Daily animation.

To better grasp what this CFM mapping/corresp. contains, let's move points linearly and at a regular speed (from source cat location to target donut loc.).

We get a different probability path that looks much more like OT. This idea is used in the rectified flow approach github.com/gnobitab/Rec...

Now, we look at the mapping/correspondance that is induced by CFM, a mapping between cat points and donut points.
To get a clearer view, let's map a stripped cat to a donut.
Color encodes the initial position of the points on the cat.

About Conditional Flow Matching (CFM) with **linear interpolation**. (1/3)

In Optimal Transport (OT) terms, this CFM induces a probability path that corresponds to the (much non-optimal) independent coupling. That means every cat point is split and sent to every donut point.

CFM Playground
Showing 3 z samples, with a deterministic equivalent of diffusion,
i.e., z = x1 and with sin/cos weighting
i.e., p(x,t|z) = 𝓝[cos(2πt) · x0 + sin(2πt) · x1, ε](x)

CFM Playground
Showing 3 z samples, with trajectories from diffusion,
i.e., z = x1 and having a noisy diffusion process towards 𝓝[0,1] (fixed)

CFM Playground
Better conditional flows, like minibatch-OT can help straighten the resulting flow.
Here we use mini-batches of 10 points on each side.
At the extreme (full batch OT) we recover optimal transport.

CFM Playground
CFM untangles even bad initial conditional flows.

CFM Playground
Showing the field u(x,t)
by flowing random points.

CFM Playground
Flowing from the mouse position (forward and backward)
i.e., Euler integration using u(x,t)

CFM Playground
The key of CFM, the "inversion"
The velocity u(x,t) at any location x,t is the average of all conditional fields *passing at this location* (here we sample a few to compute the average)
i.e., more formally u(x,t) = 𝔼_{z|x,t} [u(x,t|z)]

CFM Playground
Showing 3 z samples, with conical Gaussian path,
i.e., z = x1 and going from a Dirac at x1 to the 𝓝[0,1] (fixed)
i.e., p(x,t|z) = 𝓝[t · x1, 1-t](x)

CFM Playground
Showing 3 z samples, with linear interpolation,
i.e., z = (x0, x1) and a spiked Gaussian around (1-t) · x0 + t · x1
i.e., p(x,t|z) = 𝓝[(1-t) · x0 + t · x1, ε](x)

This caption is a lie! 😉
We actually have a 4th visual in the blog post.

Take a hot beverage ☕,
play some 🎵 www.youtube.com/watch?v=3jfI...

Welcome to the playground!
The Conditional Flow Matching playground!

Here it is. Flow matching the cat/donut pair (quickly hacked so the target distribution is not perfectly matched) (using "example 1" conditional coupling dl.heeere.com/conditional-... )