Newton's method on Riemannian manifolds and a geometric model for the human spine

By Roy L. Adler, Jean-Pierre Dedieu, J. Margulies, M. Martens, M. Shub. 2002

312 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/b956aa2c0614dbd31e30dadadbb1427b207344dd

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In Applied Mathematics

In Astronomy and Astrophysics

In Computational Mechanics

Citations in the corpus

Differential-geometric Newton method for the best rank-(R1, R2, R3) approximation of tensors (2008, methodology): “As in [1], we rely chiefly on the theory developed in [2, 4].”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, background): “149 Definition 1 [Adler et al., 2002] A mapping RX is called the retraction, if 1.”
Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups (2013, unknown): “Although the Gauss-Newton algorithm is often presented as an approximation of the Newton method, it is more interesting to see it as the exact minimization of approximated criteria obtained at each step by linearizing the residuals [52, 2].”
A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank (2013, background): “Retractionbased Riemannian optimisation, like in Adler et al. (2002), dispenses with geodesics in favour of their first-order approximations, called retractions.”
Global rates of convergence for nonconvex optimization on manifolds (2016, unknown): “The definition of retraction below can be traced back to Shub (1986) and it appears under that name in (Adler et al., 2002); see also (Absil et al.”
Low-Rank Matrix Completion by Riemannian Optimization (2012, methodology): “5) is a smooth optimization problem that can be solved by methods from Riemannian optimization as introduced, among others, by [2, 4, 20].”
Projection-like Retractions on Matrix Manifolds (2012, methodology): “A crucial reason for considering (first-order) retractions is the following: if, in Smith’s Riemannian Newton method [Smi94, Alg.”
A feasible method for optimization with orthogonality constraints (2012, methodology): “While there exist several retraction-based methods [1, 2, 3, 4, 47], ours is unique for its low computational cost.”
Optimization Methods on Riemannian Manifolds and Their Application to Shape Space (2012, methodology): “Smith proved quadratic convergence for the more general case of applying Newton’s method to find a zero of a one-form on a manifold with the Levi-Civita connection [21] (the method is stated for an unspecified affine connection in [4]), using geodesic steps.”
On the best low multilinear rank approximation of higher-order tensors (2010, methodology): “The correspondence between vectors on the tangent plane and points on the manifold is given by a retraction [65, 5 ], Fig. 5.”