Two-Sample Testing via Path-based Inference

Eshant English; Wei-Cheng Lai; Yanfeng Yang; Kenji Fukumizu; Taiji Suzuki; Christoph Lippert

stat.MLOct 5, 2026

Two-Sample Testing via Path-based Inference

Authors: Eshant English, Wei-Cheng Lai, Yanfeng Yang, Kenji Fukumizu, Taiji Suzuki, Christoph Lippert

Organizations: The University of Tokyo, Tokyo, Japan · Hasso Plattner Institute for Digital Engineering, Germany · Institute of Statistical Mathematics, Japan · The Graduate University for Advanced Studies, SOKENDAI, Japan · RIKEN Centre for Advanced Intelligence Project (AIP), Tokyo, Japan · Hasso Plattner Institute for Digital Health at the Icahn School of Medicine at Mount Sinai, USA

Abstract

Modern deep generative models are primarily studied for their ability to generate realistic samples, yet the generative dynamics they learn can also serve as objects of statistical inference. We develop this idea for two-sample testing, the problem of deciding whether the same distribution generated two finite datasets. Using stochastic interpolants, we connect both distributions to a shared Gaussian bottleneck, so that each half of the resulting path is a Gaussian channel acting on a single population. We prove that the null hypothesis holds if and only if the population denoiser, or equivalently, the velocity fields of the two halves, coincide at any single noise level, which amounts to a reflection symmetry of the path about the bottleneck. Deviations from this symmetry yield a continuum of two-sample witnesses, which we estimate via held-out regression risks on learned denoisers and velocities and aggregate along the path; under an information-theoretic weighting, the aggregated discrepancy equals the Jeffreys divergence between the noise-smoothed distributions. Calibrating the resulting statistics by permutation yields tests that are valid in finite samples for any trained networks and consistent when the fields are learned accurately. On a synthetic benchmark and three image benchmarks, the proposed tests improve power over the strongest baseline by up to 33 percentage points at an equal total sample budget, with the best choice of regression representation and path weighting depending on the data modality. These results show that generative paths provide a principled representation for statistical testing, extending stochastic-interpolant models beyond generation.

Figures & tables

Figure 1: Path-based two-sample testing with a Gaussian-bottleneck interpolant. Samples from $\rho_{0}$ are transported by the probability flow (solid curves) to the shared bottleneck $\mathcal{N}(0,\tfrac{1}{2}\mathbb{I}_{d})$ at $t=\tfrac{1}{2}$ (open circles) and onwards to $\rho_{1}$ , while dashed curves show the first half of each path mirrored under $t\mapsto 1-t$ . Left: under $\mathcal{H}_{0}$ , the path is reflection symmetric about the bottleneck, so matched slices such as $\rho_{1/4}$ and $\rho_{3/4}$ coincide, each trajectory retraces its first half, and the mirrored curves are hidden beneath the flow. Right: under $\mathcal{H}_{1}$ , the second half of each path departs from the mirror image of the first, and this asymmetry, measured through the denoiser and velocity fields at matched noise levels, is the two-sample signal.

Variant	Fitted fields	Regression target $y$	Path weight $w(s)$
PBI-Denoiser_U	$\widehat{\eta}_{s}^{0},\widehat{\eta}_{s}^{1}$	$\omega$	$1$
PBI-Denoiser_J	$\widehat{\eta}_{s}^{0},\widehat{\eta}_{s}^{1}$	$\omega$	$\lambda_{s}a_{s}/\sigma_{s}^{2}$
PBI-Velocity_U	$\widehat{u}_{s}^{0},\widehat{u}_{s}^{1}$	$a_{s}^{\prime}\omega+\sigma_{s}^{\prime}z$	$1$
PBI-Velocity_J	$\widehat{u}_{s}^{0},\widehat{u}_{s}^{1}$	$a_{s}^{\prime}\omega+\sigma_{s}^{\prime}z$	$a_{s}/(\lambda_{s}\sigma_{s}^{2})$

