Nonnegative Matrix and Tensor Factorization for Discussion Tracking - Text Mining: Classification, Clustering, and Applications - page 94

Carroll and Chang (11) published the same mathematical model under the
name Canonical Decomposition or CANDECOMP. A comprehensive review
by Kolda and Bader (22) summarizes these tensor decompositions and pro-
vides references for a wide variety of applications using them.
In the context of text analysis and mining, Acar et al. (1) used various tensor
decompositions of (user

time) data to separate different streams
of conversation in chatroom data. Several web search applications involving
tensors relied on query terms or anchor text to provide a third dimension.
Sun et al. (36) have used a three-way Tucker decomposition (38) to analyze
(user

× key word ×

web page) data for personalized web search. Kolda et
al. (23) and Kolda and Bader (21) have used PARAFAC on a (web page

× query term ×

×

web page anchor text) sparse, three-way tensor representing the web graph

with anchor-text-labeled edges to get hub/authority rankings of pages related
to (identified) topics.
Regarding use of nonnegative PARAFAC, Mørup et al. (27) have studied
its use for EEG-related applications. They used the associated multiplica-
tive update rule for a least squares and Kulbach-Leibler (KL) divergence im-
plementation of nonnegative PARAFAC, which they called NMWF-LS and
NMWF-KL, respectively. FitzGerald et al. (15) and Mørup et al. (26) both
used nonnegative PARAFAC for sound source separation and automatic music
transcription of stereo signals.
Bader, Berry, and Browne (5) described the first use of a nonnegative
PARAFAC algorithm to extract and detect meaningful discussions from email
messages. They encoded one year of messages from the Enron Email Set into
a sparse term-author-month array and found that the nonnegative decomposi-
tion was more easily interpretable through its preservation of data nonnegativ-
ity in the results. They showed that Gantt-like charts can be constructed/used
to assess the duration, order, and dependencies of focused discussions against
the progression of time. This study expands upon that work and demon-
strates the first application of a four-way term-author-recipient-day array for
the tracking of targeted threads of discussion through time.

×

5.2 Notation
Three-way and higher multidimensional arrays or tensors are denoted by
boldface Euler script letters, e.g., . An element is denoted by the requisite

number of subscripts. For example, element ( i, j, k, l ) of a fourth-order tensor
X

X

is denoted by x ijkl .