A Theory of Contrastive Learning with Natural Images

Contrastive learning relies on informative negatives to shape the representation space, yet obtaining hard negatives is costly, often requiring large batch sizes or extensive memory banks. We propose SynCo (Synthetic negatives in Contrastive learning), an approach that synthesizes hard negatives directly in the representation space from cached queue embeddings, with no additional forward passes or input-space processing. We find that six lightweight synthesis strategies, exhaustively covering the geometric, stochastic, and adversarial perturbation families, consistently improve learned representations at negligible computational cost. Although applicable to any InfoNCE-based contrastive objective, we demonstrate SynCo within the MoCo framework. On ImageNet ILSVRC-2012 linear evaluation at 200 epochs, SynCo yields improvements of +0.4% over MoCo-v2 and +1.0% over MoCHi. Unlike MoCHi, which degrades at extended pretraining schedules (underperforming MoCo-v2 by 2.4% at 800 epochs), SynCo does not: with a simple synthetic negative schedule, performance improves by +0.5% over MoCo-v2 at 800 epochs. SynCo also transfers well to a range of downstream tasks.

Contrastive learning has become a leading paradigm for self-supervised representation learning, yet the conditions under which it recovers meaningful latent geometry remain incompletely understood. We develop a measure-theoretic framework formalizing the diversity condition, a support requirement on positive-pair sampling that is necessary for isometric latent recovery. We show that the standard full-support von Mises-Fisher setting implies the satisfaction of the diversity condition and as a consequence global contrastive loss minimizers recover latent geometry up to orthogonal transformation, while restricted conditionals can make non-orthogonal maps attain strictly lower asymptotic contrastive loss. We introduce a support-corrected Information Noise Contrastive Estimation (InfoNCE) variant as a theoretical fix: this correction makes orthogonal latent space recovery achievable but does not uniquely select it. Experiments on synthetic benchmarks validate the identifiability predictions, and CIFAR-10 experiments are consistent with the qualitative prediction that architectural inductive bias becomes more important when sampling diversity is limited. Together, our results clarify how sampling mechanisms and encoder inductive bias interact in contrastive representation learning.

In this paper, we provide a computable characterization of the geometry of optimal representations in Contrastive Learning (CL) when the classes are imbalanced. When classes are balanced and the representation dimension is greater than the number of classes, it is well-known that the optimal representations exhibit Neural Collapse (NC), i.e., representations from the same class collapse to their class means and the class means form an Equiangular Tight Frame (ETF). For imbalanced classes and a large, generalized family of CL losses, we prove that the optimal representations of all samples from the same class collapse to their class means and their geometry exhibits an angular symmetry structure that is determined by the relative class proportions. In general, we show that the geometry can be determined by solving a convex optimization problem. Exploiting this symmetry structure, we analytically investigate a special case where class imbalance is extreme and prove that CL exhibits a phenomenon called Minority Collapse (MC) where all samples from the minority classes (classes with small probabilities) collapse into a single vector, whenever the class imbalance exceeds a threshold, which in turn depends on the regularity properties of the CL loss used and on the number of negative samples. Numerical results are provided to illustrate these phenomena and corroborate the theoretical results. We conclude by identifying a number of open problems.