Characteristic Learning for Provable One Step Generation

We propose the characteristic generator, a novel one-step generative model that combines the efficiency of sampling in Generative Adversarial Networks (GANs) with the stable performance of flow-based models. Our model is driven by characteristics, along which the probability density transport can be described by ordinary differential equations (ODEs). Specifically, we first estimate the underlying velocity field and use the Euler method to solve the probability flow ODE, generating discrete approximations of the characteristics. A deep neural network is then trained to fit these characteristics, creating a one-step map that pushes a simple Gaussian distribution to the target distribution. In the theoretical aspect, we provide a comprehensive analysis of the errors arising from velocity matching, Euler discretization, and characteristic fitting to establish a non-asymptotic convergence rate in the 2-Wasserstein distance under mild data assumptions. Crucially, we demonstrate that under a standard manifold assumption, this convergence rate depends only on the intrinsic dimension of data rather than the much larger ambient dimension, proving our model's ability to mitigate the curse of dimensionality. To our knowledge, this is the first rigorous convergence analysis for a flow-based one-step generative model. Experiments on both synthetic and real-world datasets demonstrate that the characteristic generator achieves high-quality and high-resolution sample generation with the efficiency of just a single neural network evaluation.