Conditional flow matching and transport paths

Fix an integer $d\ge 1$, a sigma-finite measure $\lambda$ on the Borel subsets of ${\mathbb{R}}^d$, and a Borel probability measure $\pi$ on ${\mathbb{R}}^d$. Here sigma-finite means that there are Borel sets $E_n$ with ${\mathbb{R}}^d={\bigcup }_{n\ge 1}{E}_n$ and $\lambda (E_n)<\mathrm{\infty }$ for each $n$. A conditional probability-density path with endpoint law $\pi$ is a function measurable for $\mathcal{B}([0,1])\otimes \mathcal{B}({\mathbb{R}}^d)\otimes \mathcal{B}({\mathbb{R}}^d)$ on time, state, and endpoint variables $\rho :[0,1]\times {\mathbb{R}}^d\times {\mathbb{R}}^d\to [0,\mathrm{\infty }]$ such that, for every $t\in [0,1]$, ${\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)\lambda (dx)=1$ for $\pi$-almost every endpoint $x_1$. Its mixture function is $p_t(x):={\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)\pi (d{x}_1)\mathrm{.}$ For a vector field $u_t(x\mid x_1)=(u_t^1,\dots ,u_t^d)$ measurable for the same product sigma-algebra, call it a conditional velocity field for $\rho$ when, for $\pi$-almost every $x_1$, ${\rho }_t(\cdot \mid x_1)$ and every component of ${\rho }_t(\cdot \mid x_1)u_t(\cdot \mid x_1)$ are continuously differentiable in $(t,x)$ on $(0,1)\times {\mathbb{R}}^d$, the functions ${\rho }_t$ and ${\rho }_t{u}_t^j$ are locally integrable there with respect to $dt\lambda (dx)$, and ${\int }_0^1\text{\hspace{0.17em}⁣}{\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)({\mathrm{\partial }}_t\phi (t,x)+{\sum }_{j=1}^d{u}_t^j(x\mid x_1){\mathrm{\partial }}_{x_j}\phi (t,x))\lambda (dx)dt=0$ for every continuously differentiable real function $\phi$ with compact support in $(0,1)\times {\mathbb{R}}^d$. Here $dt$ denotes the supplied Lebesgue measure on time, and local integrability means absolute integrability on every compact subset. This is the conditional continuity equation for the measures ${\rho }_t(\cdot \mid x_1)\lambda$. In the Lebesgue-reference case $\lambda (dx)=dx$, its classical form under the stated differentiability is ${\mathrm{\partial }}_t{\rho }_t(x\mid x_1)+{\sum }_{j=1}^d{\mathrm{\partial }}_{x_j}\text{\hspace{0.17em}⁣}({\rho }_t(x\mid x_1)u_t^j(x\mid x_1))=0$ on $(0,1)\times {\mathbb{R}}^d$. This unweighted divergence formula is not the convention for a general reference measure. Only the interior-time equation is imposed; no endpoint continuity or existence or uniqueness of an ODE flow is asserted. At any $(t,x)$ where $0<p_t(x)<\mathrm{\infty }$ and the numerator is a finite componentwise Lebesgue integral, the marginal velocity field is $u_t(x):=\frac{{\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)u_t(x\mid x_1)\pi (d{x}_1)}{p_t(x)}\mathrm{.}$ This definition assigns no value to the quotient where $p_t(x)=0$ or $p_t(x)=\mathrm{\infty }$, or where the numerator is not finite.

Probability measures and probability spaces

Measures on sigma-algebras

The Borel sigma-algebra

Lipman et al., §§2–3

Holderrieth and Erives, §§3.1–3.2

Santambrogio, §4.1.2, Definition 4.1: the distributional continuity equation for measures

Let $\lambda$, $\pi$, and $\rho$ satisfy the conditional probability-density path definition, and for each $t\in [0,1]$ set $p_t(x)={\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)\pi (d{x}_1)$. Then $p_t$ is a nonnegative measurable function and ${\int }_{{\mathbb{R}}^d}{p}_t(x)\lambda (dx)=1$ for every $t$; hence it is a probability density with respect to $\lambda$.

Lipman et al., §3.1

Holderrieth and Erives, §3.1

The normalization is proved locally by Tonelli; the source citations document the conditional-to-marginal mixture convention.

