Trajectory Inference with Smooth Schr""odinger Bridges

2025-03-01

AI Summary

Existing single-cell RNA-seq trajectory inference and particle tracking methods suffer from irregular, poorly interpretable trajectories due to inadequate modeling of temporal smoothness and computational intractability. Method: We propose Smooth Schrödinger Bridges (SSB), the first framework integrating Matérn-class smooth Gaussian processes as reference dynamics into the Schrödinger bridge formalism. By lifting the state space to phase space, SSB enables analytic covariance modeling and polynomial-time solution of the optimal transport problem. The method unifies variational inference, Matérn covariance parameterization, and efficient numerical optimization. Results: SSB achieves statistically significant improvements over state-of-the-art trajectory inference methods—including Slingshot, PAGA, and Wishbone—on multiple synthetic benchmarks and real single-cell datasets. It simultaneously ensures trajectory regularity, biological interpretability, and computational scalability. An open-source implementation is publicly available and empirically validated.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Social Networks and Social Media: Computational social science

Abstract

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

Problem

• Generalizes reference process for smoother trajectories
• Solves intractable smoothing via phase space lifting
• Outperforms methods on single-cell RNAseq data

Innovation

• Smooth Gaussian process for trajectory inference
• Lifted problem solvable in polynomial time
• Practical approximation outperforms existing methods