Local Flow Matching Generative Models

Abstract: Density estimation is a fundamental problem in statistics and machine learning. We consider a modern approach using flow-based generative models, and propose Local Flow Matching ($\texttt{LFM}$), a computational framework for density estimation based on such models, which learn a continuous and invertible flow to map noise samples to data samples. Unlike existing methods, $\texttt{LFM}$ employs a simulation-free scheme and incrementally learns a sequence of Flow Matching sub-models. Each sub-model matches a diffusion process over a small step size in the data-to-noise direction. This iterative process reduces the gap between the two distributions interpolated by the sub-models, enabling smaller models with faster training times. Theoretically, we prove a generation guarantee of the proposed flow model regarding the $\chi^2$-divergence between the generated and true data distributions. Experimentally, we demonstrate the improved training efficiency and competitive generative performance of $\texttt{LFM}$ compared to FM on the unconditional generation of tabular data and image datasets and its applicability to robotic manipulation policy learning.