Generative autoencoders learn compact latent representations of data distributions through jointly optimized encoder--decoder pairs. In particular, Wasserstein autoencoders (WAEs) minimize a relaxed optimal transport (OT) objective, where similarity between distributions is measured through a cost-minimizing joint distribution (OT coupling). Beyond distribution matching, neural OT methods aim to learn mappings between two data distributions induced by an OT coupling. Building on the formulation of the WAE loss, we derive a novel loss that enables sampling from OT-type couplings via two paired WAEs with shared latent space. The resulting fully parametrized joint distribution yields (i) learned cost-optimal transport maps between the two data distributions via deterministic encoders. Under cost-consistency constraints, it further enables (ii) conditional sampling from an OT-type coupling through stochastic decoders. As a proof of concept, we use synthetic data with known and visualizable marginal and conditional distributions.

翻译：生成式自编码器通过联合优化的编码器-解码器对，学习数据分布的紧凑潜在表征。特别地，Wasserstein自编码器（WAE）最小化一个松弛的最优传输（OT）目标，其中分布间的相似性通过成本最小化的联合分布（OT耦合）来度量。除了分布匹配外，神经OT方法旨在学习由OT耦合诱导的两个数据分布之间的映射。基于WAE损失的公式化，我们推导出一个新颖的损失函数，该函数通过共享潜在空间的两个配对被WAE实现对OT型耦合的采样。由此得到的完全参数化的联合分布能够：（i）通过确定性编码器学习两个数据分布之间的成本最优传输映射；在成本一致性约束下，该分布还能（ii）通过随机解码器实现OT型耦合的条件采样。作为概念验证，我们使用具有已知且可视化边际分布和条件分布的合成数据进行实验。