LeJEPA-FGP: Factorized Gaussian Process Regularization for Identifiable Joint-Embedding World Models

LeJEPA learns representations by combining predictive alignment with isotropic Gaussian regularization, but the rotational symmetry of the Gaussian target leaves learned coordinates identifiable only up to orthogonal transformations. We introduce LeJEPA-FGP, a structured extension that replaces the pointwise isotropic Gaussian embedding law with a factorized Gaussian-process law over indexed representations. Each latent channel follows its own GP covariance kernel across an index such as time, space, or pose, while remaining marginally standard Gaussian at each point. We propose FGP-SIGReg, a sketched characteristic-function regularizer for matching factorized GP embedding laws. We prove that FGP-SIGReg consistently matches finite-dimensional factorized GP distributions. Under GP-OU positive-pair transitions, alignment with exact FGP law matching forces the representation into the first Wiener chaos, yielding linear recovery of the latent GP path. Under shared index-local encoders, distinct covariance kernels reduce the residual ambiguity from orthogonal rotations to signed permutations, while repeated kernels yield block-orthogonal ambiguity. Synthetic GP experiments validate signed-permutation recovery under distinct kernels and block recovery under kernel collision.