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.

Neural collapse (N C) is a phenomenon observed in classification tasks where top-layer representations collapse into their class means, which become equinorm, equiangular and aligned with the classifiers. These behaviours -- associated with generalization and robustness -- would manifest under specific conditions: models are trained towards zero loss, with noise-free labels belonging to balanced classes, which do not outnumber the model's hidden dimension. Recent studies have explored N C in the absence of one or more of these conditions to extend and capitalize on the associated benefits of ideal geometries. Language modelling presents a curious frontier, as training by token prediction constitutes a classification task where none of the conditions exist: the vocabulary is imbalanced and exceeds the embedding dimension; different tokens might correspond to similar contextual embeddings; and large language models (LLMs) in particular are typically only trained for a few epochs. This paper empirically investigates the impact of scaling the architectures and training of causal language models (CLMs) on their progression towards N C. We find that N C properties that develop with scale (and regularization) are linked to generalization. Moreover, there is evidence of some relationship between N C and generalization independent of scale. Our work thereby underscores the generality of N C as it extends to the novel and more challenging setting of language modelling. Downstream, we seek to inspire further research on the phenomenon to deepen our understanding of LLMs -- and neural networks at large -- and improve existing architectures based on N C-related properties.

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.

Recently it has been observed that neural networks exhibit Neural Collapse (NC) during the final stage of training for the classification problem. We empirically show that multivariate regression, as employed in imitation learning and other applications, exhibits Neural Regression Collapse (NRC), a new form of neural collapse: (NRC1) The last-layer feature vectors collapse to the subspace spanned by the n principal components of the feature vectors, where n is the dimension of the targets (for univariate regression, n = 1); (NRC2) The last-layer feature vectors also collapse to the subspace spanned by the last-layer weight vectors; (NRC3) The Gram matrix for the weight vectors converges to a specific functional form that depends on the covariance matrix of the targets. After empirically establishing the prevalence of (NRC1)-(NRC3) for a variety of datasets and network architectures, we provide an explanation of these phenomena by modeling the regression task in the context of the Unconstrained Feature Model (UFM), in which the last layer feature vectors are treated as free variables when minimizing the loss function. We show that when the regularization parameters in the UFM model are strictly positive, then (NRC1)-(NRC3) also emerge as solutions in the UFM optimization problem. We also show that if the regularization parameters are equal to zero, then there is no collapse. To our knowledge, this is the first empirical and theoretical study of neural collapse in the context of regression. This extension is significant not only because it broadens the applicability of neural collapse to a new category of problems but also because it suggests that the phenomena of neural collapse could be a universal behavior in deep learning.

The spurious correlation between the background features of the image and its label arises due to that the samples labeled with the same class in the training set often co-occurs with a specific background, which will cause the encoder to extract non-semantic features for classification, resulting in poor out-of-distribution generalization performance. Although many studies have been proposed to address this challenge, the semantic and spurious features are still difficult to accurately decouple from the original image and fail to achieve high performance with deep learning models. This paper proposes a novel perspective inspired by neural collapse to solve the spurious correlation problem through the alternate execution of environment partitioning and learning semantic masks. Specifically, we propose to assign an environment to each sample by learning a local model for each environment and using maximum likelihood probability. At the same time, we require that the learned semantic mask neurally collapses to the same simplex equiangular tight frame (ETF) in each environment after being applied to the original input. We conduct extensive experiments on four datasets, and the results demonstrate that our method significantly improves out-of-distribution performance.

Many of the recent remarkable advances in computer vision and language models can be attributed to the success of transfer learning via the pre-training of large foundation models. However, a theoretical framework which explains this empirical success is incomplete and remains an active area of research. Flatness of the loss surface and neural collapse have recently emerged as useful pre-training metrics which shed light on the implicit biases underlying pre-training. In this paper, we explore the geometric complexity of a model's learned representations as a fundamental mechanism that relates these two concepts. We show through experiments and theory that mechanisms which affect the geometric complexity of the pre-trained network also influence the neural collapse. Furthermore, we show how this effect of the geometric complexity generalizes to the neural collapse of new classes as well, thus encouraging better performance on downstream tasks, particularly in the few-shot setting.

Neural Collapse (NC) was a recently discovered phenomenon that the output features and the classifier weights of the neural network converge to optimal geometric structures at the Terminal Phase of Training (TPT) under various losses. However, the relationship between these optimal structures at TPT and the classification performance remains elusive, especially in imbalanced learning. Even though it is noticed that fixing the classifier to an optimal structure can mitigate the minority collapse problem, the performance is still not comparable to the classical imbalanced learning methods with a learnable classifier. In this work, we find that the optimal structure can be designed to represent a better classification rule, and thus achieve better performance. In particular, we justify that, to achieve better classification, the features from the minor classes should align with more directions. This justification then yields a decision rule called the Generalized Classification Rule (GCR) and we also term these directions as the centers of the classes. Then we study the NC under an MSE-type loss via the Unconstrained Features Model (UFM) framework where (1) the features from a class tend to collapse to the mean of the corresponding centers of that class (named Neural Collapse to Multiple Centers (NCMC)) at the global optimum, and (2) the original classifier approximates a surrogate to GCR when NCMC occurs. Based on the analysis, we develop a strategy for determining the number of centers and propose a Cosine Loss function for the fixed classifier that induces NCMC. Our experiments have shown that the Cosine Loss can induce NCMC and has performance on long-tail classification comparable to the classical imbalanced learning methods.

Deep Equilibrium Model (DEQ), which serves as a typical implicit neural network, emphasizes their memory efficiency and competitive performance compared to explicit neural networks. However, there has been relatively limited theoretical analysis on the representation of DEQ. In this paper, we utilize the Neural Collapse (N C) as a tool to systematically analyze the representation of DEQ under both balanced and imbalanced conditions. N C is an interesting phenomenon in the neural network training process that characterizes the geometry of class features and classifier weights. While extensively studied in traditional explicit neural networks, the N C phenomenon has not received substantial attention in the context of implicit neural networks. We theoretically show that N C exists in DEQ under balanced conditions. Moreover, in imbalanced settings, despite the presence of minority collapse, DEQ demonstrated advantages over explicit neural networks. These advantages include the convergence of extracted features to the vertices of a simplex equiangular tight frame and self-duality properties under mild conditions, highlighting DEQ's superiority in handling imbalanced datasets.

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.