Characterizing conditions for uniqueness in multivariate inverse flow matching

Characterize general conditions under which the inverse flow matching problem in the multivariate setting admits a unique transport plan π in Π(p0,p1), given endpoint distributions p0 and p1 with finite exponential moments and the inverse FM inputs (the velocity field v^π or, equivalently, the interpolating distributions {p_t^π}).

Background

Beyond specific results in one-dimensional and Gaussian scenarios, the paper emphasizes that a general theory identifying conditions for uniqueness in the multivariate inverse FM problem is missing.

Developing such conditions would enable rigorous guarantees for FM-based generative AI, clarifying when recovered transport plans are unique and ensuring consistency across model distillation and training procedures.