  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.
