Matrices and Tensors in Signal Processing Set
coordinated by Gérard Favier
Volume 1
Edited by
Gérard Favier
First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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The 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: 2019945792
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-154-3
This book is part of a collection of four books about matrices and tensors, with applications to signal processing. Although the title of this collection suggests an orientation toward signal processing, the results and methods presented should also be of use to readers of other disciplines.
Writing books on matrices is a real challenge given that so many excellent books on the topic have already been written1. How then to stand out from the existing, and which Ariadne’s thread to unwind? A way to distinguish oneself was to treat in parallel matrices and tensors. Viewed as extensions of matrices with orders higher than two, the latter have many similarities with matrices, but also important differences in terms of rank, uniqueness of decomposition, as well as potentiality for representing multi-dimensional, multi-modal, and inaccurate data. Moreover, regarding the guiding thread, it consists in presenting structural foundations, then both matrix and tensor decompositions, in addition to related processing methods, finally leading to applications, by means of a presentation as self-contained as possible, and with some originality in the topics being addressed and the way they are treated.
Therefore, in Volume 2, we shall use an index convention generalizing Einstein’s summation convention, to write and to demonstrate certain equations involving multi-index quantities, as is the case with matrices and tensors. A chapter will be dedicated to Hadamard, Kronecker, and Khatri–Rao products, which play a very important role in matrix and tensor computations.
After a reminder of main matrix decompositions, including a detailed presentation of the SVD (for singular value decomposition), we shall present different tensor operations, as well as the two main tensor decompositions which will be the basis of both fundamental and applied developments, in the last two volumes. These standard tensor decompositions can be seen as extensions of matrix SVD to tensors of order higher than two. A few examples of equations for representing signal processing problems will be provided to illustrate the use of such decompositions. A chapter will be devoted to structured matrices. Different properties will be highlighted, and extensions to tensors of order higher than two will be presented. Two other chapters will concern quaternions and quaternionic matrices, on the one hand, and polynomial matrices, on the other hand.
In Volume 3, an overview of several tensor models will be carried out by taking some constraints (structural, linear dependency of factors, sparsity, and non-negativity) into account. Some of these models will be used in Volume 4, for the design of digital communication systems. Tensor trains and tensor networks will also be presented for the representation and analysis of massive data (big data). The algorithmic aspect will be taken into consideration with the presentation of different processing methods.
Volume 4 will mainly focus on tensorial approaches for array processing, wireless digital communications (first point-to-point, then cooperative), modeling and identification of both linear and nonlinear systems, as well as the reconstruction of missing data in data matrices and tensors, the so-called problems of matrix and tensor completion. For these applications, tensor-based models will be more particularly detailed. Monte Carlo simulation results will be provided to illustrate some of the tensorial methods. This will be particularly the case of semi-blind receivers recently developed for wireless communication systems.
Matrices and tensors, and more generally linear algebra and multilinear algebra, are at the same time exciting, extensive, and fundamental topics equally important for teaching and researching as for applications. It is worth noting here that the choices made for the content of the books of this collection have not been guided by educational programs, which explains some gaps compared to standard algebra treaties. The guiding thread has been rather to present the definitions, properties, concepts and results necessary for a good understanding of processing methods and applications considered in these books. In addition to the great diversity of topics, another difficulty resided in the order in which they should be addressed, due to the fact that a lot of topics overlap, certain notions or/and some results being sometimes used before they have been defined or/and demonstrated, which requires the reader to be referred to sections or chapters that follow.
Four particularities should be highlighted. The first relates to the close relationship between some of the topics being addressed, certain methods presented and recent research results, particularly with regard to tensorial approaches for signal processing. The second reflects the will to situate the results stated in their historical context, using some biographical information on certain authors being cited, as well as lists of references comprehensive enough to deepen specific results, and also to extend the biographical sources provided. This has motivated the introductory chapter entitled “Historical elements of matrices and tensors.”
