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1. Neural Collapse in Multi-label Learning with Pick-all-label Loss

   Li, Pengyu; Li, Xiao; Wang, Yutong; Qu, Qing (June 2024, International Conference on Machine Learning)

   We study deep neural networks for the multi-label classification (MLab) task through the lens of neural collapse (NC). Previous works have been restricted to the multi-class classification setting and discovered a prevalent NC phenomenon comprising of the following properties for the last-layer features: (i) the variability of features within every class collapses to zero, (ii) the set of feature means form an equi-angular tight frame (ETF), and (iii) the last layer classifiers collapse to the feature mean upon some scaling. We generalize the study to multi-label learning, and prove for the first time that a generalized NC phenomenon holds with the "pick-all-label'' formulation, which we term as MLab NC. While the ETF geometry remains consistent for features with a single label, multi-label scenarios introduce a unique combinatorial aspect we term the "tag-wise average" property, where the means of features with multiple labels are the scaled averages of means for single-label instances. Theoretically, under proper assumptions on the features, we establish that the only global optimizer of the pick-all-label cross-entropy loss satisfy the multi-label NC. In practice, we demonstrate that our findings can lead to better test performance with more efficient training techniques for MLab learning. 

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2. Are All Losses Created Equal: A Neural Collapse Perspective

   Zhou, Jinxin; You, Chong; Li, Xiao; Liu, Kangning; Liu, Sheng; Qu, Qing; Zhu, Zhihui (September 2022, NeurIPS)

   While cross entropy (CE) is the most commonly used loss function to train deep neural networks for classification tasks, many alternative losses have been developed to obtain better empirical performance. Among them, which one is the best to use is still a mystery, because there seem to be multiple factors affecting the answer, such as properties of the dataset, the choice of network architecture, and so on. This paper studies the choice of loss function by examining the last-layer features of deep networks, drawing inspiration from a recent line work showing that the global optimal solution of CE and mean-square-error (MSE) losses exhibits a Neural Collapse phenomenon. That is, for sufficiently large networks trained until convergence, (i) all features of the same class collapse to the corresponding class mean and (ii) the means associated with different classes are in a configuration where their pairwise distances are all equal and maximized. We extend such results and show through global solution and landscape analyses that a broad family of loss functions including commonly used label smoothing (LS) and focal loss (FL) exhibits Neural Collapse. Hence, all relevant losses (i.e., CE, LS, FL, MSE) produce equivalent features on training data. In particular, based on the unconstrained feature model assumption, we provide either the global landscape analysis for LS loss or the local landscape analysis for FL loss and show that the (only!) global minimizers are neural collapse solutions, while all other critical points are strict saddles whose Hessian exhibit negative curvature directions either in the global scope for LS loss or in the local scope for FL loss near the optimal solution. The experiments further show that Neural Collapse features obtained from all relevant losses (i.e., CE, LS, FL, MSE) lead to largely identical performance on test data as well, provided that the network is sufficiently large and trained until convergence. 

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3. Feature learning in deep classifiers through Intermediate Neural Collapse

   Rangamani, Akshay; Lindegaard, Marius; Galanti, Tomer; Poggio, Tomaso (February 2023, Center for Brains, Minds and Machines (CBMM))

   In this paper, we conduct an empirical study of the feature learning process in deep classifiers. Recent research has identified a training phenomenon called Neural Collapse (NC), in which the top-layer feature embeddings of samples from the same class tend to concentrate around their means, and the top layer’s weights align with those features. Our study aims to investigate if these properties extend to intermediate layers. We empirically study the evolution of the covariance and mean of representations across different layers and show that as we move deeper into a trained neural network, the within-class covariance decreases relative to the between-class covariance. Additionally, we find that in the top layers, where the between-class covariance is dominant, the subspace spanned by the class means aligns with the subspace spanned by the most significant singular vector components of the weight matrix in the corresponding layer. Finally, we discuss the relationship between NC and Associative Memories (Willshaw et al., 1969). 

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4. Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifold

   Yaras, Can; Wang, Peng; Zhu, Zhihui; Balzano, Laura; Qu, Qing (January 2022, Advances in neural information processing systems)

   When training overparameterized deep networks for classification tasks, it has been widely observed that the learned features exhibit a so-called “neural collapse” phenomenon. More specifically, for the output features of the penultimate layer, for each class the within-class features converge to their means, and the means of different classes exhibit a certain tight frame structure, which is also aligned with the last layer’s classifier. As feature normalization in the last layer becomes a common practice in modern representation learning, in this work we theoretically justify the neural collapse phenomenon under normalized features. Based on an un-constrained feature model, we simplify the empirical loss function in a multi-class classification task into a nonconvex optimization problem over the Riemannian manifold by constraining all features and classifiers over the sphere. In this context, we analyze the nonconvex landscape of the Riemannian optimization problem over the product of spheres, showing a benign global landscape in the sense that the only global minimizers are the neural collapse solutions while all other critical points are strict saddle points with negative curvature. Experimental results on practical deep networks corroborate our theory and demonstrate that better representations can be learned faster via feature normalization. Code for our experiments can be found at https://github.com/cjyaras/normalized-neural-collapse. 

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5. A Good Start Matters: Enhancing Continual Learning with Data-Driven Weight Initialization

   Harun, MY; Kanan, C (August 2025, Proc. Conference on Lifelong Learning Agents (CoLLAs))

   To adapt to real-world data streams, continual learning (CL) systems must rapidly learn new concepts while preserving and utilizing prior knowledge. When it comes to adding new information to continually-trained deep neural networks (DNNs), classifier weights for newly encountered categories are typically initialized randomly, leading to high initial training loss (spikes) and instability. Consequently, achieving optimal convergence and accuracy requires prolonged training, increasing computational costs. Inspired by Neural Collapse (NC), we propose a weight initialization strategy to improve learning efficiency in CL. In DNNs trained with mean-squared-error, NC gives rise to a Least-Square (LS) classifier in the last layer, whose weights can be analytically derived from learned features. We leverage this LS formulation to initialize classifier weights in a data-driven manner, aligning them with the feature distribution rather than using random initialization. Our method mitigates initial loss spikes and accelerates adaptation to new tasks. We evaluate our approach in large-scale CL settings, demonstrating faster adaptation and improved CL performance. 

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