Geodesics in the hyperbolic plane, forming the logo of our opensource software for geometric statistics and geometric (deep) learning: Geomstats.
Geometry and Topology in Machine Learning
Data spaces with geometric structures arise in many fields in machine learning. From (biological) shape spaces equipped with a quotient geometry to neural networks parameterized with orthogonality constraints, researchers are poised to compute with geometric objects.
As a result, Geometric Statistics, Geometric and Topological (Deep) Learning are getting more and more popular.
Our research supports topological and geometric modeling in machine learning by developing new methods and an opensource software  Geomstats  with 3 objectives:

provide educational support to learn “handson” geometry and geometric statistical learning,

foster research in geometric statistical learning by providing a platform to share algorithms; and

democratize the use of geometric statistical learning by implementing algorithms with a userfriendly API.
If you are interested by a research position in this research area, we recommend that you contribute to Geomstats to become familiar with the projects and associated opensource community.
Want to learn more?
 Contact us if you are interested in contributing! You will be joining a vibrant international collaboration of researchers across France, the U.S., India and other countries.
 Look at the research from Xavier Pennec in Geometric Statistics!
 Explore Cell Complex Neural Networks with our collaborator Mustafa Hajij!
Artist representation of the stratified geometry and quotient geometry of shape spaces.
Geometry in Shape Learning
Computing with shapes challenges the very definition of statistical learning: what does it mean to compute the mean of two (biological) shapes? How can we define learning algorithms such as regressions on shapes?
Our research explores the geometries of shape spaces and:
 investigates the properties of shape representations: shapes of sets of key points or shapes of curves, among others,
 develops quantitative methods for shape comparison relying on shape transformations,
 analyzes the uncertainty associated with statistical learning on shapes.
Check out the geometric properties of the simple  yet illustrative  space of triangles!
Artist representation of geometric and topological structures that can organize information in our brains.
Geometry of the Mind
How does our brain structure information? Which brain regions are usually coactivated, and can we represent the brain activity through geometric modeling?
Our research explores geometric representations of thoughts, analyzing for instance:
 the structure of restingstate brain activity,
 the geometry of neuronal activity in the visual cortex, or
 the electrical signals corresponding to someone's intention of action.
Relevant Publications

Miolane, N., Caorsi, M., Lupo, U., Guerard, M., Guigui, N., Mathe, J., Cabanes, Y., Reise, W., Davies, T., Leitão, A., Mohapatra, S., Utpala, S., Shailja, S., Corso, G., Liu, G., Iuricich, F., Manolache, A., Nistor, M., Bejan, M., Mihai Nicolicioiu, A., Luchian, B.A., Stupariu, M.S., Michel, F., Dao Duc, K., Abdulrahman, B., Beketov, M., Maignant, E., Liu, Z., Černý, M., Bauw, M., VelascoForero, S., Angulo, J., Long Y. ICLR 2021 Challenge for Computational Geometry & Topology: Design and Results. Workshop on Geometrical and Topologic Representation Learning (ICLR) (2021).

Miolane, N., Guigui, N., Zaatiti, H., Shewmake, C., Hajri, H., Brooks, D., Le Brigant, A., Mathe, J. Hou, B., Thanwerdas, Y., Heyder, S., Peltre, O., Koep, N., Cabanes, Y., Gerald, T. Chauchat, P., Kainz, B., Donnat, C., Holmes, S., Pennec, X. Introduction to Geometric Learning in Python with Geomstats. Conference on Scientific Computing in Python (SciPy). (2020).

Miolane, N., Guigui, N., Le Brigant, A., Mathe, J., Hou, B., Thanwerdas, Y., Heyder, S., Peltre, O., Koep, N., Cabanes, Y., Chauchat, P., Zaatiti, H., Hajri, H., Gerald, T. , Shewmake, C., Brooks, D., Kainz, B., Donnat, C., Holmes, S., Pennec, X. Geomstats: A Python package for Riemannian geometry in Machine Learning. Journal of Machine Learning Research (JMLR) (2020).

Miolane, N., Holmes, S.: Learning Weighted Submanifolds With Variational Autoencoders and Riemannian Variational Autoencoders. Conference of Computer Vision and Pattern Recognition. (CVPR) (2020).

Miolane, N., Pennec, X. Biased estimators on Quotient spaces. International Conference on Geometric Sciences of Information (GSI). 2015. (Oral presentation)

Miolane, N., Pennec, X. A survey of mathematical structures for extending 2D neurogeometry to 3D image processing. Medical Computer Vision Workshop, International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI). 2015.

Miolane, N., Pennec, X.: Computing biinvariant pseudometrics on Lie groups for consistent statistics. Entropy, 17(4), pp. 1850–1880. 2015.

Miolane, N.: Statistics on Lie groups: can we obtain a consistent framework with pseudoRiemannian metrics? Workshop on Geometrical Models in Vision, Institut Poincare, Paris. 2014. (Poster).

Miolane, N., Pennec, X. Statistics on Lie groups : A need to go beyond the pseudoRiemannian framework. International Workshop on Bayesian Inference and Maximum Entropy Methods (MaxEnt). 2014.

Miolane, N., Khanal, B.: Statistics on Lie groups for Computational Anatomy. Video for the Educational Challenge of the 17th International Conference on Medical Image Computing and Computer Assisted Intervention, MIT Boston. 2014. (Video, 1st Popular Prize).