Bilinear factorizations subject to monomial equality constraints via tensor decompositions
The Canonical Polyadic Decomposition (CPD), which decomposes a tensor into a sum of rank one terms, plays an important role in signal processing and machine learning. In this paper we extend the CPD framework to the more general case of bilinear factorizations subject to monomial equality constraint...
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| Published in | Linear algebra and its applications Vol. 621; pp. 296 - 333 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier Inc
15.07.2021
American Elsevier Company, Inc |
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| Online Access | Get full text |
| ISSN | 0024-3795 1873-1856 |
| DOI | 10.1016/j.laa.2021.03.022 |
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| Abstract | The Canonical Polyadic Decomposition (CPD), which decomposes a tensor into a sum of rank one terms, plays an important role in signal processing and machine learning. In this paper we extend the CPD framework to the more general case of bilinear factorizations subject to monomial equality constraints. This includes extensions of multilinear algebraic uniqueness conditions originally developed for the CPD. We obtain a deterministic uniqueness condition that admits a constructive interpretation. Computationally, we reduce the bilinear factorization problem into a CPD problem, which can be solved via a matrix EigenValue Decomposition (EVD). Under the given conditions, the discussed EVD-based algorithms are guaranteed to return the exact bilinear factorization. Finally, we make a connection between bilinear factorizations subject to monomial equality constraints and the coupled block term decomposition, which allows us to translate monomial structures into low-rank structures. |
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| AbstractList | The Canonical Polyadic Decomposition (CPD), which decomposes a tensor into a sum of rank one terms, plays an important role in signal processing and machine learning. In this paper we extend the CPD framework to the more general case of bilinear factorizations subject to monomial equality constraints. This includes extensions of multilinear algebraic uniqueness conditions originally developed for the CPD. We obtain a deterministic uniqueness condition that admits a constructive interpretation. Computationally, we reduce the bilinear factorization problem into a CPD problem, which can be solved via a matrix EigenValue Decomposition (EVD). Under the given conditions, the discussed EVD-based algorithms are guaranteed to return the exact bilinear factorization. Finally, we make a connection between bilinear factorizations subject to monomial equality constraints and the coupled block term decomposition, which allows us to translate monomial structures into low-rank structures. |
| Author | Sidiropoulos, Nicholaos D. Sørensen, Mikael De Lathauwer, Lieven |
| Author_xml | – sequence: 1 givenname: Mikael orcidid: 0000-0003-4337-7417 surname: Sørensen fullname: Sørensen, Mikael email: ms8tz@virginia.edu organization: University of Virginia, Dept. of Electrical and Computer Engineering, Thornton Hall 351 McCormick Road, Charlottesville, VA 22904, USA – sequence: 2 givenname: Lieven surname: De Lathauwer fullname: De Lathauwer, Lieven email: Lieven.DeLathauwer@kuleuven.be organization: Group Science, Engineering and Technology, KU Leuven - Kulak, E. Sabbelaan 53, 8500 Kortrijk, Belgium – sequence: 3 givenname: Nicholaos D. orcidid: 0000-0002-3385-7911 surname: Sidiropoulos fullname: Sidiropoulos, Nicholaos D. email: nikos@virginia.edu organization: University of Virginia, Dept. of Electrical and Computer Engineering, Thornton Hall 351 McCormick Road, Charlottesville, VA 22904, USA |
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| Keywords | Canonical polyadic decomposition Tensor Coupled decomposition 15A23 Uniqueness Monomial Eigenvalue decomposition 15A15 Block term decomposition |
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| Snippet | The Canonical Polyadic Decomposition (CPD), which decomposes a tensor into a sum of rank one terms, plays an important role in signal processing and machine... |
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| SubjectTerms | Algorithms Block term decomposition Canonical polyadic decomposition Coupled decomposition Decomposition Eigenvalue decomposition Eigenvalues Equality Factorization Linear algebra Machine learning Mathematical analysis Monomial Signal processing Tensor Tensors Uniqueness |
| Title | Bilinear factorizations subject to monomial equality constraints via tensor decompositions |
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