Digital Signal Processing Reference
In-Depth Information
where,
are positive regularization coefficients. The solution to the optimization
problem in Eq.(5) is given by proposition 1.
γ ν
,
2
:
d d
Proposition 1.
Given a positive kernel function
kR R R
×→
satisfying
(
)
( )
T
( )
are regularization coefficients, the
optimization problem in Eq.(5) turns out to a generalized eigen-problem
(
k
xz
,
=
φφ
x
z
,
γ
,
vv R
∈
+
and
ν
=
1
λ
12
(
)
)
(
)
T
T
T
1
−
λ
ΩΩ ΩΩ β K
+
λ
=
ν
+
v
1
−
λ
ΩΩ β
(6)
bb t
t
2
ww
Proof:
Using Lagrange multipliers, this optimization problem in Eq.(5) can be formu-
lated as
γ
1
v
(
)
(
)
(
)
(
)
T
T
T
T
T
T
L
β e τ β
,,
=
1
−
λ
ΩΩ ΩΩ β βKβ
+
λ
−
−
2
1
−
λ
ee τ e Ωβ
−
-
bb t
t
w
2
2
2
where
τ
is Lagrange multiplier vector. According to KKT condition, the derivative of
(
)
L
β e τ
with respect to
,,
β e τ
is computed and set zeros. Thus, we have
,,
(
)
T
(
)
δδ γ
L
=
1
−
λ
Ω Ω Ω Ω β KβτΩ
(7)
T
+
λ
T
−
+
T
=
0
β
bb t
t
w
(
)
δδ
L
e τ
=−
v
2
1
−
λ
e
=
0
,
δδ=
L
τ e Ωβ
(8)
-
=
0
w
Define
, according to Eq.(7)-(8), the optimization problem in Eq.(5) reduces to
a generalized eigen-problem expressed as
ν
=
1
λ
(
)
(
)
(
)
T
T
T
1
−
λ
ΩΩ ΩΩ β K
+
λ
=
ν
+
v
1
−
λ
ΩΩ β
■
bb t
t
2
ww
vvvv v
'
=
,
=
1
Define
, the Eq.(6) is then written as
21
2
(
)
(
)
(
)
T
T
T
1
−
λ
ΩΩ ΩΩ β
+
λ
=
vv
'
K
+
1
−
λ
ΩΩ β
(9)
bb t
t
1
ww
Let
β
be eigenvectors corresponding to the largest eigenvalue of the above
eigenproblem, the nonlinear features can be yielded via
()
1
,
,
M
T
( )
[
]
T
(
)
(
)
1
M
z
=
z
,
,
z
=
ββ
,
,
K
,
K
=
k
xx
1
,
,
,
k
x x
,
,
Md
≤
.
i
i
i
1
M
x
x
i
N
i
i
i
{
}
1
N
()
d
*
β
can be obtained by solving
Denote
Z
=
z
, after
λ
is given, the optimal
ν
,
i
i
=
the following optimization problem
(
(
)
(
)
(
)
()
()
()
d
d
z
z
d
)
}
(
)
(
)
z
CS
β
,
η
=
ax
1
−
λ
S
+
λ
S
1
−
λ
S
+
η
β Kβ
T
(10)
1
b
t
w
{
d
∈
1, 2,
,
N
(
)
where
()
d
is within-class scatter matrix associated with data set
Z
.
z
S
w
Now, we deduced LPKHDA algorithm by incorporating locality preserving into
KHDA. For the dataset
{
}
()
N
φ
x
, we refer to [14] and have
i
i
=
1
