Speaker
Description
We consider the Schrödinger bridge problem (SBP) which, given ensemble measurements of the initial and final configurations of a stochastic dynamical system and some prior knowledge on the dynamics, aims to reconstruct the “most likely” evolution of the system compatible with the data. Notably, this point of view has spurred a lot of activity in the single cell analysis community over the past few years.
Most existing literature assume Brownian reference dynamics, and are implicitly limited to modelling systems driven by the gradient of a potential energy.
We depart from this regime and consider reference processes described by a multivariate Ornstein-Uhlenbeck process with generic drift matrix $\mathbf{A}$.
When $\mathbf{A}$ is asymmetric, this corresponds to a system in which non-gradient forces are at play: this is important for applications to biological systems such as transcriptional dynamics in cells, which naturally exist out-of-equilibrium.
In the case of Gaussian marginals, we derive explicit expressions that characterise exactly the solution of both the static and dynamic Schrödinger bridge.
For general non-Gaussian marginals, we propose a simulation-free algorithm based on flow and score matching for learning a neural approximation to the Schrödinger bridge.
We demonstrate applications to a range of problems based on synthetic benchmarks and real single cell data. In particular, we highlight an iterative scheme for joint inference in the setting where both the SBP solution and $\mathbf{A}$ are unknown, and its potential applications to joint inference of dynamics and the underlying gene interaction networks.