The last two characteristics concern the presentation and illustration of properties and methods under consideration. Some will be provided without demonstration because of their simplicity or availability in numerous books thereabout. Others will be demonstrated, either for pedagogical reasons, since their knowledge should allow for better understanding the results being demonstrated, or because of the difficulty to find them in the literature, or still due to the originality of the proposed demonstrations as it will be the case, for example, of those making use of the index convention. The use of many tables should also be noted with the purpose of recalling key results while presenting them in a synthetic and comparative manner.
Finally, numerous examples will be provided to illustrate certain definitions, properties, decompositions, and methods presented. This will be particularly the case for the fourth book dedicated to applications of tensorial tools, which has been my main source of motivation. After 15 years of works dedicated to research (pioneering for some), aiming to use tensor decompositions for modeling and identifying nonlinear dynamical systems, and for designing wireless communication systems based on new tensor models, it seemed to me useful to share this experience and this scientific path for trying to make tensor tools as accessible as possible and to motivate new applications based on tensor approaches.
This first book, whose content is described below, provides an introduction to matrices and tensors based on the structures of vector spaces and tensor spaces, along with the presentation of fundamental concepts and results. In the first part (Chapters 2 and 3), a refresher of the mathematical bases related to classical algebraic structures is presented, by way of bringing forward a growing complexity of the structures under consideration, ranging from monoids to vector spaces, and to algebras. The notions of norm, inner product, and Hilbert basis are detailed in order to introduce Banach and Hilbert spaces. The Hilbertian approach, which is fundamental for signal processing, is illustrated based on two methods widely employed for signal representation and analysis, as well as for function approximation, namely, Fourier and orthogonal polynomial series.
Chapter 4 is dedicated to matrix algebra. The notions of fundamental subspaces associated with a matrix, rank, determinant, inverse, auto-inverse, generalized inverse, and pseudo-inverse are described therein. Matrix representations of linear and bilinear/sesquilinear maps are established. The effect of a change of basis is studied, leading to the definition of equivalent, similar, and congruent matrices. The notions of eigenvalue and eigenvector are then defined, ending with matrix eigendecomposition, and in some cases, with its diagonalization, which are topics to be covered in Volume 2. The case of certain structured matrices, such as symmetric/hermitian matrices and orthogonal/unitary matrices, is more specifically considered. The interpretation of eigenvalues as extrema of the Rayleigh quotient is presented, before introducing the notion of generalized eigenvalues.
In Chapter 5, we consider partitioned matrices. This type of structure is inherent to matrix products in general, and Kronecker and Khatri–Rao products in particular.
Partitioned matrices corresponding to block-diagonal and block-triangular matrices, as well as to Jordan forms are described. Next, transposition/conjugate transposition, vectorization, addition and multiplication operations, as well as Hadamard and Kronecker products, are presented for partitioned matrices. Elementary operations and associated matrices allowing the partitioned matrices to be decomposed are detailed. These operations are then utilized for block-triangularization, block-diagonalization, and block-inversion of partitioned matrices. The matrix inversion lemma, which is widely used in signal processing, is deduced from block-inversion formulae. This lemma is used to demonstrate a few inversion formulae very often encountered in calculations. Fundamental results on generalized inverse, determinant, and rank of partitioned matrices are presented. The Levinson algorithm is demonstrated using the formula for inverting a partitioned square matrix, recursively with respect to the matrix order. This algorithm, which is one of the most famous in signal processing, allows to efficiently solve the problem of parameter estimation of autoregressive (AR) models and linear predictors, recursively with respect to the order of the AR model and of the predictor, respectively. To illustrate the results of Chapter 3 relatively to orthogonal projection, it is shown that forward and backward linear predictors, optimal in the sense of the MMSE (minimum mean squared error), can be interpreted in terms of orthogonal projectors on subspaces of the Hilbert space of the second-order stationary random signals.
