rSVDdpd {rsvddpd}

Robust Singular Value Decomposition using Density Power Divergence

Description

rSVDdpd returns the singular value decomposition of a matrix with robust singular values in presence of outliers

Usage

rSVDdpd(
  X,
  alpha,
  nd = NA,
  tol = 1e-04,
  eps = 1e-04,
  maxiter = 100L,
  initu = NULL,
  initv = NULL
)

Arguments

X

matrix, whose singular value decomposition is required

alpha

numeric, robustness parameter between 0 and 1. See details for more.

nd

integer, must be lower than nrow(X) and ncol(X) both. If NA, defaults to min(nrow(X), ncol(X))

tol

numeric, a tolerance level. If the residual matrix has lower norm than this, then subsequent singular values will be taken as 0.

eps

numeric, a tolerance level for the convergence of singular vectors. If in subsequent iterations the singular vectors do not change its norm beyond this, then the iteration will stop.

maxiter

integer, upper limit to the maximum number of iterations.

initu

matrix, initializing vectors for left singular values. Must be of dimension nrow(X) \times min(nrow(X), ncol(X)). If NULL, defaults to random initialization.

initv

matrix, initializing vectors for right singular values. Must be of dimension ncol(X) \times min(nrow(X), ncol(X)). If NULL, defaults to random initialization.

Details

The usual singular value decomposition is highly prone to error in presence of outliers, since it tries to minimize the L_2 norm of the errors between the matrix X and its best lower rank approximation. While there is considerable effort to impose robustness using L_1 norm of the errors instead of L_2 norm, such estimation lacks efficiency. Application of density power divergence bridges the gap.

DPD(f|g) = \int f^{(1+\alpha)} - (1 + \frac{1}{\alpha}) \int f^{\alpha}g + \frac{1}{\alpha} \int g^{(1 + \alpha)}

The parameter alpha should be between 0 and 1, if not, then a warning is shown. Lower alpha means less robustness but more efficiency in estimation, while higher alpha means high robustness but less efficiency in estimation. The recommended value of alpha is 0.3. The function tries to obtain the best rank one approximation of a matrix by minimizing this density power divergence of the true errors with that of a normal distribution centered at the origin.

Value

A list containing different components of the decomposition X = U D V'

• d - The robust singular values, namely the diagonal entries of D.

• u - The matrix of left singular vectors U. Each column is a singular vector.

• v - The matrix of right singular vectors V. Each column is a singular vector.

References

S. Roy, A. Basu and A. Ghosh (2021), A New Robust Scalable Singular Value Decomposition Algorithm for Video Surveillance Background Modelling https://arxiv.org/abs/2109.10680

See Also

svd

Examples

X = matrix(1:20, nrow = 4, ncol = 5)
rSVDdpd(X, alpha = 0.3)