Multiply by eclairs matrix using special structure to achieve linear instead of cubic time complexity.
mult_eclairs(X, U1, dSq1, lambda, nu, alpha, sigma, transpose = FALSE)a matrix product
matrix to be transformed so *columns* are independent
orthonormal matrix with k columns representing the low rank component
eigen values so that \(U_1 diag(d_1^2) U_1^T\) is the low rank component
shrinkage parameter for the convex combination.
diagonal value of target matrix in shrinkage
exponent to be evaluated
standard deviation of each feature
logical, (default FALSE) indicating if X should be transposed first
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.