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A one-dimensional Gaussian prior-to-data conditional interpolation

Example

Assume the Axiom of Choice. Fix $a \in R$ and $σ_{\min} > 0$ , let $Z \sim N (0, 1)$ , and set $σ_{t} = 1 - (1 - σ_{\min}) t, X_{t} = σ_{t} Z + t a (0 \leq t \leq 1) .$ Then $X_{t} \sim N (t a, σ_{t}^{2})$ , with density $ρ_{t} (x ∣ a) = \frac{1}{\sqrt{2 π} σ_{t}} \exp ⁣ (- \frac{(x - t a)^{2}}{2 σ_{t}^{2}}),$ and a conditional velocity field is $u_{t} (x ∣ a) = a + \frac{σ_{\min} - 1}{σ_{t}} (x - t a) .$ For every fixed $z \in R$ , the particle path $t \mapsto σ_{t} z + t a$ is a straight line from $z$ to $a + σ_{\min} z$ . The endpoint map $T (z) = a + σ_{\min} z$ is quadratic-cost optimal from $N (0, 1)$ to $N (a, σ_{\min}^{})$ , and its squared transport cost is $a^{2} + (1 - σ_{\min})^{2}$ .

Example references

Lipman et al., §4.1 Example II

Holderrieth and Erives, §3.2 Example 8

Facts & Assumptions

Given: AC, $a \in R$ , $σ_{\min} > 0$ , and a standard normal random variable $Z$ .

[A1]

The Axiom of Choice is the assertion that every family of nonempty sets has a choice function (The Axiom of Choice); it is required by the Gaussian-law and density suppliers used by [F1].

[F1]

The Gaussian conditional affine map is quadratic-cost optimal: under AC, for $d \geq 1$ , fixed $x_{1}$ , and $σ_{\min} > 0$ , the affine conditional Gaussian path has mean $t x_{1}$ , scale $1 - (1 - σ_{\min}) t$ , the stated Gaussian density and velocity, and a quadratic-cost optimal endpoint map with cost $∥ x_{1} ∥_{2}^{2} + d (1 - σ_{\min})^{2}$ .

Verification

Given: The hypotheses in the Example.

1.1givenA1F1

Apply [F1] with $d = 1$ and $x_{1} = a$ . It gives the law and density of $X_{t}$ , the field $u_{t} (x ∣ a)$ , and the optimality of $T (z) = a + σ_{\min} z$ with cost $a^{2} + (1 - σ_{\min})^{2}$ .

2.1givenF1algebra∎

For fixed $z$ , differentiation gives $\frac{d}{d t} (σ_{t} z + t a) = (σ_{\min} - 1) z + a$ . Substituting $x = σ_{t} z + t a$ into the displayed field yields $u_{t} (x ∣ a) = a + \frac{σ_{\min} - 1}{σ_{t}} σ_{t} z = a + (σ_{\min} - 1) z,$ which is that particle derivative. Since $σ_{t} > 0$ , its dependence on $t$ is affine and the endpoints are $z$ and $a + σ_{\min} z$ .

Diagram

The graph uses $a = 1$ and $σ_{\min} = 1 / 4$ . The horizontal coordinate is time $t$ and the vertical coordinate is particle state $X_{t}$ . The three line segments are the exact paths for starting values $z = - 1, 0, 1$ , evaluated from $X_{t} = (1 - 3 t / 4) z + t$ . The plotted endpoints are the images of those starting values under $T (z) = 1 + z / 4$ ; the diagram illustrates the formulas and is not used in the verification.

diagram failed to render:
\begin{tikzpicture}[x=4cm,y=1.2cm]
  \draw[->] (-0.08,0) -- (1.08,0) node[right] {$t$};
  \draw[->] (0,-1.18) -- (0,1.38) node[above] {$X_t$};
  \draw[gray] (1,-0.04) -- (1,0.04) node[below=4pt] {$1$};
  \draw[blue,thick] (0,-1) -- (1,0.75) node[right] {$z=-1$};
  \draw[orange!85!black,thick] (0,0) -- (1,1) node[right] {$z=0$};
  \draw[green!50!black,thick] (0,1) -- (1,1.25) node[right] {$z=1$};
  \fill[blue] (0,-1) circle (1.2pt);
  \fill[orange!85!black] (0,0) circle (1.2pt);
  \fill[green!50!black] (0,1) circle (1.2pt);
\end{tikzpicture}

Source notes

The two sources present the same scalar Gaussian interpolation and its affine velocity. The optimality and cost statement here are inherited from the complete coupling proof in the Gaussian theorem, not from the diagram.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yaron Lipman et al., Flow Matching for Generative Modeling, §4.1 Example II (standard reference, not scraped)
Peter Holderrieth and Ezra Erives, Introduction to Flow Matching and Diffusion Models, §3.2 Example 8 (standard reference, not scraped)