Predicting single-cell perturbation responses requires mapping between two unpaired single-cell data distributions. Optimal transport (OT) theory provides a principled framework for constructing such mappings by minimizing transport cost. Recently, Wasserstein-2 ($W_2$) neural optimal transport solvers (\textit{e.g.}, CellOT) have been employed for this prediction task. However, $W_2$ OT relies on the general Kantorovich dual formulation, which involves optimizing over two conjugate functions, leading to a complex min-max optimization problem that converges slowly. To address these challenges, we propose a novel solver based on the Wasserstein-1 ($W_1$) dual formulation. Unlike $W_2$, the $W_1$ dual simplifies the optimization to a maximization problem over a single 1-Lipschitz function, thus eliminating the need for time-consuming min-max optimization. While solving the $W_1$ dual only reveals the transport direction and does not directly provide a unique optimal transport map, we incorporate an additional step using adversarial training to determine an appropriate transport step size, effectively recovering the transport map. Our experiments demonstrate that the proposed $W_1$ neural optimal transport solver can mimic the $W_2$ OT solvers in finding a unique and ``monotonic" map on 2D datasets. Moreover, the $W_1$ OT solver achieves performance on par with or surpasses $W_2$ OT solvers on real single-cell perturbation datasets. Furthermore, we show that $W_1$ OT solver achieves $25 \sim 45\times$ speedup, scales better on high dimensional transportation task, and can be directly applied on single-cell RNA-seq dataset with highly variable genes. Our implementation and experiments are open-sourced at \url{https://github.com/poseidonchan/w1ot}.

本研究解决了单细胞扰动响应预测中缺乏有效映射的挑战，提出了一种基于Wasserstein-1的神经最优传输求解器，简化了优化过程。通过对抗训练，我们恢复了传输地图，实验证明该方法在速度和准确性上均优于现有的Wasserstein-2求解器，对高维数据具有更好的扩展性，并成功应用于真实的单细胞RNA-seq数据集。

快速且可扩展的Wasserstein-1神经最优传输求解器用于单细胞扰动预测