conditional flow matching

Conditional Flow Matching (CFM) is a simulation-free training objective for Continuous Normalizing Flows introduced by Lipman et al. (2022). The key insight is that the intractable marginal Flow Matching loss — which requires knowledge of the marginal probability path $p_t(x)$ and its generating velocity field $u_t(x)$ — can be decomposed into tractable per-sample conditional objectives without changing the gradients.

Given conditional probability paths $p_t(x|x_1)$ with corresponding conditional velocity fields $u_t(x|x_1)$, the CFM objective is ${\mathcal{L}}_{\text{CFM}}(\theta )={\mathbb{E}}_{t\sim \mathcal{U}[0,1],x_1\sim q,x\sim p_t(\cdot |x_1)}\Vert v_{\theta }(x,t)-u_t(x|x_1){\Vert }^2$. Theorem 2 of Lipman et al. proves ${\nabla }_{\theta }{\mathcal{L}}_{\text{FM}}={\nabla }_{\theta }{\mathcal{L}}_{\text{CFM}}$, so optimizing CFM is equivalent to optimizing the intractable FM objective.

The relationship to other methods: CFM with OT conditional paths and ${\sigma }_{\min }\to 0$ reduces to the rectified flow objective $\mathbb{E}\Vert v_{\theta }(t{X}_1+(1-t)X_0,t)-(X_1-X_0){\Vert }^2$. CFM with diffusion conditional paths recovers denoising score matching. In the DSBM framework, CFM is the $\sigma =0$ (deterministic) limit of bridge matching.

Key Details

• Marginal VF $u_t(x)=\int u_t(x|x_1)\frac{p_t(x|x_1)q(x_1)}{p_t(x)}d{x}_1$ generates marginal path $p_t(x)$ (Theorem 1)
• CFM gradients equal FM gradients (Theorem 2) — enables simulation-free training
• For Gaussian paths: conditional VF has closed form $u_t(x|x_1)=\frac{{\sigma }_t^{\prime }}{{\sigma }_t}(x-{\mu }_t)+{\mu }_t^{\prime }$ (Theorem 3)
• With OT paths: $u_t(x|x_1)=(x_1-(1-{\sigma }_{\min })x)/(1-(1-{\sigma }_{\min })t)$ — constant direction in time

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