In an increasingly digitized and interconnected world, the volume of multidimensional, multimodal, heterogeneous and often incomplete data to be processed keeps growing at an unprecedented rate. This is why, today, tensors play a central role in many fields of application, particularly in signal processing and machine learning, for the representation, compression, analysis, mining, fusion and classification of data, and also for designing wireless communication systems and characterizing biomedical problems.
This is the second volume within the “Matrices and Tensors with Signal Processing” set and provides a didactic and comprehensive presentation of the main tensor operations and decompositions, with many illustrative examples.
An overview of the main matrix decompositions is given, and properties of the Hadamard, Kronecker and Khatri-Rao products are highlighted using an index convention which is very useful for tensor calculus. Several classes of tensors and tensor-based applications are briefly described. Tensor operations are detailed, including reshaping, transposition, symmetrization, inversion, pseudo-inversion, tensorization, Hankelization and different types of multiplication with tensors. Various notions of eigenvalue and singular value are also defined for tensors. The main tensor decompositions are studied in depth, along with two standard algorithms – namely ALS and HOSVD – commonly used for numerically computing tensor models. Some illustrations of tensor decompositions are provided for signals and systems modeling.
1. Matrix Decompositions.
2. Hadamard, Kronecker and Khatri–Rao Products.
3. Tensor Operations.
4. Eigenvalues and Singular Values of a Tensor.
5. Tensor Decompositions.
Gérard Favier is currently Emeritus Research Director at CNRS and I3S Laboratory in Sophia Antipolis, France. His research interests include nonlinear system modeling and identification, signal processing applications, tensor models with associated algorithms for big data processing and tensor approaches for MIMO communication systems.
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