In Chapter 6, hypermatrices and tensors are introduced in close connection with multilinear maps and multilinear forms. Hypermatrix vector spaces are first defined, along with operations such as inner product and contraction of hypermatrices – the particular case of the n-mode hypermatrix-matrix product being considered in more detail. Hypermatrices associated with multilinear forms and maps are highlighted, and symmetric hypermatrices are introduced through the definition of symmetric multilinear forms. Then, tensors of order N > 2 are defined in a formal way as elements of a tensor space, i.e., a tensor product of N vector spaces. The effect of changes to the tensor space on the coordinate hypermatrix of a tensor are analyzed. In addition, some attributes of the tensor product are described, with a focus on the so-called universal property. Following this, the notions of a rank based on the canonical polyadic decomposition (CPD) of a tensor are introduced, as well as the ranking of a tensor’s eigenvalues and singular values. These highlight the similarities and the differences between matrices, and tensors of order greater than two. Finally, the concept of tensor unfolding is illustrated via the definition of isomorphisms of tensor spaces.
I want to thank my colleagues Sylvie Icart and Vicente Zarzoso for their review of some chapters and Henrique de Morais Goulart, who co-authored Chapter 4.
GÉRARD FAVIER
August 2019
favier@i3s.unice.fr
The objective of this introduction is by no means to outline a rigorous and comprehensive historical background of the theory of matrices and tensors. Such a historical record should be the work of a historian of mathematics and would require thorough bibliographical research, including reading the original publications of authors cited to analyze and reconstruct the progress of mathematical thinking throughout years and collaborations. A very interesting illustration of this type of analysis is provided, for example, in the form of a “representation of research networks”1, over the period 1880–1907, in which are identified the interactions and influences of some mathematicians, such as James Joseph Sylvester (1814–1897), Karl Theodor Weierstrass (1815–1897), Arthur Cayley (1821–1895), Leopold Kronecker (1823–1891), Ferdinand Georg Frobenius (1849–1917), or Eduard Weyr (1852–1903), with respect to the theory of matrices, the theory of numbers (quaternions, hypercomplex numbers), bilinear forms, and algebraic structures.
Our modest goal here is to locate in time the contributions of a few mathematicians and physicists2 who have laid the foundations for the theory of matrices and tensors, and to whom we will refer later in our presentation. This choice is necessarily very incomplete.
The first studies of determinants that preceded those of matrices were conducted independently by the Japanese mathematician Seki Kowa (1642–1708) and the German mathematician Gottfried Leibniz (1646–1716), and then by the Scottish mathematician Colin Maclaurin (1698–1746) for solving 2 × 2 and 3 × 3 systems of linear equations. These works were then generalized by the Swiss mathematician Gabriel Cramer (1704–1752) for the resolution of n × n systems, leading, in 1750, to the famous formulae that bear his name, whose demonstration is due to Augustin-Louis Cauchy (1789–1857).
In 1772, Théophile Alexandre Vandermonde (1735–1796) defined the notion of determinant, and Pierre-Simon Laplace (1749–1827) formulated the computation of determinants by means of an expansion according to a row or a column, an expansion which will be presented in section 4.11.1. In 1773, Joseph-Louis Lagrange (1736–1813) discovered the link between the calculation of determinants and that of volumes. In 1812, Cauchy used, for the first time, the determinant in the sense that it has today, and he established the formula for the determinant of the product of two rectangular matrices, a formula which was found independently by Jacques Binet (1786–1856), and which is called nowadays the Binet–Cauchy formula.
In 1810, Johann Carl Friedrich Gauss (1777–1855) introduced a notation using a table, similar to matrix notation, to write a 3 × 3 system of linear equations, and he proposed the elimination method, known as Gauss elimination through pivoting, to solve it. This method, also known as Gauss–Jordan elimination method, was in fact known to Chinese mathematicians (first century). It was presented in a modern form, by Gauss, when he developed the least squares method, first published by Adrien-Marie Legendre (1752–1833), in 1805.
