Neural Rank Collapse: Weight Decay and Small Within-Class Variability
  Yield Low-Rank Bias
Emanuele Zangrando Piero Deidda Simone Brugiapaglia Nicola Guglielmi Francesco Tudisco
Recent work in deep learning has shown strong empirical and theoretical
evidence of an implicit low-rank bias: weight matrices in deep networks tend to
be approximately low-rank and removing relatively small singular values during
training or from available trained models may significantly reduce model size
while maintaining or even improving model performance. However, the majority of
the theoretical investigations around low-rank bias in neural networks deal
with oversimplified deep linear networks. In this work, we consider general
networks with nonlinear activations and the weight decay parameter, and we show
the presence of an intriguing neural rank collapse phenomenon, connecting the
low-rank bias of trained networks with networks' neural collapse properties: as
the weight decay parameter grows, the rank of each layer in the network
decreases proportionally to the within-class variability of the hidden-space
embeddings of the previous layers. Our theoretical findings are supported by a
range of experimental evaluations illustrating the phenomenon.