Wasserstein梯度流与仅前向扩散不足以实现多模态采样
Wasserstein Gradient Flows and Forward-Only Diffusion Are Not Enough for Multimodal Sampling
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中文总结 AI 辅助
本文指出基于Wasserstein梯度流和仅前向扩散的采样器因局部梯度驱动机制，在分离良好的多模态分布上存在指数级混合时间，揭示了其根本局限并呼吁发展非局部采样机制。
中文摘要 AI 辅助
基于Wasserstein梯度流（WGF）和仅前向扩散过程（FODP）的采样算法大量涌现，这些算法通常伴随有指数级快速收敛到目标分布的理论保证。这些保证常被解读为这类方法能够高效采样复杂多模态分布的证据，并常得到实证结果的支持。在本工作中，我们认为这种解读从根本上具有误导性。通过引入Jordan-Kinderlehrer-Otto（JKO）格式和Otto微积分，我们确立了规范的WGF采样动力学与过阻尼前向扩散具有相同的密度演化，因此继承了非平衡统计物理中长期理解的相同的亚稳定性和慢混合现象。我们使用两种互补工具——谱分析和平均首达时间（MFPT）分析——来分析这类采样器，并表明分离良好的多模态性可诱发与小的谱间隙和罕见的模态间跃迁相关的指数级长混合时间。对于本文研究的常用对数线性退火调度，我们发现引入中间分布并不能消除总输运时间的指数标度。这一局限性是结构性的而非实现特定的：纯局部的、梯度驱动的输运机制可能需要指数级长的时间才能将概率质量输运跨越分离良好的模态。我们认为这代表了WGF和FODP基础采样形式的一个根本局限，并激励未来开发根本非局部的机制以实现高效的多模态采样。
英文摘要
There has been a proliferation of sampling algorithms based on Wasserstein gradient flows (WGF) and forward-only diffusion processes (FODP), often accompanied by theoretical guarantees of exponentially fast convergence to the target distribution. These guarantees are frequently interpreted as evidence that such methods can efficiently sample complex multimodal distributions, often supported by empirical results. In this work, we argue that this interpretation is fundamentally misleading. By invoking the Jordan-Kinderlehrer-Otto (JKO) scheme and Otto calculus, we establish that the canonical WGF sampling dynamics and overdamped forward diffusion share the same density evolution and therefore inherit the same metastability and slow-mixing phenomena long understood in nonequilibrium statistical physics. We analyze this family of samplers using two complementary tools -- spectral analysis and mean first-passage time (MFPT) analysis -- and show that well-separated multimodality can induce exponentially long mixing times associated with small spectral gaps and rare inter-mode transitions. For the commonly adopted log-linear annealing schedule studied here, we find that introducing intermediate distributions does not remove the exponential scaling of the total transport time. The limitation is structural rather than implementation-specific: purely local, gradient-driven transport mechanisms can require exponentially long times to transport probability mass across well-separated modes. We argue that this represents a fundamental limitation of WGF- and FODP-based sampling in their standard forms, and motivates future development of fundamentally nonlocal mechanisms for efficient multimodal sampling.
发表机构
- Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
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