Best Low Multilinear Rank Approximation of Higher-Order Tensors, Based on the Riemannian Trust-Region Scheme

Ishteva, Mariya; Absil, Pierre-Antoine; Huffel, Sabine Van; Lathauwer, Lieven De

doi:10.1137/090764827

Cited by 112 publications

(88 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Algorithms for solving problem (2.2) have been proposed in [35,36,37,38,31]. In the higher-order power method of [36] each iteration rotates mass to the target subtensor by using the singular value decomposition (SVD) of the columns of one of the three matrix unfoldings of the rotated tensor corresponding to the subtensor.…”

Section: Problem (22) Is Equivalent To Maximizing (Smentioning

confidence: 99%

On best rank-2 and rank-(2,2,2) approximations of order-3 tensors

Stegeman

Friedland

2016

Linear and Multilinear Algebra

View full text Add to dashboard Cite

It is well known that a best rank-R approximation of order-3 tensors may not exist for R ≥ 2. A best rank-(R, R, R) approximation always exists, however, and is also a best rank-R approximation when it has rank (at most) R. For R = 2 and real order-3 tensors it is shown that a best rank-2 approximation is also a local minimum of the best rank-(2,2,2) approximation problem. This implies that if all rank-(2,2,2) minima have rank larger than 2, then a best rank-2 approximation does not exist. This provides an easy-to-check criterion for existence of a best rank-2 approximation. The result is illustrated by means of simulations.

show abstract

Section: Problem (22) Is Equivalent To Maximizing (Smentioning

confidence: 99%

On best rank-2 and rank-(2,2,2) approximations of order-3 tensors

Stegeman

Friedland

2016

Linear and Multilinear Algebra

View full text Add to dashboard Cite

show abstract

“…However, the theoretical justification of such methods, even for relatively simple low-rank matrix cases, is still an open problem. Recently, in order to achieve a better global convergence, an alternative version of the Riemannian Newton method called the Trust-Region scheme was developed by Absil et al [2007], Boumal and Absil [2011], Ishteva et al [2011].…”

Section: Riemannian Newton Methodsmentioning

confidence: 99%

“…Moreover, from a Riemannian geometry point of view, constrained optimization problems can often be viewed as unconstrained ones. It is therefore natural to ask whether Riemannian optimization can also help open up new research directions in conjunction with tensor networks; see for example [Ishteva et al, 2011, Kasai and Mishra, 2015, Sato et al, 2017, Steinlechner, 2016a. Riemannian optimization for tensors can be formulated in the following generalized form…”

Section: Riemannian Optimization For Low-rank Tensor Manifoldsmentioning

confidence: 99%

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives

Cichocki

Phan

Zhao

et al. 2017

FNT in Machine Learning

201

126

View full text Add to dashboard Cite

“…First, using the rankest function of Tensorlab, an upper bound of R = 4 and R = 9 was obtained for HH tensors and W tensors, respectively. Afterwards, these values were refined as follows: CPD and the low multilinear rank approximation (LMLRA) [12] were performed with increasing R, starting from R = 1, until the desired approximation was achieved. The desired approximation was defined in terms of the relative error (relerr), which was chosen here as 0.2.…”

Section: Tensor Decompositionmentioning

confidence: 99%

Nonconvulsive epileptic seizures detection using multiway data analysis

Aldana

Hunyadi

Reyes

et al. 2017

2017 25th European Signal Processing Conference (EUSIPCO)

Self Cite

View full text Add to dashboard Cite

Abstract-Nonconvulsive status epilepticus (NCSE) is observed when the patient undergoes a persistent electroencephalographic epileptic episode without physical symptoms. This condition is commonly found in critically ill patients from intensive care units and constitutes a medical emergency. This paper proposes a method to detect nonconvulsive epileptic seizures (NCES). To perform the NCES detection the electroencephalogram (EEG) is represented as a third order tensor with axes f requency × time × channels using Wavelet or Hilbert-Huang transform. The signatures obtained from the tensor decomposition are used to train five classifiers to separate between the normal and seizure EEG. Classification is performed in two ways: (1) with each signature of the different modes separately, (2) with all signatures assembled. The algorithm is tested on a database containing 139 nonconvulsive seizures. From all performed analysis, Hilbert-Huang Tensors Space and assembled signatures demonstrate to be the best features to classify between seizure and non-seizure EEG.

show abstract

Best Low Multilinear Rank Approximation of Higher-Order Tensors, Based on the Riemannian Trust-Region Scheme

Cited by 112 publications

References 34 publications

On best rank-2 and rank-(2,2,2) approximations of order-3 tensors

On best rank-2 and rank-(2,2,2) approximations of order-3 tensors

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives

Nonconvulsive epileptic seizures detection using multiway data analysis

Contact Info

Product

Resources

About