A Geometric Description of Feasible Singular Values in the Tensor Train Format

Abstract

Tree tensor networks such as the tensor train 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 the question about the feasibility of specific constellations arises: 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 tensor train (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.

On the Largest Multilinear Singular Values of Higher-Order Tensors

“…Thus, by (16) and (50)-(52), the values α i , β i , and γ i are eigenvalues of A, B, and A+B, respectively. Hence, by Horn's conjecture, (14) and (15) hold.…”

“…Horn's Conjecture has recently also been linked to singular values of matrix unfoldings in the Tensor Train format [14].…”

“…then α i , β i , and γ i satisfy the trivial equality (14) γ 1 + · · · + γ n = α 1 + · · · + α n + β 1 + · · · + β n and the list of linear inequalities…”

“…(The construction of T n r is given in Appendix A.) The inverse statement also holds: if α i , β i , and γ i satisfy (14) and (15), then there exist n × n Hermitian matrices A, B, and C such that (13) holds.…”

Stable ALS approximation in the TT-format for rank-adaptive tensor completion

“…, r µ (and can thereby also ignore that the combination of the new γ and θ might not be feasible, cf. [25] 6 Behavior of the SALSA Filter A deeper understanding of the regularization utilized by SALSA and the reason for the lower bound σ min is provided by the filter as indicated by (3.7) for the matrix case and as defined by Corollary 5.8 for tensors. Throughout this section, we assume that the sampling is such that for the minimizer in Theorem 5.6 it (approximately) holds vec(N + (j)) = F vec((L T B(j) R T )), (6.1) which is true at last for P = I (cf.…”