On the Inverse Flow Matching Problem in the One-Dimensional and Gaussian Cases

TL;DR

This paper investigates the inverse flow matching problem for distributions with finite exponential moments, establishing uniqueness in one-dimensional and Gaussian cases, with the multidimensional case remaining an open challenge.

Contribution

It provides the first uniqueness results for inverse flow matching in one-dimensional and Gaussian cases, advancing understanding in generative AI model distillation.

Findings

Uniqueness proven for 1D distributions with finite exponential moments.

Uniqueness established for Gaussian distributions.

Multidimensional case remains unresolved.

Abstract

This paper studies the inverse problem of flow matching (FM) between distributions with finite exponential moment, a problem motivated by modern generative AI applications such as the distillation of flow matching models. Uniqueness of the solution is established in two cases - the one-dimensional setting and the Gaussian case. The general multidimensional problem remains open for future studies.