SPHERE-JEPA: Spherical Prediction with Homogeneous Embeddings

A fundamental open question in self-supervised learning (SSL) is the explicit characterization of the optimal geometry of the learned representations. Recently, LeJEPA identified isotropic Gaussian embeddings as optimal for minimizing downstream prediction risk in Euclidean spaces. However, the corresponding problem for distributions supported on lower-dimensional manifolds, such as the hypersphere, remains unexplored. In this work, we demonstrate that extending this minimax analysis to smooth distributions on Riemannian manifolds fundamentally changes the optimal solution. We show that, under a worst-case formulation, both \(k\)-nearest neighbors and kernel ridge regression induce hyperspherical uniformity. More precisely, we show that uniform distributions on manifolds are optimal for \(k\)-nearest neighbors, and that the uniform distribution on the sphere is optimal for kernel ridge regression with both the exponential dot-product kernel and the linear kernel. This theoretical insight reveals a fundamental limitation of Gaussian embeddings: their non-uniform density induces anisotropic $k$-NN neighborhoods, severely biasing the estimator. To correct this, we introduce \textbf{SPHERE-JEPA}, a theoretically grounded SSL framework. We adapt LeJEPA's Cramér--Wold projection mechanism to enforce hyperspherical uniformity rather than a Gaussian prior. Empirically, SPHERE-JEPA yields significant improvements, boosting texture retrieval mAP by over 6\%, while consistently matching or outperforming LeJEPA on standard benchmarks—including a $+1.8\%$ linear probing gain on ImageNet-1K (ViT-B/14).