Tensor Spaces and Hierarchical Tensor Representations

In the present report we provide a brief introduction into recently developed hierarchical tensor representations. The new formats extend the well-known Tucker format into a hierarchical framework, by combining its favourable characteristics with low-order scaling properties. We demonstrate the basic concept of subspace approximation and higher order SVD (HOSVD), and how to extend this in a hierarchical way. We highlight that the present tensor representations are constituting smooth manifolds, and give a perspective how these properties can be used to develop numerical solvers for tensor equations and tensor optimisation problems.

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