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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)
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)
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)
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)
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)
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)
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)
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)