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- Pierre-Antoine Absil, C. G. Baker, Kyle A. Gallivan
- Foundations of Computational Mathematics
- 2007

A general scheme for trust-region methods on Riemannian manifolds is proposed. A truncated conjugate-gradient algorithm is utilized to solve the trust-region subproblems. The method is illustrated on several problems from numerical linear algebra. In particular, for computing an extreme eigenspace of a symmetric/positive-definite matrix pencil, the method… (More)

- Michel Journée, Francis R. Bach, Pierre-Antoine Absil, Rodolphe Sepulchre
- SIAM Journal on Optimization
- 2010

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 X = Y Y T , where the number of columns of Y fixes an upper bound on the rank of the positive semidefinite matrix X. It is thus very effective for solving problems that have a… (More)

- Pierre-Antoine Absil, Robert E. Mahony, B. Andrews
- SIAM Journal on Optimization
- 2005

In the early eighties Lojasiewicz [Loj84] proved that a bounded solution of a gradient flow for an analytic cost function converges to a well-defined limit point. In this paper, we show that the iterates of numerical descent algorithms, for an analytic cost function, share this convergence property if they satisfy certain natural descent conditions. The… (More)

- Nicolas Boumal, Pierre-Antoine Absil
- NIPS
- 2011

We consider large matrices of low rank. We address the problem of recovering such matrices when most of the entries are unknown. Matrix completion finds applications in recommender systems. In this setting, the rows of the matrix may correspond to items and the columns may correspond to users. The known entries are the ratings given by users to some items.… (More)

- Nicolas Boumal, Bamdev Mishra, Pierre-Antoine Absil, Rodolphe Sepulchre
- Journal of Machine Learning Research
- 2014

Optimization on manifolds is a rapidly developing branch of nonlinear optimization. Its focus is on problems where the smooth geometry of the search space can be leveraged to design efficient numerical algorithms. In particular, optimization on manifolds is well-suited to deal with rank and orthogonality constraints. Such structured constraints appear… (More)

- Paul Van Dooren, Kyle A. Gallivan, Pierre-Antoine Absil
- Appl. Math. Lett.
- 2008

We consider the problem of approximating a p × m rational transfer function H(s) of high degree by another p × m rational transfer function b H(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.

- Filippo Pompili, Nicolas Gillis, Pierre-Antoine Absil, François Glineur
- Neurocomputing
- 2014

Approximate matrix factorization techniques with both nonnegativity and orthogonality constraints, referred to as orthogonal nonnegative matrix factorization (ONMF), have been recently introduced and shown to work remarkably well for clustering tasks such as document classification. In this paper, we introduce two new methods to solve ONMF. First, we show… (More)

- Quentin Rentmeesters, Pierre-Antoine Absil
- 2011 19th European Signal Processing Conference
- 2011

This paper concerns the computation, by means of gradient and Newton methods, of the Karcher mean of a finite collection of points, both on the manifold of 3×3 rotation matrices endowed with its usual bi-invariant metric and on the manifold of 3×3 symmetric positive definite matrices endowed with its usual affine invariant metric. An explicit… (More)

- Pierre-Antoine Absil, Rodolphe Sepulchre, Paul Van Dooren, Robert E. Mahony
- SIAM J. Matrix Analysis Applications
- 2004

We propose a Newton-like iteration that evolves on the set of fixed dimensional subspaces of R n and converges locally cubically to the invariant subspaces of a symmetric matrix. This iteration is compared in terms of numerical cost and global behavior with three other methods that display the same property of cubic convergence. Moreover, we consider… (More)