cpDecomposition {scTenifoldNet}

Canonical Polyadic Decomposition

Description

Canonical Polyadic (CP) decomposition of a tensor, aka CANDECOMP/PARAFRAC. Approximate a K-Tensor using a sum of num_components rank-1 K-Tensors. A rank-1 K-Tensor can be written as an outer product of K vectors. There are a total of num_compoents *tnsr@num_modes vectors in the output, stored in tnsr@num_modes matrices, each with num_components columns. 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 CP decomposition, consult Kolda and Bader (2009).

Usage

cpDecomposition(tnsr, num_components = NULL, max_iter = 25, tol = 1e-05)

Arguments

tnsr

Tensor with K modes

num_components

the number of rank-1 K-Tensors to use in approximation

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. A progress bar is included to help monitor operations on large tensors.

Value

a list containing the following

lambdas

a vector of normalizing constants, one for each component

U

a list of matrices - one for each mode - each matrix with num_components columns

conv

whether or not resid < tol by the last iteration

norm_percent

the percent of Frobenius norm explained by the approximation

est

estimate of tnsr after compression

fnorm_resid

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

all_resids

vector containing the Frobenius norm of error for all the iterations

References

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