlsanity.com/paper/2006.08558/) (2020)
- [Neural collapse with unconstrained features](https://mlsanity.com/paper/2011.11619/) (2020)
- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [Extended Unconstrained Features Model for Exploring Deep Neural Collapse](https://mlsanity.com/paper/2202.08087/) (2022)
- [The Emerging Trends of Multi-Label Learning](https://mlsanity.com/paper/2011.11197/) (2020)
- [On the Role of Neural Collapse in Transfer Learning](https://mlsanity.com/paper/2112.15121/) (2021)
- [Inducing Neural Collapse in Imbalanced Learning: Do We Really Need a Learnable Classifier at the End of Deep Neural Network?](https://mlsanity.com/paper/2203.09081/) (2022)
- [Neural Networks are Convex Regularizers: Exact Polynomial-time Convex Optimization Formulations for Two-layer Networks](https://mlsanity.com/paper/2002.10553/) (2020)
- [On the Optimization Landscape of Neural Collapse under MSE Loss: Global Optimality with Unconstrained Features](https://mlsanity.com/paper/2203.01238/) (2022)
- [An Unconstrained Layer-Peeled Perspective on Neural Collapse](https://mlsanity.com/paper/2110.02796/) (2021)
- [ReduNet: A White-box Deep Network from the Principle of Maximizing Rate Reduction](https://mlsanity.com/paper/2105.10446/) (2021)
- [Imbalance Trouble: Revisiting Neural-Collapse Geometry](https://mlsanity.com/paper/2208.05512/) (2022)
- [ML-Decoder: Scalable and Versatile Classification Head](https://mlsanity.com/paper/2111.12933/) (2021)
- [Are All Losses Created Equal: A Neural Collapse Perspective](https://mlsanity.com/paper/2210.02192/) (2022)
- [Neural Collapse with Cross-Entropy Loss](https://mlsanity.com/paper/2012.08465/) (2020)
- [Deep Neural Collapse Is Provably Optimal for the Deep Unconstrained Features Model](https://mlsanity.com/paper/2305.13165/) (2023)
- [Neural Collapse in Deep Linear Networks: From Balanced to Imbalanced Data](https://mlsanity.com/paper/2301.00437/) (2023)
- [Neural Collapse for Unconstrained Feature Model under Cross-entropy Loss with Imbalanced Data](https://mlsanity.com/paper/2309.09725/) (2023)
- [Perturbation Analysis of Neural Collapse](https://mlsanity.com/paper/2210.16658/) (2022)
- [On the Implicit Geometry of Cross-Entropy Parameterizations for Label-Imbalanced Data](https://mlsanity.com/paper/2303.07608/) (2023)
- [A Neural Collapse Perspective on Feature Evolution in Graph Neural Networks](https://mlsanity.com/paper/2307.01951/) (2023)
- [Neural Collapse for Cross-entropy Class-Imbalanced Learning with Unconstrained ReLU Feature Model](https://mlsanity.com/paper/2401.02058/) (2024)
- [Neural Collapse in Multi-label Learning with Pick-all-label Loss](https://mlsanity.com/paper/2310.15903/) (2023)

## Abstract (arXiv)

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 kappam, 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.