Systems of Polynomial Equations, Higher-order Tensor Decompositions, and Multidimensional Harmonic Retrieval: A Unifying Framework. Part I: The Canonical Polyadic Decomposition

DOI10.1137/17M1150050zbMath1471.13061OpenAlexW3174231256MaRDI QIDQ4997839

Lieven De Lathauwer, Jeroen Vanderstukken

Publication date: 30 June 2021

Published in: SIAM Journal on Matrix Analysis and Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1137/17m1150050

zbMATH Keywords

Vandermonde matrixmultilinear algebrasystem of polynomial equationscanonical polyadic decompositionMacaulay matrixharmonic retrieval

Mathematics Subject Classification ID

Multilinear algebra, tensor calculus (15A69) Numerical computation of roots of polynomial equations (65H04) Solving polynomial systems; resultants (13P15)

Related Items (2)

A Fast Algorithm for Computing Macaulay Null Spaces of Bivariate Polynomial Systems ⋮ Systems of Polynomial Equations, Higher-Order Tensor Decompositions, and Multidimensional Harmonic Retrieval: A Unifying Framework. Part II: The Block Term Decomposition

Uses Software

• PHCpack
• SuiteSparseQR
• SuitSparseQR
• Tensorlab

Cites Work

• On the null spaces of the Macaulay matrix
• Numerical root finding via Cox rings
• Optimization-Based Algorithms for Tensor Decompositions: Canonical Polyadic Decomposition, Decomposition in Rank-$(L_r,L_r,1)$ Terms, and a New Generalization
• On the Uniqueness of the Canonical Polyadic Decomposition of Third-Order Tensors---Part I: Basic Results and Uniqueness of One Factor Matrix
• Algorithm 915, SuiteSparseQR
• A Decomposition for Three-Way Arrays
• A Multilinear Singular Value Decomposition
• Algorithm 795
• Multidimensional realisation theory and polynomial system solving
• Multidimensional Harmonic Retrieval via Coupled Canonical Polyadic Decomposition—Part I: Model and Identifiability
• Multidimensional Harmonic Retrieval via Coupled Canonical Polyadic Decomposition—Part II: Algorithm and Multirate Sampling
• Tensor Decomposition for Signal Processing and Machine Learning
• Computation of the Canonical Decomposition by Means of a Simultaneous Generalized Schur Decomposition
• Solving Polynomial Systems via Truncated Normal Forms
• Accurate solution of polynomial equations using Macaulay resultant matrices
• Numerical Polynomial Algebra
• Pencil-Based Algorithms for Tensor Rank Decomposition are not Stable
• A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization
• Almost-sure identifiability of multidimensional harmonic retrieval
• Kruskal's Permutation Lemma and the Identification of CANDECOMP/PARAFAC and Bilinear Models with Constant Modulus Constraints
• Most Tensor Problems Are NP-Hard
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