SQUEEZE: Preserving Homeomorphism and Smooth in Higher-Dimensional Flows

Rectified flow (RF) is motivated as optimizing the velocity field through trajectory crossings. However, this explanation contradicts prior work showing that exact crossings cannot occur, when the dimension exceeds 2 and the source and target distributions are continuous. We revisit this question and make three contributions. 1) By relaxing the geometric definition of a crossing, we prove that ***crossings can then occur, though their probability still drops rapidly with dimension***. 2) As a consequence, we observe ***the homeomorphism of RF breaking down in high dimensions***. This effect is hard to notice, and coloring serves as a useful indicator. 3) To address this breakdown, we propose ***Squeeze***, which corrects trajectories within the principal subspace of transport before lifting them back to the original space, thereby restoring smoothness and homeomorphism. Experiments confirm our analysis and show that ***Squeeze alleviates the high-dimensional breakdown***.