Neural Collapse versus Low-rank Bias: Is Deep Neural Collapse Really Optimal?
This challenges existing theoretical assumptions about neural collapse in deep learning, indicating it is incremental for understanding optimization in multi-layer networks.
The paper tackles the optimality of deep neural collapse (DNC) in multi-class classification with non-linear models of arbitrary depth, revealing that DNC is not optimal beyond two layers or classes due to a low-rank bias in regularization schemes, which leads to solutions with even lower rank.
Deep neural networks (DNNs) exhibit a surprising structure in their final layer known as neural collapse (NC), and a growing body of works has currently investigated the propagation of neural collapse to earlier layers of DNNs -- a phenomenon called deep neural collapse (DNC). However, existing theoretical results are restricted to special cases: linear models, only two layers or binary classification. In contrast, we focus on non-linear models of arbitrary depth in multi-class classification and reveal a surprising qualitative shift. As soon as we go beyond two layers or two classes, DNC stops being optimal for the deep unconstrained features model (DUFM) -- the standard theoretical framework for the analysis of collapse. The main culprit is a low-rank bias of multi-layer regularization schemes: this bias leads to optimal solutions of even lower rank than the neural collapse. We support our theoretical findings with experiments on both DUFM and real data, which show the emergence of the low-rank structure in the solution found by gradient descent.