Jean Feydy

Antoine Diez, Jean Feydy

Many living and physical systems such as cell aggregates, tissues or bacterial colonies behave as unconventional systems of particles that are strongly constrained by volume exclusion and shape interactions. Understanding how these constraints lead to macroscopic self-organized structures is a fundamental question in e.g. developmental biology. To this end, various types of computational models have been developed. Here, we introduce a new framework based on optimal transport theory to model particle systems with arbitrary dynamical shapes and deformability properties. Our method builds upon the pioneering work of Brenier on incompressible fluids and its recent applications to materials science. It lets us specify the shapes and volumes of individual cells and supports a wide range of interaction mechanisms, while automatically taking care of the volume exclusion constraint at an affordable numerical cost. We showcase the versatility of this approach by reproducing several classical systems in computational biology. Our Python code is freely available at https://iceshot.readthedocs.io/.

Guillaume Houry, Jean Feydy, François-Xavier Vialard

A fundamental challenge in data science is to match disparate point sets with each other. While optimal transport efficiently minimizes point displacements under a bijectivity constraint, it is inherently sensitive to rotations. Conversely, minimizing distortions via the Gromov-Wasserstein (GW) framework addresses this limitation but introduces a non-convex, computationally demanding optimization problem. In this work, we identify a broad class of distortion penalties that reduce to a simple alignment problem within a lifted feature space. Leveraging this insight, we introduce an iterative GW solver with a linear memory footprint and quadratic (rather than cubic) time complexity. Our method is differentiable, comes with strong theoretical guarantees, and scales to hundreds of thousands of points in minutes. This efficiency unlocks a wide range of geometric applications and enables the exploration of the GW energy landscape, whose local minima encode the symmetries of the matching problem.

Zhengyang Shen, Jean Feydy, Peirong Liu et al.

This work investigates the use of robust optimal transport (OT) for shape matching. Specifically, we show that recent OT solvers improve both optimization-based and deep learning methods for point cloud registration, boosting accuracy at an affordable computational cost. This manuscript starts with a practical overview of modern OT theory. We then provide solutions to the main difficulties in using this framework for shape matching. Finally, we showcase the performance of transport-enhanced registration models on a wide range of challenging tasks: rigid registration for partial shapes; scene flow estimation on the Kitti dataset; and nonparametric registration of lung vascular trees between inspiration and expiration. Our OT-based methods achieve state-of-the-art results on Kitti and for the challenging lung registration task, both in terms of accuracy and scalability. We also release PVT1010, a new public dataset of 1,010 pairs of lung vascular trees with densely sampled points. This dataset provides a challenging use case for point cloud registration algorithms with highly complex shapes and deformations. Our work demonstrates that robust OT enables fast pre-alignment and fine-tuning for a wide range of registration models, thereby providing a new key method for the computer vision toolbox. Our code and dataset are available online at: https://github.com/uncbiag/robot.

Ivan Lerner, Jean Feydy, Alexandre Kalimouttou et al.

Background: Timely, uncertainty-aware forecasting from irregular electronic health records (EHR) can support critical-care decisions, yet most approaches either impute to a grid or sacrifice interpretability. We introduce StructGP, a continuous-time multi-task Gaussian process that couples process convolutions with differentiable structure learning to uncover a sparse, ordered directed acyclic graph (DAG) of inter-variable dependencies while preserving principled uncertainty. We further propose LP-StructGP, which augments StructGP with latent pathways-shared, temporally shifted trajectories inferred via subject-specific coupling filters and a softmax gating mechanism-to capture cross-patient progression patterns. Both models are trained under sparsity and acyclicity constraints (augmented Lagrangian, Adam) using scalable low-rank updates. Results: In simulations, the approach reliably recovers ground-truth graphs (Structural Hamming Distance approaching 0 as cohorts grow) and pathway assignments (high Adjusted Rand Index). On a MIMIC-IV septic shock cohort (n=1,008; norepinephrine, creatinine, mean arterial pressure), StructGP improves short-horizon (6 h) forecasting over independent-task baselines (average RMSE 0.68 [95%CI: 0.63--0.74] vs. 0.88 [0.83-0.94]) and, with 15 additional inputs, markedly outperforms unstructured kernels (0.63 [0.58-0.69] vs. 3.02 [2.85-3.18]) with superior calibration (coverage 0.96 vs. 0.84). On the PhysioNet Challenge (12k patients, 41 variables), StructGP attains competitive accuracy (MAE 3.72e-2) relative to a state-of-the-art graph neural model while maintaining calibrated uncertainty. Conclusion: These results show that structured process convolutions with latent pathways deliver interpretable, scalable, and well-calibrated forecasting for irregular clinical time series.

