Database Reference
In-Depth Information
Chapter 9
Decomposition in Transition II:
Adaptive Tensor Factorization
Abstract We consider generalizations of the previously described SVD-based
factorization methods to a tensor framework and discuss applications to recommen-
dation. In particular, we generalize the previously introduced incremental SVD
algorithm to higher dimensions. Furthermore, we briefly address other tensor factor-
ization frameworks like CANDECOMP/PARAFAC as well as hierarchical SVD and
Tensor-Train-Decomposition.
9.1 Beyond Behaviorism: Tensor-PCA-Based CF
9.1.1 What Is a Tensor?
Historically, the concept of a tensor originated in differential geometry as a calculus
for dealing with multilinear forms on manifolds. In recent years, thought, tensor-
based approaches have made their way into numerical analysis of partial differen-
tial equations to model highly multivariate functions and, more importantly, into
data mining as formal frameworks for multimodal data. We shall address the latter
in more detail in the subsequent section after introducing the basic notions and
notations related to the concept in the following.
If we conceive of a matrix as a two-dimensional array, then, in a nutshell, a tensor
is a generalized matrix in that it may be thought of as a
d
-dimensional array, where
d
may be an arbitrary natural number. A more formal version of this definition
suffices for the purpose pursued herein.
N
d
is a
sequence of real numbers indexed by the set
n
1
...
n
d
. We denote the set of
d-mode tensors of dimensionality
(n
1
,
Definition 9.1 A (real) d-mode
tensor
of dimensionality
(n
1
,
...
,
n
d
)
∈
n
1
,
...
,
n
d
.
Multi-indexes are somewhat cumbersome to deal with. Thus, to achieve a clear
representation, we introduce the following notation.
N
d
by
...
,
n
d
)
∈
R
