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 unconstrained 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.

Weaknesses: 1) I think the main drawback of this paper is that, the design of the SEC loss is somehow too straightforward and trivial. It simply minimizes the distance between the norm of each feature and the average norm, which acts as an additional term of the existing losses. A more elegant design should be expected for the NeurIPS level. It seems the latter, but in this case how to update \mu during training? Whenever the parameters of the metric are updated, all the features are changed, and \mu should be re-calculated.

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 unconstrained 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.

Recent advances in computer vision take advantage of adversarial data augmentation to ameliorate the generalization ability of classification models. Here, we present an effective and efficient alternative that advocates adversarial augmentation on intermediate feature embeddings, instead of relying on computationally-expensive pixel-level perturbations. We propose Adversarial Feature Augmentation and Normalization (A-FAN), which (i) first augments visual recognition models with adversarial features that integrate flexible scales of perturbation strengths, (ii) then extracts adversarial feature statistics from batch normalization, and re-injects them into clean features through feature normalization. We validate the proposed approach across diverse visual recognition tasks with representative backbone networks, including ResNets and EfficientNets for classification, Faster-RCNN for detection, and Deeplab V3+ for segmentation. Extensive experiments show that A-FAN yields consistent generalization improvement over strong baselines across various datasets for classification, detection and segmentation tasks, such as CIFAR-10, CIFAR-100, ImageNet, Pascal VOC2007, Pascal VOC2012, COCO2017, and Cityspaces. Comprehensive ablation studies and detailed analyses also demonstrate that adding perturbations to specific modules and layers of classification/detection/segmentation backbones yields optimal performance. Codes and pre-trained models will be made available at: https://github.com/VITA-Group/CV_A-FAN.

We present a method for encoding game logs as numeric features in the card game Dominion. We then run the manifold learning algorithm t-SNE on these encodings to visualize the landscape of player strategies. By quantifying game states as the relative prevalence of cards in a player's deck, we create visualizations that capture qualitative differences in player strategies. Different ways of deviating from the starting game state appear as different rays in the visualization, giving it an intuitive explanation. This is a promising new direction for understanding player strategies across games that vary in length.