Large-scale problems have always been a challenge for numerical computations. An
example is the treatment of fully populated n× n matrices when n2 is close to or beyond the …

This survey provides an overview of higher-order tensor decompositions, their applications,
and available software. A tensor is a multidimensional or N-way array. Decompositions of …

There has been continued interest in seeking a theorem describing optimal low-rank
approximations to tensors of order 3 or higher that parallels the Eckart–Young theorem for …

This work was originally motivated by a classification of tensors proposed by Richard
Harshman. In particular, we focus on simple and multiple 'bottlenecks', and on 'swamps' …

The canonical polyadic and rank-(L_r,L_r,1) block term decomposition (CPD and BTD,
respectively) are two closely related tensor decompositions. The CPD and, recently, BTD are …

We consider relative error low rank approximation of tensors with respect to the Frobenius
norm. Namely, given an order-q tensor A∊ ℝ∏ i= 1 q ni, output a rank-k tensor B for which …

Abstract The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a
prespecified number of R factors and a residual array by minimizing the sum of squares of …

Compositional data are characterized by values containing relative information, and thus the
ratios between the data values are of interest for the analysis. Due to specific features of …

It has been shown that a best rank-R approximation of an order-k tensor may not exist when
R⩾ 2 and k⩾ 3. This poses a serious problem to data analysts using tensor decompositions …

The R package ThreeWay is presented and its main features are illustrated. The aim of
ThreeWay is to offer a suit of functions for handling three-way arrays. In particular, the most …