“Rybicki Press algorithm”的意思、由来-开放百科全书

The Rybicki–Press algorithm is a fast direct algorithm for inverting a matrix, whose entries are given by

, where

.^[1] It is a computational optimization of a general set of statistical methods developed to determine whether two noisy, irregularly sampled data sets are, in fact, dimensionally shifted representations of the same underlying function.^[2]^[3] The most common use of the algorithm is in the detection of periodicity in astronomical observations.^[3]

Recently, this method has been extended (Generalized Rybicki Press algorithm) for inverting matrices whose entries of the form

.^[4] The key observation in the Generalized Rybicki Press (GPP) algorithm is that the matrix

is a semi-separable matrix with rank

. More precisely, if the matrix

has a semi-separable rank is

, the cost for solving the linear system

and obtaining the determinant of the matrix scales as

, thereby making it extremely attractive for large matrices. This implementation of the GPP algorithm can be found here.^[5] The key idea is that the dense matrix

can be converted into a sparser matrix of a larger size (see figure on the right), whose sparsity structure can be leveraged to reduce the computational complexity.

The fact that matrix

is a semi-separable matrix also forms the basis for celerite^[6] library, which is a library for fast and scalable Gaussian Process (GP) Regression in one dimension^[7] with implementations in C++, Python, and Julia. The celerite method^[7] also provides an algorithm for generating samples from a high-dimensional distribution. The method has found attractive applications in a wide range of fields, especially in astronomical data analysis.^[8]^[9]

References

1. ^{{citation |last1 = Rybicki|first1 = George B.|last2 = Press|first2 = William H.|arxiv = comp-gas/9405004|doi = 10.1103/PhysRevLett.74.1060|journal = Physical Review Letters|title = Class of fast methods for processing Irregularly sampled or otherwise inhomogeneous one-dimensional data|volume = 74|issue = 7|pages = 1060–1063|year = 1995|bibcode = 1995PhRvL..74.1060R|pmid=10058924}} {{Open access}}
2. ^{{Cite journal|url = |title = Interpolation, realization, and reconstruction of noisy, irregularly sampled data|last = Rybicki|first = George B.|date = October 1992|journal = The Astrophysical Journal|doi = 10.1086/171845|pmid = |last2 = Press|first2 = William H.|bibcode = 1992ApJ...398..169R|volume=398|page=169}}{{Open access}}
3. ^¹{{Cite journal|last=MacLeod|first=C. L.|last2=Brooks|first2=K.|last3=Ivezic|first3=Z.|last4=Kochanek|first4=C. S.|last5=Gibson|first5=R.|last6=Meisner|first6=A.|last7=Kozlowski|first7=S.|last8=Sesar|first8=B.|last9=Becker|first9=A. C.|date=2011-02-10|title=Quasar Selection Based on Photometric Variability|journal=The Astrophysical Journal|volume=728|issue=1|pages=26|doi=10.1088/0004-637X/728/1/26|issn=0004-637X|arxiv=1009.2081|bibcode=2011ApJ...728...26M}}
4. ^{{Cite journal|last=Ambikasaran|first=Sivaram|date=2015-12-01|title=Generalized Rybicki Press algorithm|journal=Numerical Linear Algebra with Applications|language=en|volume=22|issue=6|pages=1102–1114|doi=10.1002/nla.2003|issn=1099-1506|arxiv=1409.7852}}
5. ^{{Cite web|url=https://github.com/sivaramambikasaran/ESS|title=sivaramambikasaran/ESS|website=GitHub|language=en|access-date=2018-04-05}}
6. ^{{Cite web|url=https://celerite.readthedocs.io/en/stable/|title=celerite — celerite 0.3.0 documentation|website=celerite.readthedocs.io|language=en|access-date=2018-04-05}}
7. ^¹{{Cite journal|last=Foreman-Mackey|first=Daniel|last2=Agol|first2=Eric|last3=Ambikasaran|first3=Sivaram|last4=Angus|first4=Ruth|date=2017|title=Fast and Scalable Gaussian Process Modeling with Applications to Astronomical Time Series|url=http://stacks.iop.org/1538-3881/154/i=6/a=220|journal=The Astronomical Journal|language=en|volume=154|issue=6|pages=220|doi=10.3847/1538-3881/aa9332|issn=1538-3881|arxiv=1703.09710|bibcode=2017AJ....154..220F}}
8. ^{{Cite journal|last=Foreman-Mackey|first=Daniel|date=2018|title=Scalable Backpropagation for Gaussian Processes using Celerite|url=http://stacks.iop.org/2515-5172/2/i=1/a=31|journal=Research Notes of the AAS|language=en|volume=2|issue=1|pages=31|doi=10.3847/2515-5172/aaaf6c|issn=2515-5172|arxiv=1801.10156|bibcode=2018RNAAS...2a..31F}}
9. ^{{Cite book|title=Handbook of Exoplanets|last=Parviainen|first=Hannu|date=2018|publisher=Springer, Cham|isbn=9783319306483|pages=1–24|language=en|doi=10.1007/978-3-319-30648-3_149-1|chapter = Bayesian Methods for Exoplanet Science|arxiv = 1711.03329}}