---
title: "Displacement Moments and Path-Terminal Separation in Brownian Interpolant Sampling"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2610.37413v1"
permalink: "https://feed.astralor.com/items/itm_77f41a3666026fdd72c55686"
published: "2026-10-27T15:30:32.000Z"
score: 54
channel: "ai"
tags: ["Displacement Moments","Path-Terminal Separation","Brownian Interpolant Sampling","Brownian","For","Borel"]
---

# Displacement Moments and Path-Terminal Separation in Brownian Interpolant Sampling

## 速览

Brownian stochastic interpolants connect paired endpoint samples, yet their sampling error can depend on displacement moments even when neither endpoint has a first moment. For arbitrary Borel couplings, we prove early-…

## 关键点

- Brownian stochastic interpolants connect paired endpoint samples, yet their sampling error can depend on displacement moments even when neither endpoint has a first moment.
- For arbitrary Borel couplings, we prove early-stopped, target-to-output KL guarantees for standard Euler sampling from the true source.
- A fourth displacement moment gives an explicit $O(N^{-1})$ bound; a second moment suffices for fixed-law convergence as drift error vanishes.

## 全文

Brownian stochastic interpolants connect paired endpoint samples, yet their sampling error can depend on displacement moments even when neither endpoint has a first moment. For arbitrary Borel couplings, we prove early-stopped, target-to-output KL guarantees for standard Euler sampling from the true source. A fourth displacement moment gives an explicit $O(N^{-1})$ bound; a second moment suffices for fixed-law convergence as drift error vanishes. For one fixed coupling with no displacement moment above two, original bridge-path KL stays of order one, comparison-path KL is of order $1/\log N$, and terminal KL is at most $C/N^2$ on the same grid. The path decomposition identifies the information about the source discarded by Markov projection. When the original bridge is Markov, current-state Euler sampling instead has a second-moment $O((M_2+d)/N)$ bound; finite-entropy Schroedinger bridges for the probability Brownian reference are included. We also construct a current-state learner with expected output KL consistency from iid endpoint pairs. Numerical diagnostics illustrate the distinct errors. Conditional diffusion identities specify a restoration interface, while image controls find no iterative advantage over cheap probabilistic predictors. Uniform terminal rates for general second-moment couplings remain open.

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