将漂移生成模型解析为最优传输梯度流，揭示其收敛本质
On the Wasserstein Gradient Flow Interpretation of Drifting Models
- 用Wasserstein梯度流视角分析漂移生成模型的优化路径
- 发现算法实际对应的是Sinkhorn散度固定点，但不具理想性质
- 该思想可推广至MMD、切片Wasserstein等其他散度的固定点
近期Deng等人（2026）提出了基于漂移的生成建模（GMD），一种新型生成任务框架。本文从Wasserstein梯度流（WGF）角度分析GMD，即在概率测度空间中以最优传输几何为基底的功能泛函最速下降路径。与以往WGF研究不同，GMD可视为直接逼近特定WGF流的固定点。我们证明了三个主要结果：第一，Deng等人（2026）提出的某算法对应于在KL散度上施加Parzen平滑时的WGF极限点；第二，Deng等人实际实现的算法虽与基于Sinkhorn散度的WGF固定点相似，但缺乏后者的某些理想性质；第三，相同思路可扩展至其他WGF的极限点，包括最大均值差异（MMD）、切片Wasser斯坦距离及GAN判别器函数。
原文摘要 · Abstract (English)
Recently, Deng et al. (2026) proposed Generative Modeling via Drifting (GMD), a novel framework for generative tasks. This note presents an analysis of GMD through the lens of Wasserstein Gradient Flows (WGF), i.e., the path of steepest descent for a functional in the space of probability measures, equipped with the geometry of optimal transport. Unlike previous WGF-based contributions, GMD can be thought of as directly targeting a fixed point of a specific WGF flow. We demonstrate three main results: first, that one algorithm proposed by Deng et al. (2026) corresponds to finding the limiting point of a WGF on the KL divergence, with Parzen smoothing on the densities. Second, that the algorithm actually implemented by Deng et al. (2026) corresponds to a different procedure, which bears some resemblance to the fixed point of a WGF on the Sinkhorn divergence, but lacks certain desirable properties of the latter. Third, the same same idea can be extended to the limiting point of other WGFs, including the Maximum Mean Discrepancy (MMD), the sliced Wasserstein distance, and GAN critic functions.
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