Gao, Hoogeboom, Heek, De Bortoli, Murphy and Salimans (2024) — https://diffusionflow.github.io/
A blog post, filed under a url: because it has no arXiv id or DOI
(ADR-009’s last resort). The page uses the ICLR Blogposts 2025 template,
but acceptance to that track could not be confirmed, so treat it as
self-published. Read as text. The interactive figures (sliders and
animations) were not seen, and they are where the post’s only empirical
content sits.
Key takeaways
The thesis, in its own words: “diffusion models and Gaussian flow matching are the same, although different model specifications can lead to different network outputs and sampling schedules”. It covers only “the common special case that the source distribution used with flow matching corresponds to a Gaussian”. It also declines to take a side: “Our purpose is not to recommend one approach over another”.
Forward process. Diffusion’s z_t = α_t x + σ_t ε and flow matching’s
z_t = (1−t) x + t ε coincide when the noise is Gaussian and
α_t = 1−t, σ_t = t.
Sampling: DDIM is the flow-matching Euler step. DDIM can be rewritten as
z̃_s = z̃_t + [output]·(η_s − η_t), with one reparametrization per network
output (x̂, ε̂, or the flow-matching vector field û). With the
flow-matching schedule the û row reduces to the flow-matching update:
“Diffusion with DDIM sampler == Flow matching sampler (Euler)”. The post
qualifies this twice. Under a higher-order solver “the network output can
make a difference”. And DDIM is invariant to a linear rescaling of α_t, σ_t, which is “not true for other samplers”.
Training: every output is a weighted ε-MSE. The loss is written
E[w(λ_t) · dλ/dt · ‖ε̂ − ε‖²], with the weighting and the schedule term
kept separate on purpose. The four outputs give ε̂: 1, x̂: e^{−λ},
v̂: α_t²(e^{−λ}+1)² and û: (e^{−λ/2}+1)². “Flow matching weighting ==
diffusion weighting of v-MSE loss + cosine noise schedule”. That identity is
not derived here. The post cites Kingma & Gao (2023) App. D.2–3 for it,
now LIT-692. That derivation holds for the unshifted cosine only.
The training schedule is the least important of the three choices. The loss “is invariant to the training noise schedule … only related to the endpoints”, and the schedule “might still affect the variance of the Monte Carlo estimator”. This is THEORY-027’s invariance restated in weighted form. “One can choose completely different noise schedules for training and sampling.”
“Straightness” is a misnomer. “Flow matching schedule is only straight for a model predicting a single point. For realistic distributions, other schedules can give straighter paths.” The support is a 1-D Gaussian toy: VP is straighter for wide distributions, FM for narrow ones.
Stochastic sampling applies to both. One DDPM step is “exactly equivalent” to a DDIM step of twice the size followed by renoising. The renoised fraction is EDM’s churn. This corrects “the common belief that flow matching is always deterministic”.
What flow matching adds is two specifications, both left untested. The
two are the û output and the 1−t, t sampling schedule. “It would be
interesting to investigate the importance of these two model specifications
empirically … which we leave to future work.”
Traps
- “The same model family” is not “the choice doesn’t matter”. The post’s closing bullets name the two specifications that can matter, network output and sampling schedule. It says the output “may also affect the training dynamics”.
- Three sentences in it are assertions, not measurements. “The weighting function is the most important part of the loss” is argued perceptually and never tested. “SD3 weighting … is very similar to the EDM weighting” is two curves plotted side by side, with no models trained. And “empirically” refers to those plotted curves.
- Anyone citing this post for the weighting identity is citing a pointer. The derivation is in LIT-692.
- Gaussian source only. Non-Gaussian flow matching, on manifolds or with learned sources, is outside the claim.
Standing in the anthology
Filed from the reading-time triage of 2026-09-25 as the account that joins LIT-630 (flow matching) and LIT-038 (DDIM). Those two were held as separate lines, and a practice sat between them that says “straight” beats “curved” and admits it does not know why.
It sources THEORY-106: under a Gaussian source, a comparison
labelled “flow matching against diffusion” or “straight against curved”
changes the weighting, the network output and the sampling schedule together.
That is the decomposition SOTA-266’s Conditions say is missing
(“No mechanism has been isolated”). It supplies a decomposition, not an
intervention. The measured result behind SOTA-266 stands.
Two follow-ups this note made visible. Kingma & Gao, which carries the weighting identity, is now filed as LIT-692, and this post extends it. And LIT-626’s phrase “flow matching beats diffusion” describes one specification of one family beating another, and should say which.