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◦
The symbol
denotes the tensor outer product,
⎛
⎞
A
11
B
11
···
A
11
B
m
1
⎝
⎠
.
.
.
.
.
A
1
◦
B
1
=
.
A
m
1
B
11
···
A
m
1
B
m
1
∗
The symbol
denotes the Hadamard (i.e., elementwise) matrix product,
⎛
⎝
⎞
⎠
A
11
B
11
···
A
1
n
B
1
n
.
.
.
.
.
A
∗
B
=
.
A
m
1
B
m
1
···
A
mn
B
mn
And the symbol
denotes the Khatri-Rao product (columnwise Kronecker)
(35),
B
=
A
1
⊗
B
n
,
A
B
1
···
A
n
⊗
where the symbol
denotes the Kronecker product.
The concept of
matricizing
or
unfolding
is simply a rearrangement of the
entries of
⊗
into a matrix. We will follow the notation used in (35), but
alternate notations exist. For a four-way array
X
X
of size
m
×
n
×
p
×
q
,the
notation
X
(
m×npq
)
represents a matrix of size
m
npq
in which the
n
-index
runs the fastest over the columns and
p
the slowest. Many other permutations,
such as
X
(
q×mnp
)
, are possible by changing the row index and the fastest-to-
slowest column indices.
The norm of a tensor,
×
, is the square root of the sum of squares of all
its elements, which is the same as the Frobenius norm of any of the various
matricized arrays.
X
5.3 Tensor Decompositions and Algorithms
While the original PARAFAC algorithm was presented for three-way arrays,
it generalizes to higher-order arrays (22). Earlier text analysis work using
PARAFAC in (5) focused on the three-way case, but here we present the
four-way case because our application also pertains to four-way data.
Suppose we are given a tensor
X
of size
m
×
n
×
p
×
q
and a desired
approximation rank
r
. The goal is to decompose
as a sum of vector outer
products as shown in
Figure 5.1
for the three-way case. It is convenient to
group all
r
vectors together in factor matrices
A, B, C,
and
D
,eachhaving
r
columns. The following mathematical expressions of this model use different
X
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