Displacement Moments and Path-Terminal Separation in Brownian Interpolant Sampling

Zhong Han

cs.LGOct 27, 2026

Displacement Moments and Path-Terminal Separation in Brownian Interpolant Sampling

Authors: Zhong Han

Organizations: Department of Biomedical Engineering, College of Design and Engineering, National University of Singapore

Abstract

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.

Figures & tables

Result	Moments and score/density conditions	Drift input	Initialization	Discretization rate
Liu et al. (2025), Thm. 4.3	Eighth displacement moment; interpolant regularity; no endpoint/joint score condition	$(t,x)$	Interior start; $K_{\rm init}$ retained	EM; $O(N^{-1})$
Koike (2026), Thm. 3.1	Point source; second target moment $m_{2}$ ; no target smoothness assumption	$(t,x)$	Gaussian at $t_{0}>0$ ; explicit $t_{0}(d+m_{2})/2$ term	Brownian EM; $O((d+m_{2})/N)$ at fixed stopping parameters
Gentiloni Silveri et al. (2026), Thms. 2–3	Eighth endpoint moments; conditional-source density and score condition H4	$(t,x)$	True source at $t=0$ ; $K_{\rm init}=0$	EM; $O(N^{-1})$ , with explicit $d^{3}$ term
Benton et al. (2024), App. C, Thm. 2	Data second moment $m_{2}$ ; no target density or score-smoothness condition	$(t,x)$	Standard Gaussian; $K_{\rm init}\lesssim(d+m_{2})e^{-2T}$	EI; $O((d+m_{2})/N)$ at fixed $T,\delta$
Theorem 3.1 , $p=4$	Arbitrary Borel coupling; $M_{4}<\infty$ ; no endpoint/joint score condition	$(t,x)$	True source at $t=0$ ; $K_{\rm init}=0$	EM; $O((M_{4}+d^{2})/N)$
Corollary 5.1	Arbitrary Borel coupling; $M_{2}<\infty$ ; no endpoint/joint score condition	$(t,x,x_{0})$	True source at $t=0$ ; $K_{\rm init}=0$	EM; joint KL $O((M_{2}+d)/N)$

Table 1: Selected early-stopped KL guarantees. Rates describe discretization at fixed stopping parameters and noise level; learned-drift/score error and the initialization contribution in column four are omitted from the rate. All results require a weighted

L^{2}

approximation bound under the true intermediate laws. “No score condition” means no endpoint/coupling score regularity assumption. The DFM dimension terms are not bounds uniform over unnormalized moments and scores.

K_{\rm init}

denotes the KL from the true starting law to the chosen starting law;

m_{2}

below is a data second moment, whereas

M_{p}

is our displacement moment. EI denotes an exponential integrator.

Figure 1: Source information and terminal accuracy. (a) Three path laws on one grid: Markov projection preserves every grid marginal but discards the source information in ( 12 ); Euler replaces each projected kernel by a Gaussian. The colored orders are analytic results for the fixed strict-

M_{2}

coupling of Theorem 4.1 , and the terminal order is an upper bound only. (b) A different, bounded coupling,

X_{0}=X_{1}\in\{-1,1\}

, illustrates how projection changes source–terminal pairing while preserving the marginals. Color encodes the source; thick paths end across zero from their source. Percentages use all 6000 paths per law. This illustration does not test Theorem 4.1 ; the simulation protocol is given in Section U.3 .

Figure 2: General bounds can greatly exceed terminal error in the nonlinear Student model. All curves use the geometric grid; symbols are numerical evaluations and segments guide the eye. Only the exact-law

M_{2}

expression at

N=64

is interval-certified (Appendix S.2 ). Truncated path integrals and

\Psi_{15}

are diagnostics, not certified KL values. The full-space certificate in the text uses a different, uniform grid. Profiles and refinement appear in the appendix, Figure 3 . This model differs from Theorem 4.1 .

