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.

Geometry of the Space of Partitioned Networks: A Unified Theoretical and Computational Framework

Interactions and relations between objects may be pairwise or higher-order in nature, and so network-valued data are ubiquitous in the real world. The "space of networks", however, has a complex structure that cannot be adequately described using conventional statistical tools. We introduce a measure-theoretic formalism for modeling generalized network structures such as graphs, hypergraphs, or graphs whose nodes come with a partition into categorical classes. We then propose a metric that extends the Gromov-Wasserstein distance between graphs and the co-optimal transport distance between hypergraphs. We characterize the geometry of this space, thereby providing a unified theoretical treatment of generalized networks that encompasses the cases of pairwise, as well as higher-order, relations. In particular, we show that our metric is an Alexandrov space of non-negative curvature, and leverage this structure to define gradients for certain functionals commonly arising in geometric data analysis tasks. We extend our analysis to the setting where vertices have additional label information, and derive efficient computational schemes to use in practice. Equipped with these theoretical and computational tools, we demonstrate the utility of our framework in a suite of applications, including hypergraph alignment, clustering and dictionary learning from ensemble data, multi-omics alignment, as well as multiscale network alignment.