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Third-Order Tensors as Operators on Matrices: : A Theoretical and Computational Framework with Applications in Imaging

Authors:

Misha E. Kilmer,

Randy C. HooverAuthors Info & Claims

SIAM Journal on Matrix Analysis and Applications, Volume 34, Issue 1

Pages 148 - 172

https://doi.org/10.1137/110837711

Published: 01 January 2013 Publication History

Abstract

Recent work by Kilmer and Martin [Linear Algebra Appl., 435 (2011), pp. 641--658] and Braman [Linear Algebra Appl., 433 (2010), pp. 1241--1253] provides a setting in which the familiar tools of linear algebra can be extended to better understand third-order tensors. Continuing along this vein, this paper investigates further implications including (1) a bilinear operator on the matrices which is nearly an inner product and which leads to definitions for length of matrices, angle between two matrices, and orthogonality of matrices, and (2) the use of t-linear combinations to characterize the range and kernel of a mapping defined by a third-order tensor and the t-product and the quantification of the dimensions of those sets. These theoretical results lead to the study of orthogonal projections as well as an effective Gram--Schmidt process for producing an orthogonal basis of matrices. The theoretical framework also leads us to consider the notion of tensor polynomials and their relation to tensor eigentuples defined in the recent article by Braman. Implications for extending basic algorithms such as the power method, QR iteration, and Krylov subspace methods are discussed. We conclude with two examples in image processing: using the orthogonal elements generated via a Golub--Kahan iterative bidiagonalization scheme for object recognition and solving a regularized image deblurring problem.

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Cited By

Cao ZWei YXie P(2025)An L-DEIM induced high order tensor interpolatory decompositionJournal of Computational and Applied Mathematics10.1016/j.cam.2024.116143453:COnline publication date: 1-Jan-2025
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https://dl.acm.org/doi/10.1145/3696669
Liang TShen QWang SChen YZhang GChen J(2024)Data Completion-Guided Unified Graph Learning for Incomplete Multi-View ClusteringACM Transactions on Knowledge Discovery from Data10.1145/366429018:8(1-23)Online publication date: 9-May-2024
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Index Terms

Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Linear algebra algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on matrices

Index terms have been assigned to the content through auto-classification.

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Published In

cover image SIAM Journal on Matrix Analysis and Applications

SIAM Journal on Matrix Analysis and Applications Volume 34, Issue 1

2013

281 pages

ISSN:0895-4798

DOI:10.1137/sjmael.34.1

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© 2013, Society for Industrial and Applied Mathematics.

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2013

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Cited By

Cao ZWei YXie P(2025)An L-DEIM induced high order tensor interpolatory decompositionJournal of Computational and Applied Mathematics10.1016/j.cam.2024.116143453:COnline publication date: 1-Jan-2025
https://dl.acm.org/doi/10.1016/j.cam.2024.116143
Yu MTang ZLiang XZhang XLi ZZhang X(2024)Robust Hashing with Deep Features and Meixner Moments for Image Copy DetectionACM Transactions on Multimedia Computing, Communications, and Applications10.1145/369666920:12(1-23)Online publication date: 23-Sep-2024
https://dl.acm.org/doi/10.1145/3696669
Liang TShen QWang SChen YZhang GChen J(2024)Data Completion-Guided Unified Graph Learning for Incomplete Multi-View ClusteringACM Transactions on Knowledge Discovery from Data10.1145/366429018:8(1-23)Online publication date: 9-May-2024
https://dl.acm.org/doi/10.1145/3664290
Gu ZFeng SLi ZYuan JLiu J(2024)NOODLE: Joint Cross-View Discrepancy Discovery and High-Order Correlation Detection for Multi-View Subspace ClusteringACM Transactions on Knowledge Discovery from Data10.1145/365330518:6(1-23)Online publication date: 29-Apr-2024
https://dl.acm.org/doi/10.1145/3653305
Ji HXie KWen JZhang QXie GLiang W(2024)FineMon: An Innovative Adaptive Network Telemetry Scheme for Fine-Grained, Multi-Metric Data Monitoring with Dynamic Frequency Adjustment and Enhanced Data RecoveryProceedings of the ACM on Management of Data10.1145/36392672:1(1-26)Online publication date: 26-Mar-2024
https://dl.acm.org/doi/10.1145/3639267
Ji JFeng SLi YBaeza-Yates RBonchi F(2024)Tensorized Unaligned Multi-view Clustering with Multi-scale Representation LearningProceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining10.1145/3637528.3671689(1246-1256)Online publication date: 25-Aug-2024
https://dl.acm.org/doi/10.1145/3637528.3671689
Wang ZSo HZoubir A(2024)Low-Rank Tensor Completion via Novel Sparsity-Inducing RegularizersIEEE Transactions on Signal Processing10.1109/TSP.2024.342427272(3519-3534)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1109/TSP.2024.3424272
Cai BLu GLi HSong W(2024)Tensorized Scaled Simplex Representation for Multi-View ClusteringIEEE Transactions on Multimedia10.1109/TMM.2024.335564926(6621-6631)Online publication date: 18-Jan-2024
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Qin YTang ZWu HFeng G(2024)Flexible Tensor Learning for Multi-View Clustering With Markov ChainIEEE Transactions on Knowledge and Data Engineering10.1109/TKDE.2023.330562436:4(1552-1565)Online publication date: 1-Apr-2024
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Qin YPu NWu H(2024)Elastic Multi-View Subspace Clustering With Pairwise and High-Order CorrelationsIEEE Transactions on Knowledge and Data Engineering10.1109/TKDE.2023.329349836:2(556-568)Online publication date: 1-Feb-2024
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