Matrix and Tensor Decompositions in Signal Processing, Volume 2 E-Book

Gérard Favier

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 521

Veröffentlichungsjahr: 2021

Beschreibung

The second volume will deal with a presentation of the main matrix and tensor decompositions and their properties of uniqueness, as well as very useful tensor networks for the analysis of massive data. Parametric estimation algorithms will be presented for the identification of the main tensor decompositions. After a brief historical review of the compressed sampling methods, an overview of the main methods of retrieving matrices and tensors with missing data will be performed under the low rank hypothesis. Illustrative examples will be provided.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Introduction

I.1. What are the advantages of tensor approaches?

I.2. For what uses?

I.3. In what fields of application?

I.4. With what tensor decompositions?

I.5. With what cost functions and optimization algorithms?

I.6. Brief description of content

1 Matrix Decompositions

1.1. Introduction

1.2. Overview of the most common matrix decompositions

1.3. Eigenvalue decomposition

1.4. URV

H

decomposition

1.5. Singular value decomposition

1.6. CUR decomposition

2 Hadamard, Kronecker and Khatri–Rao Products

2.1. Introduction

2.2. Notation

2.3. Hadamard product

2.4. Kronecker product

2.5. Kronecker sum

2.6. Index convention

2.7. Commutation matrices

2.8. Relations between the

diag

operator and the Kronecker product

2.9. Khatri–Rao product

2.10. Relations between vectorization and Kronecker and Khatri–Rao products

2.11. Relations between the Kronecker, Khatri–Rao and Hadamard products

2.12. Applications

3 Tensor Operations

3.1. Introduction

3.2. Notation and particular sets of tensors

3.3. Notion of slice

3.4. Mode combination

3.5. Partitioned tensors or block tensors

3.6. Diagonal tensors

3.7. Matricization

3.8. Subspaces associated with a tensor and multilinear rank

3.9. Vectorization

3.10. Transposition

3.11. Symmetric/partially symmetric tensors

3.12. Triangular tensors

3.13. Multiplication operations

3.14. Inverse and pseudo-inverse tensors

3.15. Tensor decompositions in the form of factorizations

3.16. Inner product, Frobenius norm and trace of a tensor

3.17. Tensor systems and homogeneous polynomials

3.18. Hadamard and Kronecker products of tensors

3.19. Tensor extension

3.20. Tensorization

3.21. Hankelization

4 Eigenvalues and Singular Values of a Tensor

4.1. Introduction

4.2. Eigenvalues of a tensor of order greater than two

4.3. Best rank-one approximation

4.4. Orthogonal decompositions

4.5. Singular values of a tensor

5 Tensor Decompositions

5.1. Introduction

5.2. Tensor models

5.3. Examples of tensor models

Appendix Random Variables and Stochastic Processes

A1.1. Introduction

A1.2. Random variables

A1.3. Discrete-time random signals

A1.4. Application to system identification

References

Index

End User License Agreement

List of Illustrations

Introduction

Figure I.1.

Third-order PARAFAC model

Figure I.2.

Third-order Tucker model

Figure I.3.

Fourth-order TT model

Chapter 3

Figure 3.1.

Fibers of a third-order tensor

Figure 3.2.

Matrix slices of a third-order tensor

Figure 3.3.

Unfolding

B

J

1×

J

2

I

3

.

Appendix Random Variables and Stochastic Processes

Figure A1.1.

Symmetry regions of the third-order cumulant

List of Tables

Introduction

Table I.1.

Signal and image tensors

Table I.2.

Other fields of application

Table I.3.

Parametric complexity of the CPD, TD, and TT decompositions

Table I.4.

Cost functions for model estimation and recovery of missing data

Chapter 1

Table 1.1.

The most common matrix decompositions

Table 1.2.

Eigenvalues of a real square matrix

Table 1.3.

Eigenvalues and extrema of the Rayleigh quotient

Table 1.4.

Eigenvalue properties of certain matrix classes

Table 1.5.

Eigenvalues of positive/negative (semi-)definite matrices

Table 1.6.

Equivalent and similar matrices

Table 1.7.

Fundamental subspaces of

A ∈

I

×

J

Table 1.8.

Relations between the fundamental subspaces

Table 1.9.

Relations between left and right singular vectors

Table 1.10.

Definition and properties of the SVD

Table 1.11.

Relations between the left and right singular vectors

Table 1.12.

SVD and fundamental subspaces

Table 1.13.

Computation of an SVD and of the Moore-Penrose pseudo-inverse

Table 1.14.

Matrix norms

Table 1.15.

Rank-

k

approximation and approximation errors

Table 1.16.

Least squares estimator

Chapter 2

Table 2.1.

Hadamard, Kronecker, and Khatri-Rao products and Kronecker sum

Table 2.2.