Table 1: The four PBI variants, each combining a regression representation with a path weight and run with Algorithms 1 , 2 and 3 . The two Jeffreys weights give the same population target, the Jeffreys divergence of Proposition 7 up to path truncation. All weights are positive and bounded on

I_{\tau}

Figure 2: Test power at level

\kappa=0.05

as a function of the per-population sample size

n

(

n_{\mathrm{tr}}=n_{\mathrm{te}}=n

; every method receives

4n

observations in total). MNIST, OrganAMNIST and Fashion-MNIST use class contamination at a fraction

0.1

\to

8, left

\to

right lung, sneaker

\to

ankle boot). Lines show the mean over five training seeds (50 test sets each); shaded bands show

\pm 1

standard deviation across seeds for the PBI variants. Highlighted baselines are those that are strongest at some sample size; the remaining baselines are shown in light grey. The HDGM axis is logarithmic.

Figure 3: Empirical Type-I error at level

\kappa=0.05

, with both samples drawn from

\rho_{0}

, as a function of the per-population sample size

n

(

n_{\mathrm{tr}}=n_{\mathrm{te}}=n

). Lines show the mean rejection rate over five training seeds (50 test sets each); shaded bands show

\pm 1

standard deviation across seeds for the PBI variants, and the dotted line marks

\kappa

. Baselines are styled as in Figure 2 , and the HDGM axis is logarithmic. The PBI variants remain near the nominal level on all four benchmarks, as guaranteed by Proposition 9 .

Panel of Figure 4 — Figure 4: Samples from the fitted MNIST models at $n_{\mathrm{tr}}=n_{\mathrm{te}}=50$ per population, for (a) the denoiser and (b) the velocity representation. Rows show, from top to bottom, held-out observations from $\rho_{0}$ (digit 3), samples from the $\rho_{0}$ model, held-out observations from $\rho_{1}$ (90% digit 3, 10% digit 8) and samples from the $\rho_{1}$ model; columns are unpaired examples, so a row need not contain a target-class image. Samples are generated with the probability flow ODE with $150$ steps. Although the samples are visibly noisy, models trained at this sample size already achieve the power in Table 4 .

Panel of Figure 5 — Figure 5: Samples from the fitted OrganAMNIST models at $n_{\mathrm{tr}}=n_{\mathrm{te}}=100$ per population, for (a) the denoiser and (b) the velocity representation. Rows show, from top to bottom, held-out observations from $\rho_{0}$ (Left Lung), samples from the $\rho_{0}$ model, held-out observations from $\rho_{1}$ (90% Left Lung, 10% Right Lung) and samples from the $\rho_{1}$ model; columns are unpaired examples, so a row need not contain a target-class image. Samples are generated with the probability flow ODE with $150$ steps. Although the samples are visibly noisy, models trained at this sample size already achieve the power in Table 6 .

Panel of Figure 6 — Figure 6: Samples from the fitted Fashion-MNIST models at $n_{\mathrm{tr}}=n_{\mathrm{te}}=50$ per population, for (a) the denoiser and (b) the velocity representation. Rows show, from top to bottom, held-out observations from $\rho_{0}$ (Sneaker), samples from the $\rho_{0}$ model, held-out observations from $\rho_{1}$ (90% Sneaker, 10% Ankle Boot) and samples from the $\rho_{1}$ model; columns are unpaired examples, so a row need not contain a target-class image. Samples are generated with the probability flow ODE with $150$ steps. Although the samples are visibly noisy, models trained at this sample size already achieve the power in Table 8 .

