Estelle Massart

PhD student in Applied mathematics, Université catholique de Louvain

I am a teaching assistant in the mathematical engineering department (which is part of the ICTEAM institute ) at UCL .

My research is about data fitting problems on manifolds. More specifically, I am mainly working on averaging and interpolation problems on the manifold of positive (semi-)definite matrices.

My PhD advisors are Julien Hendrickx and Pierre-Antoine Absil.

Research projects

  • Fitting on the set of fixed-rank positive-semidefinite matrices

    Many applications involve covariance matrices. Often (when the underlying process is high-dimensional), those matrices are low-rank. A common approach consists then in truncating the rank of the matrices to a same common value. The data are then represented as points on the manifold of fixed-rank positive-semidefinite matrices.
    In collaboration with Pierre-Yves Gousenbourger , we have proposed some algorithms for curve fitting on manifolds (see this paper), and we have used positive-semidefinite (PSD) matrices interpolation for wind field estimation here. The wind field in a given area is represented by a Gaussian process, characterized by an average field and a low-rank covariance matrix. To estimate of the wind field for some external parameter values (e.g., prevailing wind direction or amplitude), we precompute the average field and the covariance matrix for a grid of parameter values, and we recover the target mean field and covariance by interpolation.

  • Quotient geometry of the manifold of fixed-rank positive-semidefinite matrices

    This recent preprint contains a detailed description of the manifold of fixed-rank positive-semidefinite matrices seen as a quotient of the set of full-rank rectangular matrices by the orthogonal group. In particular, we obtain expressions for the Riemannian logarithm and the injectivity radius. We show that the injectivity radius at a given point C is equal to the square root of the smallest eigenvalue of the matrix C . The resulting Riemannian distance coincides with the Wasserstein distance between centered degenerate Gaussians with corresponding low-rank covariance matrices.

  • Averaging positive definite matrices: faster convergence than SGD

    We have proposed an incremental gradient descent algorithm for computing the Riemannian barycenter on the manifold of positive-definite matrices (endowed with its classical affine-invariant metric). The algorithm is equipped with a deterministic shuffling process, resulting on average in a faster convergence that the well-known stochastic gradient algorithm. The algorithm is described in this paper. We have also used here an incremental algorithm for mean computation, in the framework of adaptative classifier design for EEG signals. Finally, this work presents a decentralized algorithm for mean computation on the set of positive-definite matrices, based on ideas from consensus theory.