FM (-OT) is not necessarily straight when you map to anything more than a single point (=more than delta peak).

Doing naive distillation with noise/image pairs will give straight paths (on any diffusion/FM model).

After training doing Reflow from rectified flow will also indeed straighten paths.

Finally, something that is not obvious at all, and requires some digging / equation re-writing. SD3's (arxiv.org/abs/2403.03206) Flow matching weighting is very similar to EDM's weighting (arxiv.org/abs/2206.00364).

Above FM does not actually look that straight, an often claimed feature of FM. What's going on?

- FM schedule (often omitted "OT") is straight to a single point (possibly with tiny noise).
- Unfortunately, that does not guarantee straightness between distributions.

DDIM vs Flow Matching.
Note whatever schedule (alpha/sigma, VP, VE, FM) we pick, DDIM always ends up at the same spot:
- DDIM is invariant to alpha/sigma rescalings
- With FM schedule, sampling with either DDIM and Euler (=what FM uses) is the same.

This is a really nice blogpost by
@RuiqiGao and team that I enjoyed being a part of. My favorite key learnings are:
- DDIM sampler == flow matching sampling
- (Not) straight?
- SD3 weighting (Esser, Rombach, et al) is very similar to the EDM weighting (Karras, et al).
👇