stochastic interpolant

A stochastic interpolant is a continuous-time stochastic process $x_t=I(t,x_0,x_1)+\gamma (t)z$ that bridges between two arbitrary probability densities ${\rho }_0$ and ${\rho }_1$ in finite time $t\in [0,1]$. Here $I:[0,1]\times {\mathbb{R}}^d\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ is a smooth interpolant function satisfying $I(0,x_0,x_1)=x_0$ and $I(1,x_0,x_1)=x_1$; $\gamma :[0,1]\to {\mathbb{R}}_{\ge 0}$ controls the latent noise level with $\gamma (0)=\gamma (1)=0$ and $\gamma (t)>0$ for $t\in (0,1)$; $(x_0,x_1)\sim \nu$ is drawn from a coupling with marginals ${\rho }_0,{\rho }_1$; and $z\sim \mathcal{N}(0,I_d)$ is an independent Gaussian.

The framework introduced by Albergo, Boffi, and Vanden-Eijnden (2023) proves that the density $\rho (t)$ of $x_t$ is absolutely continuous, strictly positive, and satisfies both a transport equation ${\partial }_t\rho +\nabla \cdot (b\rho )=0$ and forward/backward Fokker-Planck equations with any diffusion coefficient $ϵ(t)\ge 0$. The velocity $b(t,x)=\mathbb{E}[{\dot{x}}_t|x_t=x]$ and score $s(t,x)=\nabla \log \rho (t,x)=-{\gamma }^{-1}(t)\mathbb{E}[z|x_t=x]$ are learned via simple quadratic objectives.

The connection to the Lévy random bridge framework of Shelley & Mengütürk is direct: both construct processes bridging between distributions, but stochastic interpolants add a third ingredient — the latent noise $\gamma (t)z$ — that is absent in the random bridge construction. Setting $\gamma =0$ recovers rectified flow (linear interpolant without noise). The key design space is the triple $(\alpha ,\beta ,\gamma )$ for the spatially linear case $x_t=\alpha (t)x_0+\beta (t)x_1+\gamma (t)z$, which includes score-based diffusion ($\alpha =\sqrt{1-t^2}$, $\beta =t$, $\gamma =0$ one-sided) and the variance-preserving constraint ${\alpha }^2+{\beta }^2+{\gamma }^2=1$ as special cases.

Key Details

• Velocity: $b(t,x)=\mathbb{E}[{\partial }_{t}I(t,x_0,x_1)+\dot{\gamma }(t)z|x_t=x]$, learned via quadratic loss ${\mathcal{L}}_b$
• Score: $s(t,x)=-{\gamma }^{-1}(t)\mathbb{E}[z|x_t=x]$, learned via quadratic loss ${\mathcal{L}}_s$
• Denoiser: ${\eta }_z(t,x)=\mathbb{E}[z|x_t=x]$, related to score by $s=-{\gamma }^{-1}{\eta }_z$
• Three generative models from one interpolant: ODE ($\dot{X}=b$), forward SDE ($dX=b_{F}dt+\sqrt{2ϵ}dW$), backward SDE — all sharing the same $\rho (t)$
• $\gamma (t)$ smooths: eliminates spurious intermediate modes present in pure linear interpolation
• Schrödinger bridge recovery: optimizing over $I$ in the Benamou-Brenier hydrodynamic formulation recovers the SB (Theorem 41)
• The coupling $\nu$ can be independent (${\rho }_0\otimes {\rho }_1$), OT-coupled, or data-adapted

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concept