Learning non-equilibrium diffusions with Schrödinger bridges: from exactly solvable to simulation-free

We consider the Schr\"odinger bridge problem 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. Most existing literature assume Brownian reference dynamics and are implicitly limited to potential-driven dynamics. We depart from this regime and consider reference processes described by a multivariate Ornstein-Uhlenbeck process with generic drift matrix $\mathbf{A} \in \mathbb{R}^{d \times d}$. When $\mathbf{A}$ is asymmetric, this corresponds to a non-equilibrium system with non-conservative forces at play: this is important for applications to biological systems, which are naturally exist out-of-equilibrium. In the case of Gaussian marginals, we derive explicit expressions that characterise the solution of both the static and dynamic Schr\"odinger bridge. For general marginals, we propose mvOU-OTFM, a simulation-free algorithm based on flow and score matching for learning the Schr\"odinger bridge. In application to a range of problems based on synthetic and real single cell data, we demonstrate that mvOU-OTFM achieves higher accuracy compared to competing methods, whilst being significantly faster to train.

Non-negative matrix and tensor factorisations with a smoothed Wasserstein loss

Non-negative matrix and tensor factorisations are a classical tool in machine learning and data science for finding low-dimensional representations of high-dimensional datasets. In applications such as imaging, datasets can often be regarded as distributions in a space with metric structure. In such a setting, a Wasserstein loss function based on optimal transportation theory is a natural choice since it incorporates knowledge about the geometry of the underlying space. We introduce a general mathematical framework for computing non-negative factorisations of matrices and tensors with respect to an optimal transport loss, and derive an efficient method for its solution using a convex dual formulation. We demonstrate the applicability of this approach with several numerical examples.