Conditional and marginal flow-matching objectives differ by a constant

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.