Zero Flux: Flow-Based Comparison of High-Dimensional Discrete Distributions

Leyang Wang; Yakun Wang; Song Liu; Taiji Suzuki

stat.MLOct 1, 2026

Zero Flux: Flow-Based Comparison of High-Dimensional Discrete Distributions

Authors: Leyang Wang, Yakun Wang, Song Liu, Taiji Suzuki

Organizations: Department of Mathematical Informatics, The University of Tokyo, Japan · Center for Advanced Intelligence Project, RIKEN, Japan · School of Mathematics, University of Bristol, United Kingdom

Abstract

Comparing two high-dimensional discrete distributions has always been a challenging task due to the exponentially growing state space and complex changes in interactions. A recent work suggests comparing distributions through a vector field trained using flow matching between two continuous distributions. The resulting vector field at mid-point vanishes if and only if two distributions identical. However, such a flow-based criterion does not naturally apply to discrete distributions. We extend this principle to the discrete domain and introduce the \emph{Zero Flux} criterion, a discrepancy based on local probability fluxes. Under independent coupling, we show that all local probability fluxes vanish at the midpoint if and only if two distributions are the same. This discrepancy decomposes the joint distributional difference into smaller, local contributions and can be efficiently estimated from samples. We establish finite sample error bounds for our estimator. Experiments on synthetic and real categorical data demonstrate reliable recovery of sparse dependence signals and stable tracking of distribution shifts in high dimensions.

Figures & tables

Figure 1: Local probability fluxes at the midpoint. Each surrounding state differs from $\mathbf{z}$ in one coordinate. Blue and orange arrows represent opposing posterior contributions from the two endpoints; green arrows show the resulting fluxes. All fluxes vanish for identical distributions (left), while non-identical distributions produce nonzero fluxes at some midpoint states (right).

Panel of Figure 2 — Figure 2: Recovery of sparse dependence changes. Both results show that Zero Flux closely recovers the analytical oracle field as $d$ increases while Zero Flow lose accuracy in direction and magnitude.

	Markov 8d		Markov 128d		HMM 128d		Binarized MNIST 784d
Method	AUROC $\uparrow$	FPR@95% TPR $\downarrow$	AUROC $\uparrow$	FPR@95% TPR $\downarrow$	AUROC $\uparrow$	FPR@95% TPR $\downarrow$	AUROC $\uparrow$	FPR@95% TPR $\downarrow$
MMD	0.794	0.455	0.690	0.695	0.532	0.855	0.891	0.255
Deep MMD	0.797	0.440	0.661	0.665	0.632	0.690	0.631	0.810
Classifier-JS	0.977	0.075	0.424	0.945	0.516	0.940	0.661	0.980
VAE-ELBO	0.695	0.575	0.363	0.990	0.415	0.985	0.674	0.800
VAE-IWAE	0.670	0.590	0.387	0.985	0.436	0.980	0.703	0.715

Table 1: Quantitative Comparisons on Distribution Contamination. LRD uses a fitted first order Markov likelihood on both Markov and HMM data, and an autoregressive likelihood on binarized MNIST. The best results are highlighted in bold, and the second-best results are underlined.

Figure 3: Contamination-response curves. Results are averaged over 50 independent runs; shaded bands show one standard deviation. Each curve is rescaled by its mean value at $20\%$ contamination.

Method	C-index $\uparrow$	Pearson $r(s,D)$ $\uparrow$	NVI $\downarrow$
MMD	$0.930\pm 0.045$	$0.952\pm 0.035$	$0.096\pm 0.077$
MMD-FUSE	$0.982\pm 0.028$	$0.940\pm 0.065$	$0.002\pm 0.002$
Cramér’s $V$	$0.999\pm 0.004$	$0.952\pm 0.010$	$0.001\pm 0.003$
Zero Flux	$0.994\pm 0.016$	$0.953\pm 0.040$	$0.002\pm 0.008$

Table 2: UCI categorical data results.

