A Newton-Grassmann Method for Computing the Best Multilinear Rank-(r1, r2, r3) Approximation of a Tensor

By Lars Eldén, Berkant Savas. 2009

163 citations recorded by the source corpus.

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In Computational Mathematics

Citations in the corpus

Moment Component Analysis: An Illustration With International Stock Markets (2018, background): “Recently, alternative derivatives-based algorithms have been also proposed in the multilinear algebra literature (see Eldén and Savas 2009).”
Krylov-Type Methods for Tensor Computations (2010, background): “The exact relations of the entries of A to the three diﬀerent matricizations can be found in [9].”
Scalable Tensor Decompositions for Multi-aspect Data Mining (2008, methodology): “Of note, Eldén and Savas [10] have proposed a Newton-Grassmann optimization approach.”
Differential-geometric Newton method for the best rank-(R1, R2, R3) approximation of tensors (2008, background): “Another manifold-based Newton algorithm has recently been proposed in [25].”
Riemannian Optimization on Tensor Products of Grassmann Manifolds: Applications to Generalized Rayleigh-Quotients (2010, methodology): “ ···×r Xr, X † j Xj = Im j,j= 1,...,r. Thus, solving (3.9) is equivalent to solving the maximization problem max X 1,...,X r kA ×1 X1 ×2 ···×r Xrk 2, (3.11) with X† j Xj = Im j, j = 1,...,r, see e.g. [8]. Using vec−operation and Kronecker product language, one has vec(A ×1 X1 ×2 ··· ×r Xr) = vec(A) ”
Linear algebra for tensor problems (2009, unknown): “[6]).”
Quasi-Newton Methods on Grassmannians and Multilinear Approximations of Tensors (2009, methodology): “Further mathematical details concerning tensors, tensor ranks, and tensor approximations as well as the counterpart for symmetric tensors may be found in [8, 19, 24].”
Wedderburn Rank Reduction and Krylov Subspace Method for Tensor Approximation. Part 1: Tucker Case (2010, methodology): “For tensor given as full array of elements, multilinear SVD provides a quasi-optimal Tucker approximation which further can be refined by different iterative methods such that Tucker-ALS [8, 9], NewtonGra mann [10, 11], etc.”
Krylov-type methods for tensor computations I (2013, background): “org/link/?SML/30/1084/1 [9] Eldén, L.”
Variational calculus with sums of elementary tensors of fixed rank (2012, unknown): “It suggests that the approach might be competitive with optimization algorithms for alternative tensor formats as the Tucker Format [30] or the HT [17] format (with the TT format [26–28] as special case), for which the development is also still in progress: For the Tucker format, one obtains an opti”