Let $\lambda$, $\pi$, and $\rho$ be as in the conditional probability-path definition, and let $u$ be a jointly measurable vector field. Suppose that for $\pi$-almost every $x_1$, $\rho$ and each component of $\rho u$ are continuously differentiable in $(t,x)$ and satisfy the classical coordinate equation ${\mathrm{\partial }}_t{\rho }_t+{\sum }_{j=1}^d{\mathrm{\partial }}_{x_j}({\rho }_t{u}_t^j)=0$ on $(0,1)\times {\mathbb{R}}^d$. This is an explicit algebraic hypothesis; for general $\lambda$ it is not the definition of measure transport. Suppose also that for every compact $K\subset (0,1)\times {\mathbb{R}}^d$ there is a nonnegative $G_K\in L^1(\pi )$ which, for $\pi$-almost every $x_1$, bounds on $K$ the absolute values of ${\rho }_t(x\mid x_1)$, ${\mathrm{\partial }}_t{\rho }_t(x\mid x_1)$, each component of ${\rho }_t(x\mid x_1)u_t(x\mid x_1)$, and each derivative ${\mathrm{\partial }}_{x_j}({\rho }_t{u}_t^j)(x\mid x_1)$. If $p_t(x)>0$ on $(0,1)\times {\mathbb{R}}^d$, define $J_t(x):={\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)u_t(x\mid x_1)\pi (d{x}_1),u_t(x):=\frac{J_t(x)}{p_t(x)}\mathrm{.}$ Then ${\mathrm{\partial }}_t{p}_t$ and each indicated flux derivative ${\mathrm{\partial }}_{x_j}{J}_t^j$ exist and are continuous locally in $(t,x)$, and the marginal continuity equation ${\mathrm{\partial }}_t{p}_t(x)+{\sum }_{j=1}^d{\mathrm{\partial }}_{x_j}\text{\hspace{0.17em}⁣}(p_t(x)u_t^j(x))=0$ holds on $(0,1)\times {\mathbb{R}}^d$. The domination hypothesis is a sufficient Leibniz condition for the time and spatial derivatives to pass through the $x_1$ integral.

Lipman et al., §3.1 and Appendix A

Holderrieth and Erives, §3.2

The source proofs explicitly require differentiation under the endpoint integral. This item states local integrable domination for every derivative it exchanges and derives those exchanges from the mean value theorem and dominated convergence.

Let $d\ge 1$, fix $x_1\in {\mathbb{R}}^d$, and let $t\mapsto {\mu }_t(x_1)\in {\mathbb{R}}^d$ and $t\mapsto {\sigma }_t(x_1)\in (0,\mathrm{\infty })$ be continuously differentiable on $[0,1]$, with one-sided derivatives at the endpoints. For $z\in {\mathbb{R}}^d$, define the affine map ${\psi }_t(z\mid x_1):={\sigma }_t(x_1)z+{\mu }_t(x_1)\mathrm{.}$ For each $t$, this map is a diffeomorphism with inverse ${\psi }_t^{-1}(x\mid x_1)=(x-{\mu }_t(x_1))\mathrm{/}{\sigma }_t(x_1)$. The unique vector field satisfying $\frac{d}{dt}{\psi }_t(z\mid x_1)=u_t({\psi }_t(z\mid x_1)\mid x_1)$ for every $z\in {\mathbb{R}}^d$ is $u_t(x\mid x_1)=\frac{{\dot{\sigma }}_t(x_1)}{{\sigma }_t(x_1)}(x-{\mu }_t(x_1))+{\dot{\mu }}_t(x_1)\mathrm{.}$

Lipman et al., §4 Theorem 3 and Appendix A

Holderrieth and Erives, §3.2 Example 10

Fjelde, Mathieu, and Dutordoir, Gaussian conditional path derivation

The proof is the explicit affine inversion and differentiation argument. The cited paper and two independent treatments were inspected for the same formula and schedule convention.

Let $\mu$ be a Borel probability measure on ${\mathbb{R}}^d$ with finite second moment, mean $\int x\mu (dx)=0$, and $m_2:=\int \mathrm{\parallel }x{\mathrm{\parallel }}_2^2\mu (dx)<\mathrm{\infty }$. Fix $a\in {\mathbb{R}}^d$ and $s>0$, define $T(x):=a+sx$, and let $\nu$ be the law of $T(X)$ when $X$ has law $\mu$. For every Borel probability measure $\gamma$ on ${\mathbb{R}}^d\times {\mathbb{R}}^d$ whose coordinate marginals are $\mu$ and $\nu$, ${\int }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}\mathrm{\parallel }y-x{\mathrm{\parallel }}_2^2\gamma (dx,dy)\ge \mathrm{\parallel }a{\mathrm{\parallel }}_2^2+(1-s{)}^2{m}_2\mathrm{.}$ The law of $(X,T(X))$ under $\mu$ attains equality. Thus $T$ is an optimal transport map for the squared Euclidean cost between $\mu$ and its translated dilation $\nu$.

