Rybicki Press algorithm: Difference between revisions

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{{Notability|date=July 2015}}

[[File:Extended_Sparse_Matrix.png|thumb|Extended Sparse Matrix arising from a <math>10 \times 10</math> semi-separable matrix whose semi-separable rank is <math>4</math>.]]

The '''Rybicki–Press algorithm''' is a fast direct algorithm for inverting a matrix, whose entries are given by <math>A(i,j) = \exp(-a \vert t_i - t_j \vert)</math>, where <math>a \in \mathbb{R}</math>.<ref>{{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|page = 1060|title = Class of fast methods for processing Irregularly sampled or otherwise inhomogeneous one-dimensional data|volume = 74|year = 1995|bibcode = 1995PhRvL..74.1060R|pmid=10058924}} {{Open access}}</ref> 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.<ref>{{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}}</ref><ref name=":0">{{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|url=http://arxiv.org/abs/1009.2081|journal=The Astrophysical Journal|volume=728|issue=1|pages=26|doi=10.1088/0004-637X/728/1/26|issn=0004-637X}}</ref> The most common use of the algorithm is in the detection of periodicity in astronomical observations<ref name=":0" />.

Recently, this has been extended ('''Generalized Rybicki Press algorithm''') for inverting matrices whose entries of the form <math>A(i,j) = \sum_{k=1}^p a_k \exp(-\beta_k \vert t_i - t_j \vert)</math><ref>{{Cite journal|last=Ambikasaran|first=Sivaram|date=2015-12-01|title=Generalized Rybicki Press algorithm|url=https://onlinelibrary.wiley.com/doi/pdf/10.1002/nla.2003|journal=Numerical Linear Algebra with Applications|language=en|volume=22|issue=6|pages=1102–1114|doi=10.1002/nla.2003|issn=1099-1506}}</ref>. The key observation for the Generalized Rybicki Press (GPP) algorithm is that the matrix <math>A</math> is a semi-separable matrix with rank <math>p</math>. More precisely, if the matrix <math>A \in \mathbb{R}^{n\times n}</math> has a semi-separable rank is <math>p</math>, the cost for solving the linear system <math>Ax=b</math> and obtain the determinant of the matrix scales as <math>\mathcal{O}\left(p^2n \right)</math>, thereby making it extremely attractive for large matrices. This implementation of the GPP algorithm can be found here<ref>{{Cite web|url=https://github.com/sivaramambikasaran/ESS|title=sivaramambikasaran/ESS|website=GitHub|language=en|access-date=2018-04-05}}</ref>. The key idea is that the dense matrix <math>A</math> can be converted into a sparser matrix of a larger size (see figure on the right), whose structure can be leveraged to reduce the computational cost.

The fact that matrix <math>A</math> is a semi-separable matrix also forms the basis for celerite<ref>{{Cite web|url=http://celerite.readthedocs.io/en/stable/|title=celerite — celerite 0.3.0 documentation|website=celerite.readthedocs.io|language=en|access-date=2018-04-05}}</ref> library, which is a library for fast and scalable Gaussian Process (GP) Regression in one dimension<ref name=":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}}</ref> with implementations in C++, Python, and Julia. The celerite method<ref name=":1" /> 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<ref>{{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}}</ref><ref>{{Cite book|url=https://link.springer.com/referenceworkentry/10.1007/978-3-319-30648-3_149-1|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.pdf}}</ref>.