Information Technology Reference
In-Depth Information
1.6.1 Space Spanned by Intensity Functions
In the introduction of the mCI kernel the usual dot product in
L
2
(
T
)
, the space of
square integrable intensity functions defined on
, was utilized. The definition of
the inner product in this space provides an intuitive understanding to the reasoning
involved.
L
2
(
λ
s
i
(
T
is clearly a Hilbert space with inner product
and norm defined in Equations (1.12) and (1.13). Notice that the span of this space
contains also elements (functions) that may not be valid intensity functions since, by
definition, intensity functions are always nonnegative. However, since our interest
is mainly on the evaluation of the inner product this is of no consequence. The key
limitation, however, is that
L
2
(
λ
s
i
(
t
)
,
t
∈T
)
⊂
L
2
(
T
)
is
not
an RKHS. This should be clear
because elements in this space are functions defined on
t
)
,
t
∈T
)
T
, whereas elements in the
RKHS
H
I
must be functions defined on
S
(
T
)
.
Despite the differences, the spaces
L
2
(
λ
s
i
(
t
)
,
t
∈T
)
and
H
I
are closely related.
In fact,
L
2
H
I
are congruent. This congruence can be verified
explicitly since there is clearly a one-to-one mapping,
(
λ
s
i
(
t
)
,
t
∈T
)
and
(
)
∈
(
λ
(
)
,
∈T
)
←→
Λ
(
)
∈H
,
λ
t
L
2
t
t
s
s
i
s
i
s
i
I
and, by definition of the mCI kernel,
s
j
)=
λ
s
i
,
λ
s
j
L
2
(
T
)
=
Λ
s
i
,
Λ
s
j
I
(
s
i
,
H
I
.
(1.19)
A direct implication of the basic congruence theorem is that the two spaces have the
same dimension [20].
1.6.2 Induced RKHS
In Section 1.5.1 it was shown that the mCI kernel is symmetric and positive definite
(Properties 1.1 and 1.3, respectively) and consequently, by the Moore-Aronszajn
theorem [1], there exists a Hilbert space
H
I
in which the mCI kernel evaluates the
inner product and is a reproducing kernel (Property 1.4). This means that
I
(
s
i
, ·
)
∈
H
I
for any
s
i
∈S
(
T
)
and, for any
ξ
∈H
I
, the reproducing property holds
ξ
,
I
(
s
i
, ·
)
H
I
=
ξ
(
s
i
)
.
(1.20)
As a result the kernel trick follows:
s
j
)=
I
s
j
, ·
)
I
(
s
i
,
(
s
i
, ·
)
,
I
(
H
I
.
(1.21)
Written in this form, it is easy to verify that the point in
H
I
corresponding to a spike
train
s
i
∈S
(
T
)
is
I
(
s
i
, ·
)
. In other words, given any spike train
s
i
∈S
(
T
)
, this spike
∈H
train is mapped to
Λ
I
, given explicitly (although unknown in closed form) as
s
i
=
(
, ·
)
Λ
I
s
i
. Then Equation (1.21) can be restated in the more usual form
s
i
