Improved Uniqueness Conditions for Canonical Tensor Decompositions with Linearly Dependent Loadings

Alwin Stegeman; Tam T. T. Lam

doi:10.1137/110847275

Improved Uniqueness Conditions for Canonical Tensor Decompositions with Linearly Dependent Loadings

Alwin Stegeman^*, Tam T. T. Lam

^*Corresponding author for this work

Heymans Institute for Psychological Research

Research output: Contribution to journal › Article › Academic › peer-review

10 Citations (Scopus)

260 Downloads (Pure)

Abstract

In this paper, we derive improved uniqueness conditions for a constrained version of the canonical order-3 tensor decomposition, also known as Candecomp/Parafac (CP). CP decomposes a three-way array into a prespecified number of outer product arrays. The constraint is that some vectors forming the outer product arrays are linearly dependent according to a prespecified pattern. This is known as the PARALIND family of decompositions. We provide both uniqueness conditions and partial uniqueness conditions for PARALIND, and show that these are improved and more precise variants of existing conditions. Our results are illustrated by means of examples.

Original language	English
Pages (from-to)	1250-1271
Number of pages	22
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	33
Issue number	4
DOIs	https://doi.org/10.1137/110847275
Publication status	Published - 2012

Keywords

PARALIND
CONFAC
parafac
candecomp
tensor decomposition
uniqueness
INDEPENDENT COMPONENT ANALYSIS
LOW-RANK APPROXIMATION
DS-CDMA SIGNALS
CANDECOMP/PARAFAC MODEL
DIVERGING COMPONENTS
ORDER TENSOR
3-WAY ARRAYS
BLOCK TERMS
X-2 ARRAYS
SYSTEMS

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@article{c60c906c52454f6db0b93da99ec05451,

title = "Improved Uniqueness Conditions for Canonical Tensor Decompositions with Linearly Dependent Loadings",

abstract = "In this paper, we derive improved uniqueness conditions for a constrained version of the canonical order-3 tensor decomposition, also known as Candecomp/Parafac (CP). CP decomposes a three-way array into a prespecified number of outer product arrays. The constraint is that some vectors forming the outer product arrays are linearly dependent according to a prespecified pattern. This is known as the PARALIND family of decompositions. We provide both uniqueness conditions and partial uniqueness conditions for PARALIND, and show that these are improved and more precise variants of existing conditions. Our results are illustrated by means of examples.",

keywords = "PARALIND, CONFAC, parafac, candecomp, tensor decomposition, uniqueness, INDEPENDENT COMPONENT ANALYSIS, LOW-RANK APPROXIMATION, DS-CDMA SIGNALS, CANDECOMP/PARAFAC MODEL, DIVERGING COMPONENTS, ORDER TENSOR, 3-WAY ARRAYS, BLOCK TERMS, X-2 ARRAYS, SYSTEMS",

author = "Alwin Stegeman and Lam, {Tam T. T.}",

year = "2012",

doi = "10.1137/110847275",

language = "English",

volume = "33",

pages = "1250--1271",

journal = "SIAM Journal on Matrix Analysis and Applications",

number = "4",

}

TY - JOUR

T1 - Improved Uniqueness Conditions for Canonical Tensor Decompositions with Linearly Dependent Loadings

AU - Stegeman, Alwin

AU - Lam, Tam T. T.

PY - 2012

Y1 - 2012

N2 - In this paper, we derive improved uniqueness conditions for a constrained version of the canonical order-3 tensor decomposition, also known as Candecomp/Parafac (CP). CP decomposes a three-way array into a prespecified number of outer product arrays. The constraint is that some vectors forming the outer product arrays are linearly dependent according to a prespecified pattern. This is known as the PARALIND family of decompositions. We provide both uniqueness conditions and partial uniqueness conditions for PARALIND, and show that these are improved and more precise variants of existing conditions. Our results are illustrated by means of examples.

AB - In this paper, we derive improved uniqueness conditions for a constrained version of the canonical order-3 tensor decomposition, also known as Candecomp/Parafac (CP). CP decomposes a three-way array into a prespecified number of outer product arrays. The constraint is that some vectors forming the outer product arrays are linearly dependent according to a prespecified pattern. This is known as the PARALIND family of decompositions. We provide both uniqueness conditions and partial uniqueness conditions for PARALIND, and show that these are improved and more precise variants of existing conditions. Our results are illustrated by means of examples.

KW - PARALIND

KW - CONFAC

KW - parafac

KW - candecomp

KW - tensor decomposition

KW - uniqueness

KW - INDEPENDENT COMPONENT ANALYSIS

KW - LOW-RANK APPROXIMATION

KW - DS-CDMA SIGNALS

KW - CANDECOMP/PARAFAC MODEL

KW - DIVERGING COMPONENTS

KW - ORDER TENSOR

KW - 3-WAY ARRAYS

KW - BLOCK TERMS

KW - X-2 ARRAYS

KW - SYSTEMS

U2 - 10.1137/110847275

DO - 10.1137/110847275

M3 - Article

VL - 33

SP - 1250

EP - 1271

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 4

ER -

Improved Uniqueness Conditions for Canonical Tensor Decompositions with Linearly Dependent Loadings

Abstract

Keywords

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