用概率电路定义最优传输距离，可高效计算并优化模型参数。
Optimal Transport for Probabilistic Circuits
- 将最优传输的耦合测度限制为概率电路，构建新距离度量。
- 通过求解一系列小规模线性规划，实现距离的高效计算。
- 可直接获取最优传输方案，适合需要精准分布匹配的场景。
我们提出一种针对概率电路（PCs）的新最优传输框架。尽管近期已证明某些类别的概率电路间分布的散度可高效计算，但目前尚无方法计算由概率电路表示的概率分布之间的Wasserstein距离。本文提出一种受限于概率电路的Wasserstein型距离，并设计算法通过求解一系列小型线性规划来计算该距离，同时推导出保证计算可行的电路条件。此外，我们证明可从线性规划解中轻松恢复两个概率电路间的最优传输计划。最后，研究了概率电路与数据集间的经验Wasserstein距离，并提出一种高效的迭代算法以优化电路参数来最小化该距离。
原文摘要 · Abstract (English)
We introduce a novel optimal transport framework for probabilistic circuits (PCs). While it has been shown recently that divergences between distributions represented as certain classes of PCs can be computed tractably, to the best of our knowledge, there is no existing approach to compute the Wasserstein distance between probability distributions given by PCs. We propose a Wasserstein-type distance that restricts the coupling measure of the associated optimal transport problem to be a probabilistic circuit. We then develop an algorithm for computing this distance by solving a series of small linear programs and derive the circuit conditions under which this is tractable. Furthermore, we show that we can easily retrieve the optimal transport plan between the PCs from the solutions to these linear programs. Lastly, we study the empirical Wasserstein distance between a PC and a dataset, and show that we can estimate the PC parameters to minimize this distance through an efficient iterative algorithm.
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