CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Global continuous-flow log-density error from state and quadrature bounds

Statement

Assume Countable Choice and the hypotheses of A continuous flow changes log density by minus its divergence. Let $q (t, x) : = - {div}_{x} f (t, x)$ , and suppose a numerical state method and scalar quadrature satisfy the hypotheses of Global divergence-integral error separates state and quadrature error, with numerical initial value satisfying $∣ {\hat{ℓ}}_{0} - \log p_{0} (x (0)) ∣ \leq C_{ℓ, 0} h^{r}$ . Then the numerical scalar values ${\hat{ℓ}}_{n}$ approximate the actual flow log densities and $\max_{0 \leq n \leq N} ∣ {\hat{ℓ}}_{n} - \log p_{t_{n}} (x (t_{n})) ∣ \leq C_{ℓ, 0} h^{r} + T K_{q} K_{x} h^{p} + D T h^{m},$ with the constants and orders from that theorem.

Facts & Assumptions

Given: Countable Choice, a C¹ flow of diffeomorphisms with positive initial density, and numerical state and quadrature updates satisfying the separate error hypotheses.

[A1]

Countable Choice is assumed only for the change-of-variables step that identifies the flow density (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Along the flow, $\frac{d}{d t} \log p_{t} (x (t)) = - {div}_{x} f (t, x (t))$ (A continuous flow changes log density by minus its divergence).

[F2]

For the divergence integrand and numerical updates in the Statement, the grid error is at most $C_{ℓ, 0} h^{r} + T K_{q} K_{x} h^{p} + D T h^{m}$ (Global divergence-integral error separates state and quadrature error).

[F3]

A solution of the scalar initial-value problem $ℓ^{'} = q (t, x (t))$ satisfies $ℓ (t) = ℓ (0) + \int_{0}^{t} q (s, x (s)) d s$ (A first-order initial value problem is equivalent to its Volterra integral equation).

Proof

1.1A1F1F3given

By [F1], the exact log density along the trajectory solves the scalar IVP $ℓ^{'} (t) = q (t, x (t))$ with $ℓ (0) = \log p_{0} (x (0))$ . Applying [F3] on each step shows that its increment from $t_{n}$ to $t_{n + 1}$ is exactly $I_{n} = \int_{t_{n}}^{t_{n + 1}} q (s, x (s)) d s$ , so the exact scalar grid sequence in [F2] is $ℓ_{n} = \log p_{t_{n}} (x (t_{n}))$ .

2.1F2step 1.1∎

Apply [F2] to the numerical state and scalar updates. The exact grid sequence is the log density by step 1.1, so the stated bound is precisely the numerical log-density error.

Depends on

Used by

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Sources

Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud, Neural Ordinary Differential Equations (standard reference, not scraped)
George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshminarayanan, Normalizing Flows for Probabilistic Modeling and Inference (standard reference, not scraped)