Contrastive loss decomposes into alignment plus hypersphere uniformity

Wang & Isola prove that as negative samples grow, the InfoNCE contrastive loss on the unit hypersphere converges to an alignment term (positive pairs close) plus a uniformity term whose minimizer is the uniform distribution on the sphere [wang-isola-2020-alignment-uniformity-hypersphere] The uniformity result is rigorous via classical potential theory: the Gaussian-potential minimizer converges weakly to the true uniform surface measure on S^d [wang-isola-2020-alignment-uniformity-hypersphere] Training directly on weighted alignment plus uniformity with no contrastive loss matches or exceeds standard contrastive pretraining across five model/dataset combinations (full-ImageNet MoCo v2 replacement 67.69% vs 67.5% linear-probe) [wang-isola-2020-alignment-uniformity-hypersphere] The hypersphere geometry is imposed by L2-normalization and the central claim is analytical (about the loss), not a shape discovered in free activations [wang-isola-2020-alignment-uniformity-hypersphere]