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

2025-12-29

AI Summary

This work addresses the inverse problem of flow matching (FM): given a pair of target distributions, can the underlying transport flow be uniquely reconstructed? Focusing on the one-dimensional and Gaussian settings under finite exponential moment assumptions, we establish existence and uniqueness of the inverse FM solution. Our analysis leverages tools from measure-theoretic probability, stochastic differential equations, and exponential moment estimates, combined with the explicit structure of monotone transport maps in one dimension and closed-form properties of Gaussian distributions. To the best of our knowledge, this is the first rigorous proof of invertibility—i.e., existence and uniqueness—for these fundamental FM classes. The result provides the first theoretically guaranteed foundation for knowledge distillation and lightweight compression of FM models, filling a critical gap in the FM literature where inverse problem analysis—particularly regarding existence and uniqueness—has been absent.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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.

Problem

• Studies inverse flow matching for generative AI applications
• Establishes uniqueness in one-dimensional and Gaussian cases
• Leaves general multidimensional problem open for future research

Innovation

• Establishes uniqueness in one-dimensional flow matching
• Proves solution uniqueness for Gaussian distributions
• Leaves multidimensional cases open for future research