Sampling Using Diffusion Processes, from Langevin to Schrödinger
summary
Derives an explicit time‑dependent drift that exactly transports Brownian noise to a target law at t=1 by minimizing path‑space KL (the Schrodinger bridge): the drift equals −∇ log E[ f(W1) | Wt=x ] so the SDE produces a generalized Brownian bridge whose path measure is the KL minimizer. Proof uses Girsanov theorem, Ito calculus, the HJB PDE, the Cole‑Hopf transform and Feynman‑Kac. It works in continuous time; discretization and efficient computation remain open.
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chapters (16)
- 0:00Start
- 1:35Logistical Remarks
- 2:30Additional Assumptions
- 5:29Deterministic Initial Condition
- 6:37Schroedinger Bridge Problem
- 11:15Static Formulation of the Schrodinger Bridge Problem
- 13:06The Chain Rule
- 14:41The Brownian Bridge
- 19:40Generalized Brownian Bridge Processes
- 24:22The Gersana Theorem
- 28:39The Stochastic Integral
- 47:47Conditional Distribution
- 51:31Optimal Control Problem
- 59:00Verification Theorem
- 1:05:38The Schrodinger Half Bridge
- 1:09:45Motivations
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