Characterization of the general induced probability path

Characterize the general probability path induced by the Euclidean-distance-based (L_{2,1}) Flow Matching population vector field, beyond the special cases analyzed in the paper.

Background

The Euclidean-distance-based Flow Matching objective has a conditional geometric median as its population minimizer, whereas standard squared-loss Flow Matching has a conditional expectation. The standard argument that the learned field generates the prescribed interpolation relies on the tower property of conditional expectations.

The authors explain that no analogous property is known for geometric medians. Consequently, they cannot generally identify the probability path generated by the L_{2,1} population field, although they derive results for symmetric distributions, Gaussian targets, and selected contamination settings.

References

While this result yields a probabilistic interpretation of the $L_{2,1}$-regression objective, we are unable to give a characterization of the general induced probability path. The proof for standard Flow Matching relies on the tower property of conditional expectations, which to our knowledge does not have a counterpart for the geometric median. Consequently, the standard Flow Matching argument does not show that the $L_{2,1}$-population field generates the prescribed interpolation, and our analysis does not provide a general explicit description.

— Euclidean Distance-Based Flow Matching  (2610.18534 - Heilmann et al., 13 Oct 2026) in Section 3, subsection “Comparison with standard Flow Matching”