Conference abstracts

Session A1 - Approximation Theory

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The Geometrical Description of Feasible Singular Values in the Tensor Train Format

Sebastian Kraemer

IGPM at RWTH Aachen University, Germany   -   kraemer@igpm.rwth-aachen.de

Tensors have grown in importance and are applied to an increasing number of fields. Crucial in this regard are tensor formats, such as the widespread Tensor Train (TT) decomposition, which represent low rank tensors. This multivariate TT-rank and accordant $d-1$ tuples of singular values are based on different matricizations of the same $d$-dimensional tensor. While the behavior of these singular values is as essential as in the matrix case $(d=2)$, here the question about the $\textit{feasibility}$ of specific TT-singular values arises: for which prescribed tuples exist correspondent tensors and how is the topology of this set of feasible values? $\\$ This work is largely based on a connection that we establish to eigenvalues of sums of hermitian matrices. After extensive work spanning several centuries, that problem, known for the Horn Conjecture, was basically resolved by Knutson and Tao through the concept of so called $\textit{honeycombs}$. We transfer and expand their and earlier results on that topic and thereby find that the sets of squared, feasible TT-singular values are geometrically described by polyhedral cones, resolving our problem setting to the largest extend. Besides necessary inequalities, we also present a linear programming algorithm to check feasibility as well as a simple heuristic, but quite reliable, parallel algorithm to construct tensors with prescribed, feasible singular values.

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