Problem
Research questions and friction points this paper is trying to address.
high-dimensional discrete distributions
distribution comparison
flow matching
probability flux
discrepancy
Zero Flux: Flow-Based Comparison of High-Dimensional Discrete Distributions
🤖 AI Summary
This study addresses the challenge of comparing high-dimensional discrete distributions, where state spaces grow exponentially and existing flow matching methods are not directly applicable to discrete domains. To overcome this, the work extends continuous flow matching principles to discrete spaces for the first time by introducing a zero-flux criterion based on local probability flux. Specifically, it establishes a distribution equivalence test by nullifying midpoint flux under independent coupling, decomposing joint distribution discrepancies into local contributions to enable efficient estimation with finite-sample error bounds. Experimental results demonstrate that the proposed method reliably recovers sparse dependency signals and robustly tracks distribution shifts in high-dimensional settings, offering a novel paradigm for discrete distribution comparison.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
High-dimensional discrete distributions
Flow matching
Zero Flux criterion
Local probability fluxes
Distribution comparison
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