UOTIP: Unbalanced Optimal Transport Map for Unpaired Inverse Problems

TL;DR

UOTIP introduces a novel unbalanced optimal transport-based method for unpaired image inverse problems, demonstrating robustness and state-of-the-art results across various scenarios.

Contribution

The paper proposes UOTIP, a new unbalanced optimal transport framework that handles unpaired data, noise, and class imbalance in inverse problems, with theoretical guarantees.

Findings

UOTIP achieves state-of-the-art performance on benchmark inverse problems.

The method is robust to multi-level noise and class imbalance.

Theoretical analysis guarantees existence and uniqueness of the transport map.

Abstract

We investigate unpaired image inverse problems, a challenging setting where only independent, non-paired sets of noisy measurements and clean target signals are available for training. We propose a novel inverse problem solver based on Unbalanced Optimal Transport, called Unbalanced Optimal Transport Map for Inverse Problems (UOTIP). Our method formulates the reconstruction task, predicting clean target signals from noisy measurements, as learning a UOT Map from noisy measurement distribution to clean signal distribution by incorporating a likelihood-based cost function. By relaxing the exact marginal constraint, the UOT framework provides key advantages to our model: robustness to multi-level observation noise, adaptability to class imbalance between noisy and clean datasets, and generalizability to diverse noise-type scenarios. Furthermore, we theoretically demonstrate that incorporating a quadratic cost term ensures the existence and uniqueness of the transport map by satisfying the twist condition, even for ill-posed inverse problems. Our experiments demonstrate that UOTIP achieves state-of-the-art performance on unpaired image inverse problem benchmarks, across linear and nonlinear inverse problems.