We define the hierarchical singular value decomposition (SVD) for tensors of order d ≥ 2. 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 (H-Tucker) which requires only O((d - 1)k3 + dnk) parameters, where d is the order of the tensor, n the size of the modes, and k the (hierarchical) rank. The 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 O((d-1)k4+dnk2) and the attainable accuracy is just 2-3 digits less than machine precision. Copyright © 2010 Society for Industrial and Applied Mathematics.
CITATION STYLE
Grasedyck, L. (2009). Hierarchical singular value decomposition of tensors. SIAM Journal on Matrix Analysis and Applications, 31(4), 2029–2054. https://doi.org/10.1137/090764189
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