Uniformity Is Relative: A Maximum-Entropy Principle Across Representation Geometries

A fundamental objective of self-supervised learning is to learn representations that preserve semantic consistency while occupying the representation space in a sufficiently dispersed yet non-concentrated manner. Existing notions of uniformity are tied to particular representation spaces, leaving unclear how to define an appropriate reference distribution across different representation spaces. We resolve this by defining uniformity as maximum entropy with respect to intrinsic Riemannian volume under a fixed geodesic scale. This makes uniformity a joint property of geometry and scale, recovering the canonical spherical and Euclidean references while yielding a distinct curved-space target. Meanwhile, tangent-space Gaussian transplantation misses curvature-dependent volume growth, leaving an irreducible Kullback–Leibler divergence gap even after scale retuning. A radial-angular decomposition yields tractable minibatch objectives, and experiments verify the predicted curvature-dependent mismatch in learned geometry, reveal its effects on radial and hierarchical organization, and demonstrate complementary roles for intrinsic uniformity and cross-view alignment.