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Fast Randomized Singular Value Decomposition using R

Randomized singular value decomposition (rsvd) is a fast probabilistic algorithm that can be used to compute the near optimal low-rank singular value decomposition of massive data sets with high accuracy. The key idea is to compute a compressed representation of the data to capture the essential information. This compressed representation can then be used to obtain the low-rank singular value decomposition decomposition. The rsvd package provides one of the fastest routines for low-rank matrix approximations in R, as far as we know.
The computational advantage becomes pronounced with an increasing matrix dimension (here target-rank k=50):

The singular value decomposition plays a central role in data analysis and scientific computing. The SVD is also widely used for computing (randomized) principal component analysis (PCA), a linear dimensionality reduction technique. Randomized PCA (rpca) uses the approximated singular value decomposition to compute the most significant principal components. This package also includes a function to compute (randomized) robust principal component analysis (RPCA). In addition several plot functions are provided. See for further details: “Randomized Matrix Decompositions using R”.

SVD example: Image compression

library(rsvd)
data(tiger)

# Image compression using randomized SVD
s <- rsvd(tiger, k=150)
tiger.re = s$u %*% diag(s$d) %*% t(s$v) # reconstruct image

# Display orginal and reconstrucuted image
par(mfrow=c(1,2))
image(tiger, col = gray((0:255)/255))
image(tiger.re, col = gray((0:255)/255))

Here are the results:

and the speedup gained over the base SVD function:

library(microbenchmark)

timing_svd <- microbenchmark(
  'SVD' = svd(tiger, nu=150, nv=150),
  'rSVD' = rsvd(tiger, k=150),
  times=50)

print(timing_svd, unit='s')

Installation

Install the rsvd package via CRAN

install.packages("rsvd")

You can also install the development version from GitHub using devtools:

devtools::install_github("erichson/rsvd")

The source packge can be obtained here: CRAN: rsvd.

New in Version 1.0.2

• Several small issues are fixed.
• Thanks to Aaron Lun, who has fixed a bug in the rsvd function that occured when nu=0 or nv=0.

New in Version 1.0.0

• Support for non-default matrix types to deal with large-scale matrices that are held on file, added by Aaron Lun.
• Fixed a bug which occured runninig rpca with k=1 and retx=TRUE, discovered by Will.

References

• Erichson NB, Voronin S, Brunton SL, Kutz JN (2019). Randomized Matrix Decompositions Using R. Journal of Statistical Software, 89(11), 1–48. doi: 10.18637/jss.v089.i11.
• Sergey Voronin, Per-Gunnar Martinsson. RSVDPACK: Subroutines for computing partial singular value decompositions via randomized sampling on single core, multi core, and GPU architectures. (2015)
• Nathan Halko, et al. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. (2011)

Cite as

@Article{,
    title = {Randomized Matrix Decompositions Using {R}},
    author = {N. Benjamin Erichson and Sergey Voronin and Steven L.
      Brunton and J. Nathan Kutz},
    journal = {Journal of Statistical Software},
    year = {2019},
    volume = {89},
    number = {11},
    pages = {1--48},
    doi = {10.18637/jss.v089.i11},
}