Several determinants of special matrices are designated by the names of their authors, such as Vandermonde’s, Cauchy’s, Hilbert’s, and Sylvester’s determinants. The latter of whom used the word “matrix” for the first time in 1850, to designate a rectangular table of numbers. The presentation of the determinant of an nth-order square matrix as an alternating n-linear function of its n column vectors is due to Weierstrass and Kronecker, at the end of the 19th century.
The foundations of the theory of matrices were laid in the 19th century around the following topics: determinants for solving systems of linear equations, representation of linear transformations and quadratic forms (a topic which will be addressed in detail in Chapter 4), matrix decompositions and reductions to canonical forms, that is to say, diagonal or triangular forms such as the Jordan (1838–1922) normal form with Jordan blocks on the diagonal, introduced by Weierstrass, the block-triangular form of Schur (1875–1941), or the Frobenius normal form that is a block-diagonal matrix, whose blocks are companion matrices.
A history of the theory of matrices in the 19th century was published by Thomas Hawkins3 in 1974, highlighting, in particular, the contributions of the British mathematician Arthur Cayley, seen by historians as one of the founders of the theory of matrices. Cayley laid the foundations of the classical theory of determinants4 in 1843. He then developed matrix computation5 by defining certain matrix operations as the product of two matrices, the transposition of the product of two matrices, and the inversion of a 3 × 3 matrix using cofactors, and by establishing different properties of matrices, including, namely, the famous Cayley–Hamilton theorem which states that every square matrix satisfies its characteristic equation. This result highlighted for the fourth order by William Rowan Hamilton (1805–1865), in 1853, for the calculation of the inverse of a quaternion, was stated in the general case by Cayley in 1857, but the demonstration for any arbitrary order is due to Frobenius, in 1878.
An important part of the theory of matrices concerns the spectral theory, namely, the notions of eigenvalue and characteristic polynomial. Directly related to the integration of systems of linear differential equations, this theory has its origins in physics, and more particularly in celestial mechanics for the study of the orbits of planets, conducted in the 18th century by mathematicians, physicists, and astronomers such as Lagrange and Laplace, then in the 19th century by Cauchy, Weierstrass, Kronecker, and Jordan.
The names of certain matrices and associated determinants are those of the mathematicians who have introduced them. This is the case, for example, for Alexandre Théophile Vandermonde (1735–1796) who gave his name to a matrix whose elements on each row (or each column) form a geometric progression and whose determinant is a polynomial. It is also the case for Carl Jacobi (1804–1851) and Ludwig Otto Hesse (1811–1874), for Jacobian and Hessian matrices, namely, the matrices of first- and second-order partial derivatives of a vector function, whose determinants are called Jacobian and Hessian, respectively. The same is true for the Laplacian matrix or Laplace matrix, which is used to represent a graph. We can also mention Charles Hermite (1822–1901) for Hermitian matrices, related to the so-called Hermitian forms (see section 4.15). Specific matrices such as Fourier (1768–1830) and Hadamard (1865–1963) matrices are directly related to the transforms of the same name. Similarly, Householder (1904–1993) and Givens (1910–1993) matrices are associated with transformations corresponding to reflections and rotations, respectively. The so-called structured matrices, such as Hankel (1839–1873) and Toeplitz (1881–1943) matrices, play a very important role in signal processing.
Matrix decompositions are widely used in numerical analysis, especially to solve systems of equations using the method of least squares. This is the case, for example, of EVD (eigenvalue decomposition), SVD (singular value decomposition), LU, QR, UD, Cholesky (1875–1918), and Schur (1875–1941) decompositions, which will be presented in Volume 2.
