Neural Collapse under Gradient Flow on Shallow ReLU Networks for Orthogonally Separable Data

• URL: http://arxiv.org/abs/2510.21078v1
• Date: Fri, 24 Oct 2025 01:36:19 GMT
• Title: Neural Collapse under Gradient Flow on Shallow ReLU Networks for Orthogonally Separable Data
• Authors: Hancheng Min, Zhihui Zhu, René Vidal,
• Abstract summary: We show that gradient flow on a two-layer ReLU network for classifying orthogonally separable data provably exhibits Neural Collapse (NC)<n>We reveal the role of the implicit bias of the training dynamics in facilitating the emergence of NC.
• Score: 52.737775129027575
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Among many mysteries behind the success of deep networks lies the exceptional discriminative power of their learned representations as manifested by the intriguing Neural Collapse (NC) phenomenon, where simple feature structures emerge at the last layer of a trained neural network. Prior works on the theoretical understandings of NC have focused on analyzing the optimization landscape of matrix-factorization-like problems by considering the last-layer features as unconstrained free optimization variables and showing that their global minima exhibit NC. In this paper, we show that gradient flow on a two-layer ReLU network for classifying orthogonally separable data provably exhibits NC, thereby advancing prior results in two ways: First, we relax the assumption of unconstrained features, showing the effect of data structure and nonlinear activations on NC characterizations. Second, we reveal the role of the implicit bias of the training dynamics in facilitating the emergence of NC.

Related papers

• Neural Collapse Beyond the Unconstrained Features Model: Landscape, Dynamics, and Generalization in the Mean-Field Regime [17.726510286383697]
  Neural Collapse is a phenomenon where the last-layer representations of a well-trained neural network converge to a highly structured geometry.<n>In this paper, we focus on its first (and most basic) property, known as NC1.<n>We analyze NC1 in a three-layer neural network, with the first two layers operating in the mean-field regime and followed by a linear layer.
  arXiv  Detail & Related papers  (2025-01-31T12:57:29Z)
• Beyond Unconstrained Features: Neural Collapse for Shallow Neural Networks with General Data [0.8594140167290099]
  Neural collapse (NC) is a phenomenon that emerges at the terminal phase of the training of deep neural networks (DNNs) We provide a complete characterization of when the NC occurs for two or three-layer neural networks.
  arXiv  Detail & Related papers  (2024-09-03T12:30:21Z)
• Supervised Contrastive Representation Learning: Landscape Analysis with Unconstrained Features [33.703796571991745]
  Recent findings reveal that overparameterized deep neural networks, trained beyond zero training, exhibit a distinctive structural pattern at the final layer. These results indicate that the final-layer outputs in such networks display minimal within-class variations.
  arXiv  Detail & Related papers  (2024-02-29T06:02:45Z)
• Deep Neural Collapse Is Provably Optimal for the Deep Unconstrained Features Model [21.79259092920587]
  We show that in a deep unconstrained features model, the unique global optimum for binary classification exhibits all the properties typical of deep neural collapse (DNC) We also empirically show that (i) by optimizing deep unconstrained features models via gradient descent, the resulting solution agrees well with our theory, and (ii) trained networks recover the unconstrained features suitable for DNC.
  arXiv  Detail & Related papers  (2023-05-22T15:51:28Z)
• Neural Collapse in Deep Linear Networks: From Balanced to Imbalanced Data [12.225207401994737]
  We show that complex systems with massive amounts of parameters exhibit the same structural properties when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of vectors.
  arXiv  Detail & Related papers  (2023-01-01T16:29:56Z)
• Extended Unconstrained Features Model for Exploring Deep Neural Collapse [59.59039125375527]
  Recently, a phenomenon termed "neural collapse" (NC) has been empirically observed in deep neural networks. Recent papers have shown that minimizers with this structure emerge when optimizing a simplified "unconstrained features model" In this paper, we study the UFM for the regularized MSE loss, and show that the minimizers' features can be more structured than in the cross-entropy case.
  arXiv  Detail & Related papers  (2022-02-16T14:17:37Z)
• Topological obstructions in neural networks learning [67.8848058842671]
  We study global properties of the loss gradient function flow. We use topological data analysis of the loss function and its Morse complex to relate local behavior along gradient trajectories with global properties of the loss surface.
  arXiv  Detail & Related papers  (2020-12-31T18:53:25Z)
• Dual-constrained Deep Semi-Supervised Coupled Factorization Network with Enriched Prior [80.5637175255349]
  We propose a new enriched prior based Dual-constrained Deep Semi-Supervised Coupled Factorization Network, called DS2CF-Net. To ex-tract hidden deep features, DS2CF-Net is modeled as a deep-structure and geometrical structure-constrained neural network. Our network can obtain state-of-the-art performance for representation learning and clustering.
  arXiv  Detail & Related papers  (2020-09-08T13:10:21Z)
• Modeling from Features: a Mean-field Framework for Over-parameterized Deep Neural Networks [54.27962244835622]
  This paper proposes a new mean-field framework for over- parameterized deep neural networks (DNNs) In this framework, a DNN is represented by probability measures and functions over its features in the continuous limit. We illustrate the framework via the standard DNN and the Residual Network (Res-Net) architectures.
  arXiv  Detail & Related papers  (2020-07-03T01:37:16Z)
• Revisiting Initialization of Neural Networks [72.24615341588846]
  We propose a rigorous estimation of the global curvature of weights across layers by approximating and controlling the norm of their Hessian matrix. Our experiments on Word2Vec and the MNIST/CIFAR image classification tasks confirm that tracking the Hessian norm is a useful diagnostic tool.
  arXiv  Detail & Related papers  (2020-04-20T18:12:56Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.