Lipman et al., §4.1

Peyré and Cuturi, §2.6 Remark 2.31

The cited flow-matching paper states the Gaussian special case. The local proof above establishes the more general centered finite-second-moment dilation case directly by Cauchy–Schwarz and does not import Brenier's theorem.

Let $d,q\ge 1$, let $\lambda$ be a sigma-finite measure on the Borel subsets of ${\mathbb{R}}^d$, let $\pi$ be a Borel probability measure on ${\mathbb{R}}^d$, and let $\rho$ be a conditional probability-density path relative to $(\lambda ,\pi )$ and $u^c$ a jointly measurable vector field. Set $p_t(x):={\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)\pi (d{x}_1),J_t(x):={\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)u_t^c(x\mid x_1)\pi (d{x}_1)\mathrm{.}$ Assume $0<p_t(x)<\mathrm{\infty }$ and $J_t(x)$ is finite on $[0,1]\times {\mathbb{R}}^d$, and define $u_t^m(x):=J_t(x)\mathrm{/}{p}_t(x)$. Suppose that for $\pi$-almost every $x_1$, $\rho$ and each component of $\rho u^c$ are continuously differentiable in $(t,x)$ and satisfy ${\mathrm{\partial }}_t{\rho }_t+{\mathrm{div}}_x({\rho }_t{u}_t^c)=0$ on $(0,1)\times {\mathbb{R}}^d$. This classical coordinate equation is a separate algebraic hypothesis, not a measure-transport convention for general $\lambda$. Suppose also that for every compact $K\subset (0,1)\times {\mathbb{R}}^d$ there is a nonnegative $G_K\in L^1(\pi )$ that bounds on $K$, for $\pi$-almost every $x_1$, the absolute values of ${\rho }_t(x\mid x_1)$, ${\mathrm{\partial }}_t{\rho }_t(x\mid x_1)$, each component of ${\rho }_t{u}_t^c$, and each corresponding spatial flux derivative.

Let $\mathrm{\Theta }\subseteq {\mathbb{R}}^q$ be open, and let $v:[0,1]\times {\mathbb{R}}^d\times \mathrm{\Theta }\to {\mathbb{R}}^d$ be jointly measurable and continuously differentiable in $\theta$, with jointly measurable parameter derivative. Define ${\mathcal{L}}_C(\theta ):={\int }_0^1\text{\hspace{0.17em}⁣}{\int }_{{\mathbb{R}}^d}\text{\hspace{0.17em}⁣}{\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)\mathrm{\parallel }v(t,x;\theta )-u_t^c(x\mid x_1){\mathrm{\parallel }}_2^2\pi (d{x}_1)\lambda (dx)dt,$ and ${\mathcal{L}}_M(\theta ):={\int }_0^1\text{\hspace{0.17em}⁣}{\int }_{{\mathbb{R}}^d}{p}_t(x)\mathrm{\parallel }v(t,x;\theta )-u_t^m(x){\mathrm{\parallel }}_2^2\lambda (dx)dt\mathrm{.}$ Assume ${\mathcal{L}}_C(\theta )$ and ${\mathcal{L}}_M(\theta )$ are finite for every $\theta \in \mathrm{\Theta }$, and define $C:={\int }_0^1\text{\hspace{0.17em}⁣}{\int }_{{\mathbb{R}}^d}\text{\hspace{0.17em}⁣}{\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)\mathrm{\parallel }{u}_t^c(x\mid x_1)-u_t^m(x){\mathrm{\parallel }}_2^2\pi (d{x}_1)\lambda (dx)dt,$ and assume $C<\mathrm{\infty }$. For every compact $H\subset \mathrm{\Theta }$ and coordinate $j$, suppose there are nonnegative integrable majorants $G_{H,j}^C$ on $dt\lambda (dx)\pi (d{x}_1)$ and $G_{H,j}^M$ on $dt\lambda (dx)$ such that for every $\theta \in H$, outside fixed null sets, $2{\rho }_t(x\mid x_1)\mid ⟨{\mathrm{\partial }}_{{\theta }_j}v(t,x;\theta ),v(t,x;\theta )-u_t^c(x\mid x_1)⟩\mid \le G_{H,j}^C(t,x,x_1),$ and $2p_t(x)\mid ⟨{\mathrm{\partial }}_{{\theta }_j}v(t,x;\theta ),v(t,x;\theta )-u_t^m(x)⟩\mid \le G_{H,j}^M(t,x)\mathrm{.}$ Then both objectives are finite and continuously differentiable on $\mathrm{\Theta }$, and ${\mathcal{L}}_C(\theta )={\mathcal{L}}_M(\theta )+C$. In particular, $C$ is independent of $\theta$ and ${\mathrm{\nabla }}_{\theta }{\mathcal{L}}_C(\theta )={\mathrm{\nabla }}_{\theta }{\mathcal{L}}_M(\theta )$. Under the stated classical coordinate PDE and local domination, $(p,u^m)$ also satisfies ${\mathrm{\partial }}_{t}p+{\mathrm{div}}_x(p{u}^m)=0$ on $(0,1)\times {\mathbb{R}}^d$. Uniform boundedness of $u^c,u^m,v,D_{\theta }v$ on compact parameter neighborhoods is a sufficient special case of the integrability and derivative-majorant assumptions above.

