This page is under construction

The Schrödinger Bridge Problem

The Schrödinger Bridge Problem is a weak version of the Optimal Transport

Flow Matching and The Conditioning Trick

Flow Matching^[1] introduce the conditioning trick. The conditioning trick is first derived in the context of speeding up continuous flow model but its ability for allowing simulation-free training is later used in training Schrödinger Bridges^[3]. In short, the conditioning trick can be seen in the simplification of the flow-matching loss:

The Diffusion Process

The diffusion equation or also known as the Fokker-Planck equation is described by a hyperbolic PDE:

This diffusion process can also be described by the corresponding SDE:

The link between these two equation can be seen through the proof of the famous Feynmann-Kac’s formula.

References

[0] SB-FBSDE: Likelihood Training of Schrödinger Bridge using Forward-Backward SDEs Theory
[1] Flow Matching for Generative Modeling
[2] I²SB: Image-to-Image Schrödinger Bridge