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A mixture of straight Gaussian paths has a nonlinear marginal velocity

Example

Assume the Axiom of Choice and fix $a > 0$ and $σ_{\min} > 0$ . Let the endpoint law assign probability $1 / 2$ to each endpoint $x_{1} = - a$ and $x_{1} = a$ . For each endpoint, use the one-dimensional Gaussian path with $σ_{t} = 1 - (1 - σ_{\min}) t, X_{t} = σ_{t} Z + t x_{1}, Z \sim N (0, 1) .$ For either endpoint and each fixed $Z = z$ , the conditional particle trajectory is a straight line. The marginal density is $p_{t} (x) = \frac{1}{2 \sqrt{2 π} σ_{t}} [\exp ⁣ (- \frac{(x - t a)^{2}}{2 σ_{t}^{2}}) + \exp ⁣ (- \frac{(x + t a)^{2}}{2 σ_{t}^{2}})],$ and its marginal velocity field, defined by averaging the conditional velocity against the endpoint density at fixed $(t, x)$ , is $u_{t}^{m} (x) = \frac{σ_{\min} - 1}{σ_{t}} x + \frac{a}{σ_{t}} \tanh ⁣ (\frac{a t x}{σ_{t}^{2}}), \tanh (y) : = \frac{\exp (y) - \exp (- y)}{\exp (y) + \exp (- y)} .$ The denominator in the definition of $\tanh$ is positive for every real input. The density is strictly positive, and $\partial_{t} p_{t} (x) + \partial_{x} (p_{t} (x) u_{t}^{m} (x)) = 0$ on $(0, 1) \times R$ . For every $t \in (0, 1)$ , $x \mapsto u_{t}^{m} (x)$ is nonlinear.

Example references

Lipman et al., §4.1 Example II and the qualification that a conditional OT path does not make the marginal field optimal

Holderrieth and Erives, §§3.2–3.3

Facts & Assumptions

Given: AC, $a > 0$ , $σ_{\min} > 0$ , the two-point endpoint law, and the Gaussian conditional path in the Example.

[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 each fixed endpoint $x_{1}$ the mean-linear Gaussian path has the displayed density and affine conditional velocity, and its density and flux satisfy the conditional continuity equation.

[F2]

Probability paths and velocity fields: the marginal density is $p_{t} (x) = \int ρ_{t} (x ∣ x_{1}) π (d x_{1})$ and its velocity, where $0 < p_{t} (x) < \infty$ and the flux is finite, is $u_{t}^{m} (x) = \int ρ_{t} u_{t}^{c} d π / p_{t} (x)$ .

[F3]

A conditional mixture is a probability path: the endpoint mixture of normalized conditional densities is a probability density at each time.

[F4]

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value: a continuous real-valued function on a compact metric space is bounded.

[F5]

Conditional continuity equations pass to the marginal path: if the conditional PDE derivatives admit the stated local integrable endpoint majorant, then the marginal density and flux satisfy the continuity equation.

[F6]

The real exponential function and the number $e$ by a power series: the real exponential is defined on every real input by its convergent power series.

[F7]

The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ : $\exp (r + s) = \exp (r) \exp (s)$ for all real $r, s$ .

[F8]

The exponential function is strictly increasing: $r < s$ implies $\exp (r) < \exp (s)$ .

[F9]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $\exp (- r) = 1 / \exp (r)$ and $\exp (r) > 0$ for every real $r$ .

Verification

Given: The hypotheses in the Example.

1.1givenA1F1F3F4F5

Apply [F1] to the two endpoints $x_{1} = a$ and $x_{1} = - a$ . Their conditional densities are positive and continuous in $(t, x)$ , as are their time derivatives, fluxes, and spatial flux derivatives; $σ_{t} > 0$ throughout $[0, 1]$ . Hence for each compact $K \subset (0, 1) \times R$ the finitely many such functions are bounded on $K$ by [F4]. Taking a common bound as the constant function $G_{K}$ gives an element of $L^{1} (π)$ because $π$ has total mass one. Thus [F5] applies. The mixture density is normalized by [F3] and is strictly positive because it is a sum of two positive Gaussian densities.