Hadamard product of Hermitian positive (semi-)definite matrices

Table 2.3.

Basic relations satisfied by the Hadamard product

Table 2.4.

The

diag

operator and the Hadamard product

Table 2.5.

Basic relations satisfied by the Kronecker product of vectors

Table 2.6.

Rank-one matrices and Kronecker products of vectors

Table 2.7.

Other decompositions of

(u ⊗ v)(x ⊗ y)

H

Table 2.8.

Relations involving Kronecker products of vectors

Table 2.9.

Properties of the multiple Kronecker product

Table 2.10.

Identities involving matrix-vector Kronecker products

Table 2.11.

Rank, trace, determinant, and spectrum of a Kronecker product

Table 2.12.

Kronecker product of positive/negative definite matrices

Table 2.13.

Structural properties of the Kronecker product

Table 2.14.

Kronecker products of vectors and index convention

Table 2.15.

Kronecker products of matrices and index convention

Table 2.16.

Commutation matrices, vectorization, and Kronecker product

Table 2.17.

Khatri-Rao product and index convention

Table 2.18.

Relations between the Hadamard, Khatri-Rao, and Kronecker products

Table 2.19. Relations between the Kronecker, Khatri-Rao, and Hadamard products o...

Table 2.20. Other relations between the Kronecker, Khatri-Rao, and Hadamard prod...

Table 2.21.

Basic relations

Table 2.22.

Vectorization formulae

Table 2.23.

Partial derivatives of various functions

Table 2.24.

Derivatives of a scalar function and a vector function

Table 2.25.

Derivatives of a matrix function with respect to a matrix variable

Table 2.26.

Definitions of

in partitioned form

Table 2.27.

Other derivatives

Table 2.28.

Derivatives of matrix traces

Chapter 3

Table 3.1.

Notation for sets of indices and dimensions

Table 3.2.

Various sets of tensors

Table 3.3.

Multilinear forms and associated tensors

Table 3.4.

Multilinear forms and associated homogeneous polynomials

Table 3.5.

Notation for tensors

Table 3.6.

Matricization by index blocks for different sets of tensors

Table 3.7.

Block-wise transposition

Table 3.8.

Hermitian/symmetric tensors by blocks of order

P

Table 3.9.

Block symmetrized tensor

χ

of a third-order tensor

Table 3.10.

Outer products of matrices and tensors

Table 3.11.

Scalar elements of outer products

Table 3.12.

Matricized and vectorized forms of a rank-one third-order tensor

Table 3.13.

Mode-

p

products of a third-order tensor with a matrix

Table 3.14.

Matrix products and mode-

p

product

Table 3.15.

Different types of multiplication with tensors

Table 3.16.

Orthogonality and idempotence properties

Table 3.17.

Inverses of orthogonal/unitary tensors

Table 3.18.

Generalized inverses

Table 3.19.

Operations with the Einstein product

Table 3.20.

Properties of the Moore-Penrose pseudo-inverse

Table 3.21.

LS solutions of tensor linear systems

Table 3.22.

Matrix operations

Table 3.23.

Tensor operations

Chapter 4

Table 4.1.

Eigenpairs and generalized eigenpairs for tensors

Table 4.2.

Properties of hypercubic symmetric tensors

Chapter 5

Table 5.1.

Third-order Tucker model

Table 5.2.

Tucker-(2,3) and Tucker-(1,3) models

Table 5.3.

Third-order PARAFAC model

Table 5.4.

Fourth-order PARAFAC model

Table 5.5.

Variants of the third-order PARAFAC model

Table 5.6.

k

-rank of certain matrices and essential uniqueness of a third-order ...

Appendix Random Variables and Stochastic Processes

Table A1.1.

Some definitions for jointly distributed r.v.s

x

and

y

Table A1.2.

Definitions of uncorrelated and orthogonal r.v.s

x

and

y

Table A1.3.

Properties of r.v.s

Table A1.4. Definitions and properties of the second-order statistics of real ra...

Table A1.5.

Definitions and properties for real random signals

x

(

k

)

and

y

(

k

)

Table A1.6. Definitions and properties of second-order statistics for stationary...

Table A1.7.

Second-order statistics of the output of a linear system

Table A1.8.

Factors of the polyspectra

Table A1.9.

Bispectra and trispectra of the output of a linear system

Guide

Cover

Table of Contents

Title Page

Copyright

Introduction

Begin Reading

Appendix Random Variables and Stochastic Processes

References

Index

Other titles from iSTE in Digital Signal and Image Processing

End User License Agreement

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Matrices and Tensors with Signal Processing Set

coordinated byGérard Favier

Volume 2

Matrix and Tensor Decompositions in Signal Processing

Gérard Favier

First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK

www.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.wiley.com

© ISTE Ltd 2021The rights of Gérard Favier to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2021938218

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-155-0