Block Row Kronecker-Structured Linear Systems With a Low-Rank Tensor Solution
Several problems in compressed sensing and randomized tensor decomposition can be formulated as a structured linear system with a constrained tensor as the solution. In particular, we consider block row Kronecker-structured linear systems with a low multilinear rank multilinear singular value decomp...
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| Format: | Article |
| Language: | English |
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Frontiers Media S.A.
2022-03-01
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| Series: | Frontiers in Applied Mathematics and Statistics |
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| Online Access: | https://www.frontiersin.org/articles/10.3389/fams.2022.832883/full |
| _version_ | 1818775276879872000 |
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| author | Stijn Hendrikx Stijn Hendrikx Lieven De Lathauwer Lieven De Lathauwer |
| author_facet | Stijn Hendrikx Stijn Hendrikx Lieven De Lathauwer Lieven De Lathauwer |
| author_sort | Stijn Hendrikx |
| collection | DOAJ |
| description | Several problems in compressed sensing and randomized tensor decomposition can be formulated as a structured linear system with a constrained tensor as the solution. In particular, we consider block row Kronecker-structured linear systems with a low multilinear rank multilinear singular value decomposition, a low-rank canonical polyadic decomposition or a low tensor train rank tensor train constrained solution. In this paper, we provide algorithms that serve as tools for finding such solutions for a large, higher-order data tensor, given Kronecker-structured linear combinations of its entries. Consistent with the literature on compressed sensing, the number of linear combinations of entries needed to find a constrained solution is far smaller than the corresponding total number of entries in the original tensor. We derive conditions under which a multilinear singular value decomposition, canonical polyadic decomposition or tensor train solution can be retrieved from this type of structured linear systems and also derive the corresponding generic conditions. Finally, we validate our algorithms by comparing them to related randomized tensor decomposition algorithms and by reconstructing a hyperspectral image from compressed measurements. |
| first_indexed | 2024-12-18T10:54:28Z |
| format | Article |
| id | doaj.art-41991d44d3f847e5beaf55f48be2134f |
| institution | Directory Open Access Journal |
| issn | 2297-4687 |
| language | English |
| last_indexed | 2024-12-18T10:54:28Z |
| publishDate | 2022-03-01 |
| publisher | Frontiers Media S.A. |
| record_format | Article |
| series | Frontiers in Applied Mathematics and Statistics |
| spelling | doaj.art-41991d44d3f847e5beaf55f48be2134f2022-12-21T21:10:23ZengFrontiers Media S.A.Frontiers in Applied Mathematics and Statistics2297-46872022-03-01810.3389/fams.2022.832883832883Block Row Kronecker-Structured Linear Systems With a Low-Rank Tensor SolutionStijn Hendrikx0Stijn Hendrikx1Lieven De Lathauwer2Lieven De Lathauwer3Dynamical Systems, Signal Processing and Data Analytics (STADIUS), Department of Electrical Engineering (ESAT), KU Leuven, Leuven, BelgiumGroup Science, Engineering and Technology, KU Leuven Kulak, Kortrijk, BelgiumDynamical Systems, Signal Processing and Data Analytics (STADIUS), Department of Electrical Engineering (ESAT), KU Leuven, Leuven, BelgiumGroup Science, Engineering and Technology, KU Leuven Kulak, Kortrijk, BelgiumSeveral problems in compressed sensing and randomized tensor decomposition can be formulated as a structured linear system with a constrained tensor as the solution. In particular, we consider block row Kronecker-structured linear systems with a low multilinear rank multilinear singular value decomposition, a low-rank canonical polyadic decomposition or a low tensor train rank tensor train constrained solution. In this paper, we provide algorithms that serve as tools for finding such solutions for a large, higher-order data tensor, given Kronecker-structured linear combinations of its entries. Consistent with the literature on compressed sensing, the number of linear combinations of entries needed to find a constrained solution is far smaller than the corresponding total number of entries in the original tensor. We derive conditions under which a multilinear singular value decomposition, canonical polyadic decomposition or tensor train solution can be retrieved from this type of structured linear systems and also derive the corresponding generic conditions. Finally, we validate our algorithms by comparing them to related randomized tensor decomposition algorithms and by reconstructing a hyperspectral image from compressed measurements.https://www.frontiersin.org/articles/10.3389/fams.2022.832883/fulltensordecompositioncompressed sensing (CS)randomizedKroneckerlinear system |
| spellingShingle | Stijn Hendrikx Stijn Hendrikx Lieven De Lathauwer Lieven De Lathauwer Block Row Kronecker-Structured Linear Systems With a Low-Rank Tensor Solution Frontiers in Applied Mathematics and Statistics tensor decomposition compressed sensing (CS) randomized Kronecker linear system |
| title | Block Row Kronecker-Structured Linear Systems With a Low-Rank Tensor Solution |
| title_full | Block Row Kronecker-Structured Linear Systems With a Low-Rank Tensor Solution |
| title_fullStr | Block Row Kronecker-Structured Linear Systems With a Low-Rank Tensor Solution |
| title_full_unstemmed | Block Row Kronecker-Structured Linear Systems With a Low-Rank Tensor Solution |
| title_short | Block Row Kronecker-Structured Linear Systems With a Low-Rank Tensor Solution |
| title_sort | block row kronecker structured linear systems with a low rank tensor solution |
| topic | tensor decomposition compressed sensing (CS) randomized Kronecker linear system |
| url | https://www.frontiersin.org/articles/10.3389/fams.2022.832883/full |
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