This paper introduces the use of unbalanced optimal transport methods as a similarity measure for diffeomorphic matching of imaging data. The similarity measure is a key object in diffeomorphic registration methods that, together with the regularization on the deformation, defines the optimal deformation. Most often, these similarity measures are local or non local but simple enough to be computationally fast. We build on recent theoretical and numerical advances in optimal transport to propose fast and global similarity measures that can be used on surfaces or volumetric imaging data. This new similarity measure is computed using a fast generalized Sinkhorn algorithm. We apply this new metric in the LDDMM framework on synthetic and real data, fibres bundles and surfaces and show that better matching results are obtained.

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This paper extends the formulation of Sinkhorn divergences to the unbalanced setting of arbitrary positive measures, providing both theoretical and algorithmic advances. Sinkhorn divergences leverage the entropic regularization of Optimal Transport (OT) to define geometric loss functions. They are differentiable, cheap to compute and do not suffer from the curse of dimensionality, while maintaining the geometric properties of OT, in particular they metrize the weak * convergence. Extending these divergences to the unbalanced setting is of utmost importance since most applications in data sciences require to handle both transportation and creation/destruction of mass. This includes for instance problems as diverse as shape registration in medical imaging, density fitting in statistics, generative modeling in machine learning, and particles flows involving birth/death dynamics. Our first set of contributions is the definition and the theoretical analysis of the unbalanced Sinkhorn divergences. They enjoy the same properties as the balanced divergences (classical OT), which are obtained as a special case. Indeed, we show that they are convex, differentiable and metrize the weak * convergence. Our second set of contributions studies generalized Sinkkhorn iterations, which enable a fast, stable and massively parallelizable algorithm to compute these divergences. We show, under mild assumptions, a linear rate of convergence, independent of the number of samples, i.e. which can cope with arbitrary input measures. We also highlight the versatility of this method, which takes benefit from the latest advances in term of GPU computing, for instance through the KeOps library for fast and scalable kernel operations.

Proteins’ biological functions are defined by the geometric and chemical structure of their 3D molecular surfaces. Recent works have shown that geometric deep learning can be used on mesh-based representations of proteins to identify potential functional sites, such as binding targets for potential drugs. Unfortunately though, the use of meshes as the underlying representation for protein structure has multiple drawbacks including the need to pre-compute the input features and mesh connectivities. This becomes a bottleneck for many important tasks in protein science.In this paper, we present a new framework for deep learning on protein structures that addresses these limitations. Among the key advantages of our method are the computation and sampling of the molecular surface on-the-fly from the underlying atomic point cloud and a novel efficient geometric convolutional layer. As a result, we are able to process large collections of proteins in an end-to-end fashion, taking as the sole input the raw 3D coordinates and chemical types of their atoms, eliminating the need for any hand-crafted pre-computed features.To showcase the performance of our approach, we test it on two tasks in the field of protein structural bioinformatics: the identification of interaction sites and the prediction of protein-protein interactions. On both tasks, we achieve state-of-the-art performance with much faster run times and fewer parameters than previous models. These results will considerably ease the deployment of deep learning methods in protein science and open the door for end-to-end differentiable approaches in protein modeling tasks such as function prediction and design.

The data fidelity term is a key component of shape registration pipelines: computed at every step, its gradient is the vector field that drives a deformed model towards its target. Unfortunately, most classical formulas are at most semi-local: their gradients saturate and stop being informative above some given distance, with appalling consequences on the robustness of shape analysis pipelines. In this paper, we build on recent theoretical advances on Sinkhorn entropies and divergences [6] to present a unified view of three fidelities between measures that alleviate this problem: the Energy Distance from statistics; the (weighted) Hausdorff distance from computer graphics; the Wasserstein distance from Optimal Transport theory. The ε-Hausdorff and ε-Sinkhorn divergences are positive fidelities that interpolate between these three quantities, and we implement them through efficient, freely available GPU routines. They should allow the shape analyst to handle large deformations without hassle.

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We present a new multiscale algorithm for solving regularized Optimal Transport problems on the GPU, with a linear memory footprint. Relying on Sinkhorn divergences which are convex, smooth and positive definite loss functions, this method enables the computation of transport plans between millions of points in a matter of minutes. We show the effectiveness of this approach on brain tractograms modeled either as bundles of fibers or as track density maps. We use the resulting smooth assignments to perform label transfer for atlas-based segmentation of fiber tractograms. The parameters-blur and reachof our method are meaningful, defining the minimum and maximum distance at which two fibers are compared with each other. They can be set according to anatomical knowledge. Furthermore, we also propose to estimate a probabilistic atlas of a population of track density maps as a Wasserstein barycenter. Our CUDA implementation is endowed with a user-friendly PyTorch interface, freely available on the PyPi repository (pip install geomloss) and at www.kernel-operations.io/geomloss.