Provable Emergence of Deep Neural Collapse and Low-Rank Bias in $L^2$-Regularized Nonlinear Networks

Abstract

We present a unified theoretical framework connecting the first property of Deep Neural Collapse (DNC1) to the emergence of implicit low-rank bias in nonlinear networks trained with $L^2$ weight decay regularization. Our main contributions are threefold. First, we derive a quantitative relation between the Total Cluster Variation (TCV) of intermediate embeddings and the numerical rank of stationary weight matrices. In particular, we establish that, at any critical point, the distance from a weight matrix to the set of rank-$K$ matrices is bounded by a constant times the TCV of earlier-layer features, scaled inversely with the weight-decay parameter. Second, we prove global optimality of DNC1 in a constrained representation-cost setting for both feedforward and residual architectures, showing that zero TCV across intermediate layers minimizes the representation cost under natural architectural constraints. Third, we establish a benign landscape property: for almost every interpolating initialization there exists a continuous, loss-decreasing path from the initialization to a globally optimal, DNC1-satisfying configuration. Our theoretical claims are validated empirically; numerical experiments confirm the predicted relations among TCV, singular-value structure, and weight decay. These results indicate that neural collapse and low-rank bias are intimately linked phenomena arising from the optimization geometry induced by weight decay.