

Preprint 27/2009
Hierarchical Singular Value Decomposition of Tensors
Lars Grasedyck
Contact the author: Please use for correspondence this email.
Submission date: 26. Jun. 2009 (revised version: March 2010)
Pages: 29
published in: SIAM journal on matrix analysis and applications, 31 (2010) 4, p. 2029-2054
DOI number (of the published article): 10.1137/090764189
Bibtex
Keywords and phrases: SVD, Tucker, Tensor
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Abstract:
We define the hierarchical singular value decomposition (SVD)
for tensors of order . 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 (
-Tucker) which requires only
data, where d is the order of the tensor,
n the size of the modes and k the rank. The
-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
and the attainable accuracy is just 2-3 digits less than machine precision.