Optimal Shrinkage of Singular Values — optshrink

Performs optimal shrinkage of singular values according to a specified loss function, given the shape parameter beta and noise level sigma. This function adjusts the singular values of a matrix to reduce noise and enhance signal components based on the Marčenko-Pastur distribution.

Usage

optshrink_impl(singvals, beta, loss, sigma)

Arguments

singvals: A numeric vector of singular values to be shrunk.
beta: A numeric value between 0 and 1 representing the aspect ratio of the matrix (n/d), where n is the number of columns and d is the number of rows.
loss: A character string specifying the type of loss function to use for shrinkage. Must be one of "fro" (Frobenius norm), "op" (Operator norm), or "nuc" (Nuclear norm).
sigma: A positive numeric value representing the noise level to be used in the shrinkage process.

Value

A numeric vector of shrunk singular values.

Details

The function applies different shrinkage strategies based on the specified loss function:

"fro": Uses Frobenius norm shrinkage, which minimizes the mean squared error.
"op": Uses Operator norm shrinkage, which targets the largest singular value.
"nuc": Uses Nuclear norm shrinkage, which adjusts the sum of singular values while penalizing those close to zero.

Internal helper functions are used to compute the adjusted singular values according to the specified loss type:

opt_fro_shrink: Calculates shrinkage based on the Frobenius norm.
opt_op_shrink: Calculates shrinkage based on the Operator norm.
opt_nuc_shrink: Calculates shrinkage based on the Nuclear norm.