A randomized algorithm for the decomposition of matrices
Given an m × n matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying A T to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is efficient when...
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| Vydané v: | Applied and computational harmonic analysis Ročník 30; číslo 1; s. 47 - 68 |
|---|---|
| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Elsevier Inc
2011
|
| Predmet: | |
| ISSN: | 1063-5203, 1096-603X |
| On-line prístup: | Získať plný text |
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| Shrnutí: | Given an
m
×
n
matrix
A and a positive integer
k, we describe a randomized procedure for the approximation of
A with a matrix
Z of rank
k. The procedure relies on applying
A
T
to a collection of
l random vectors, where
l is an integer equal to or slightly greater than
k; the scheme is efficient whenever
A and
A
T
can be applied rapidly to arbitrary vectors. The discrepancy between
A and
Z is of the same order as
l
m
times the
(
k
+
1
)
st greatest singular value
σ
k
+
1
of
A, with negligible probability of even moderately large deviations. The actual estimates derived in the paper are fairly complicated, but are simpler when
l
−
k
is a fixed small nonnegative integer. For example, according to one of our estimates for
l
−
k
=
20
, the probability that the spectral norm
‖
A
−
Z
‖
is greater than
10
(
k
+
20
)
m
σ
k
+
1
is less than
10
−
17
. The paper contains a number of estimates for
‖
A
−
Z
‖
, including several that are stronger (but more detailed) than the preceding example; some of the estimates are effectively independent of
m. Thus, given a matrix
A of limited numerical rank, such that both
A and
A
T
can be applied rapidly to arbitrary vectors, the scheme provides a simple, efficient means for constructing an accurate approximation to a singular value decomposition of
A. Furthermore, the algorithm presented here operates reliably independently of the structure of the matrix
A. The results are illustrated via several numerical examples. |
|---|---|
| Bibliografia: | ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
| ISSN: | 1063-5203 1096-603X |
| DOI: | 10.1016/j.acha.2010.02.003 |