提出新型粒子采样方法，高效处理高维分布推断。
Radon--Wasserstein Gradient Flows for Interacting-Particle Sampling in High Dimensions
- 基于径向-沃尔什几何设计梯度流，利用一维投影降低计算复杂度。
- 每步计算成本在粒子数和维度上均为线性，支持高维应用。
- 适合高维概率分布采样，尤其适用于需要快速收敛的场景。
KL散度的梯度流（如福克-普朗克方程和斯坦变分梯度下降）可将分布演化至目标密度，该密度仅知归一化常数。本文提出新的KL散度梯度流，具备显著优势：在高维下可实现精确的相互作用粒子近似，且每步计算成本在粒子数与维度上均呈线性增长。该方法基于新定义的运输型黎曼几何——径向-沃尔什几何（Radon--Wasserstein geometry）及其正则化版本（RRW geometry），通过径向变换使梯度流速度仅依赖于一维投影，从而借助快速傅里叶变换高效计算所需的一维卷积。我们还提供了数值实验，评估算法性能并比较收敛行为与量化效果。理论方面，证明了流的适定性及RRW流的长期收敛性。
原文摘要 · Abstract (English)
Gradient flows of the Kullback--Leibler (KL) divergence, such as the Fokker--Planck equation and Stein Variational Gradient Descent, evolve a distribution toward a target density known only up to a normalizing constant. We introduce new gradient flows of the KL divergence with a remarkable combination of properties: they admit accurate interacting-particle approximations in high dimensions, and the per-step cost scales linearly in both the number of particles and the dimension. These gradient flows are based on new transportation-based Riemannian geometries on the space of probability measures: the Radon--Wasserstein geometry and the related Regularized Radon--Wasserstein (RRW) geometry. We define these geometries using the Radon transform so that the gradient-flow velocities depend only on one-dimensional projections. This yields interacting-particle-based algorithms whose per-step cost follows from efficient Fast Fourier Transform-based evaluation of the required 1D convolutions. We additionally provide numerical experiments that study the performance of the proposed algorithms and compare convergence behavior and quantization. Finally, we prove some theoretical results including well-posedness of the flows and long-time convergence guarantees for the RRW flow.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。