Nonnegative Matrix and Tensor Factorization for Discussion Tracking - Text Mining: Classification, Clustering, and Applications - page 108

4−way Results

1

Forwarding of the Texas A&M Aggies fight song
from E. Bass to 4 other possible alumni

0.5

0

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

FIGURE 5.6 : Forwarding of Texas A&M school fight song.

may be an alum and the four recipients went to a football game and asked
“what is everyone singing?” Exposing that type of social interaction is an ad-
vantage for four-way analysis over the three-way analysis without recipients.

5.6 Visualizing Results of the NMF Clustering
The previous sections demonstrate the value of three-way and four-way
tensor decompositions. Yet it is either very cumbersome or often impossible
to visualize these higher-dimensional tensors. Figures 5.4 -5.6 are attempts at
visualizing the information provided by the tensors, yet they are somewhat
limited in scope. As an alternative, in this section, we resort to the standard
two-way (or matrix) decomposition to help us visualize some of the patterns
uncovered by the three-way and higher decompositions. In general, one can
always easily visualize any two dimensions of an n -way tensor decomposition
by considering the matrix associated with those dimensions as created by the
tensor decomposition. In this spirit, we discuss a tool for visualizing clusters
in two-way factors.
It is well known (9) that the nonnegative matrix factorization (NMF) can
be used to cluster items in a collection. For instance, if the data matrix
is a term-by-document matrix X , which has been factored with the NMF as
X = AB , then the rows of ¯ can be used to cluster terms, while the columns of
¯ can be used to cluster documents. As a result, terms and documents are, in
some sense, clustered independently. There are two main types of clustering:
hard clustering and soft clustering. Hard clustering means that items (in
this case, terms and documents) can belong to only one cluster, whereas in
soft clustering items are allowed to belong to multiple clusters, perhaps with
varying weights for these multiple assignments. If hard clustering is employed,
then cluster assignment is easy. Term i belongs to cluster j if A ( i, j )isthe
maximum element in the i th row of A . Similarly, document k belongs to
cluster l if B ( l, k ) is the maximum element in the k th column of B .
Once cluster assignments are available (by either hard or soft clustering), a