Flows don't cross in high dimension

Teodora Reu, Sixtine Dromigny, Michael M. Bronstein, Francisco Vargas

Track: long paper (up to 8 pages)

Keywords: Rectified Flows, CFM, Variance

TL;DR: Study into correlation between variance of gradients and intersections of interpolating lines.

Abstract: Conditional Flow Matching (CFM) has emerged as a competitive framework for generative modeling, yet persistent concerns about trajectory crossings and their impact on gradient variance have influenced the development, of a new framework Rectify Flows. In this work, we rigorously analyze these assumptions through theoretical and empirical lenses. First, we prove that in high-dimensional spaces ($d > 2$), interpolating trajectories between source-target pairs almost surely never cross—a zero-measure phenomenon contradicting low-dimensional intuition. Second, we derive closed-form expressions for gradient variance under Gaussian distributions, revealing that suboptimal deterministic couplings (e.g., rotation-based pairings) incur dimension-dependent variance scaling. Empirically, we demonstrate that while 2D rotations inducing crossings amplify gradient noise, this effect diminishes linearly with dimension rather than abruptly vanishing. We also identify time-dependent variance patterns ($t \to 1$) uncorrelated with crossings, suggesting additional variance sources in CFM optimization.

Submission Number: 15