On the Wasserstein Gradient Flow Interpretation of Drifting Models

TL;DR

This paper analyzes the Generative Modeling via Drifting (GMD) framework through Wasserstein Gradient Flows, revealing its connection to fixed points of various optimal transport-based flows and comparing different algorithms.

Contribution

It provides a novel interpretation of GMD as targeting fixed points of Wasserstein Gradient Flows, extending analysis to multiple divergences and algorithms.

Findings

One algorithm corresponds to the limiting point of a WGF on KL divergence with Parzen smoothing.

The implemented algorithm resembles a fixed point of a WGF on Sinkhorn divergence but lacks some properties.

The approach extends to other WGFs like MMD, sliced Wasserstein, and GAN critic functions.

Abstract

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.