Link with Diffusion Models

Thu, 09 Jul 2026 21:00:00 GMT

In this section, we will establish a connection between the mathematical objects that are utilized to construct a narrowly continuous probability path between two distributions and . The objects under consideration are the velocity field and the score function, appearing in FM and DM formulations respectively. The following theorem is of main interest.

Theorem 1 . Let be a probability space, and be two -valued independent random variables defined on that probability space such that and . Also, let denote an arbitrary FM schedule, and the velocity field that defines the “standard” FM-ODE. For any , let denote the density of the stochastic interpolant, . One has,

To give a formal proof of the preceding theorem, we will make use of the following lemma.

Lemma 1 . Under the assumptions of Theorem 1 denote

the so-called MMSE denoiser of at level . One has,

Proof. One writes,

since . Using this equation, we deduce that,

Now observe that,

Therefore,

and a proper rearrangement of the terms in this equation implies the desired result.

Proof. Theorem 1

Starting by the definition of the velocity field as a conditional expectation we derive the following relation between itself and the MMSE denoiser.

By Lemma 1, we have,

Notice that,

Taking the logarithm, differentiating with respect to , and substituting in Equation 2 we get,

Substituting Equation 3 in Equation 1, one writes,

which is the desired expression.

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Link with Diffusion Models