ProjSDD {ddpca}

Projection onto the Symmetric Diagonally Dominant Cone

Description

Given a matrix C, this function outputs the projection of C onto the cones of symmetric diagonally domimant matrices using Dykstra's projection algorithm.

Usage

ProjSDD(A, max_iter_SDD = 20, eps = NA)

Arguments

A

Input matrix of size n\times n

max_iter_SDD

Maximal number of iterations of the Dykstra's projection algorithm

eps

The iterations will stop either when the Frobenious norm of difference matrix between two updates is less than eps or after max_iter_SDD steps. If set to NA, then no check will be done during iterations and the iteration will stop after max_iter_SDD steps. Default is NA.

Details

This function projects the input matrix C of size n\times n onto the cones of symmetric diagonally domimant matrices defined as

\{A = (a_{ij})_{1\le i\le n, 1\le j\le n} : a_{ij} = a_{ji}, a_{jj} \ge \sum_{k\not=j} |a_{jk}| \quad \textrm{for all} \quad 1\le j\le n, 1\le i\le n \}

It makes use of Dykstra's algorithm, which is a variation of iterative projection algorithm. The two key steps are projection onto the diagonally domimant cone by calling function ProjDD and projection onto the symmetric matrix cone by simple symmetrization.

More details can be found in Mendoza, M., Raydan, M. and Tarazaga, P., 1998. Computing the nearest diagonally dominant matrix.

Value

A n\times n symmetric diagonally dominant matrix

Author(s)

Fan Yang <[e-mail]>

References

Mendoza, M., Raydan, M. and Tarazaga, P., 1998. Computing the nearest diagonally dominant matrix. Numerical linear algebra with applications, 5(6), pp.461-474.

Ke, Z., Xue, L. and Yang, F., 2019. Diagonally Dominant Principal Component Analysis. Journal of Computational and Graphic Statistics, under review.

See Also

ProjDD

Examples

ProjSDD(matrix(runif(100),nrow=10))