Fields	Corruption $C$	$\rho_{0}$	$\rho_{1}$ matched	$\rho_{1}$ corrupted	U	J	P@ $k$	Mirror	Rej.
		$\bar{A}$			AUROC
Exact	new cluster	$-0.005$	$-0.005$	$0.134$	0.999	0.998	0.99	1.000	1.0
	shifted mode	$-0.012$	$-0.008$	$0.207$	0.979	0.981	0.78	0.942	1.0
	deformed mode	$-0.005$	$-0.006$	$0.138$	0.904	0.922	0.44	0.796	0.7
Fitted	new cluster	$-0.006$	$-0.005$	$0.139$	0.982	1.000	0.89	0.990	0.9
	shifted mode	$-0.010$	$-0.010$	$0.204$	0.982	0.981	0.79	0.925	0.9

Table 12: Observation-level localisation under structural corruption,

\rho_{1}=0.9\,\rho_{0}+0.1\,C

, with 500 held-out observations per group, averaged over 10 seeds.

\bar{A}

is the mean asymmetry (denoiser, uniform weight) of observations from

\rho_{0}

and of uncorrupted and corrupted observations from

\rho_{1}

. AUROC (uniform and Jeffreys weights) and P@

k

(uniform weight) measure how well

A

ranks the corrupted observations within the

\rho_{1}

sample; random ranking gives AUROC

0.5

and P@

k=0.10

. Mirror is the AUROC of the geometric mismatch

|X_{0}-\omega|

, computed on 30 corrupted and 90 uncorrupted observations per seed. Rej. is the rejection rate of the test at

\kappa=0.05

with

R=999

permutations.

Figure 7: Observation-level localisation ( $\rho_{1}=0.9\,\rho_{0}+0.1\,C$ , 500 held-out observations per group.) (a)–(c) Held-out $\rho_{1}$ observations coloured by the asymmetry $A(\omega)=-\widehat{\Phi}^{\mathrm{den}}_{U}(\omega)$ , computed with the exact denoisers for one seed; positive values mean the $\rho_{1}$ denoiser reconstructs $\omega$ better than the $\rho_{0}$ denoiser. Black edges mark corrupted observations, and dashed ellipses show the $2\sigma$ region of $C$ . For the deformed mode, uncorrupted points near the shared centre also score highly, so only the tails are localised. (d) AUROC for ranking the corrupted observations within the $\rho_{1}$ sample, averaged over 10 seeds, for $A$ and for the geometric mirror mismatch $|X_{0}-\omega|$ , with exact (filled) and fitted (open) fields. Setup and full results: Sections 5 and 12 .

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
$\rho_{0},\rho_{1}$	Unknown data densities; $\mathcal{H}_{0}\colon\rho_{0}=\rho_{1}$
$\mathcal{D}_{\ell}=\{x_{\ell}^{(i)}\}_{i=1}^{n_{\ell}}$	Dataset drawn i.i.d. from $\rho_{\ell}$ , $\ell\in\{0,1\}$
$\kappa$	Significance level (often denoted $\alpha$ in the testing literature)
$x_{t},\rho_{t}$	Stochastic interpolant and its density at time $t\in[0,1]$
$z\sim\mathcal{N}(0,\mathbb{I}_{d})$	Gaussian latent variable
$s=\|t-\tfrac{1}{2}\|$	Radial coordinate, the distance from the Gaussian bottleneck

Appendix

Table 13: Summary of notation.

Figure 8: Test power on HDGM at level

\kappa=0.05

against the per-population sample size

n

(

n_{\mathrm{tr}}=n_{\mathrm{te}}=n

), with all ten methods shown individually; a condensed version appears in Figure 2 . Lines show the mean over five training seeds (50 test sets each); shaded bands show

\pm 1

standard deviation across seeds; numerical values are in Table 2 .

Figure 9: Empirical Type-I error on HDGM at level

\kappa=0.05

, with both samples drawn from

\rho_{0}

, against the per-population sample size

n

(

n_{\mathrm{tr}}=n_{\mathrm{te}}=n

). Lines show the mean over five training seeds (50 test sets each); shaded bands show

\pm 1

standard deviation across seeds; the dotted line marks

\kappa

. The PBI variants stay close to the nominal level, as guaranteed by Proposition 9 .

