Digital Signal Processing Reference
In-Depth Information
Section
14.7
draws conclusions and provides perspectives for improvement of the
proposed system.
14.1.4 Notations
In the sequel, uppercase bold letters denote matrices,
I
is the identity matrix, and
E
is the matrix full of ones. Lowercase bold letters denote vectors, and
e
is the vector
full of ones. A bold zero
0
represents either a null matrix or vector. Lowercase plain
letters such as
n
R
++
denote respectively the sets of
non-negative and of positive scalars. The element-wise multiplication and division
between two matrices
A
and
B
are denoted respectively by a circled times
A
,
m
,
r
denote scalars.
R
+
and
⊗
B
B
. The element-wise power
p
of
A
is denoted by
A
.
p
, and the
and a fraction bar
A
/
element-wise square-root of
A
can alternatively be denoted by
√
A
. Element-wise
inequalities between
A
and
B
are simply written as
A
≤
B
. The transpose of
A
is denoted by
A
. The non-negative matrices
A
+
and
A
−
denote respectively the
positive and negative parts of
A
defined as follows:
a
ij
if
a
ij
>
0
−
a
ij
if
a
ij
<
0
a
ij
=
a
ij
=
and
otherwise
.
(14.1)
0
otherwise
0
14.2 Related Background
In this section, we first introduce the standard NMF problems and algorithms. We
then provide an overview of NMF techniques in applications to audio event detection.
14.2.1 Non-Negative Matrix Factorization
The standard NMF model is a low-rank approximation technique for unsupervised
multivariate data analysis. Given an
n
×
m
non-negative matrix
V
and a positive
integer
r
<
min
(
n
,
m
)
, NMF tries to factorize
V
into an
n
×
r
non-negative matrix
W
and an
r
×
m
non-negative matrix
H
such that:
V
≈
WH
.
(14.2)
The multivariate data to decompose are stacked into
V
, whose columns represent
the different observations, and whose rows represent the different variables. Each
column
v
j
of
V
can then be expressed as
v
j
Wh
j
=
i
h
ij
w
i
, where
w
i
and
h
j
are respectively the
i
-th column of
W
and the
j
-th column of
H
. The columns of
≈
