A Projected Newton-type Algorithm for Rank - revealing Nonnegative Block - Term Tensor Decomposition

Eleftherios Kofidis, Paris V. Giampouras, Athanasios A. Rontogiannis

Abstract: The block-term tensor decomposition (BTD) model has been receiving increasing attention as a quite flexible way to capture the structure of 3-dimensional data that can be naturally viewed as the superposition of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R$</tex> block terms of multilinear rank ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{r}, L_{r}, 1), r=1,2,\ldots,R$</tex> . Versions with nonnegativity constraints, especially relevant in applications like blind source separation problems, have only recently been proposed and they all share the need to have an a-priori knowledge of the number of block terms, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R$</tex> , and their individual ranks, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{i}$</tex> . Clearly, the latter requirement may severely limit their practical applicability. Building upon earlier work of ours on unconstrained BTD model selection and computation, we develop for the first time in this paper a method for nonnegative BTD approximation that is also rank-revealing. The idea is to impose column sparsity jointly on the factors and successively estimate the ranks as the numbers of factor columns of non-negligible magnitude. This is effected with the aid of nonnegative alternating iteratively reweighted least squares, implemented via projected Newton updates for increased convergence rate and accuracy. Simulation results are reported that demonstrate the effectiveness of our method in accurately estimating both the ranks and the factors of the nonnegative least squares BTD approximation.