README.md
In rsvddpd: Robust Singular Value Decomposition using Density Power Divergence

rsvddpd

The R package rsvddpd is an acronym for Robust Singular Value Decomposition using Density Power Divergence. As the name suggests, the package mainly concerns with a special function for performing SVD in a robust way in presence of outliers. The details of the algorithm can be found in the paper https://arxiv.org/abs/2109.10680.

There are 3 primary functions in the package.

rSVDdpd - Performs the robust SVD of a numeric matrix X.
simSVD - Simulates different scenarios to compare performances of SVD algorithms under outlier contamination.
cv.alpha - It computes the optimal robustness parameter α for the rSVDdpd algorithm based on the data matrix X.

You can install the development version from GitHub with:

# install.packages("devtools")
devtools::install_github("subroy13/rsvddpd")

Use the following to install the development version with manuals and vignettes, which provides useful information about the structure of the function.

devtools::install_github("subroy13/rsvddpd", build_opts = c("--no-resave-data"), build_manual = TRUE, build_vignettes = TRUE)

This is a basic example usages which shows the need for the package.

library(rsvddpd)

X <- matrix(1:20, nrow = 4, ncol = 5)
svd(X)
#> $d
#> [1] 5.352022e+01 2.363426e+00 4.870683e-15 7.906968e-16
#> 
#> $u
#>            [,1]       [,2]        [,3]       [,4]
#> [1,] -0.4430188 -0.7097424 -0.52426094  0.1585890
#> [2,] -0.4798725 -0.2640499  0.81721984  0.1793091
#> [3,] -0.5167262  0.1816426 -0.06165685 -0.8343851
#> [4,] -0.5535799  0.6273351 -0.23130204  0.4964870
#> 
#> $v
#>             [,1]        [,2]       [,3]       [,4]
#> [1,] -0.09654784  0.76855612 -0.6000256  0.1704800
#> [2,] -0.24551564  0.48961420  0.5577664 -0.5560862
#> [3,] -0.39448345  0.21067228  0.2312115  0.1606664
#> [4,] -0.54345125 -0.06826963  0.2643802  0.6650059
#> [5,] -0.69241905 -0.34721155 -0.4533325 -0.4400661

As you can see, the first two singular values are 53.5 and 2.36, and the third and fourth singular values are very small positive reals.

Let us see what happens when you contaminate just one entry of the matrix by a large value say 100.

X[2, 3] <- 100
svd(X)
#> $d
#> [1] 1.070340e+02 3.617861e+01 2.200002e+00 1.851858e-15
#> 
#> $u
#>            [,1]       [,2]         [,3]          [,4]
#> [1,] -0.1472125 -0.4893994  0.816938282  2.672612e-01
#> [2,] -0.9548191  0.2971284  0.005940614 -1.110223e-16
#> [3,] -0.1753739 -0.5614500 -0.105644232 -8.017837e-01
#> [4,] -0.1894546 -0.5974754 -0.566935489  5.345225e-01
#> 
#> $v
#>             [,1]       [,2]         [,3]          [,4]
#> [1,] -0.03121244 -0.1097166 -0.798115312  3.357170e-01
#> [2,] -0.08603093 -0.2591083 -0.524844308 -6.516983e-01
#> [3,] -0.94371346  0.3306537 -0.008548386  1.387779e-16
#> [4,] -0.19566792 -0.5578918  0.021697699  6.122270e-01
#> [5,] -0.25048641 -0.7072836  0.294968703 -2.962457e-01

Note that, the first singular value changes drastically, being 107, while second and third singular values 36.1 and 2.2 respectively. However, such error is very common in practice, and can pose serious problem in many statistical estimation techniques. rSVDdpd solves the problem as shown in the following code.

rSVDdpd(X, alpha = 0.3)
#> $d
#> [1] 5.355990e+01 2.358915e+00 1.492008e-01 6.694858e-11
#> 
#> $u
#>           [,1]       [,2]       [,3]          [,4]
#> [1,] 0.4426825 -0.7124356  0.4743827  2.672615e-01
#> [2,] 0.4810583 -0.2588203 -0.8376126  2.697441e-07
#> [3,] 0.5163450  0.1804013  0.2408039 -8.017834e-01
#> [4,] 0.5531753  0.6268197  0.1240144  5.345228e-01
#> 
#> $v
#>            [,1]        [,2]       [,3]          [,4]
#> [1,] 0.09646637  0.77032520  0.2174706 -5.827481e-01
#> [2,] 0.24532578  0.49133794  0.2200041  7.031679e-01
#> [3,] 0.39606323  0.20319890 -0.8954575  5.552961e-07
#> [4,] 0.54304936 -0.06663662  0.2250709  2.214907e-01
#> [5,] 0.69191099 -0.34562390  0.2276043 -3.419085e-01

Author

Subhrajyoty Roy, Indian Statistical Institute, Kolkata (Package author and Maintainer)
Ayanendranath Basu, Indian Statistical Institute, Kolkata
Abhik Ghosh, Indian Statistical Institute, Kolkata