UOTIP: Unbalanced Optimal Transport Map for Unpaired Inverse Problems

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.

Many image inverse problems, such as image restoration, require recovering clean images from noisy or incomplete observations. However, existing methods often rely on paired training data, where each noisy observation is matched with its corresponding clean image. In practice, collecting such paired data is often expensive or impossible. In this work, we propose a new framework that can solve image inverse problems using only unpaired collections of noisy observations and clean images, without requiring paired examples. Our method is based on Unbalanced Optimal Transport, which incorporates inverse problem structures through cost function design while robustly aligning noisy observation distribution and clean image distribution. Experiments show that the proposed approach works effectively across various image inverse problems and noise conditions. Our framework could provide a practical alternative for real-world inverse problems where paired datasets are difficult to obtain or noisy measurements are not perfectly matched with clean signals.