What I worked on

Data are usually indirect observations of an underlying process, and the observation distorts its geometry. Standard geometric tools inherit that distortion. We instead learn the process geometry, its carré du champ (local covariance), from samples of data perturbations, such as the steps of a time series.

This recovers the geometry of the process, and the samples needed to learn its tangent spaces grow with the intrinsic dimension of the system, not with that of the observation. The transformation law also lets the geometry be learned from several observations of the same process, even in different spaces or modalities, and pushed forward to new ones, such as a latent space.

Many complex systems, brain networks among them, are organized by directed, multi-way interactions that pairwise graphs miss. We introduced a notion of higher-order directionality, and neural networks that process information along these directed, possibly asymmetric higher-order interactions: directed simplicial neural networks, and semi-simplicial neural networks for directed higher-order motifs, with applications to brain activity decoding. We also contributed to the open-source libraries TopoX and TopoBench, and to the topological deep learning challenges.

Equivariant, multimodal re-assembly of fragmented 3D objects, and the expressivity of piecewise-affine networks — how many affine regions trained networks actually use as tasks become harder.