Deep neural networks (DNNs) exhibit a surprising structure in their final layer known as neural collapse (NC), and a growing body of works is 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 either linear models, the last 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 the 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.

Deep neural networks (DNNs) exhibit a surprising structure in their final layer known as neural collapse (NC), and a growing body of works is 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 either linear models, the last 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 the low-rank bias of multi-layer regularization schemes. This bias leads to optimal solutions of even lower rank than the neural collapse.

Modern deep neural networks have been observed to exhibit a simple structure in their final layer features and weights, commonly referred to as neural collapse. This phenomenon has also been noted in layers beyond the final one, an extension known as deep neural collapse. Recent findings indicate that such a structure is generally not optimal in the deep unconstrained feature model, an approximation of an expressive network. This is attributed to a low-rank bias induced by regularization, which favors solutions with lower-rank than those typically associated with deep neural collapse. In this work, we extend these observations to the cross-entropy loss and analyze how the low-rank bias influences various solutions. Additionally, we explore how this bias induces specific structures in the singular values of the weights at global optima. Furthermore, we examine the loss surface of these models and provide evidence that the frequent observation of deep neural collapse in practice, despite its suboptimality, may result from its higher degeneracy on the loss surface.

We study the implicit bias towards low-rank weight matrices when training neural networks (NN) with Weight Decay (WD). We prove that when a ReLU NN is sufficiently trained with Stochastic Gradient Descent (SGD) and WD, its weight matrix is approximately a rank-two matrix. Empirically, we demonstrate that WD is a necessary condition for inducing this low-rank bias across both regression and classification tasks. Our work differs from previous studies as our theoretical analysis does not rely on common assumptions regarding the training data distribution, optimality of weight matrices, or specific training procedures. Furthermore, by leveraging the low-rank bias, we derive improved generalization error bounds and provide numerical evidence showing that better generalization can be achieved. Thus, our work offers both theoretical and empirical insights into the strong generalization performance of SGD when combined with WD.

The $L_{2}$-regularized loss of Deep Linear Networks (DLNs) with more than one hidden layers has multiple local minima, corresponding to matrices with different ranks. In tasks such as matrix completion, the goal is to converge to the local minimum with the smallest rank that still fits the training data. While rank-underestimating minima can be avoided since they do not fit the data, GD might get stuck at rank-overestimating minima. We show that with SGD, there is always a probability to jump from a higher rank minimum to a lower rank one, but the probability of jumping back is zero. More precisely, we define a sequence of sets $B_{1}\subset B_{2}\subset\cdots\subset B_{R}$ so that $B_{r}$ contains all minima of rank $r$ or less (and not more) that are absorbing for small enough ridge parameters $\lambda$ and learning rates $\eta$: SGD has prob. 0 of leaving $B_{r}$, and from any starting point there is a non-zero prob. for SGD to go in $B_{r}$.