Figure 10: Test power on MNIST at level

\kappa=0.05

against the per-population sample size

n

(

n_{\mathrm{tr}}=n_{\mathrm{te}}=n

), with all ten methods shown individually; a condensed version appears in Figure 2 . Lines show the mean over five training seeds (50 test sets each); shaded bands show

\pm 1

standard deviation across seeds; numerical values are in Table 4 .

Figure 11: Empirical Type-I error on MNIST at level

\kappa=0.05

, with both samples drawn from

\rho_{0}

, against the per-population sample size

n

(

n_{\mathrm{tr}}=n_{\mathrm{te}}=n

). Lines show the mean over five training seeds (50 test sets each); shaded bands show

\pm 1

standard deviation across seeds; the dotted line marks

\kappa

. The PBI variants stay close to the nominal level, as guaranteed by Proposition 9 .

Figure 12: Test power on OrganAMNIST at level

\kappa=0.05

against the per-population sample size

n

(

n_{\mathrm{tr}}=n_{\mathrm{te}}=n

), with all ten methods shown individually; a condensed version appears in Figure 2 . Lines show the mean over five training seeds (50 test sets each); shaded bands show

\pm 1

standard deviation across seeds; numerical values are in Table 6 .

Figure 13: Empirical Type-I error on OrganAMNIST at level

\kappa=0.05

, with both samples drawn from

\rho_{0}

, against the per-population sample size

n

(

n_{\mathrm{tr}}=n_{\mathrm{te}}=n

). Lines show the mean over five training seeds (50 test sets each); shaded bands show

\pm 1

standard deviation across seeds; the dotted line marks

\kappa

. The PBI variants stay close to the nominal level, as guaranteed by Proposition 9 .

Figure 14: Test power on Fashion-MNIST at level

\kappa=0.05

against the per-population sample size

n

(

n_{\mathrm{tr}}=n_{\mathrm{te}}=n

), with all ten methods shown individually; a condensed version appears in Figure 2 . Lines show the mean over five training seeds (50 test sets each) and shaded bands

\pm 1

standard deviation across seeds; numerical values are in Table 8 .

Figure 15: Empirical Type-I error on Fashion-MNIST at level

\kappa=0.05

, with both samples drawn from

\rho_{0}

, against the per-population sample size

n

(

n_{\mathrm{tr}}=n_{\mathrm{te}}=n

). Lines show the mean over five training seeds (50 test sets each) and shaded bands

\pm 1

standard deviation across seeds; the dotted line marks

\kappa

. The PBI variants stay close to the nominal level, as guaranteed by Proposition 9 .

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Zero-Flow Two-Sample Tests

Yakun Wang, Leyang Wang, Song Liu, Taiji Suzuki

University of Bristol · RIKEN AIP · University of Tokyo & RIKEN AIP

Two-Sample Testing

Open paper AskPDF arXiv

Motivated by the success of modern flow-based generative models in modeling complex data, we study two-sample testing through the lens of flow-based methods. We propose the Zero-Flow Two-Sample Test (ZF2ST), built on the zero-flow criterion, which characterizes distributional equality through a time-reversal antisymmetry of a learnable velocity field. We extend this criterion and further develop the Zero-Flow Discrepancy, an identifying discrepancy that controls the Wasserstein distance, and derive a variational representation in terms of a witness function. This representation naturally leads to a witness-based test whose power is governed by the signal-to-noise ratio (SNR), allowing direct power maximization for witness learning. ZF2ST learns the witness on one data split and performs testing on held-out samples, thereby maintaining Type-I error control and admitting a simple asymptotic null distribution. Experimentally, ZF2ST performs competitively across various synthetic and real-world benchmarks, while showing particularly strong performance in distinguishing image distributions from different sources.

Abstract

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