Master's Thesis Analysis of 2 × 2 × 2 Tensors Ana Rovi

115 Pages·2010·1.27 MB·English

Master’s Thesis
Analysis of 2 × 2 × 2 Tensors
Ana Rovi
LiTH - MAT - INT - A - - 2010/01 - - SE

Analysis of 2 × 2 × 2 Tensors
MAI Mathematics, Link¨opings Universitet
Universidad Nacional de Educaci´on a Distancia. Espan˜a
Ana Rovi
LiTH - MAT - INT - A - - 2010/01 - - SE
Master’s Thesis: 30 ETCS
Supervisor: G¨oran Bergqvist,
MAI Mathematics, Link¨opings Universitet
Examiner: G¨oran Bergqvist,
MAI Mathematics, Link¨opings Universitet
Link¨oping: June 2010

Datum
Avdelning, Institution
Date
Division, Department
Matematiska Institutionen June 2010
581 83 LINKO¨PING
SWEDEN
Spr˚ak Rapporttyp ISBN
Language Report category
ISRN
Svenska/Swedish Licentiatavhandling
LiTH - MAT - INT - A - - 2010/01 - - SE
x Engelska/English x Examensarbete
C-uppsats Serietitel och serienummer ISSN
0348-2960
D-uppsats Titleofseries,numbering
O¨vrigrapport
URL f¨or elektronisk version
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-
56762
Titel
Analysis of 2×2×2 Tensors
Title
Fo¨rfattare Ana Rovi
Author
Sammanfattning
Abstract
The question about how to determine the rank of a tensor has been widely studied
in the literature. However the analytical methods to compute the decomposition of
tensors have not been so much developed even for low-rank tensors.
In this report we present analytical methods for finding real and complex PARAFAC
decompositions of 2×2×2 tensors before computing the actual rank of the tensor.
These methods are also implemented in MATLAB.
We also consider the question of how best lower-rank approximation gives rise to
problems of degeneracy, and give some analytical explanations for these issues.
Nyckelord
Tensordecomposition. PARAFACdecomposition. AlternatingLeastSquares. Tensor
Keyword
Rank. Typicalandgenericrank. Bestlower-rankapproximation. TensorToolboxfor
MATLAB. Degeneracy. Uniqueness. Tensor classification
,

To my parents

Abstract
The question about how to determine the rank of a tensor has been widely
studied in the literature. However the analytical methods to compute the de-
composition of tensors have not been so much developed even for low-rank
tensors.
In this report we present analytical methods for finding real and complex
PARAFACdecompositionsof2×2×2tensorsbeforecomputingtheactualrank
of the tensor. These methods are also implemented in MATLAB.
We also consider the question of how best lower-rank approximation gives
rise to problems of degeneracy, and give some analytical explanations for these
issues.
Keywords: Tensor decomposition. PARAFAC decomposition. Alternating
Least Squares. Tensor Rank. Typical and generic rank. Best lower-rank
approximation. Tensor Toolbox for MATLAB. Degeneracy. Uniqueness.
Tensor classification
Rovi, 2010. i
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