Pierre-Antoine Absil

Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in Rn. In these formulas, p-planes are represented as the column space of n×p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications – computing an invariant subspace of a matrix and the mean of subspaces – are worked out.

Trust-Region Methods on Riemannian Manifolds

A general scheme for trust-region methods on Riemannian manifolds is proposed and analyzed. Among the various approaches available to (approximately) solve the trust-region subproblems, particular attention is paid to the truncated conjugate-gradient technique. The method is illustrated on problems from numerical linear algebra.

H2-optimal model reduction of MIMO systems

We consider the problem of approximating a p × m rational transfer function H(s) of high degree by another p × m rational transfer function bH(s) of much smaller degree. We derive the gradients of the H2-norm of the approximation error and show how stationary points can be described via tangential interpolation.

Low-Rank Optimization on the Cone of Positive Semidefinite Matrices

We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable X. This algorithm rests on the factorization X = Y Y T , where the number of columns of Y fixes the rank of X. It is thus very effective for solving programs that have a low rank solution. The factorization X = Y Y T evokes a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second order optimization method. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. The efficiency of the proposed algorithm is illustrated on two applications: the maximal cut of a graph and the sparse principal component analysis problem.We propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices. This algorithm relies on the factorization

Projection-like Retractions on Matrix Manifolds

X=YY^T

A Grassmann--Rayleigh Quotient Iteration for Computing Invariant Subspaces

, where the number of columns of

Best Low Multilinear Rank Approximation of Higher-Order Tensors, Based on the Riemannian Trust-Region Scheme

Y

On the stable equilibrium points of gradient systems

fixes an upper bound on the rank of the positive semidefinite matrix

Two algorithms for orthogonal nonnegative matrix factorization with application to clustering

X

Nonlinear analysis of cardiac rhythm fluctuations using DFA method

. It is thus very effective for solving problems that have a low-rank solution. The factorization