Trajectory Inference with Smooth Schr\"odinger Bridges

Motivated by applications in trajectory inference and particle tracking, we introduce Smooth Schr\"odinger Bridges. Our proposal generalizes prior work by allowing the reference process in the Schr\"odinger Bridge problem to be a smooth Gaussian process, leading to more regular and interpretable trajectories in applications. Though na\"ively smoothing the reference process leads to a computationally intractable problem, we identify a class of processes (including the Mat\'ern processes) for which the resulting Smooth Schr\"odinger Bridge problem can be lifted to a simpler problem on phase space, which can be solved in polynomial time. We develop a practical approximation of this algorithm that outperforms existing methods on numerous simulated and real single-cell RNAseq datasets. The code can be found at https://github.com/WanliHongC/Smooth_SB