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

1 INTRODUCTION

• High-dimensional discrete comparison is challenging because state spaces grow exponentially and differences may involve complex joint dependence.Existing approaches represent differences through particular kernels, geometries, or statistical models, making performance sensitive to how discrete structure is modelled.Source: §1 · p. 1
• Zero Flux extends the Zero Flow criterion to discrete data, characterizing bridge evolution with local probability fluxes between states differing in one coordinate.Under independent coupling, all fluxes vanish at the midpoint if and only if the endpoint distributions are identical.Source: §1 · p. 1
• The resulting overall discrepancy quantifies distributional differences and can be estimated from samples.The paper also establishes finite-sample bounds for its neural estimator.Source: §1 · p. 2
• Local signed fluxes decompose joint changes into interpretable contributions, revealing changes in word associations even when marginal word frequencies are similar.Their direction and strength depend on surrounding coordinates.Source: §1 · p. 2

2 BACKGROUND

• The setup uses independently coupled endpoints drawn from distributions on a finite alphabet.The endpoint variables are denoted X0 and X1, with X0 distributed as P0 and X1 as P1.Source: §2 · p. 2

2.1 ZERO FLOW CRITERION

• Zero Flow uses endpoint-exchange symmetry along a path to characterize equality, reducing at the midpoint to a vanishing velocity field.The cited works apply the principle to representation learning, two-sample testing, and one-step generative models.Source: §2.1 · p. 2

2.2 DISCRETE FLOW MATCHING

• Discrete Flow Matching forms intermediate states by independently copying each coordinate from one of the two endpoints, while retaining dependence through their shared endpoint pair.The resulting path has the endpoint distributions at t = 0 and t = 1; at t = 1/2, exchanging endpoints leaves the conditional bridge unchanged.Source: §2.2 · p. 2

3.1 LOCAL PROBABILITY FLUX AND THE CONTINUITY EQUATION

• Local probability fluxes describe net probability transfer between states differing in one coordinate, replacing a vector field that categorical states do not intrinsically support.Signed fluxes across coordinate replacements sum to the net outflow from a state, giving the discrete bridge a continuity equation.Source: §3.1 · p. 3

3.2 MIDPOINT ZERO FLUX CRITERION

• The midpoint criterion uses agreement between endpoint posterior probabilities given the complete midpoint state; flux direction and magnitude can vary with surrounding coordinates.At t = 0.5, the bridge’s endpoint-exchange symmetry gives the simplified flux expression.Source: §3.2 · pp. 3–4
• Under independent coupling, the endpoint distributions are identical if and only if every local flux vanishes at the midpoint.The figure illustrates one-coordinate neighboring states and shows zero fluxes for identical distributions versus nonzero fluxes at some midpoint states for differing distributions.Source: Figure 1 · p. 3, §3.2 · p. 4
• Agreement of endpoint posteriors at every supported midpoint extends inductively from univariate marginals to equality of the full joint distributions.The proof uses factorization of the conditional bridge and independence of the endpoints.Source: §3.2 · p. 4
• The criterion detects differences in joint dependence even when all univariate marginals match, while context-dependent replacement directions show how those differences vary.The paper then uses the criterion to construct a sample-based discrepancy that preserves coordinate-wise localization and directional information.Source: §3.2 · p. 4

4 A SAMPLE-BASED ZERO FLUX DISCREPANCY

• The method normalizes local bridge fluxes, then averages their summed squares under the midpoint distribution to form the population discrepancy.Squaring before aggregation prevents opposite-signed fluxes from cancelling.Source: §4.1 · pp. 4–5, §4 · p. 4

4.1 NORMALIZED FLUX AND POPULATION DISCREPANCY

• At the midpoint, the discrepancy is zero if and only if the endpoint distributions are identical.It is symmetric in the endpoint distributions.Source: §4.1 · p. 5

