← 研踪 PaperTrailarXiv:2610.37413 · v1
Displacement Moments and Path-Terminal Separation in Brownian Interpolant Sampling
首次提交 2026-10-27 23:30
版本更新 2026-10-27 23:30
雷达观测 2026-10-29 00:34
摘要
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.
作者备注：56 pages, 4 figures; 8-page main text and complete appendices
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v1 · 2026-10-27 23:30 · 最新观测版本
Displacement Moments and Path-Terminal Separation in Brownian Interpolant Sampling
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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