Applications for Geomstats that illustrate more involved uses of the package.

We consider the fMRI data from the 2014 MLSP Schizophrenia Classification challenge, consisting of the resting-state fMRIs of 86 patients split into two balanced categories: control vs people suffering schizophrenia.

We approach the classification task by using a SVM classifier on pairwise-similarities between brain connectomes, represented as the SPD matrices of their regularized Laplacians. The similarities are computed with the affine-invariant Riemannian distance on the manifold of SPD matrices.

pip3 install geomstats==1.7

# from the root of your unziped directory
cd brain_connectome
pip3 install -r requirements.txt
python3 spd_fmri.py

We consider a robot arm in this application. In robotics, it is common to control a manipulator in Cartesian space rather than configuration space. We generate a geodesic on SO(3) between the initial orientation of the robot arm and its desired final orientation. We use the generated trajectory as an input to the robot controller.

The robotics application is a little bit more involved. It has its own README.md file and corresponding instructions. You will need a C++ compilation toolchain.

pip3 install geomstats==1.7

# from the root of your unziped directory
cat robotics/README.md

We consider the training of MNIST with weights constrained on the Hypersphere. The optimization step has been modified in keras such that the stochastic gradient descent is done on the manifold through the Exponential Map.

The deep learning application requires a tensorflow patch and installing a modified version of keras.

pip3 install geomstats==1.7

# from the root of your unziped directory
cat deep_learning/README.md

This example trains a pose estimation network using a SE3 Geodesic Loss function.

pip3 install geomstats==1.7

# from the root of your unziped directory
cat se3_pose_estimation/README.md

This application illustrates the modulation of the speed of convergence of the empirical Fréchet mean with the curvature in spheres and hyperbolic spaces described in the arXiv paper 1906.07418. The application is actually the source-code to create the figures of this paper.

The variance of the Fréchet mean FM_n of a sample of n IID random variables of variance Var is decreasing more slowly in a sphere than in a Euclidean space, and more quickly in a Hyperbolic space. Two scripts of this application compare the speed of convergence in spheres and hyperbolic spaces to the classical one in Euclidean spaces by computing the modulation factor alpha = Var( FM_n) / ( n * Var) for synthetic IID n-samples drawn from isotropic distributions on hyper-spheres (bubble distributions) of different radii. The last script computes the modulation factor for bootstrap n-samples drawn from a few real-world empirical distributions on the sphere S2 taken from the book of Fisher, Lewis and Embleton 1987.

pip3 install geomstats==1.18

# from the root of your unziped directory
cat modulation_empirical_frechet_mean/README.md

Enjoy :)