Perturbation Analysis of Neural Collapse

TL;DR

This paper introduces a perturbation-based model to analyze neural collapse in deep neural networks, capturing near-collapse behavior and the effects of regularization, supported by theoretical analysis and practical experiments.

Contribution

It proposes a new model that accounts for near-collapse features in neural networks and provides perturbation analysis results that previous idealized models could not achieve.

Findings

Reduction in within-class feature variability near collapse

Insights into how regularization affects feature collapse

Theoretical validation supported by practical experiments

Abstract

Training deep neural networks for classification often includes minimizing the training loss beyond the zero training error point. In this phase of training, a "neural collapse" behavior has been observed: the variability of features (outputs of the penultimate layer) of within-class samples decreases and the mean features of different classes approach a certain tight frame structure. Recent works analyze this behavior via idealized unconstrained features models where all the minimizers exhibit exact collapse. However, with practical networks and datasets, the features typically do not reach exact collapse, e.g., because deep layers cannot arbitrarily modify intermediate features that are far from being collapsed. In this paper, we propose a richer model that can capture this phenomenon by forcing the features to stay in the vicinity of a predefined features matrix (e.g., intermediate features). We explore the model in the small vicinity case via perturbation analysis and establish results that cannot be obtained by the previously studied models. For example, we prove reduction in the within-class variability of the optimized features compared to the predefined input features (via analyzing gradient flow on the "central-path" with minimal assumptions), analyze the minimizers in the near-collapse regime, and provide insights on the effect of regularization hyperparameters on the closeness to collapse. We support our theory with experiments in practical deep learning settings.