Stable Truncation and Root-Independent Normalization of Tree Tensor Networks

A ubiquitous tool in numerical linear algebra is the singular value decomposition, which can be lifted to higher order tensors and provides a useful mechanic for quasi-optimal approximations. The stable computation of a therefor required hierarchical singular value decomposition is non-trivial if it is to be efficiently achieved only via data-sparse, hierarchical low rank representations. Even if the initial representation is stable, the standard approach for the decomposition based on Gram matrices involves a squaring of singular values which will typically reduce the attainable accuracy. We provide an efficient and stable variant of the hierarchical singular value decomposition related to (root-independent) normal forms of tensors that can be computed in linear complexity of the dimension or order of the tensor and is accurate up to machine precision. Numerical tests highlight the higher accuracy of the introduced approach.