# Multilinear Operators for Higher-order Decompositions

### Abstract

We propose two new multilinear operators for expressing the matrix compositions that are needed in the Tucker and PARAFAC (CANDECOMP) decompositions. The first operator, which we call the Tucker operator, is shorthand for performing an n-mode matrix multiplication for every mode of a given tensor and can be employed to consisely express the Tucker decomposition. The second operator, which we call the Kruskal operator, is shorthand for the sum of the outer-products of the columns of N matrices and allows a divorce from a matricized representation and a very consise expression of the PARAFAC decomposition. We explore the properties of the Tucker and Kruskal operators independently of the related decompositions. Additionally, we provide a review of the matrix and tensor operations that are frequently used in the context of tensor decompositions.

Type
Publication
Tech. Rep., Sandia National Laboratories
Date
Tags
Citation
T. G. Kolda. Multilinear Operators for Higher-order Decompositions. Tech. Rep. No. SAND2006-2081, Sandia National Laboratories, 2006. https://doi.org/10.2172/923081

### Keywords

PARAFAC, CANDECOMP, Tucker, HOSVD, multilinear algebra, higher-order factor analysis, Khatri-Rao product

There is an error in Example 3.3.

### BibTeX

@techreport{SAND2006-2081,
author = {Tamara G. Kolda},
title = {Multilinear Operators for Higher-order Decompositions},
number = {SAND2006-2081},
institution = {Sandia National Laboratories},
month = {April},
year = {2006},
doi = {10.2172/923081},
url = {http://www.osti.gov/scitech/biblio/923081},
urldate = {2014-04-17},
}