Shape space of triangles: in this space, each point represents a triangle. By Elodie Maignant, Geomstats.

## Shape Spaces

We design geometric techniques to study shape spaces.

What does it mean to compute the average of two or more shapes, for example the average of two brain shapes? How can we design machine learning algorithms, such as regressions, on shapes?

Our research explores and implements shape spaces to:

- investigate the properties of shape representations: shapes of sets of key points, shapes of curves, among others,
- develop quantitative methods for shape comparison,
- analyze the uncertainty associated with statistics, machine learning and deep learning on shapes.

Paper spotlight:

Nina Miolane, Susan Holmes, and Xavier Pennec. "Template shape estimation: correcting an asymptotic bias." *SIAM Journal on Imaging Sciences* 10.2 (2017): 808-844.

Geodesic in the space of one-dimensional normal distributions, which is a Riemannian manifold with hyperbolic geometry.

## Information Geometry

We design geometric techniques to study randomness and probability theory.

What is the average of two, or more, probability distributions, such as the average of two Poisson distributions corresponding to spiking neurons?

Our research explores and implements the geometries of probability distributions.

Paper Spotlight:

Alice Le Brigant, Jules Deschamps, Antoine Collas, and Nina Miolane. "Parametric Information Geometry with the Package Geomstats". *Transactions of Mathematical Software (TOMS). *.