Conditional Unbalanced Optimal Transport Maps: An Outlier-Robust Framework for Conditional Generative Modeling

Conditional Optimal Transport (COT) has emerged as a principled framework for conditional generative modeling by learning transport maps between conditional distributions. However, COT's hard distribution-matching constraints make it highly sensitive to outliers. This is a critical limitation in conditional settings, where per-condition data sparsity amplifies the impact of outliers. To address this, we introduce the Conditional Unbalanced Optimal Transport (CUOT) framework, which relaxes conditional distribution-matching constraints through Csiszár divergence penalties. We establish a rigorous formulation of the CUOT problem and derive its dual and semi-dual formulations. Building on this theory, we propose Conditional Unbalanced Optimal Transport Maps (CUOTM), an outlier-robust conditional generative model built upon a triangular $c$-transform parameterization. Our experiments on 2D synthetic and image-scale datasets demonstrate that CUOTM achieves superior outlier robustness and competitive distribution-matching performance compared to existing COT-based baselines, while maintaining high sampling efficiency.