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Post-doctorat 12 mois au Gipsa-lab "New approach beyond Riemannian geometry for machine learning algorithms on Stiefel manifolds"

9 Mai 2022


Catégorie : Post-doctorant


Title : New approach beyond Riemannian geometry for machine learning algorithms on Stiefel manifolds.

Duration : 1 year

Location : Grenoble

We are looking for a postdoctoral candidate who will join the AMELIORATE research project, a collaboration team of researchers from GIPSA-lab and LJK in Grenoble and from L2S in Paris-Saclay.

 

Efficiently dealing with high-dimensional data is currently a major issue in many applications of machine learning. The AMELIORATE project aims at contributing to solve this problem by exploiting the structures and geometric properties that might be contained in the data. Indeed, in many cases, data actually belong to a lower dimensional subspace (often a manifold) of the global ambient space. The geometrical structure can be leveraged to achieve machine learning tasks. In recent works, such a geometrical machine learning approach has obtained competitive results in various applications – e.g., electroencephalography, hyperspectral, computer vision. This success relies on employing a well-chosen Riemannian geometry – in particular, the resulting Riemannian distance. However, the Riemannian distance can only be found for a small number of manifolds – mainly for covariance matrices and subspaces. This drastically limits the potential applications of this approach. For instance, the Riemannian distance on the Stiefel manifold – rectangular orthogonal matrices – is unknown. It is quite unfortunate as it is encountered in many situations. The objective of the AMELIORATE project is to overcome this issue and propose efficient geometrical machine learning algorithms, especially on the Stiefel manifold. To do so, we will provide original solutions based on new objects beyond Riemannian geometry.

Three different tasks will be considered in the project. The first one will consist in studying a new alternative geometry for the Stiefel manifold, and propose the associated Fréchet mean. This will allow us to develop well-adapted machine learning methods on this manifold. The second task will focus on extending R-barycenters’ theory based on divergences. The corresponding obtained means will be studied and resulting geometrical machine learning algorithms will be obtained and tested. The third task will explore higher order approximations of the Riemannian exponential and logarithm, the connected metric and resulting Fréchet mean and machine learning algorithms. The classification/clustering algorithms to be developped in the AMELIORATE project will be tested and validated on simulated datasets. Application to real dataset could be considered but may be a long term perspective. Great care will be taken to make avalaible to the community the developed algorithms by mean of a toolbox.

Applications (CV and cover letter describing your motivation and qualifications for the position) should be sent to
nicolas.le-bihan@gipsa-lab.fr, florent.bouchard@l2s.centralesupelec.fr
and salem.said@univ-grenoble-alpes.fr