Representations of quasi-Newton matrices and their use in limited memory methods
Mathematical Programming: Series A and B
Decomposition of quantics in sums of powers of linear forms
Signal Processing - Special issue on higher order statistics
Matrix computations (3rd ed.)
A Modified Cholesky Algorithm Based on a Symmetric Indefinite Factorization
SIAM Journal on Matrix Analysis and Applications
The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
A Multilinear Singular Value Decomposition
SIAM Journal on Matrix Analysis and Applications
On the Best Rank-1 and Rank-(R1,R2,. . .,RN) Approximation of Higher-Order Tensors
SIAM Journal on Matrix Analysis and Applications
Adaptive Eigenvalue Computations Using Newton's Method on the Grassmann Manifold
SIAM Journal on Matrix Analysis and Applications
On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors
SIAM Journal on Matrix Analysis and Applications
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
Algorithm 862: MATLAB tensor classes for fast algorithm prototyping
ACM Transactions on Mathematical Software (TOMS)
Trust-Region Methods on Riemannian Manifolds
Foundations of Computational Mathematics
Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem
SIAM Journal on Matrix Analysis and Applications
Symmetric Tensors and Symmetric Tensor Rank
SIAM Journal on Matrix Analysis and Applications
Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time
SIAM Journal on Matrix Analysis and Applications
Optimization Algorithms on Matrix Manifolds
Optimization Algorithms on Matrix Manifolds
SIAM Journal on Matrix Analysis and Applications
Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays
SIAM Journal on Scientific Computing
Tensor Decompositions and Applications
SIAM Review
SIAM Journal on Matrix Analysis and Applications
A New Truncation Strategy for the Higher-Order Singular Value Decomposition
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
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In this paper we proposed quasi-Newton and limited memory quasi-Newton methods for objective functions defined on Grassmannians or a product of Grassmannians. Specifically we defined BFGS and limited memory BFGS updates in local and global coordinates on Grassmannians or a product of these. We proved that, when local coordinates are used, our BFGS updates on Grassmannians share the same optimality property as the usual BFGS updates on Euclidean spaces. When applied to the best multilinear rank approximation problem for general and symmetric tensors, our approach yields fast, robust, and accurate algorithms that exploit the special Grassmannian structure of the respective problems and which work on tensors of large dimensions and arbitrarily high order. Extensive numerical experiments are included to substantiate our claims.