Block Term Decompositions - Block Term Decompositions of Higher-Order Tensors: Uniqueness, Computation and Links with Structured Matrices

From 01-01-2014 to 31-12-2017

Description

Higher-order tensors are the natural generalizations of vectors (first order) and matrices (secondorder). They may be imagined as multi-way arrays of numbers. Decomposition of a higher order tensor in rank-1 terms is unique under conditions that do not have a matrix counterpart. We have recently introduced Block Term Decompositions (BTDs) of higher-order tensors. In BTDs the terms are not necessarily rank-1. Instead, they may have low multilinear rank, which means that their columns (rows, . . . ) are restricted to a low-dimensional, but not necessarily one dimensional,vector space. A large part of the project will concern the study of the uniqueness of BTDs and the derivation of algorithms for their computation. In a somewhat unexpected manner, the work on algorithms may give insight in the complexity of the computation of amatrix product. The uniqueness properties of tensor decompositions make them essential tools for the separation of signals that are observed together. Matrix-based thinking has so far led one to consideral most exclusively components that are rank-1. This fundamental assumption may actually be questioned. The project will lay the foundations for more general BTD-based signal separation. We will make use of connections between hypothesized signal structure (such as exponential polynomial, rational function, . . . ) on one hand and properties of structured matrix representation(such as Hankel, Loewner, . . . ) on the other hand.

Team

• Marc Van Barel, Co-promoter (External)
• Lieven De Lathauwer, Promoter

Financing

Funding: FWO - Research Foundation - Flanders

Program/Grant Type: FWO Research Grant - FWO Research Grant