Geodesics in the hyperbolic plane, forming the logo of our open-source 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 open-source software --- Geomstats --- with 3 objectives:

  • provide educational support to learn “hands-on” 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 user-friendly 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 open-source 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!


geometric art

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!


brain schematics

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 co-activated, and can we represent the brain activity through geometric modeling?

Our research explores geometric representations of thoughts, analyzing for instance:

  • the structure of resting-state brain activity,
  • the geometry of neuronal activity in the visual cortex, or
  • the electrical signals corresponding to someone's intention of action.


Check out this method that allows to detect someone's intentions of movement using geometric modeling!


Relevant Publications