hooi {tensr}

Calculate the higher-order orthogonal iteration (HOOI).

Description

This function will calculate the best rank r (where r is a vector) approximation (in terms of sum of squared differences) to a given data array.

Usage

hooi(X, r, tol = 10^-6, print_fnorm = FALSE, itermax = 500)

Arguments

X

An array of numerics.

r

A vector of integers. This is the given low multilinear rank of the approximation.

tol

A numeric. Stopping criterion.

print_fnorm

Should updates of the optimization procedure be printed? This number should get larger during the optimizaton procedure.

itermax

The maximum number of iterations to run the optimization procedure.

Details

Given an array X, this code will find a core array G and a list of matrices with orthonormal columns U that minimizes fnorm(X - atrans(G, U)). If r is equal to the dimension of X, then it returns the HOSVD (see hosvd).

For details on the HOOI see Lathauwer et al (2000).

Value

G An all-orthogonal core array.

U A vector of matrices with orthonormal columns.

Author(s)

David Gerard.

References

De Lathauwer, L., De Moor, B., & Vandewalle, J. (2000). On the best rank-1 and rank-(r_1, r_2,..., r_n) approximation of higher-order tensors. SIAM Journal on Matrix Analysis and Applications, 21(4), 1324-1342.

Examples

## Generate random data.
p <- c(2, 3, 4)
X <- array(stats::rnorm(prod(p)), dim = p)

## Calculate HOOI
r <- c(2, 2, 2)
hooi_x <- hooi(X, r = r)
G <- hooi_x$G
U <- hooi_x$U

## Reconstruct the hooi approximation.
X_approx <- atrans(G, U)
fnorm(X - X_approx)