Phase Collapse in Neural Networks

Florentin Guth, John Zarka, Stéphane Mallat

Keywords: phase collapse, neural collapse, concentration, classification, imagenet, deep networks, complex networks, sparsity in deep networks

Abstract: Deep convolutional classifiers linearly separate image classes and improve accuracy as depth increases. They progressively reduce the spatial dimension whereas the number of channels grows with depth. Spatial variability is therefore transformed into variability along channels. A fundamental challenge is to understand the role of non-linearities together with convolutional filters in this transformation. ReLUs with biases are often interpreted as thresholding operators that improve discrimination through sparsity. This paper demonstrates that it is a different mechanism called \emph{phase collapse} which eliminates spatial variability while linearly separating classes. We show that collapsing the phases of complex wavelet coefficients is sufficient to reach the classification accuracy of ResNets of similar depths. However, replacing the phase collapses with thresholding operators that enforce sparsity considerably degrades the performance. We explain these numerical results by showing that the iteration of phase collapses progressively improves separation of classes, as opposed to thresholding non-linearities.

One-sentence Summary: The classification accuracy of CNNs mostly relies on the mechanism of phase collapses to eliminate spatial variability and linearly separate class means.

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2110.05283/code)