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

Same-instance self-supervised learning (SSL) learns representations by enforcing consistency across two views of the same underlying instance. This principle alone, however, does not determine which downstream tasks remain recoverable from the learned representation. We study this question through semantic recoverability, defined as the amount of a task's posterior score captured by the represented function space. We show that, for centered and whitened representations, recoverability exactly determines directional class-distance-normalized variance (CDNV), controls few-shot nearest-centroid classification, and governs the strength of task-relevant semantic directions. The population linear probe and centroid axis coincide, and multiple well-recovered tasks approach a factorial centroid geometry. We then analyze a canonical two-view SSL objective and show that its population optimum spans the leading cross-view-stable modes of the associated two-view operator. This yields a closed-form spectral characterization of semantic recoverability: a downstream task is preserved to the extent that its posterior lies in the selected spectral subspace. We validate these predictions on synthetic and real datasets across several SSL methods, testing the predicted relationships among recoverability, directional geometry, spectral structure, and few-shot transfer. Together, these results give a task-level account of what information survives same-instance SSL and how the retained information appears in downstream geometry and transfer.

Spectral features can remain optimal under strongly uneven task preferences when they retain the directions most useful to the tasks. In a local-task model, expected probing gain depends on the task prior only through its covariance $Λ$: with $B$ recording dependence between views, $k$ spectral features span the leading eigenspace of $BB^\top$, while task-optimal features span that of $BΛB^\top$. We derive alignment-dependent regret bounds, matching worst-case lower bounds for a flat leading spectrum, and bounds using the leading $2k$ directions with a spectral-tail term; a spectral gap makes the bound quadratic in small anisotropy. We also bound the imbalance from random preference patterns and from averaging independent tasks with an isotropic population covariance. With known $Λ$, changing one term of the spectral contrastive loss selects task-optimal features; with a labelled task bank, we propose a diagnostic and test correction without retraining. Correction reduces synthetic held-out regret from $0.858$ to $0.003$ with $2000$ tasks when less task-relevant directions dominate, and on CIFAR-100 lowers empirical regret by $0.031$--$0.053$ on held-out fine-label tasks using the same images. With a small task bank, however, correction can hurt.

Self-supervised representation learning is central to modern machine learning because it extracts structured latent features from unlabeled data and enables robust transfer across tasks and domains. However, it can suffer from representation collapse, a widely observed failure mode in which embeddings lose discriminative structure and distinct inputs become indistinguishable. To understand the mechanisms that drive collapse and the ingredients that prevent it, we introduce a minimal embedding-only model whose gradient-flow dynamics and fixed points can be analyzed in closed form, using a classification-representation setting as a concrete playground where collapse is directly quantified through the contraction of label-embedding geometry. We illustrate that the model does not collapse when the data are perfectly classifiable, while a small fraction of frustrated samples that cannot be classified consistently induces collapse through an additional slow time scale that follows the early performance gain. Within the same framework, we examine collapse prevention by adding a shared projection head and applying stop-gradient at the level of the training dynamics. We analyze the resulting fixed points and develop a dynamical mean-field style self-consistency description, showing that stop-gradient enables non-collapsed solutions and stabilizes finite class separation under frustration. We further verify empirically that the same qualitative dynamics and collapse-prevention effects appear in a linear teacher-student model, indicating that the minimal theory captures features that persist beyond the pure embedding setting.