Hard-thresholding operator — hard

hard_threshold() implements the hard-thresholding operator on a given matrix D, making D sparser: elements of D whose absolute value are less than a given threshold thresh are set to 0, i.e. \(D[|D| < thresh] = 0\).

This is used in the non-convex PCP function rrmc() to provide a non-convex replacement for the prox_l1() method used in the convex PCP function root_pcp(). It is used to iteratively model the sparse S matrix with the help of an adaptive threshold (thresh changes over the course of optimization).

Usage

hard_threshold(D, thresh)

Arguments

D: The input data matrix.
thresh: The scalar-valued hard-threshold acting on D such that D[i, j] = 0 when abs(D[i, j]) < thresh, and D[i, j] = D[i, j] otherwise.

Value

The hard-thresholded matrix.

Examples

set.seed(42)
D <- matrix(rnorm(25), 5, 5)
S <- hard_threshold(D, thresh = 1)
D
#>            [,1]        [,2]       [,3]       [,4]       [,5]
#> [1,]  1.3709584 -0.10612452  1.3048697  0.6359504 -0.3066386
#> [2,] -0.5646982  1.51152200  2.2866454 -0.2842529 -1.7813084
#> [3,]  0.3631284 -0.09465904 -1.3888607 -2.6564554 -0.1719174
#> [4,]  0.6328626  2.01842371 -0.2787888 -2.4404669  1.2146747
#> [5,]  0.4042683 -0.06271410 -0.1333213  1.3201133  1.8951935
S
#>          [,1]     [,2]      [,3]      [,4]      [,5]
#> [1,] 1.370958 0.000000  1.304870  0.000000  0.000000
#> [2,] 0.000000 1.511522  2.286645  0.000000 -1.781308
#> [3,] 0.000000 0.000000 -1.388861 -2.656455  0.000000
#> [4,] 0.000000 2.018424  0.000000 -2.440467  1.214675
#> [5,] 0.000000 0.000000  0.000000  1.320113  1.895193