Just as matrices and matrix computation play a fundamental role in linear algebra, tensors and tensor computation are at the origin of multilinear algebra. It was in the 19th century that tensor analysis first appeared, along with the works of German mathematicians Georg Friedrich Bernhard Riemann6 (1826–1866) and Elwin Bruno Christoffel (1829–1900) in geometry (non-Euclidean), introducing the index notation and notions of metric, manifold, geodesic, curved space, curvature tensor, which gave rise to what is today called Riemannian geometry and differential geometry.
It was the Italian mathematician Gregorio Ricci-Curbastro (1853–1925) with his student Tullio Levi-Civita (1873–1941) who were the founders of the tensor calculus, then called absolute differential calculus7, with the introduction of the notion of covariant and contravariant components, which was used by Albert Einstein (1879–1955) in his theory of general relativity, in 1915.
Tensor calculus originates from the study of the invariance of quadratic forms under the effect of a change of coordinates and, more generally, from the theory of invariants initiated by Cayley8, with the introduction of the notion of hyperdeterminant which generalizes matrix determinants to hypermatrices. Refer to the article by Crilly9 for an overview of the contribution of Cayley on the invariant theory. This theory was developed by Jordan and Kronecker and involved controversy10 between these two authors, then continued by David Hilbert (1862–1943), Elie Joseph Cartan (1869–1951), and Hermann Klaus Hugo Weyl (1885–1955), for algebraic forms (or homogeneous polynomials), or for symmetric tensors11. A historical review of the theory of invariants was made by Dieudonné and Carrell12.
This property of invariance vis-à-vis the coordinate system characterizes the laws of physics and, thus, mathematical models of physics. This explains that tensor calculus is one of the fundamental mathematical tools for writing and studying equations that govern physical phenomena. This is the case, for example, in general relativity, in continuum mechanics, for the theory of elastic deformations, in electromagnetism, thermodynamics, and so on.
The word tensor was introduced by the German physicist Woldemar Voigt (1850–1919), in 1899, for the geometric representation of tensions (or pressures) and deformations in a body, in the areas of elasticity and crystallography. Note that the word tensor was introduced independently by the Irish mathematician, physicist and astronomer William Rowan Hamilton (1805–1865), in 1846, to designate the modulus of a quaternion13.
As we have just seen in this brief historical overview, tensor calculus was used initially in geometry and to describe physical phenomena using tensor fields, facilitating the application of differential operators (gradient, divergence, rotational, and Laplacian) to tensor fields14.
Thus, we define the electromagnetic tensor (or Maxwell’s (1831–1879) tensor) describing the structure of the electromagnetic field, the Cauchy stress tensor, and the deformation tensor (or Green–Lagrange deformation tensor), in continuum mechanics, and the fourth-order curvature tensor (or Riemann–Christoffel tensor) and the third-order torsion tensor (or Cartan tensor15) in differential geometry.
After their introduction as computational and representation tools in physics and geometry, tensors have been the subject of mathematical developments related to polyadic decomposition (Hitchcock 1927) aiming to generalize dyadic decompositions, that is to say, matrix decompositions such as SVD.
Then emerged their applications as tools for the analysis of three-dimensional data generalizing matrix analysis to sets of matrices, viewed as arrays of data characterized by three indices. We can mention here the works of pioneers in factor analysis by Cattell16 and Tucker17 in psychometrics (Cattell 1944; Tucker 1966), and Harshman18 in phonetics (Harshman 1970) who have introduced Tucker’s and PARAFAC (parallel factors) decompositions. This last one was proposed independently by Carroll and Chang (1970), under the name of canonical decomposition (CANDECOMP), following the publication of an article by Wold (1966), with the objective to generalize the (Eckart and Young 1936) decomposition, that is, SVD, to arrays of order higher than two. This decomposition was then called CP (for CANDECOMP/PARAFAC) by Kiers (2000). For an overview of tensor methods applied to data analysis, the reader should consult the books by Coppi and Bolasco (1989) and Kroonenberg (2008).
