Hierarchical Singular Value Decomposition of Tensors

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.