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Matrix and Tensor Decompositions in Signal Processing, Volume 2
Gérard Favier
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eBook - ePub
Matrix and Tensor Decompositions in Signal Processing, Volume 2
Gérard Favier
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About This Book
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.
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1
Matrix Decompositions
1.1. Introduction
The goal of this chapter is to give an overview of the most important matrix decompositions, with a more detailed presentation of the eigenvalue decomposition (EVD) and singular value decomposition (SVD), as well as some of their applications. Matrix decompositions (also called factorizations) play a key role in matrix computation, in particular, for computing the pseudo-inverse of a matrix (see section 1.5.4), the low-rank approximation of a matrix (see section 1.5.7), the solution of a system of linear equations using the least squares (LS) method (see section 1.5.9), or for parametric estimation of nonlinear models using the ALS method, as illustrated in Chapter 5 with the estimation of tensor models.
Matrix decompositions have two goals. The first is to factorize a given matrix with structured factor matrices that are easier to invert, and the second is to reduce the dimensionality, in order to reduce both the memory capacity required to store the data and the computational cost of the data processing algorithms. After giving a brief overview of the most common decompositions, we will recall a few results about the eigenvalues of a matrix, and then present the EVD decomposition of a square matrix. The use of this decomposition will be illustrated by computing the powers of...