Beyond Implicit Mapping: Advancing Generative Models Through Smoothed Optimal Transport

Optimal transport (OT) has gained significant attention in deep learning as a powerful mathematical tool for transforming distributions. Specifically, in deep generative models, the incorporation of OT helps address issues such as training instability, vanishing gradients, and mode collapse. However, in these models, most of the OT mappings learned by neural networks are typically implicit, making it difficult to explicitly model the relationship between the source and target domains. This limitation reduces the interpretability of the model and hinders its applicability in conditional generation tasks. To address this issue, we introduce Nesterov's smoothing technique to smooth the Brenier potential, enabling the derivation of an explicit OT mapping that serves as the foundation for constructing an advanced generative model. The proposed model offers the following advantages. First, it explicitly captures the mapping between the source and target domains, thereby enhancing the interpretability of the generative process and enabling a novel pathway for conditional sample generation based on a smoothed approximation of OT mapping. Second, the model can generate new samples directly through an explicit OT mapping, eliminating the need for interpolation and rejection sampling commonly seen in traditional methods, thereby improving generation efficiency. Moreover, extensive experiments show that our proposed model achieves superior performance in both unconditional and conditional generation tasks.