Théo Uscidda

Optimal transport (OT) theory has reshaped the field of generative modeling: Combined with neural networks, recent \textitNeural OT (N-OT) solvers use OT as an inductive bias, to focus on “thrifty” mappings that minimize average displacement costs. This core principle has fueled the successful application of N-OT solvers to high-stakes scientific challenges, notably single-cell genomics. N-OT solvers are, however, increasingly confronted with practical challenges: while most N-OT solvers can handle squared-Euclidean costs, they must be repurposed to handle more general costs; their reliance on deterministic Monge maps as well as mass conservation constraints can easily go awry in the presence of outliers; mapping points \textitacross heterogeneous spaces is out of their reach. While each of these challenges has been explored independently, we propose a new framework that can handle, natively, all of these needs. The \textitgenerative entropic neural OT (GENOT) framework models the conditional distribution \pi_\varepsilon(\*y|\*x) of an optimal \textitentropic coupling \pi_\varepsilon, using conditional flow matching. GENOT is generative, and can transport points \textitacross spaces, guided by sample-based, unbalanced solutions to the Gromov-Wasserstein problem, that can use any cost. We showcase our approach on both synthetic and single-cell datasets, using GENOT to model cell development, predict cellular responses, and translate between data modalities.

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.

Optimal transport (OT) theory has been been used in machine learning to study and characterize maps that can push-forward efficiently a probability measure onto another. Recent works have drawn inspiration from Brenier’s theorem, which states that when the ground cost is the squared-Euclidean distance, the “best” map to morph a continuous measure in \mathcalP(\Rd) into another must be the gradient of a convex function. To exploit that result, [Makkuva+ 2020, Korotin+2020] consider maps T=∇f_θ, where f_θ is an input convex neural network (ICNN), as defined by Amos+2017, and fit θ with SGD using samples. Despite their mathematical elegance, fitting OT maps with ICNNs raises many challenges, due notably to the many constraints imposed on θ; the need to approximate the conjugate of f_θ; or the limitation that they only work for the squared-Euclidean cost. More generally, we question the relevance of using Brenier’s result, which only applies to densities, to constrain the architecture of candidate maps fitted on samples. Motivated by these limitations, we propose a radically different approach to estimating OT maps: Given a cost c and a reference measure ρ, we introduce a regularizer, the Monge gap M_ρ^c(T) of a map T. That gap quantifies how far a map T deviates from the ideal properties we expect from a c-OT map. In practice, we drop all architecture requirements for T and simply minimize a distance (e.g., the Sinkhorn divergence) between T♯μ and ν, regularized by M_ρ^c. We study M_ρ^c, and show how our simple pipeline outperforms significantly other baselines in practice.