Trust-Region Methods on Riemannian Manifolds

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

525 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c845e699ac8ba3d1a8b927f439004431440a83fa

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Citations in the corpus

Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview (2018, methodology): “Trust-region methods (TRM): At the tth iterate, this method minimizes a quadratic approximation of the loss function in a local neighborhood, where the quadratic approximation is typically based on a second-order Taylor expansion at the current iterate [169], [170].”
On-Manifold Preintegration for Real-Time Visual--Inertial Odometry (2015, background): “Section III-C reviews optimization on manifolds, following standard references [45].”
Optimization Methods on Riemannian Manifolds and Their Application to Shape Space (2012, methodology): “For a smooth retraction Rx we do not necessarily have D(2)fRx(0) = D(2)f(x), however, this holds at stationary points of f [1, 2].”
A Geometric Analysis of Phase Retrieval (2016, background): “Their tCG solver uses the gradient as the initial search direction, which does not ensure that the TRM solver can escape from saddle points [1,2].”
Geodesic finite elements on simplicial grids (2012, methodology): “[15] have proposed the Riemannian trust-region method, which generalizes the standard trust-region approach to nonlinear configuration spaces while retaining the convergence properties of regular trust-region methods.”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, background): “Methods of Riemannian Optimization (RO) have been recently a subject of great interest in data analytics communities; see, for example, [Absil et al., 2007, 2008, Bento et al., 2016, Bonnabel, 2013, Cambier and Absil, 2016].”
A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank (2013, background): “However, most global convergence theory for optimisation on Riemannian manifolds assumes completeness; see, e.g., Ferreira & Svaiter (2002); Dedie et al. (2003); Li & Wang (2006); Absil et al. (2007); Alvarez et al. (2008)— although Yang (2006) does not.”
Low-Rank Matrix Completion by Riemannian Optimization (2012, methodology): “In [44], for example, it was shown that the second-order Riemannian trust-region algorithm of [1] performs very well in combination with our embedded submanifold geometry.”
Proximal point method for a special class of nonconvex functions on Hadamard manifolds (2008, background): “It has been frequently done in recent years, with a theoretical purpose and also to obtain effective algorithms; see [1], [3], [10], [11], [16], [19], [20], [23] and [25].”
Manopt, a matlab toolbox for optimization on manifolds (2013, methodology): “Currently, Riemannian trust-regions (based on (Absil et al. , 2007)) and conjugate-gradients are implemented (with preconditioning), as well as steepest-descent and a couple derivative free schemes.”