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Geometric description of low-rank tensor manifolds as principal fiber bundles and applications
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The idea of a high-dimensional tensor calculus invokes at several points problems defined on sets of tensors of some fixed rank parameter. These problems include static ones like finding the best approximation of a tensor by one of lower rank, or dynamic ones like projected ODEs. To design and investigate algorithms for these tasks, an explicit knowledge of the differential geometric properties of the set of considered tensors at hand is helpful, if not inevitable. This reaches from the pure information that this set has a differentiable manifold structure to explicit charts and tangent vectors, all of which should inherit a data sparse representation. While the standard canonical format does not provide a smooth structure in general, the tensor formats based on subspace ranks, such as the Tucker, the hierachical Tucker, and the TT format, share nice properties with matrix manifolds of fixed rank. A description of them as principal fiber bundles, which we will explain in detail, therefore appears natural. The setting of Lie groups and fiber bundles may seem unusual to numerical analysis, but leads to a general viewpoint, which for instance makes the local convergence theory of such algorithms like alternating least squares quite transparent, independent of the chosen tensor format.This talk is partially based on joint work with Thorsten Rohwedder and Bart Vandereycken.