Figure 4: Monthly Zero Flux discrepancy between the weighted joint policy response distributions of self-identified Democrats and Republicans. Dashed lines mark selected major events. Shaded bands show standard deviation of independent runs.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Discrete probability flow. Each node represents a state in $\mathcal{V}^{d}$ ; the shown links change one coordinate. The highlighted arrow depicts $\mathcal{F}_{t,i}(\mathbf{x},v)>0$ from $\mathbf{x}$ to $\mathbf{x}^{i\to v}$ . Other arrows show local probability transfer.

Method	$d$	Sign $\uparrow$	Cosine $\uparrow$	Amp $\to 1$
Zero Flux	8	$1.000{\pm}0.000$	$0.995{\pm}0.001$	$0.991{\pm}0.016$
	24	$1.000{\pm}0.000$	$0.993{\pm}0.004$	$1.003{\pm}0.043$
	48	$1.000{\pm}0.000$	$0.987{\pm}0.010$	$1.005{\pm}0.036$
	64	$1.000{\pm}0.000$	$0.983{\pm}0.005$	$0.999{\pm}0.041$
	128	$1.000{\pm}0.000$	$0.977{\pm}0.006$	$1.004{\pm}0.032$
	256	$1.000{\pm}0.000$	$0.819{\pm}0.170$	$0.976{\pm}0.016$

Appendix

Table 3: Sparse Field Recovery on

G

. Mean

\pm

sd over 3 seeds. Sign and cosine: higher is better. Amp: closer to 1 is better. Budgets are not matched across

d

: 5k steps (

d\leq 48

), 20k (

d=64

), 40k (

d=128

), 80k (

d=256

n=20{,}000

\eta=0.8

Figure 6: Sparse Field Recovery Visualization Overall Cube across Dimensions.

Figure 7: Contamination detection on binary Markov chains (MM), hidden Markov models (HMM), and binarized MNIST. The first and third rows show AUROC; the second and fourth show FPR@95% TPR. Metrics compare 50 runs at each positive contamination level with 50 uncontaminated runs. Higher AUROC and lower FPR indicate better detection. Dashed lines indicate an AUROC of 0.5, corresponding to random guessing.

	Mean AUROC $\uparrow$
Setting	Zero Flux	Zero Flow	MMD	Deep MMD	Clf-JS	LRD	ELBO	IWAE
Markov $d=8$	0.874	0.958	0.794	0.797	0.977	0.982	0.695	0.670
Markov $d=64$	0.889	0.854	0.738	0.765	0.394	1.000	0.519	0.557
Markov $d=128$	0.920	0.861	0.690	0.661	0.424	1.000	0.363	0.387
HMM $d=64$	0.665	0.520	0.603	0.709	0.474	0.433	0.399	0.435
HMM $d=128$	0.702	0.563	0.532	0.632	0.516	0.522	0.415	0.436

Appendix

Table 4: Mean detection metrics over

\varepsilon\in\{0.02,0.05,0.1,0.2\}

, with

50

seeds and

n=1000

. Each score is compared with the

50

null runs at

\varepsilon=0

in the same setting. AUROC is higher-better; FPR at TPR

0.95

is lower-better. The best entry in each row is in bold, and the second-best is underlined. LRD on the Markov models and HMMs is the closed-form stay-Markov likelihood; LRD on MNIST is the causal Transformer.

Figure 9: Sensitivity to interaction order on the binary parity family. Curves show population discrepancies and learned estimates; shaded bands indicate one standard deviation across five seeds.

Figure 10: Comparison of monthly Nationscape trajectories for Zero Flux, mixture MMD, and median MMD. Each method’s trajectory is standardized to zero mean and unit variance across months. Curves show means over three seeds; shaded bands show one standard deviation. Dashed lines mark selected major events.

Figure 11: Policy-item decomposition of the monthly Zero Flux discrepancy between Democrats and Republicans in Nationscape. Each cell shows the contribution of one item in one month, averaged over three seeds. Brighter colors indicate larger contributions on a common scale.

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Jul 23, 2026, cs.LGJ/K move · Enter open · S save

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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Zero-Flow Two-Sample Tests

Discrete Flow Matching: Convergence Guarantees Under Minimal Assumptions

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