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

Alexander Korotin, Gudmund Pammer

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.

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

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.
Paper Structure (2 theorems, 10 equations)

This paper contains 2 theorems, 10 equations.

Key Result

Theorem 1

Consider distributions $p_0,p_1\in\mathcal{P}_{\rm exp}(\mathbb{R}^{D})$. Let $\pi,\pi'\!\in\!\Pi(p_0,p_1)$ be two transport plans between $p_0,p_1$. If for all $t\in[0,1]$$p_{t}^{\pi}\!=\!p_{t}^{\pi'}$, then $\pi\!=\!\pi'$.

Theorems & Definitions (4)

  • Theorem 1: On the Uniqueness of the Solution to the Inverse FM Problem in the One-Dimensional Case
  • proof : Proof
  • Theorem 2: On the Uniqueness of the Gaussian Solution to the Inverse FM Problem in the Multivariate Gaussian Case
  • proof : Proof