Simultaneous Neural Optimal Transport

We propose an implicit neural formulation of optimal transport that eliminates adversarial min--max optimization and multi-network architectures commonly used in existing approaches. Our key idea is to parameterize a single potential in the Kantorovich dual and reformulate the associated c-transform as a proximal fixed-point problem. This yields a stable single-network framework in which dual feasibility is enforced exactly through proximal optimality conditions rather than adversarial training. Despite the inner fixed-point computation, gradients can be computed without differentiating through the fixed-point iterations, enabling efficient training without requiring implicit differentiation. We further establish convergence of stochastic gradient descent. The resulting framework is efficient, scalable, and broadly applicable: it simultaneously recovers forward and backward transport maps and naturally extends to class-conditional settings. Experiments on high-dimensional Gaussian benchmarks, physical datasets, and image translation tasks demonstrate strong transport accuracy together with improved training stability and favorable computational and memory efficiency.

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.

Alignment plays a fundamental role in many machine learning problems, such as multi-network analysis, multimodal learning, and point cloud registration. Recent works increasingly leverage optimal transport (OT) for distributional alignment, whose effectiveness largely depends on sparse supervision that is hard or costly to obtain in practice. Existing works, however, largely overlook how to actively acquire high-quality supervision to improve their alignment performance under OT frameworks. In this paper, we propose a principled active alignment framework for optimal transport alignment called AvAtar. We quantify the informativeness of a candidate by measuring its gradient-based impact on the global alignment result, computed as the gradient propagation from the global alignment result to all possible supervisions of the candidate through the entropy-regularized OT formulation. While differentiating through OT is challenging given its constrained nature, we leverage the adjoint-state method to reformulate the computation to a linear system solvable by the conjugate gradient method with linear complexity and guaranteed convergence. By encoding the global alignment result via effective utility functions, AvAtar is applicable to general alignment problems under the OT framework. Extensive experiments on three representative alignment tasks demonstrate the effectiveness, scalability, and generalizability of the proposed AvAtar.