Euclidean Distance-Based Flow Matching

Flow Matching has recently become popular as a technique for generative modeling with applications in many domains. In the standard Flow Matching formulation, we interpret the vector field as the solution of a least-squares regression problem, which is known to be sensitive to contamination. Therefore we leverage the viewpoint of robust statistics and replace the squared Euclidean distance with the Euclidean distance as a regression objective. This seemingly minor change strongly affects the properties of Flow Matching. We provide a probabilistic interpretation of the new regression target as a conditional geometric median and use this characterization to study distributional robustness of the Euclidean Distance-based Flow Matching. We analyze the mathematical properties of this changed objective and the effect of a contaminated target distribution on the involved conditional velocity distributions. We discuss the challenge of directly characterizing the distribution that our method induces. In specific contamination settings we show theoretically that our method yields an improvement for severe contamination. Finally, we present numerical experiments conducted on the CIFAR-10 image dataset where a proportion of the samples has been contaminated. Our Euclidean distance-based Flow Matching shows a clear advantage in scenarios of severe contamination of the target distribution.