tucker {rTensor}

Tucker Decomposition

Description

The Tucker decomposition of a tensor. Approximates a K-Tensor using a n-mode product of a core tensor (with modes specified by ranks) with orthogonal factor matrices. If there is no truncation in one of the modes, then this is the same as the MPCA, mpca. If there is no truncation in all the modes (i.e. ranks = tnsr@modes), then this is the same as the HOSVD, hosvd. This is an iterative algorithm, with two possible stopping conditions: either relative error in Frobenius norm has gotten below tol, or the max_iter number of iterations has been reached. For more details on the Tucker decomposition, consult Kolda and Bader (2009).

Usage

tucker(tnsr, ranks = NULL, max_iter = 25, tol = 1e-05)

Arguments

tnsr

Tensor with K modes

ranks

a vector of the modes of the output core Tensor

max_iter

maximum number of iterations if error stays above tol

tol

relative Frobenius norm error tolerance

Details

Uses the Alternating Least Squares (ALS) estimation procedure also known as Higher-Order Orthogonal Iteration (HOOI). Intialized using a (Truncated-)HOSVD. A progress bar is included to help monitor operations on large tensors.

Value

a list containing the following:

Z

the core tensor, with modes specified by ranks

U

a list of orthgonal factor matrices - one for each mode, with the number of columns of the matrices given by ranks

conv

whether or not resid < tol by the last iteration

est

estimate of tnsr after compression

norm_percent

the percent of Frobenius norm explained by the approximation

fnorm_resid

the Frobenius norm of the error fnorm(est-tnsr)

all_resids

vector containing the Frobenius norm of error for all the iterations

Note

The length of ranks must match tnsr@num_modes.

References

T. Kolda, B. Bader, "Tensor decomposition and applications". SIAM Applied Mathematics and Applications 2009.

See Also

hosvd, mpca

Examples

tnsr <- rand_tensor(c(4,4,4,4))
tuckerD <- tucker(tnsr,ranks=c(2,2,2,2))
tuckerD$conv 
tuckerD$norm_percent
plot(tuckerD$all_resids)