CorrFlow: Exact and Permutation-Equivariant Flow Matching on the Correlation Elliptope

Correlation matrices combine nonlinear positive-definiteness constraints with a simultaneous row-column permutation symmetry. Their generation therefore involves two distinct problems. First, every output must remain on the correlation elliptope, irrespective of how its variables are ordered. Second, the generator must learn the domain-specific distribution occupied by empirical data inside that valid space. Existing financial image generators mix these problems: they learn ordered matrix entries and repair invalid draws afterward. We introduce CorrFlow, which separates them. A generalized Fisher transform maps full-rank correlation matrices bijectively to unconstrained coordinates, guaranteeing valid outputs without projection. A graph flow-matching network learns the empirical distribution in those coordinates and is permutation equivariant: reordering the variables only reorders the output. We compare conditional CorrGAN, image diffusion, and CorrFlow on $80\times80$ Taiwan equity matrices using five seeds per neural method. CorrFlow produces $100\%$ valid draws without repair and obtains feature sliced-Wasserstein distance $.712\pm.141$, versus $4.373\pm.255$ for CorrGAN and $4.645\pm.195$ for image diffusion after their invalid outputs are projected back to the correlation space. It also improves this distance over historical lookup, while lookup retains higher local coverage. The result isolates the benefit of combining exact coordinates with an architecture that does not depend on an arbitrary asset ordering, illustrating how geometry and symmetry can be separated from distribution learning.