Flow Matching for Symmetric Positive Definite Matrices using Wrapped Gaussians

Conditional Flow Matching (CFM) has emerged as a state-of-the-art paradigm for generative modeling, yet its extension to non-Euclidean geometries remains an open challenge. Current Riemannian CFM methods rely strictly on geodesic interpolation or on pullback to Euclidean spaces, limiting their expressivity. In this work, we propose a new method for Riemannian CFM, coined \emph{WrappedCFM}, that leverages the wrapped Gaussian distribution to construct conditional probability paths. We focus our development on the manifold of Symmetric Positive Definite (SPD) matrices denoted $\P_d$, a core component of information geometry. Our geometric method respects the intrinsic structure of $\P_d$ and relies neither on extrinsic Euclidean approximations nor on geodesic interpolation. Through both synthetic and real-world experiments, we demonstrate that WrappedCFM accurately models complex distributions on $\P_d$ and achieves competitive performance.