4.2 GLOBAL COMPARISON AND LOCAL DECOMPOSITION

• Coordinate-wise discrepancies show where the global difference is concentrated, while signed fluxes indicate each local change’s direction and magnitude.The same coordinate replacement can behave differently with different remaining coordinates.Source: §4.2 · p. 5

4.3 EMPIRICAL ESTIMATION

• When differing coordinates S are independent of identically distributed remaining coordinates, those irrelevant coordinates leave the global discrepancy and fluxes on S unchanged.The empirical estimator uses neural networks to predict each endpoint’s coordinate from the complete midpoint state, then estimates coordinate and overall discrepancies from evaluation midpoints.Source: §4.2 · p. 5, §4.3 · pp. 5–6

• The error bound accounts for endpoint and midpoint sampling fluctuations as well as posterior estimation error.The latter is measured by average excess cross entropy of learned posteriors relative to exact posteriors.Source: §5 · p. 6
• For categorical posterior networks, the empirical discrepancy is bounded above by 8d.Source: §5 · p. 6
• When P0 = P1, exact normalized fluxes vanish at every midpoint with positive probability, so the empirical bound depends only on posterior-estimation error.The deterministic posterior-error bound applies even when networks are trained and evaluated on the same samples.Source: §5 · p. 6

• Kernel and feature-based discrepancies compare distributions through differences in feature mean embeddings.Source: §6 · p. 7
• Classifier and density-ratio methods assess population distinguishability or estimate relative density, while model-based approaches rely on probabilistic fits.Likelihood ratios use explicit likelihoods; VAEs and IWAEs provide tractable variational surrogates.Source: §6 · p. 7
• Zero Flux links local posterior balance under the midpoint distribution to equality of the complete endpoint distributions, connecting discrete local-structure methods with flow dynamics.Source: §6 · p. 7

7.1 SPARSE FIELD RECOVERY IN HIGH DIMENSIONS

• As irrelevant coordinates are added, Zero Flux maintains accurate directions and near-target average magnitudes, while both coordinate-wise Zero Flow methods lose accuracy.The distributions share univariate marginals and differ only in dependence on coordinates S = {1, 2, 3}; added coordinates have identical independent uniform laws.Source: §7.1 · pp. 7–8

7.2 TRACKING STRUCTURED DISTRIBUTION SHIFTS

• Zero Flux improves on longer Markov sequences, achieves the best detection performance on the HMM and binarized MNIST, and generally increases with contamination.On the HMM, it detects longer-range dependence changes where Markov-based LRD performs near chance; some baselines have nearly flat or non-monotonic response curves.Source: Figure 3 · p. 8, Table 1 · p. 8, §7.2 · p. 9

7.3 TRACKING DEPENDENCE DEGRADATION IN CATEGORICAL TABULAR DATA

• Zero Flux closely tracks dependence degradation in UCI categorical data, approaching Cramér’s V in severity ordering and exhibiting few downward reversals.MMD-FUSE also tracks the path reliably, while fixed-kernel MMD shows more ordering violations.Source: §7.3 · p. 9, Table 2 · p. 9

7.4 TRACING PARTISAN DISTRIBUTIONAL SEPARATION OVER TIME IN U.S. NATIONSCAPE

• Nationscape measurements show lower partisan separation around COVID-19’s onset and an increase after George Floyd’s murder; the border wall item contributes most in every month.The analysis uses survey-weighted joint distributions over a fixed 14-item policy response vector, grouped into 19 monthly bins.Source: §7.4 · pp. 9–10, Figure 4 · p. 10

8 CONCLUSIONS

• Zero Flux characterizes equality of discrete distributions through local probability fluxes under independent coupling.The paper presents it as a native discrete method.Source: §8 · p. 10
• The local-flux criterion is developed into a statistical discrepancy estimated through categorical posterior estimation.The resulting procedure is described as an effective sample-based discrepancy.Source: §8 · p. 10
• The paper establishes finite-sample guarantees and demonstrates recovery of sparse dependence changes and reliable comparison of high-dimensional categorical distributions.Source: §8 · p. 10

