Tensors for data processing : theory, methods, and applications

Tensors for data processing : theory, methods, and applications /

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Bibliographic Details
Main Author: Liu, Yipeng (Author)
Corporate Author: ProQuest (Firm)
Format: Electronic eBook
Language:English
Published: San Diego : Elsevier Science & Technology, [2021]
Subjects:
Tensor algebra > Data processing.
Online Access:Connect to this title online (unlimited simultaneous users allowed; 325 uses per year)
Table of Contents:
  • Front Cover
  • Tensors for Data Processing
  • Copyright
  • Contents
  • List of contributors
  • Preface
  • 1 Tensor decompositions: computations, applications, and challenges
  • 1.1 Introduction
  • 1.1.1 What is a tensor?
  • 1.1.2 Why do we need tensors?
  • 1.2 Tensor operations
  • 1.2.1 Tensor notations
  • 1.2.2 Matrix operators
  • 1.2.3 Tensor transformations
  • 1.2.4 Tensor products
  • 1.2.5 Structural tensors
  • 1.2.6 Summary
  • 1.3 Tensor decompositions
  • 1.3.1 Tucker decomposition
  • 1.3.2 Canonical polyadic decomposition
  • 1.3.3 Block term decomposition
  • 1.3.4 Tensor singular value decomposition
  • 1.3.5 Tensor network
  • 1.3.5.1 Hierarchical Tucker decomposition
  • 1.3.5.2 Tensor train decomposition
  • 1.3.5.3 Tensor ring decomposition
  • 1.3.5.4 Other variants
  • 1.4 Tensor processing techniques
  • 1.5 Challenges
  • References
  • 2 Transform-based tensor singular value decomposition in multidimensional image recovery
  • 2.1 Introduction
  • 2.2 Recent advances of the tensor singular value decomposition
  • 2.2.1 Preliminaries and basic tensor notations
  • 2.2.2 The t-SVD framework
  • 2.2.3 Tensor nuclear norm and tensor recovery
  • 2.2.4 Extensions
  • 2.2.4.1 Nonconvex surrogates
  • 2.2.4.2 Additional prior knowledge
  • 2.2.4.3 Multiple directions and higher-order tensors
  • 2.2.5 Summary
  • 2.3 Transform-based t-SVD
  • 2.3.1 Linear invertible transform-based t-SVD
  • 2.3.2 Beyond invertibility and data adaptivity
  • 2.4 Numerical experiments
  • 2.4.1 Examples within the t-SVD framework
  • 2.4.2 Examples of the transform-based t-SVD
  • 2.5 Conclusions and new guidelines
  • References
  • 3 Partensor
  • 3.1 Introduction
  • 3.1.1 Related work
  • 3.1.2 Notation
  • 3.2 Tensor decomposition
  • 3.2.1 Matrix least-squares problems
  • 3.2.1.1 The unconstrained case
  • 3.2.1.2 The nonnegative case
  • 3.2.1.3 The orthogonal case
  • 3.2.2 Alternating optimization for tensor decomposition
  • 3.3 Tensor decomposition with missing elements
  • 3.3.1 Matrix least-squares with missing elements
  • 3.3.1.1 The unconstrained case
  • 3.3.1.2 The nonnegative case
  • 3.3.2 Tensor decomposition with missing elements: the unconstrained case
  • 3.3.3 Tensor decomposition with missing elements: the nonnegative case
  • 3.3.4 Alternating optimization for tensor decomposition with missing elements
  • 3.4 Distributed memory implementations
  • 3.4.1 Some MPI preliminaries
  • 3.4.1.1 Communication domains and topologies
  • 3.4.1.2 Synchronization among processes
  • 3.4.1.3 Point-to-point communication operations
  • 3.4.1.4 Collective communication operations
  • 3.4.1.5 Derived data types
  • 3.4.2 Variable partitioning and data allocation
  • 3.4.2.1 Communication domains
  • 3.4.3 Tensor decomposition
  • 3.4.3.1 The unconstrained and the nonnegative case
  • 3.4.3.2 The orthogonal case
  • 3.4.3.3 Factor normalization and acceleration
  • 3.4.4 Tensor decomposition with missing elements
  • 3.4.4.1 The unconstrained case

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