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Nonlinear Student interaction with finite

M_{2}

and infinite

M_{3},M_{4}

. Density and signed-error profiles use geometric

N=32

; the cumulative-window diagnostic uses the same fixed model and true tilted initialization at

\Delta x=0.01

. Blue circles and solid lines denote uniform grids, and blue squares and dashed lines geometric grids. Points are evaluated values and segments guide the eye. Panel (b) shows

\max_{k}|\Delta x\sum_{j\leq k}(q_{j}-\rho_{j})|

, with both indices restricted to the evaluation window

[-15,15]

; it is not a full-space CDF error. The largest relative change of

\Psi_{15}

from

\Delta x=0.02

0.01

5.70\,10^{-4}

; this sensitivity check does not certify full-space KL. Appendix S specifies the computations; the separate uniform-grid certificate is summarized in Figure 2 .

Comparator	Score	Gain (%)	$10^{3}L$	Pass
P-BP	CRPS U	$+6.860$	$+3.0528$	yes
D-BP	CRPS U	$+6.044$	$+2.6663$	yes
R-EM	CRPS U	$-0.872$	$-0.7538$	no
RR	CRPS U	$-0.124$	$-1.1119$	no
Member shuffle	ES V	$+0.0837$	$+0.0394$	no
RR	ES U	$-0.7409$	$-1.7435$	no

Appendix

Table 2: Urban100: comparator minus R-BP.

L

is the one-sided six-comparison-adjusted absolute lower bound, multiplied by

10^{3}

. Only the first two rows satisfy the full fixed rule.

ES U contrast	2000 updates	6000 updates
RR $-$ R-EM	$-7.027\;[-7.871,-6.099]$	$-6.755\;[-7.617,-5.814]$
RR $-$ R-BP	$-5.905\;[-6.569,-5.183]$	$-5.238\;[-5.846,-4.586]$
D-EM $-$ R-EM	$+0.909\;[+0.698,+1.123]$	$+0.757\;[+0.591,+0.929]$
D-BP $-$ R-BP	$+0.586\;[+0.376,+0.796]$	$-0.229\;[-0.439,-0.015]$
EM interaction	$-5.429\;[-6.577,-4.275]$	$-5.725\;[-6.794,-4.653]$
BP interaction	$-3.108\;[-4.516,-1.742]$	$-2.788\;[-3.991,-1.628]$

Appendix

Table 3: Spatial study: all fixed contrasts, multiplied by

10^{3}

, with descriptive 95% original-image intervals. Positive values favor the R sampler; interaction is

(\mathrm{RR}-\mathrm{R})_{\rm spatial}-(\mathrm{RR}-\mathrm{R})_{\rm pixel}

. Near parameter matching does not equalize optimization or identify a causal architecture effect.

Figure 4: Source-information loss without heavy tails. The bounded coupling has

M_{2}=0

. Numerical projection and comparison-path losses add to the original-path KL; the dotted horizontal line is the separately evaluated continuous drift energy. All points are numerical quadrature, with segments guiding the eye.

Stochastic Interpolants

Stochastic interpolants learn continuous probability paths that transform noise or one distribution into another.

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Counterfactual Generation via Flow Matching: Coupling-Sensitive End-to-End Rates

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Department of Applied Mathematics and Statistics, Johns Hopkins University. · Amazon · Department of Statistics, University of California, Davis.

Flow Matching Causal Counterfactual Generation Flow-Based Generative Modeling Conditional Generative Modeling Counterfactual Learning

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Counterfactual generation seeks to sample outcomes under a hypothetical intervention or decision using observational data collected under the factual assignment mechanism. We develop a flow-matching approach that combines a sample-split, doubly robust training objective with a learned coupling between observed source outcomes and target outcomes drawn from a fitted conditional outcome model. To enable finite-step generation, we leverage a score-corrected stochastic sampler based on a Gaussian-smoothed interpolation. Our main theoretical contribution is a coupling-sensitive KL bound for constant-step Euler discretization: the error is controlled by moments of the source--target displacement under the chosen coupling, rather than by global uniform regularity of the velocity field, and has near-linear dependence on the ambient dimension. We also establish finite-sample non-parametric guarantees for the learned velocity and score fields when both the conditional outcome model and the source-target coupling are estimated from data. These bounds separate approximation, coupling-replacement, nuisance-estimation, generalization, and Monte Carlo errors and, combined with the sampler analysis, yield an end-to-end guarantee for counterfactual generation. Experiments on synthetic and semi-synthetic image benchmarks support the coupling-dependent theory and show that, at finite discretization budgets, the stochastic sampler can outperform the corresponding deterministic ODE sampler.

Abstract

Stochastic Interpolants

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