统一解释扩散模型与流匹配的几何本质，揭示其路径差异。
The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space
- 用Wasserstein空间几何统一两类生成模型的原理
- 扩散模型走自由能梯度下降路径，流匹配走最优传输测地线
- 相同起点终点，路径不同，流匹配采样更少步数
概率测度空间 $\mathcal{P}_2(\mathbb{R}^d)$ 在二次Wasserstein距离 $W_2$ 下构成完备度量空间，并根据Otto理论成为（形式）黎曼流形，其测地线为最优传输插值。在此流形上，自由能 $F(\rho) = \text{KL}(\rho || \pi)$ 的梯度流即为Fokker-Planck方程，其隐式欧拉离散化为JKO方案。这正是扩散模型的几何基础：前向过程沿自由能下降，每一步去噪即一次JKO步骤，从而统一了DDPM、DDIM、NCSN/SMLD和能量匹配。另一变分原理下的测地线——由Benamou-Brenier公式定义的最小作用路径——恰好是流匹配所学习的最优传输路径。固定两端点沿测地线生成，可转化为确定性常微分方程，采样步数大幅减少。将两类模型置于同一流形，明确揭示其关系：扩散模型为初值问题，沿梯度流；流匹配为边值问题，沿测地线；二者终点相同，路径不同。
原文摘要 · Abstract (English)
The space $\mathcal{P}_2(\mathbb{R}^d$) of probability measures with finite second moment carries a natural geometry: the quadratic Wasserstein distance W_2 makes it a complete metric space and, following Otto, a (formal) Riemannian manifold whose geodesics are the optimal-transport interpolations. On this manifold, the gradient flow of the free energy F(rho) = KL(rho || π) is exactly the Fokker-Planck equation, and its implicit-Euler discretization is the JKO scheme. This is the geometry underlying diffusion models: the forward process descends the free energy, and each denoising step realizes one JKO step, which recovers DDPM, DDIM, NCSN/SMLD, and Energy Matching; this is one scheme, not separate theories. The same manifold supports a second variational principle. Its geodesics - the minimum-action curves of the Benamou-Brenier formula - are precisely the optimal-transport paths that Flow Matching learns. Fixing both endpoints and following the geodesic, generation becomes a deterministic ODE along a straight line, hence far fewer sampling steps. Placing both families of models on one manifold makes their relationship exact: diffusion follows a free-energy gradient flow, an initial-value problem; optimal-transport Flow Matching follows a Wasserstein geodesic, a boundary-value problem. The two reach the same endpoints along different paths.
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