A Multilinear Singular Value Decomposition
SIAM Journal on Matrix Analysis and Applications
Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem
SIAM Journal on Matrix Analysis and Applications
Enhanced Line Search: A Novel Method to Accelerate PARAFAC
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Tensor Decompositions and Applications
SIAM Review
Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions
SIAM Journal on Scientific Computing
Sequential unfolding SVD for tensors with applications in array signal processing
IEEE Transactions on Signal Processing
Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
A New Truncation Strategy for the Higher-Order Singular Value Decomposition
SIAM Journal on Scientific Computing
The Alternating Linear Scheme for Tensor Optimization in the Tensor Train Format
SIAM Journal on Scientific Computing
Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems
SIAM Journal on Matrix Analysis and Applications
An equi-directional generalization of adaptive cross approximation for higher-order tensors
Applied Numerical Mathematics
Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats
Computers & Mathematics with Applications
Approximation rates for the hierarchical tensor format in periodic Sobolev spaces
Journal of Complexity
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We define the hierarchical singular value decomposition (SVD) for tensors of order $d\geq2$. This hierarchical SVD has properties like the matrix SVD (and collapses to the SVD in $d=2$), and we prove these. In particular, one can find low rank (almost) best approximations in a hierarchical format ($\mathcal{H}$-Tucker) which requires only $\mathcal{O}((d-1)k^3+dnk)$ parameters, where $d$ is the order of the tensor, $n$ the size of the modes, and $k$ the (hierarchical) rank. The $\mathcal{H}$-Tucker format is a specialization of the Tucker format and it contains as a special case all (canonical) rank $k$ tensors. Based on this new concept of a hierarchical SVD we present algorithms for hierarchical tensor calculations allowing for a rigorous error analysis. The complexity of the truncation (finding lower rank approximations to hierarchical rank $k$ tensors) is in $\mathcal{O}((d-1)k^4+dnk^2)$ and the attainable accuracy is just 2-3 digits less than machine precision.