Learning measure-to-measure mappings is a crucial task in machine learning, featured prominently in generative modeling. Recent years have witnessed a surge of techniques that draw inspiration from optimal transport (OT) theory. Combined with neural network models, these methods collectively known as \textit{Neural OT} use optimal transport as an inductive bias: such mappings should be optimal w.r.t. a given cost function, in the sense that they are able to move points in a thrifty way, within (by minimizing displacements) or across spaces (by being isometric). This principle, while intuitive, is often confronted with several practical challenges that require adapting the OT toolbox: cost functions other than the squared-Euclidean cost can be challenging to handle, the deterministic formulation of Monge maps leaves little flexibility, mapping across incomparable spaces raises multiple challenges, while the mass conservation constraint inherent to OT can provide too much credit to outliers. While each of these mismatches between practice and theory has been addressed independently in various works, we propose in this work an elegant framework to unify them, called \textit{generative entropic neural optimal transport} (GENOT). GENOT can accommodate any cost function; handles randomness using conditional generative models; can map points across incomparable spaces, and can be used as an \textit{unbalanced} solver. We evaluate our approach through experiments conducted on various synthetic datasets and demonstrate its practicality in single-cell biology. In this domain, GENOT proves to be valuable for tasks such as modeling cell development, predicting cellular responses to drugs, and translating between different data modalities of cells.

翻译：学习测度间映射是机器学习中的关键任务，尤其在生成建模中占有重要地位。近年来，受最优输运（OT）理论启发的技术大量涌现。这些方法统称为《神经最优输运》（Neural OT），通过神经网络模型将最优输运作为归纳偏置：映射应相对于给定代价函数是最优的——即能够以节俭方式移动点，在空间内通过最小化位移实现，或在跨空间时通过等距映射实现。该原则虽直观，但常面临若干实践挑战，需对OT工具箱进行适应性调整：平方欧氏代价之外的代价函数可能难以处理，Monge映射的确定性公式缺乏灵活性，跨不可比空间的映射带来多重挑战，而OT固有的质量守恒约束可能对异常值赋予过多权重。尽管上述理论与实践的错配已在各项研究中被独立解决，本文提出一个优雅的统一框架——《生成熵神经最优输运》（GENOT）。GENOT可适配任意代价函数，利用条件生成模型处理随机性，支持跨不可比空间映射，并可作为非平衡求解器使用。我们通过在多种合成数据集上的实验评估该方法，并在单细胞生物学中验证其实用性。在该领域，GENOT证明对细胞发育建模、预测药物响应及跨细胞数据模态转化等任务具有重要价值。