tucker {tensorregress}

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 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, Vol. 51, No. 3 (September 2009), pp. 455-500. URL: https://www.jstor.org/stable/25662308

See Also

hosvd

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)