Average gradient outer product as a mechanism for deep neural collapse

TL;DR

This paper introduces a data-dependent explanation for Deep Neural Collapse (DNC) based on the average gradient outer product (AGOP), demonstrating its role in feature learning and collapse phenomena in deep neural networks.

Contribution

It proposes the AGOP as a mechanism for DNC, introduces the Deep RFM model, and provides both empirical and theoretical evidence linking AGOP to neural collapse.

Findings

DNC occurs in Deep RFM due to AGOP-based projections

Theoretical explanation of DNC via kernel learning and asymptotic analysis

Singular vectors of weights relate to within-class variability collapse

Abstract

Deep Neural Collapse (DNC) refers to the surprisingly rigid structure of the data representations in the final layers of Deep Neural Networks (DNNs). Though the phenomenon has been measured in a variety of settings, its emergence is typically explained via data-agnostic approaches, such as the unconstrained features model. In this work, we introduce a data-dependent setting where DNC forms due to feature learning through the average gradient outer product (AGOP). The AGOP is defined with respect to a learned predictor and is equal to the uncentered covariance matrix of its input-output gradients averaged over the training dataset. The Deep Recursive Feature Machine (Deep RFM) is a method that constructs a neural network by iteratively mapping the data with the AGOP and applying an untrained random feature map. We demonstrate empirically that DNC occurs in Deep RFM across standard settings as a consequence of the projection with the AGOP matrix computed at each layer. Further, we theoretically explain DNC in Deep RFM in an asymptotic setting and as a result of kernel learning. We then provide evidence that this mechanism holds for neural networks more generally. In particular, we show that the right singular vectors and values of the weights can be responsible for the majority of within-class variability collapse for DNNs trained in the feature learning regime. As observed in recent work, this singular structure is highly correlated with that of the AGOP.