Database Reference
In-Depth Information
1. Complexity:
m ¼ O
(
k
q
) for some
q
N,
i.e.,
m
should increase at most
∈
polynomially with the order
k
.
p
s
ss
0
X
^
2. Approximation:
m
u
s
β
c
ss
0
β
should be small with respect to
β
∈
s
ss
0
an appropriate norm
kk
.
3. Consistency: Each slice obtained by fixing the index corresponding to the first
mode should be a row-stochastic matrix.
In a nutshell, fulfilling the above requirements amounts to reducing complexity
under preservation of as much information as possible.
Let us consider the case where (the matricization of) the core tensor is
taken to be
:¼ U
þ
P
ðÞ
,
C
ðÞ
where
U
+
denotes the Moore-Penrose pseudo-inverse (MPP) of
U
with respect to
the canonical inner product, i.e.,
1
U
T
1
U
þ
:¼ U
T
U
β
∈
m
U
T
¼
G
β
:
ð
10
:
6
Þ
The complexity requirement may be controlled by a suitable choice of the
partition. As regards approximation, it follows from the definition of the MPP
that the factorization is a minimizer of
F
¼
F
:
P U
1
C
P U
1
C
min
mss
C
∈R
Finally, consistency follows from the fact that each 1 slice of
U
1
C
is a convex
combination of row-stochastic matrices.
Besides its fulfilling our requirements, a crucial case for this factorization
approach can be made in virtue of its intuitive interpretation. Specifically, the
second identity in equation (
10.6
) reveals that the factorization is an algebraic
representation of replacing each slice by the centroid of the slices of
P
in the
corresponding class of the partition. This is the procedure of vector quantization
that we already had used in Sect.
8.6
.
Thus, in case we already know the transition probabilities
P
, we can apply the
k
-
means algorithm to find the best partition
G
β
β
∈
m
for a given partition number
m
.
Here,
k ¼ m
is used in the
k
-means algorithm, and the slices of
P
are treated as
vectors, i.e., vectorization is applied to the slice matrices in a similar way as
matricization to tensors.
Example 10.1
In the following, we shall illustrate the proposed factorization by
means of a simple example. For the sake of simplicity, we shall ignore the actions
