conditional probability path

A conditional probability path $p_t(x|x_1)$ is a time-dependent probability density function conditioned on a data sample $x_1$, satisfying $p_0(x|x_1)=p(x)$ (a simple prior, e.g., $\mathcal{N}(0,I)$) and $p_1(x|x_1)\approx {\delta }_{x_1}$ (concentrated at the data point). Marginalizing over data gives the marginal probability path $p_t(x)=\int p_t(x|x_1)q(x_1)d{x}_1$, which interpolates between the prior and the data distribution.

This construction, introduced in Flow Matching for Generative Modeling, is the key to making flow matching tractable: rather than constructing the intractable marginal path directly, one specifies simple per-sample conditional paths and aggregates. The family of Gaussian conditional paths $p_t(x|x_1)=\mathcal{N}(x|{\mu }_t(x_1),{\sigma }_t(x_1{)}^{2}I)$ encompasses diffusion paths (VE: ${\mu }_t=x_1$, ${\sigma }_t={\sigma }_{1-t}$; VP: ${\mu }_t={\alpha }_{1-t}{x}_1$, ${\sigma }_t=\sqrt{1-{\alpha }_{1-t}^2}$) and OT paths (${\mu }_t=t{x}_1$, ${\sigma }_t=1-(1-{\sigma }_{\min })t$) as special cases.

The connection to stochastic interpolants: the conditional path construction is equivalent to specifying the interpolant $x_t=I(t,x_0,x_1)+\gamma (t)z$. The one-sided interpolant with $\gamma (t)=0$ gives CFM without latent noise; the two-sided interpolant generalizes to arbitrary couplings between ${\rho }_0$ and ${\rho }_1$.

Key Details

• Gaussian conditional paths: $p_t(x|x_1)=\mathcal{N}(x|{\mu }_t(x_1),{\sigma }_t^{2}I)$ with boundary conditions ${\mu }_0=0,{\sigma }_0=1,{\mu }_1=x_1,{\sigma }_1={\sigma }_{\min }$
• OT paths produce straight trajectories with constant-direction VFs
• Diffusion paths produce curved trajectories that overshoot and backtrack
• The conditional VF generating a Gaussian path is $u_t(x|x_1)=\frac{{\sigma }_t^{\prime }}{{\sigma }_t}(x-{\mu }_t)+{\mu }_t^{\prime }$

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