On the Inverse Flow Matching Problem in the One-Dimensional and Gaussian Cases (2512.23265v1)

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.

• The paper establishes rigorous uniqueness in the inverse flow matching problem for one-dimensional and Gaussian settings.
• It employs analytic techniques using characteristic functions and linear velocity fields to uniquely recover transport plans.
• The findings justify FM-based model distillation and highlight challenges in extending these results to higher dimensions and non-Gaussian cases.

The Inverse Flow Matching Problem: Uniqueness in One-Dimensional and Gaussian Settings

Introduction

This paper investigates the inverse flow matching (FM) problem, a critical task arising in the context of generative AI and optimal transport. While FM has become an underpinning methodology for state-of-the-art generative models, especially for constructing efficient distillation and trajectory-based generation mechanisms, theoretical aspects concerning the inverse problem—specifically, the uniqueness of underlying transport plans—have remained unresolved except in special cases. This work rigorously characterizes conditions for solution uniqueness in the one-dimensional case and for (multivariate) Gaussian distributions, thereby providing foundational results for the reliability and reproducibility of FM-based generative approaches.

Problem Formulation

The study considers distributions with finite exponential moment. Given distributions $p_0$ and $p_1$ on $\mathbb{R}^{D}$, and a (potentially non-unique) transport plan $\pi \in \Pi(p_0, p_1)$ (i.e., a joint law with prescribed marginals), flow matching constructs for each $t \in (0,1)$, a random variable $X_t = (1-t)X_0 + t X_1$ and a family of intermediate distributions $p_t^{\pi}$. The forward FM problem seeks a velocity field $v^{\pi}$ transporting mass according to the continuity equation parameterized by $\pi$.

The inverse FM problem aims to recover the original transport plan $\pi$ given the marginals $p_0$, $p_1$, and the velocity field $v^{\pi}$ (or the associated collection $\{p_t^{\pi}\}$). This question is consequential for interpretation, distillation, and analysis of learned generative models that internally leverage FM or rectified flows.

One-Dimensional Case: Uniqueness Result

The authors establish that, for $D=1$, the inverse FM problem has a unique solution. The main technical apparatus consists of analyzing the sequence $\{p_t^{\pi}\}$, which, through the characteristic function, encodes the complete joint law $(X_0, X_1)$.

The core result, proven with functional analytic arguments, shows that the family of intermediate marginals $p_t^{\pi}$, defined for all $t \in [0,1]$, uniquely determines the joint characteristic function $\phi_{\pi}(\xi_0,\xi_1)$. Analyticity of $\phi_{\pi}$ (enforced by the exponential moment condition) then yields global uniqueness of the initial coupling $\pi$. Thus, any two transport plans yielding the same interpolants $\{p_t^{\pi}\}$ must coincide.

Gaussian Case: Uniqueness via Velocity Field

For the multivariate Gaussian setting, the plan $\pi$ is parametrized by the cross-covariance $S$ in the joint Gaussian $(X_0, X_1)$. The authors demonstrate that, given $p_0 = \mathcal{N}(\mu_0,\Sigma_0)$ and $p_1 = \mathcal{N}(\mu_1,\Sigma_1)$, the velocity field $v_0^{\pi}$ at $t=0$ uniquely identifies the cross-covariance $S$. Since $v_0^{\pi}$ reduces to a linear map involving $S$ (explicitly computable), the uniqueness of $S$ translates directly to the uniqueness of $\pi$ within the class of Gaussian couplings.

This result certifies that for any two Gaussian transport plans between fixed marginals, if they induce the same initial velocity field, then their induced FM processes and joint distributions must be identical.

Theoretical and Practical Implications

These theoretical guarantees have several important implications. First, the established uniqueness results provide justification for learning and deploying FM-based distillation in one-dimensional and Gaussian regimes; practitioners can be confident that the generative process inferred from velocity fields (or interpolated marginals) corresponds, up to natural equivalences, to a single underlying transport plan.

Second, in high-dimensional or non-Gaussian cases, the general inverse FM problem may admit multiple solutions, potentially impacting model interpretability and reproducibility. The results in this paper delineate boundaries where generative models based on FM are theoretically well-posed.

On the practical side, methods that require plan recovery, such as generator distillation and efficient flow-based inference schemes {(Huang et al., 2024)}, benefit directly from these findings. Robustness in solution structure also implies stability of downstream observables and generated samples.

Limitations and Prospects for Future Work

The general multidimensional, non-Gaussian inverse FM problem remains open. It is currently unknown under what wider conditions (if any) uniqueness can be expected or if classification of all compatible plans is possible. The methods employed for the proven cases, leveraging characteristic functions and Gaussian parameterizations, do not trivially generalize to arbitrary distributions in higher dimensions. Progress in this direction would have significant implications for the theoretical foundation of FM-based generative algorithms in high-dimensional, realistic data regimes.

Conclusion

This paper delivers rigorous uniqueness results for the inverse flow matching problem in the one-dimensional and Gaussian cases. These findings provide essential theoretical foundations for FM-based generative modeling, ensuring solution identifiability in key scenarios pertinent to applications. The open questions regarding the multidimensional case highlight important directions for further research at the intersection of optimal transport, generative modeling, and applied probability.

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Reference: "On the Inverse Flow Matching Problem in the One-Dimensional and Gaussian Cases" (2512.23265)