A self-supervised encoder is trained once, frozen, and reused through lightweight probes on tasks nobody named at training time; the practitioner's question is when the off-the-shelf features are good enough and when they need fixing. Canonical correlation analysis, HGR maximal correlation, and the population optimum of the spectral contrastive loss all return the top-k singular subspace of a cross-view dependence operator, justified by isotropy: if the task prior has no directional preference, that subspace is universally optimal. We show isotropy is the wrong hypothesis. The prior enters the transfer risk only through the task covariance Λ=E[ΔΔ⊤], and only through its compression onto the operator's leading singular directions; what matters is not whether Λ is isotropic but whether its preferred directions are ordered consistently with the operator's spectrum. We prove matching two-sided rates---worst-case regret is exactly 1−1/κ(Λ), refines to 1−Ak for an alignment coefficient Ak, localizes to the top-2k subspace, becomes second order under a spectral gap, and is improvable by no task-agnostic representation---and show why alignment is generic: incoherent preferences cancel in high dimension, and T diverse tasks force α=O(dx/T), a quantitative account of why task diversity, not symmetry, makes self-supervised features transfer. The governing statistics cost O(kdx2), and when they signal misalignment a one-line reweighting of the positive-pair term provably restores exact optimality. The result is a diagnostic that answers the practitioner's question from a small labelled budget and refuses when the task bank cannot support the width requested; on controlled data it takes a regret of 0.86 down to 0.003, and on a CIFAR-100 encoder it correctly predicts that no correction is needed.
We argue that learning visual representations without labels requires a training signal jointly complete across three non-overlapping objectives: semantic invariance across augmented views, patch-level spatial prediction, and representational non-degeneracy. We formalize these as the observation, prediction, and regularization principles and prove (i) that combining observation and prediction without regularization admits the constant encoder as a global minimizer under negative-free alignment; (ii) that the two objectives are gradient-complementary and structurally non-conflicting at the encoder output; and (iii) that the momentum encoder converges to the same fixed point as the online encoder and provides no collapse guarantee at convergence. Contrastive alignment provides only self-limiting collapse resistance, formalized via an explicit gradient-decay argument. Dropping prediction withholds the spatial training signal by construction; dropping observation forfeits cross-view semantic invariance by construction; at the scale we study, no pair substitutes for the third. Every major self-supervised method is a special case of a single unified energy decomposition. We pair every theoretical claim with a controlled experiment, including a patch-retrieval evaluation for the spatial consequence of prediction.
Self-supervised learning has achieved remarkable empirical success in learning robust representations without explicit labels, most recently demonstrated within the framework of Joint-Embedding Predictive Architectures (JEPA). However, a fundamental question remains: what analytical principles drive these encoders toward specific distributional states? In this paper, we demonstrate that the preference for normal distributions in self-supervised encoders is a direct consequence of the Information Bottleneck (IB) principle. By recasting the IB objective as a rate-distortion problem over the predictive manifold, we provide a theoretical basis for why optimal, target-neutral, latent representations should tend towards isotropic Gaussian states. Under this framework, we show that latent representations correspond to soft clustering of inputs sharing similar predictive distributions, organized within a natural simplex structure. This perspective unifies a wide range of existing supervised and less-supervised objectives and provides a principled explanation for commonly used regularization schemes. Furthermore, we derive practical loss objectives that approximate this structure and demonstrate their effectiveness on standard benchmarks. Ultimately, our framework offers a geometric lens to understanding representation collapse and it establishes a mathematical system for regularization strategies to be used to ensure high-entropy, informative embeddings in modern self-supervised models.
Self-supervised pre-training, where large corpora of unlabeled data are used to learn representations for downstream fine-tuning, has become a cornerstone of modern machine learning. While a growing body of theoretical work has begun to analyze this paradigm, existing bounds leave open the question of how sharp the current rates are, and whether they accurately capture the complex interaction between pre-training and fine-tuning. In this paper, we address this gap by developing an asymptotic theory of pre-training via two-stage M-estimation. A key challenge is that the pre-training estimator is often identifiable only up to a group symmetry, a feature common in representation learning that requires careful treatment. We address this issue using tools from Riemannian geometry to study the intrinsic parameters of the pre-training representation, which we link with the downstream predictor through a notion of orbit-invariance, precisely characterizing the limiting distribution of the downstream test risk. We apply our main result to several case studies, including spectral pre-training, factor models, and Gaussian mixture models, and obtain substantial improvements in problem-specific factors over prior art when applicable.