Nathan Kessler, Robin Magnet, Jean Feydy

Smoothing a signal based on local neighborhoods is a core operation in machine learning and geometry processing. On well-structured domains such as vector spaces and manifolds, the Laplace operator derived from differential geometry offers a principled approach to smoothing via heat diffusion, with strong theoretical guarantees. However, constructing such Laplacians requires a carefully defined domain structure, which is not always available. Most practitioners thus rely on simple convolution kernels and message-passing layers, which are biased against the boundaries of the domain. We bridge this gap by introducing a broad class of smoothing operators, derived from general similarity or adjacency matrices, and demonstrate that they can be normalized into diffusion-like operators that inherit desirable properties from Laplacians. Our approach relies on a symmetric variant of the Sinkhorn algorithm, which rescales positive smoothing operators to match the structural behavior of heat diffusion. This construction enables Laplacian-like smoothing and processing of irregular data such as point clouds, sparse voxel grids or mixture of Gaussians. We show that the resulting operators not only approximate heat diffusion but also retain spectral information from the Laplacian itself, with applications to shape analysis and matching.

Jean Feydy, Pierre Roussillon, Alain Trouvé et al.

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 reach -- of 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.

Benjamin Charlier, Jean Feydy, Joan Alexis Glaunès et al.

The KeOps library provides a fast and memory-efficient GPU support for tensors whose entries are given by a mathematical formula, such as kernel and distance matrices. KeOps alleviates the major bottleneck of tensor-centric libraries for kernel and geometric applications: memory consumption. It also supports automatic differentiation and outperforms standard GPU baselines, including PyTorch CUDA tensors or the Halide and TVM libraries. KeOps combines optimized C++/CUDA schemes with binders for high-level languages: Python (Numpy and PyTorch), Matlab and GNU R. As a result, high-level "quadratic" codes can now scale up to large data sets with millions of samples processed in seconds. KeOps brings graphics-like performances for kernel methods and is freely available on standard repositories (PyPi, CRAN). To showcase its versatility, we provide tutorials in a wide range of settings online at \url{www.kernel-operations.io}.

Thibault Séjourné, Jean Feydy, François-Xavier Vialard et al.

Optimal transport induces the Earth Mover's (Wasserstein) distance between probability distributions, a geometric divergence that is relevant to a wide range of problems. Over the last decade, two relaxations of optimal transport have been studied in depth: unbalanced transport, which is robust to the presence of outliers and can be used when distributions don't have the same total mass; entropy-regularized transport, which is robust to sampling noise and lends itself to fast computations using the Sinkhorn algorithm. This paper combines both lines of work to put robust optimal transport on solid ground. Our main contribution is a generalization of the Sinkhorn algorithm to unbalanced transport: our method alternates between the standard Sinkhorn updates and the pointwise application of a contractive function. This implies that entropic transport solvers on grid images, point clouds and sampled distributions can all be modified easily to support unbalanced transport, with a proof of linear convergence that holds in all settings. We then show how to use this method to define pseudo-distances on the full space of positive measures that satisfy key geometric axioms: (unbalanced) Sinkhorn divergences are differentiable, positive, definite, convex, statistically robust and avoid any "entropic bias" towards a shrinkage of the measures' supports.

Jean Feydy, Benjamin Charlier, François-Xavier Vialard et al.

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.