Square root and inverse square root of a PSD kernel

Computes the matrix square root and inverse square root of a positive semi-definite kernel using eigendecomposition with optional regularization.

Usage

kernel_roots(K, jitter = 1e-10)

Arguments

K

Symmetric positive semi-definite matrix

jitter

Small ridge parameter to ensure numerical stability (default 1e-10)

Value

A list containing:

Khalf

The matrix square root K^(1/2) with same dimensions as K

Kihalf

The matrix inverse square root K^(-1/2) with same dimensions as K

evals

Eigenvalues (after jitter adjustment)

evecs

Eigenvectors matrix

Details

For a positive semi-definite matrix K, this function computes K^(1/2) and K^(-1/2) using the eigendecomposition K = V * diag(lambda) * V', where K^(1/2) = V * diag(sqrt(lambda)) * V' and K^(-1/2) = V * diag(1/sqrt(lambda)) * V'. The jitter parameter adds a small ridge to eigenvalues to prevent numerical issues with near-zero eigenvalues.

Examples

K <- matrix(c(2, 1, 1, 2), 2, 2)
result <- kernel_roots(K)
print(dim(result$Khalf))  # 2x2
#> [1] 2 2