What Does Neural Collapse Indicate for Generalization?: A Singular Learning Theory Perspective

Neural Collapse (NC), which describes the emergence of highly structured class representations in the terminal phase of training, has been widely studied in connection with generalization. However, its practical role remains unclear, as several studies report that NC can hinder generalization in downstream tasks. Existing analyses have mainly focused on NC at the level of representation geometry, leaving unclear what this geometric simplification implies for the complexity of the underlying model. We study this question through Singular Learning Theory (SLT), which quantifies local singular complexity using the finite-temperature empirical Local Learning Coefficient (LLC). In this work, we establish a theoretical bridge between NC and SLT by characterizing local singular complexity through the LLC under exact and approximate NC. Empirically, we validate the predicted stability of local LLC under small deviations from exact-NC and the changes in local loss geometry near exact-NC in Polynomial Neural Networks, and observe similar complexity-related trends across broader architectures. This complexity-based view also offers a new perspective on in-domain generalization under NC. Furthermore, we show that deterministic NC statistics can serve as efficient sampling-free indicators of changes in model complexity during training.