In optimal transport (OT), a Monge map is known as a mapping that transports a source distribution to a target distribution in the most cost-efficient way. Recently, multiple neural estimators for Monge maps have been developed and applied in diverse unpaired domain translation tasks, e.g. in single-cell biology and computer vision. However, the classic OT framework enforces mass conservation, which makes it prone to outliers and limits its applicability in real-world scenarios. The latter can be particularly harmful in OT domain translation tasks, where the relative position of a sample within a distribution is explicitly taken into account. While unbalanced OT tackles this challenge in the discrete setting, its integration into neural Monge map estimators has received limited attention. We propose a theoretically grounded method to incorporate unbalancedness into any Monge map estimator. We improve existing estimators to model cell trajectories over time and to predict cellular responses to perturbations. Moreover, our approach seamlessly integrates with the OT flow matching (OT-FM) framework. While we show that OT-FM performs competitively in image translation, we further improve performance by incorporating unbalancedness (UOT-FM), which better preserves relevant features. We hence establish UOT-FM as a principled method for unpaired image translation.

SUMMARY: Spatial omics technologies are increasingly leveraged to characterize how disease disrupts tissue organization and cellular niches. While multiple methods to analyze spatial variation within a sample have been published, statistical and computational approaches to compare cell spatial organization across samples or conditions are mostly lacking. We present GraphCompass, a comprehensive set of omics-adapted graph analysis methods to quantitatively evaluate and compare the spatial arrangement of cells in samples representing diverse biological conditions. GraphCompass builds upon the Squidpy spatial omics toolbox and encompasses various statistical approaches to perform cross-condition analyses at the level of individual cell types, niches, and samples. Additionally, GraphCompass provides custom visualization functions that enable effective communication of results. We demonstrate how GraphCompass can be used to address key biological questions, such as how cellular organization and tissue architecture differ across various disease states and which spatial patterns correlate with a given pathological condition. GraphCompass can be applied to various popular omics techniques, including, but not limited to, spatial proteomics (e.g. MIBI-TOF), spot-based transcriptomics (e.g. 10× Genomics Visium), and single-cell resolved transcriptomics (e.g. Stereo-seq). In this work, we showcase the capabilities of GraphCompass through its application to three different studies that may also serve as benchmark datasets for further method development. With its easy-to-use implementation, extensive documentation, and comprehensive tutorials, GraphCompass is accessible to biologists with varying levels of computational expertise. By facilitating comparative analyses of cell spatial organization, GraphCompass promises to be a valuable asset in advancing our understanding of tissue function in health and disease. .

The relevance of optimal transport methods to machine learning has long been hindered by two salient limitations. First, the $O(n^3)$ computational cost of standard sample-based solvers (when used on batches of $n$ samples) is prohibitive. Second, the mass conservation constraint makes OT solvers too rigid in practice: because they must match \textit{all} points from both measures, their output can be heavily influenced by outliers. A flurry of recent works in OT has addressed these computational and modelling limitations, but has resulted in two separate strains of methods: While the computational outlook was much improved by entropic regularization, more recent $O(n)$ linear-time \textit{low-rank} solvers hold the promise to scale up OT further. On the other hand, modelling rigidities have been eased owing to unbalanced variants of OT, that rely on penalization terms to promote, rather than impose, mass conservation. The goal of this paper is to merge these two strains, to achieve the promise of \textit{both} versatile/scalable unbalanced/low-rank OT solvers. We propose custom algorithms to implement these extensions for the linear OT problem and its Fused-Gromov-Wasserstein generalization, and demonstrate their practical relevance to challenging spatial transcriptomics matching problems.