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←Flow Matching for Generative Modelingpage 7 / 24
FLOW MATCHING7
3   Conditional Flow Matching
Constructing the target field ut directly is intractable: the marginal path pt(x) has no closed form. Conditioning on a single datum x1 fixes this.
ut(x) = ∫ ut(x | x1) pt(x|x1)q(x1)pt(x) dx1 (7)
Remarkably, regressing onto the conditional field yields the same gradients as the intractable marginal one.
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Friend
reading page 7 with you
friend
Before eq. (7) — why can't we just train vθ on the marginal field directly?
you
There's no closed form for u_t(x) — it's an integral over every path.
friend
Right. So what does conditioning on x1 buy us computationally?
you're typing…
Explain it back, or ask…
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the friend asks. you answer. it checks. — not a summary box
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Reading hard things, made warm and easy.

Whatever you’ve been meaning to read — a paper, a textbook, a story — you can finally enjoy it, one calm page at a time, with someone friendly right beside you.

01
A friend on every page.
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One page at a time, at your pace.

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Drop in a PDF
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It turns it around
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It checks you — or just reads along
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cites 1 − β₁ᵗ correctly — because it can see it
DENOISING DIFFUSION · 4
We define the forward process variances β1,…,βT and write αt = 1 − βt, ᾱt = ∏s≤t αs.
q(xt | x0) = 𝒩(xt; √ᾱt x0, (1−ᾱt)I) (4)
For the first step this reduces to a variance of 1 − β1t, which the schedule keeps small.
friend
On this page the schedule uses 1 − β₁ᵗfor the first step — note that's β₁ raised to the t, not β at step 1. Eq. (4) hinges on that distinction. Want to see why?
you
…oh. I'd been reading that as β₁.
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Who it's for

For anyone staring down a document too dense to read straight through — a paper, a textbook, a spec, a contract you actually have to understand.

It started with ML and math papers read solo at 1am, with no advisor down the hall to ask. Turns out the same loop works on anything you need to truly understand.

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