Question regarding entropic OT

[Luciennnnnnn]

Hi, I have read the paper "Simulation-free Schr"odinger bridges via score and flow matching", interesting work and very nice presentation!

After read through it, I have some questions.

According to Proposition 2.1, the solution to the Schrödinger bridge problem is derived from the entropic OT plan. However, in Appendices A.1 and D.2, you mention that the entropic OT plan does not perform as well as exact optimal transport in settings with smaller batch sizes and higher dimensions. According to Algorithm 4 in the appendix, the sigma in the entropic OT plan is the same as the sigma in p(x|x_0, x_1). When we use exact optimal transport, does this mean that the sigma in p(x|x_0, x_1) needs to be set to 0? (this will ruin the stochasticity however). Additionally, what is the motivation or practical guidance for setting sigma?

Furthermore, I noticed that you always use larger batch sizes in your experiments. However, I am more concerned with high-dimensional image tasks, such as unpaired super-resolution, where we typically use larger networks and relatively smaller batch sizes like (16, 32 or 64). Can we expect SF2M to perform well under these settings? If minibatch OT does not perform well, and we degrade to the independent coupling of p(x_0) and p(x_1), does SF2M imply very slow convergence and potential poor performance compared to CycleGAN? CycleGAN seems to ensure more reliable convergence due to its explicit consistency loss.

Any replies or answers are appreciated!

Best regards,
Xin