symplectic geometry

Beyond Isotropy in JEPAs: Hamiltonian Geometry and Symplectic Prediction

This paper critiques the standard practice of regularizing Joint-Embedding Predictive Architecture (JEPA) encoders toward isotropic Gaussian marginals, showing that this Euclidean symmetry assumption incurs a quantifiable 'price of isotropy' and that no geometry-independent fixed marginal target is universally canonical. The authors prove that oracle one-view marginals do not identify the view-to-view predictive coupling, arguing structural bias should enter the cross-view coupling instead. They introduce HamJEPA, which encodes views as phase-space states and uses a learned Hamiltonian leapfrog map for view-to-view prediction, with symplectic coupling identified as the key driver of gains. HamJEPA outperforms SIGReg on CIFAR-100 by up to +6.45 kNN@20 and +10.64 linear-probe points at 80 epochs, with similar improvements on ImageNet-100.