Efficient Low-Rank Approximation of Matrices Based on Randomized Pivoted Decomposition
Given a matrix <inline-formula><tex-math notation="LaTeX">\bf A</tex-math></inline-formula> with numerical rank <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>, the two-sided orthogonal decomposition (TSOD)...
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| Vydáno v: | IEEE transactions on signal processing Ročník 68; s. 3575 - 3589 |
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| Jazyk: | angličtina |
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2020
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| ISSN: | 1053-587X, 1941-0476 |
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| Abstract | Given a matrix <inline-formula><tex-math notation="LaTeX">\bf A</tex-math></inline-formula> with numerical rank <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>, the two-sided orthogonal decomposition (TSOD) computes a factorization <inline-formula><tex-math notation="LaTeX">{\bf A} = {\bf UDV}^T</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">{\bf U}</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">{\bf V}</tex-math></inline-formula> are orthogonal, and <inline-formula><tex-math notation="LaTeX">{\bf D}</tex-math></inline-formula> is (upper/lower) triangular. TSOD is rank-revealing as the middle factor <inline-formula><tex-math notation="LaTeX">{\bf D}</tex-math></inline-formula> reveals the rank of <inline-formula><tex-math notation="LaTeX">\bf A</tex-math></inline-formula>. The computation of TSOD, however, is demanding. In this paper, we present an algorithm called randomized pivoted TSOD (RP-TSOD), where the middle factor is lower triangular. Key in our work is the exploitation of randomization, and RP-TSOD is primarily devised to efficiently construct an approximation to a low-rank matrix. We provide three different types of bounds for RP-TSOD: (i) we furnish upper bounds on the error of the low-rank approximation, (ii) we bound the <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula> approximate principal singular values, and (iii) we derive bounds for the canonical angles between the approximate and the exact singular subspaces. Our bounds describe the characteristics and behavior of our proposed algorithm. Through numerical tests, we show the effectiveness of the devised bounds as well as our proposed algorithm. |
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| AbstractList | Given a matrix [Formula Omitted] with numerical rank [Formula Omitted], the two-sided orthogonal decomposition (TSOD) computes a factorization [Formula Omitted], where [Formula Omitted] and [Formula Omitted] are orthogonal, and [Formula Omitted] is (upper/lower) triangular. TSOD is rank-revealing as the middle factor [Formula Omitted] reveals the rank of [Formula Omitted]. The computation of TSOD, however, is demanding. In this paper, we present an algorithm called randomized pivoted TSOD (RP-TSOD), where the middle factor is lower triangular. Key in our work is the exploitation of randomization, and RP-TSOD is primarily devised to efficiently construct an approximation to a low-rank matrix. We provide three different types of bounds for RP-TSOD: (i) we furnish upper bounds on the error of the low-rank approximation, (ii) we bound the [Formula Omitted] approximate principal singular values, and (iii) we derive bounds for the canonical angles between the approximate and the exact singular subspaces. Our bounds describe the characteristics and behavior of our proposed algorithm. Through numerical tests, we show the effectiveness of the devised bounds as well as our proposed algorithm. Given a matrix <inline-formula><tex-math notation="LaTeX">\bf A</tex-math></inline-formula> with numerical rank <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>, the two-sided orthogonal decomposition (TSOD) computes a factorization <inline-formula><tex-math notation="LaTeX">{\bf A} = {\bf UDV}^T</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">{\bf U}</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">{\bf V}</tex-math></inline-formula> are orthogonal, and <inline-formula><tex-math notation="LaTeX">{\bf D}</tex-math></inline-formula> is (upper/lower) triangular. TSOD is rank-revealing as the middle factor <inline-formula><tex-math notation="LaTeX">{\bf D}</tex-math></inline-formula> reveals the rank of <inline-formula><tex-math notation="LaTeX">\bf A</tex-math></inline-formula>. The computation of TSOD, however, is demanding. In this paper, we present an algorithm called randomized pivoted TSOD (RP-TSOD), where the middle factor is lower triangular. Key in our work is the exploitation of randomization, and RP-TSOD is primarily devised to efficiently construct an approximation to a low-rank matrix. We provide three different types of bounds for RP-TSOD: (i) we furnish upper bounds on the error of the low-rank approximation, (ii) we bound the <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula> approximate principal singular values, and (iii) we derive bounds for the canonical angles between the approximate and the exact singular subspaces. Our bounds describe the characteristics and behavior of our proposed algorithm. Through numerical tests, we show the effectiveness of the devised bounds as well as our proposed algorithm. |
| Author | Chen, Jie Kaloorazi, Maboud F. |
| Author_xml | – sequence: 1 givenname: Maboud F. orcidid: 0000-0003-3385-9244 surname: Kaloorazi fullname: Kaloorazi, Maboud F. email: kaloorazi@nwpu.edu.cn organization: Center of Intelligent Acoustics and Immersive Communications, School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, China – sequence: 2 givenname: Jie orcidid: 0000-0003-2306-8860 surname: Chen fullname: Chen, Jie email: dr.jie.chen@ieee.org organization: Center of Intelligent Acoustics and Immersive Communications, School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, China |
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| Snippet | Given a matrix <inline-formula><tex-math notation="LaTeX">\bf A</tex-math></inline-formula> with numerical rank <inline-formula><tex-math... Given a matrix [Formula Omitted] with numerical rank [Formula Omitted], the two-sided orthogonal decomposition (TSOD) computes a factorization [Formula... |
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| SubjectTerms | Acoustics Algorithms Approximation Approximation algorithms Decomposition dimension reduction image recovery low-rank approximation Machine learning algorithms Mathematical analysis Matrix decomposition Randomization randomized numerical linear algebra rank-revealing factorization Signal processing algorithms Sparse matrices Subspaces Surges Upper bounds |
| Title | Efficient Low-Rank Approximation of Matrices Based on Randomized Pivoted Decomposition |
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