Lipman et al., §3.2 Theorem 2 and Appendix A proof

Holderrieth and Erives, §3.3 Theorem 12 and complete direct proof

Lipman et al. state Theorem 2 under positive marginal density and give the Appendix A argument using squared-norm expansion, the mixture density, the marginal conditional-average formula, and Fubini; they explicitly assume sufficient decay and bounded $u_t,v_t,{\mathrm{\nabla }}_{\theta }{v}_t$. Holderrieth and Erives give the complete direct expansion in §3.3, Theorem 12, including the parameter-independent squared-error term. This item states exact integrability and dominated-derivative hypotheses, of which bounded fields on compact parameter neighborhoods are a sufficient case, and gives a local endpoint majorant for the marginal continuity equation. Thus no informal tail-decay phrase is needed: the proof uses absolute integrability and no integration by parts at spatial infinity. It makes no optimizer, training-convergence, or sample-quality claim.

Assume the Axiom of Choice. Let $d\ge 1$, fix $x_1\in {\mathbb{R}}^d$, and choose ${\sigma }_{\min }>0$. Let $Z$ have the standard multivariate normal law $N_d(0,I_d)$. For $t\in [0,1]$, define ${\mu }_t(x_1):=t{x}_1,{\sigma }_t:=1-(1-{\sigma }_{\min })t,{\psi }_t(z\mid x_1):={\sigma }_{t}z+{\mu }_t(x_1),$ and let $P_t(\cdot \mid x_1)$ be the law of ${\psi }_t(Z\mid x_1)$. Then $P_t(\cdot \mid x_1)$ is the Gaussian probability path with mean ${\mu }_t(x_1)$ and covariance ${\sigma }_t^2{I}_d$, and its Lebesgue density is ${\rho }_t(x\mid x_1)=(2\pi {\sigma }_t^2{)}^{-d\mathrm{/}2}\exp \text{\hspace{0.17em}⁣}(-\frac{\mathrm{\parallel }x-{\mu }_t(x_1){\mathrm{\parallel }}_2^2}{2{\sigma }_t^2})\mathrm{.}$ A conditional velocity field for this path is $u_t(x\mid x_1)=\frac{{\sigma }_{\min }-1}{{\sigma }_t}(x-t{x}_1)+x_1\mathrm{.}$ For each endpoint $x_1$, these densities and fields form a conditional path in the probability-path definition; hence they can be mixed against any Borel endpoint law. At $t=0$ the law is $N_d(0,I_d)$, and at $t=1$ it is the law of $x_1+{\sigma }_{\min }Z$, a Gaussian with mean $x_1$ and covariance ${\sigma }_{\min }^{}{I}_d$. The affine endpoint map $T(z)=x_1+{\sigma }_{\min }z$ is optimal for squared Euclidean cost between these two laws: every coupling has cost at least $\mathrm{\parallel }{x}_1{\mathrm{\parallel }}_2^2+d(1-{\sigma }_{\min }{)}^2,$ and the law of $(Z,T(Z))$ attains this value.

Lipman et al., §4 Theorem 3, §4.1, and Appendix A

Peyré and Cuturi, §2.6 Remark 2.31