Multiply by eclairs matrix — mult_eclairs • decorrelate

Multiply by eclairs matrix using special structure to achieve linear instead of cubic time complexity.

Usage

mult_eclairs(X, U1, dSq1, lambda, nu, alpha, sigma, transpose = FALSE)

Arguments

X: matrix to be transformed so *columns* are independent
U1: orthonormal matrix with k columns representing the low rank component
dSq1: eigen values so that \(U_1 diag(d_1^2) U_1^T\) is the low rank component
lambda: shrinkage parameter for the convex combination.
nu: diagonal value of target matrix in shrinkage
alpha: exponent to be evaluated
sigma: standard deviation of each feature
transpose: logical, (default FALSE) indicating if X should be transposed first

Value

a matrix product

Details

Let \(\Sigma = U_1 diag(d_1^2) U_1^T * (1-\lambda) + diag(\nu\lambda, p)\), where \(\lambda\) shrinkage parameter for the convex combination between a low rank matrix and the diagonal matrix with values \(\nu\).

Evaluate \(X \Sigma^\alpha\) using special structure of the eclairs decomposition in \(O(k^2p)\) when there are \(k\) components in the decomposition.