研究WFR梯度流分裂算法的顺序影响，发现合理选择可更快收敛到目标分布。
An operator splitting analysis of Wasserstein--Fisher--Rao gradient flows
- 通过算子分裂法分步求解Wasserstein与Fisher-Rao流
- 特定步长与顺序下，分裂方案比精确解收敛更快
- 首次获得WFR流的对数凹性保持与精确衰减界
Wasserstein-Fisher-Rao（WFR）梯度流作为一种强大的采样工具，结合了纯Wasserstein（W）和纯Fisher-Rao（FR）梯度流的优势。现有算法隐式采用算子分裂技术数值逼近WFR偏微分方程，即先计算一步W流，再执行FR流（或反之）。本文研究算子评估顺序对结果的影响，并提供定量分析。令人意外的是，通过合理选择步长与顺序，分裂方案可在模型时间上快于精确的WFR流。本文推导出两种分裂方案在单步时间内的变分公式，并分析何时应优先采用W-FR分裂而非FR-W分裂。作为重要进展，本文证明了WFR梯度流保持对数凹性，并首次获得其精确衰减速率边界。
原文摘要 · Abstract (English)
Wasserstein-Fisher-Rao (WFR) gradient flows have been recently proposed as a powerful sampling tool that combines the advantages of pure Wasserstein (W) and pure Fisher-Rao (FR) gradient flows. Existing algorithmic developments implicitly make use of operator splitting techniques to numerically approximate the WFR partial differential equation, whereby the W flow is evaluated over a given step size and then the FR flow (or vice versa). This works investigates the impact of the order in which the W and FR operator are evaluated and aims to provide a quantitative analysis. Somewhat surprisingly, we show that with a judicious choice of step size and operator ordering, the split scheme can converge to the target distribution faster than the exact WFR flow (in terms of model time). We obtain variational formulae describing the evolution over one time step of both splitting schemes and investigate in which settings the W-FR split should be preferred to the FR-W split. As a step towards this goal we show that the WFR gradient flow preserves log-concavity and obtain the first sharp decay bound for WFR flow.
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