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Tree tensor networks such as the tensor train (TT) format are a common tool for high-dimensional problems. The associated multivariate rank and accordant tuples of singular values are based on different matricizations of the same tensor. While the behavior of such is as essential as in the matrix case, here questions about the feasibility of specific constellations arise: which prescribed tuples can be realized as singular values of a tensor, and what is this feasible set? We first show the equivalence of the tensor feasibility problem (TFP) to the quantum marginal problem (QMP). In higher dimensions, in case of the TT-format, the conditions for feasibility can be decoupled. By present results for three dimensions for the QMP, it then follows that the tuples of squared, feasible TT-singular values form polyhedral cones. We further establish a connection to eigenvalue relations of sums of Hermitian matrices, which in turn are described by sets of interlinked, so-called honeycombs, as they have been introduced by Knutson and Tao. Besides a large class of universal, necessary inequalities as well as the vertex description for a special, simpler instance, we present a linear programming algorithm to check feasibility and a simple, heuristic algorithm to construct representations of tensors with prescribed, feasible TT-singular values in parallel. [ABSTRACT FROM AUTHOR]

Given an n-by-n Hermitian matrix A and a real number λ, index i is said to be Parter (resp., neutral, downer) if the multiplicity of λ as an eigenvalue of the principal submatrix A(i) is one more (resp., the same, one less) than that in A. In case the multiplicity of λ in A is at least 2 and the graph of A is a tree, there are always Parter vertices. Our purpose here is to advance the classification of vertices and, in particular, to relate classification to the combinatorial structure of eigenspaces. Some general results are given and then used to deduce some rather specific facts not otherwise easily observed. Examples are given. [ABSTRACT FROM AUTHOR]