1.2givenF2F6F7F9algebra

Put $A_{+} (t, x) : = \exp ⁣ (- \frac{(x - t a)^{2}}{2 σ_{t}^{2}}), A_{-} (t, x) : = \exp ⁣ (- \frac{(x + t a)^{2}}{2 σ_{t}^{2}}) .$ The common Gaussian normalizing factor cancels in the endpoint-weighted mean, so [F2] gives $m_{t} (x) : = \frac{\int x_{1} ρ_{t} (x ∣ x_{1}) π (d x_{1})}{p_{t} (x)} = a \frac{A_{+} (t, x) - A_{-} (t, x)}{A_{+} (t, x) + A_{-} (t, x)} .$ Expanding the two squares gives $A_{+} / A_{-} = \exp (2 a t x / σ_{t}^{2})$ by [F7] and [F9]. Multiplying numerator and denominator by $\exp (- a t x / σ_{t}^{2})$ therefore yields $m_{t} (x) = a \tanh (a t x / σ_{t}^{2})$ .

2.1step 1.2F1F2algebra

For endpoint $x_{1} \in {- a, a}$ , [F1] gives $u_{t}^{c} (x ∣ x_{1}) = c x + (1 - t c) x_{1}, c : = \frac{σ_{\min} - 1}{σ_{t}} .$ Taking the endpoint-weighted mean and using step 1.2 gives $u_{t}^{m} (x) = c x + (1 - t c) m_{t} (x)$ . Since $σ_{t} = 1 + t (σ_{\min} - 1)$ , we have $σ_{t} - t (σ_{\min} - 1) = 1$ and hence $1 - t c = 1 / σ_{t}$ . Substitution gives the marginal velocity formula in the Example.

3.1step 2.1F6F7F8F9algebra∎

Fix $t \in (0, 1)$ and put $β : = a / σ_{t} > 0$ and $k : = a t / σ_{t}^{2} > 0$ . The formula for $\tanh$ , the addition formula [F7], and the reciprocal identity [F9] give $\tanh (- y) = - \tanh (y)$ , so $u_{t}^{m}$ is odd. For $y > 0$ , [F7] gives $\exp (0) = \exp (0)^{2}$ , and positivity [F9] therefore gives $\exp (0) = 1$ ; strict increase [F8] now yields $r : = \exp (2 y) > 1$ . Put $s : = (r - 1) / (r + 1) \in (0, 1)$ . Rewriting $\tanh (y)$ and $\tanh (2 y)$ using [F7, F9] gives $\tanh (y) = s, \tanh (2 y) = \frac{r^{2} - 1}{r^{2} + 1} = \frac{2 s}{1 + s^{2}} < 2 s .$ Thus for every $x > 0$ , step 2.1 gives $u_{t}^{m} (2 x) - 2 u_{t}^{m} (x) = β (\tanh (2 k x) - 2 \tanh (k x)) < 0.$ An odd affine function must be linear through the origin, and every such function has $u (2 x) - 2 u (x) = 0$ . Hence $u_{t}^{m}$ is not affine and is nonlinear.

Source notes

The cited sources give the straight conditional Gaussian paths and the conditional-to-marginal velocity formula. The explicit two-endpoint mixture calculation is local. Its conclusion concerns nonlinearity of the marginal velocity; it makes no claim that the marginal field is a globally optimal transport field.

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Sources

Yaron Lipman et al., Flow Matching for Generative Modeling, §4.1 Example II and the note on marginal optimal transport (standard reference, not scraped)
Peter Holderrieth and Ezra Erives, Introduction to Flow Matching and Diffusion Models, §§3.2–3.3 (standard reference, not scraped)