From the early 1990s, tensor analysis, also called multi-way analysis, has also been widely used in chemistry, and more specifically in chemometrics (Bro 1997). Refer to, for example, the book by Smilde et al. (2004) for a description of various applications in chemistry.
In parallel, at the end of the 1980s, statistic “objects,” such as moments and cumulants of order higher than two, have naturally emerged as tensors (McCullagh 1987). Tensor-based applications were then developed in signal processing for solving the problem of blind source separation using cumulants (Cardoso 1990, 1991; Cardoso and Comon 1990). The book by Comon and Jutten (2010) outlines an overview of methods for blind source separation.
In the early 2000s, tensors were used for modeling digital communication systems (Sidiropoulos et al. 2000a), for array processing (Sidiropoulos et al. 2000b), for multi-dimensional harmonics recovery (Haardt et al. 2008; Jiang et al. 2001; Sidiropoulos 2001), and for image processing, more specifically for face recognition (Vasilescu and Terzopoulos 2002). The field of wireless communication systems has then given rise to a large number of tensor models (da Costa et al. 2018; de Almeida and Favier 2013; de Almeida et al. 2008; Favier et al. 2012a; Favier and de Almeida 2014b; Favier et al. 2016). These models will be covered in a chapter of Volume 3. Tensors have also been used for modeling and parameter estimation of dynamic systems both linear (Fernandes et al. 2008, 2009a) and nonlinear, such as Volterra systems (Favier and Bouilloc 2009a, 2009b, 2010) or Wiener-Hammerstein systems (Favier and Kibangou 2009a, 2009b; Favier et al. 2012b; Kibangou and Favier 2008, 2009, 2010), and for modeling and estimating nonlinear communication channels (Bouilloc and Favier 2012; Fernandes et al. 2009b, 2011; Kibangou and Favier 2007). These different tensor-based applications in signal processing will be addressed in Volume 4.
Many applications of tensors also concern speech processing (Nion et al. 2010), MIMO radar (Nion and Sidiropoulos 2010), and biomedical signal processing, particularly for electroencephalography (EEG) (Cong et al. 2015; de Vos et al. 2007; Hunyadi et al. 2016), and electrocardiography (ECG) signals (Padhy et al. 2018), magnetic resonance imaging (MRI) (Schultz et al. 2014), or hyperspectral imaging (Bourennane et al. 2010; Velasco-Forero and Angulo 2013), among many others. Today, tensors viewed as multi-index tables are used in many areas of application for the representation, mining, analysis, and fusion of multi-dimensional and multi-modal data (Acar and Yener 2009; Cichocki 2013; Lahat et al. 2015; Morup 2011).
A very large number of books address linear algebra and matrix calculus, for example: Gantmacher (1959), Greub (1967), Bellman (1970), Strang (1980), Horn and Johnson (1985, 1991), Lancaster and Tismenetsky (1985), Noble and Daniel (1988), Barnett (1990), Rotella and Borne (1995), Golub and Van Loan (1996), Lütkepohl (1996), Cullen (1997), Zhang (1999), Meyer (2000), Lascaux and Théodor (2000), Serre (2002), Abadir and Magnus (2005), Bernstein (2005), Gourdon (2009), Grifone (2011), and Aubry (2012).
For multilinear algebra and tensor calculus, there are much less reference books, for example: Greub (1978), McCullagh (1987), Coppi and Bolasco (1989), Smilde et al. (2004), Kroonenberg (2008), Cichocki et al. (2009), and Hackbusch (2012). For an introduction to multilinear algebra and tensors, see Ph.D. theses by de Lathauwer (1997) and Bro (1998). The following synthesis articles can also be consulted: (Bro 1997; Cichocki et al. 2015; Comon 2014; Favier and de Almeida 2014a; Kolda and Bader 2009; Lu et al. 2011; Papalexakis et al. 2016; Sidiropoulos et al. 2017).