Directional Neural Collapse Explains Few-Shot Transfer in Self-Supervised Learning

Frozen self-supervised representations often transfer well with only a few labels across many semantic tasks. We argue that a single geometric quantity, directional CDNV (decision-axis variance), sits at the core of two favorable behaviors: strong few-shot transfer within a task, and low interference across many tasks. We show that both emerge when variability along class-separating directions is small. First, we prove sharp non-asymptotic multiclass generalization bounds for downstream classification whose leading term is the directional CDNV. The bounds include finite-shot corrections that cleanly separate intrinsic decision-axis variability from centroid-estimation error. Second, we link decision-axis collapse to multitask geometry: for independent balanced labelings, small directional CDNV across tasks forces the corresponding decision axes to be nearly orthogonal, helping a single representation support many tasks with minimal interference. Empirically, across SSL objectives, directional CDNV collapses during pretraining even when classical CDNV remains large, and our bounds closely track few-shot error at practical shot sizes. Additionally, on synthetic multitask data, we verify that SSL learns representations whose induced decision axes are nearly orthogonal.

Self-supervised learning can produce representations that transfer well to new tasks, even with very few labeled examples, but the reason has remained unclear. Our paper shows that strong transfer does not require making all examples within a class nearly identical. Instead, what matters is reducing variation specifically along the directions used to separate classes, a phenomenon we call directional neural collapse. We provide theory showing that directional class-distance-normalized variance predicts few-shot transfer performance and experiments showing it appears across many self-supervised methods (SimCLR, VICReg, MAE, IJEPA, DINOv2, CLIP, and SigLIP), models, and datasets. We also show that different tasks (for example, color and shape) become encoded in nearly orthogonal directions, helping one representation support many tasks at once.