A BACKGROUND ON DISCRETE FLOW MATCHING

• DFM uses an independent endpoint coupling and a coordinate-wise linear bridge to construct a CTMC whose rates average endpoint-conditioned jumps and whose signed local edge fluxes describe probability transfer.At t = 0.5, Zero Flux estimates both endpoint posteriors and uses their difference to quantify local distributional change.Source: A BACKGROUND ON DISCRETE FLOW MATCHING · pp. 13–14

B CONTINUITY EQUATION FOR THE COORDINATE-WISE BRIDGE

• For the coordinate-wise bridge, the path marginal’s derivative is expressed through local fluxes, yielding a continuity equation that sums incoming and outgoing probability transfer.The flux-indicator identity supports the continuity-equation result.Source: B CONTINUITY EQUATION FOR THE… · pp. 16–17

C MIDPOINT EDGE ZERO FLUX CRITERION

• Under independent endpoint coupling, zero midpoint edge flux implies posterior coordinate symmetry, which in turn yields equality of product moments across every coordinate subset.The product-moment equivalence follows by induction from the posterior symmetry condition.Source: C MIDPOINT EDGE ZERO FLUX CRITERION · pp. 18, 21
• On the finite product space, agreement of all product moments implies P0 = P1.The proof reduces equality of distributions to agreement at every point of V^d.Source: C MIDPOINT EDGE ZERO FLUX CRITERION · p. 21

C.1 PROOF OF THEOREM 3.4

• Posterior coordinate symmetry implies agreement of all product moments, which in turn yields P0 = P1.The proof uses Lemmas C.1–C.3 to connect symmetry, moment equality, and distribution equality.Source: §C.1 · p. 21
• If P0 = P1, independent identically distributed endpoints and a swap-invariant midpoint bridge make the two directions cancel, completing the converse.The argument uses endpoint-swap invariance to establish the zero-flux direction.Source: §C.1 · p. 22

C.2 PROOF OF PROPOSITION 4.1

• Endpoint exchange preserves the midpoint law and swaps the endpoint posteriors, reversing flux signs while leaving their squared expectation unchanged.Consequently, the discrepancy is symmetric in the two endpoint distributions.Source: §C.2 · p. 22
• If the discrepancy is zero, all midpoint fluxes vanish and the endpoint distributions are identical; identical endpoints also give zero discrepancy.The proof establishes both directions of the separation claim.Source: §C.2 · p. 22
• With a non-exchangeable coupling, equal endpoint marginals can still yield discrepancy 2, so arbitrary coupling does not preserve separation.The proof’s example uses a uniform three-state endpoint distribution and a directed modulo-3 pairing.Source: §C.2 · p. 22

C.3 PROOF OF PROPOSITION 4.2

• Adding independent irrelevant coordinates gives zero discrepancy contributions on those coordinates while preserving the contributions on the signal set.The restricted midpoint bridge has the law of the bridge between the endpoint distributions restricted to the signal coordinates.Source: §C.3 · p. 23

D PROOFS FOR THE THEORETICAL ANALYSIS

• The theoretical construction independently couples endpoint vectors, then uses independent fair coordinate masks to form a shared midpoint state.For finite-sample construction, independent endpoint datasets are paired with replacement and each draw receives a fresh mask.Source: D PROOFS FOR THE THEORETICAL ANALYSIS · pp. 23–24

D.1 INEQUALITIES USED IN THE PROOF

• The proof bounds the plug-in discrepancy using posterior errors, then controls sampling fluctuations from endpoint pools and random bridge draws.The argument separates deterministic plug-in stability from concentration of the exact empirical discrepancy.Source: §D.1 · pp. 25–26
• One concentration event supports the guarantee simultaneously for every pair of categorical heads, including heads trained on the same samples.The event depends on the exact flux and sampling scheme, while the plug-in inequality is deterministic once samples and heads are fixed.Source: §D.1 · p. 27

