Convergence analysis of OT-Flow for sample generation
This provides theoretical assurances for OT-Flow, addressing a gap in generative modeling, but it is incremental as it focuses on proofs for an existing method.
The paper tackles the lack of rigorous theoretical convergence proofs for deep generative models by establishing convergence results for OT-Flow, showing Γ-convergence to an optimal transport problem as a regularization parameter goes to infinity and convergence of discrete loss functions as sample size increases.
Deep generative models aim to learn the underlying distribution of data and generate new ones. Despite the diversity of generative models and their high-quality generation performance in practice, most of them lack rigorous theoretical convergence proofs. In this work, we aim to establish some convergence results for OT-Flow, one of the deep generative models. First, by reformulating the framework of OT-Flow model, we establish the $Γ$-convergence of the formulation of OT-flow to the corresponding optimal transport (OT) problem as the regularization term parameter $α$ goes to infinity. Second, since the loss function will be approximated by Monte Carlo method in training, we established the convergence between the discrete loss function and the continuous one when the sample number $N$ goes to infinity as well. Meanwhile, the approximation capability of the neural network provides an upper bound for the discrete loss function of the minimizers. The proofs in both aspects provide convincing assurances for OT-Flow.