Uniqueness of the inverse flow matching solution (general case)

Determine whether, in the general multidimensional setting with endpoint distributions p0 and p1 on R^D having finite exponential moments and with the inverse flow matching inputs (specifically the velocity field v^π and the associated interpolating distributions {p_t^π}), the transport plan π in Π(p0,p1) is uniquely determined.

Background

The paper defines the inverse flow matching problem: given p0, p1, and the FM outputs obtained using some plan π (the velocity field v^π and, equivalently, the interpolating family {p_t^π}), the task is to recover π. While uniqueness is proven in special cases—one-dimensional distributions and multivariate Gaussian distributions—the general case is not settled.

This question is central to both theory and practice, including the distillation of flow-based generative models. Establishing whether π is uniquely determined by the inverse FM inputs would provide stronger theoretical guarantees for methods that rely on recovering or distilling transport plans from learned velocity fields.