E EXPERIMENTAL IMPLEMENTATION DETAILS

• Endpoint observations are sampled independently with replacement, midpoint coordinates copy either endpoint with probability one half, and evaluation uses fresh pairings and masks.The cross-entropy objective is averaged over coordinates.Source: E EXPERIMENTAL IMPLEMENTATION DETAILS · p. 27

E.1 SPARSE FIELD RECOVERY

• The sparse-field experiment varies dimension from 8 to 256, adding independent Bernoulli(1/2) coordinates beyond the first three signal coordinates.Recovery is assessed using sign and cosine similarity, plus amplitude; reported values are means and one standard deviation across three seeds.Source: §E.1 · pp. 28, 30
• The cube visualization averages edge values over the remaining coordinates; its vertices denote signal states and its edges change one signal coordinate.Table 3 reports mean ± sd over three seeds; sign and cosine are higher-is-better, while amplitude is closer-to-1-is-better, and budgets vary by dimension.Source: Table 3 · p. 28, §E.1 · p. 30

E.2 CONTAMINATION EXPERIMENTS

• Contamination detection compares reference and contaminated binary Markov chains, HMMs, and binarized MNIST using AUROC and FPR@95% TPR across 50 runs per contamination level.The table reports mean metrics over contamination levels 0.02, 0.05, 0.1, and 0.2, with 50 seeds and 1,000 observations; the supplied passages do not give the metric values.Source: Table 4 · p. 31, §E.2 · p. 31, Figure 7 · p. 32

E.3 UCI CATEGORICAL JOINT-DEPENDENCE BENCHMARK

• The UCI benchmark progressively separates feature blocks by independent row permutations, preserving univariate marginals and within-block joint distributions until all columns are independently rematched.Severity is the fraction of feature pairs assigned to different blocks; evaluation measures trajectory ordering, linear association with severity, and successive downward violations.Source: §E.3 · pp. 35–36

E.4 NATIONSCAPE PARTISAN DISCREPANCY TRAJECTORY

• Nationscape analysis pools weighted Democratic and Republican respondents into 19 monthly samples from July 2019 through January 2021, with median retained counts of 2,774 and 2,416, respectively.Method trajectories are standardized separately across months for shape comparison.Source: §E.4 · pp. 36–37

F.1 INDUCTIVE GEOMETRY

• Zero Flux follows its population discrepancy through interaction order 4, while coordinate-wise Zero Flow declines more gradually and full-state Zero Flow stays nearly constant but underestimates it.At interaction order 5, Zero Flux shows greater relative variability; overall, it is sensitive to higher-order differences with stronger relative emphasis on lower-order interactions than coordinate-wise Zero Flow.Source: §F.1 · p. 38

F.1.1 IMPLEMENTATION DETAILS

• For parity experiments, each endpoint pool contains 500 observations; methods share pools, and each score averages 100,000 midpoint evaluations per interaction order and seed.Zero Flux and coordinate-wise Zero Flow use a Transformer, while full-state Zero Flow uses a one-hot state encoding and an MLP; the reported comparison subtracts null from alternative scores.Source: §F.1.1 · pp. 38–39

F.2 TRACING PARTISAN DISTRIBUTIONAL SEPARATION OVER TIME IN U.S. NATIONSCAPE

• In January 2021, Zero Flux rises sharply while mixture MMD and median MMD decline in the standardized Nationscape trajectories.The Zero Flux increase is described as qualitatively more consistent with deepening partisan divisions around the 2020 election and its aftermath.Source: §F.1.1 · p. 39, Figure 10 · p. 40
• The border wall contributes most in every month; the January rise spans government insurance, college, and gun restrictions, while the wall contribution changes little.The policy-item decomposition conditions posterior comparisons on the complete midpoint state.Source: §F.1.1 · p. 39, Figure 11 · p. 41