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.
Simplex-to-Euclidean Bijections for Categorical Flow Matching
Department of Computer Science, University of Helsinki（赫尔辛基大学计算机科学系）；Department of Computer Science, Aalto University（阿莱大学计算机科学系）；ELLIS Institute Finland（芬兰ELLIS研究所）
The Lie We Tell: Correcting the Euclidean Fallacy in Vision Language Action Policies via Score Matching on Tangent Space
National Taiwan University（台湾大学）
针对扩散视觉-语言-动作策略将SE(3)位姿表示为平坦R^12向量导致的欧几里得谬误，提出Lie Diffuser Actor (LDA)框架，通过左不变SDE注入噪声、在切空间预测分数并利用指数映射回缩样本，从根本上消除流形漂移、保证坐标框架等变性和测地线最优性，在CALVIN ABC→D上平均任务长度从3.27提升至3.51。
A Dual Basis Approach for Structured Robust Euclidean Distance Geometry
Department of Statistics and Data Science University of Central Florida（统计与数据科学系 佛罗里达中央大学）；Department of Mathematics Tufts University（数学系 塔夫茨大学）；Department of Statistics and Data Science Department of Computer Science University of Central Florida（统计与数据科学系 计算机科学系 佛罗里达中央大学）
