arXiv:2605.26900v1 Announce Type: new 
Abstract: 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 SPHERE-JEPA, a theoretically grounded SSL framework. We adapt LeJEPA's Cram{\'e}r-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).

تم نشر دراسة جديدة بعنوان 'SPHERE-JEPA: Spherical Prediction with Homogeneous Embeddings' تتناول الهندسة المثلى للتمثيلات المتعلمة في التعلم الذاتي الإشراف. توسع هذه الدراسة النتائج السابقة لـ LeJEPA، حيث تُظهر أن التوزيعات المتجانسة على المتنوعات الريمانية، وخاصة على الكرة الفائقة، هي الأمثل لطرق الجيران الأقرب k والانحدار عبر القمة.

Se ha publicado un nuevo estudio titulado 'SPHERE-JEPA: Spherical Prediction with Homogeneous Embeddings', que aborda la geometría óptima de las representaciones aprendidas en el aprendizaje auto-supervisado. La investigación amplía los hallazgos previos de LeJEPA, demostrando que las distribuciones uniformes en variedades riemannianas, particularmente en la hiperesfera, son óptimas para los métodos de k-vecinos más cercanos y la regresión de crestas.

Une nouvelle étude intitulée 'SPHERE-JEPA: Spherical Prediction with Homogeneous Embeddings' a été publiée, abordant la géométrie optimale des représentations apprises dans l'apprentissage auto-supervisé. La recherche prolonge les résultats précédents de LeJEPA, démontrant que les distributions uniformes sur les variétés riemanniennes, en particulier sur l'hypersphère, sont optimales pour les méthodes des k-plus proches voisins et de la régression par crête.

A new study titled 'SPHERE-JEPA: Spherical Prediction with Homogeneous Embeddings' has been released, addressing the optimal geometry of learned representations in self-supervised learning. The research extends previous findings by LeJEPA, demonstrating that uniform distributions on Riemannian manifolds, particularly the hypersphere, are optimal for k-nearest neighbors and kernel ridge regression methods.