Schrodinger Bridge

Schrödinger Bridge Problem

Let (R) be a reference path measure on continuous trajectories (C([0,1]; \mathbb{R}^d)), corresponding to the diffusion process [ dZ_t = b(Z_t,t)\,dt + dB_t, \qquad Z_0 \sim r_0, ] where (B_t) is standard Brownian motion and (r_0) is the initial distribution of the reference process.

Given two probability distributions (p_0) and (p_1) on (\mathbb{R}^d), the Schrödinger Bridge problem consists of finding a path measure (P) such that [ P_0 = p_0, \qquad P_1 = p_1, ] and minimizing the Kullback–Leibler divergence with respect to the reference measure (R): [ \min_{P:\,P_0=p_0,\;P_1=p_1} \mathrm{KL}(P \,|\, R). ]

When the reference dynamics is Brownian motion (i.e. (b \equiv 0)), the optimal solution (P^\star) is absolutely continuous with respect to (R) and corresponds to a controlled diffusion of the form [ dZ_t = u^\star(Z_t,t)\,dt + dB_t, ] where the control drift (u^\star) is chosen to match the prescribed initial and terminal distributions (p_0) and (p_1), while remaining closest to the reference dynamics in the relative entropy sense.

Questions:

  1. Should the initial distribution for the reference dynamics be $p_0$ as well?

  2. If i look at the forward process of diffusion models (OU process, VPSDE, VESDE), then I think the SB will be the one that reaches to $p_noise$ at terminal time T, and then if i look at the reverse process, then SB will be the one that reaches $p_data$ at terminal time 0 and then we can do this recursively. Alg :

  3. Take the forward process : dx_t = -x_t dt + dB_t
  4. learn a SB to P_T = noise at time = 1
  5. let that process be dx_t = (-x_t + b)dt + dB_t
  6. then take the reverse process dx_t = -x_t + \sigma \sigma^T \nabla \log p(x_t)) +dB_t and then learn SB to go to p_{data}
  7. and then rinse repeat?