Neural collapse (NC) describes the elegant geometric structure that sample features and classifier vectors exhibit when a classification model's loss reaches its minimum. Class imbalance, however, generally distorts standard NC. In this paper, we theoretically reformulate NC on imbalanced datasets and in open-set settings. Our analysis is based on an optimization problem over class-wise averaged losses, rather than the global averaged losses used in prior NC studies. We prove that when class-wise averaged losses attain their minima, standard NC is achieved on balanced or imbalanced datasets, for both cross-entropy (CE) and binary cross-entropy (BCE) losses. We further extend the analysis to an open-set-inspired scenario, where some classes contain no training samples yet are still assigned classifier vectors in the losses. We find that in this scenario, when class-wise averaged losses reach their minima, standard NC still emerges on the known classes with training samples, while the classifier vectors of unknown classes without training samples overlap with each other and become orthogonal to those of the known classes. Finally, we analyze Softmax- and Sigmoid-based contrastive losses re-balanced by inverse sample frequency, proving that at their minima on balanced or imbalanced datasets, sample features of different classes likewise form standard NC.

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