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)...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:IEEE transactions on signal processing Jg. 68; S. 3575 - 3589
Hauptverfasser: Kaloorazi, Maboud F., Chen, Jie
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 2020
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
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
ISSN:1053-587X, 1941-0476
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung: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.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2020.3001399