Information Technology Reference
In-Depth Information
and corresponds to the
usual dot product from
L
2
, this property is a direct consequence of the properties
inherited from
L
2
. More specifically, Property 1.1 guaranties the mCI kernel is a
valid inner product.
Because the mCI kernel operates on elements of
L
2
(
T
)
Property 1.2.
For any set of
n
≥
1 spike trains, the mCI kernel matrix
⎡
⎣
⎤
⎦
,
)
(
,
)
...
(
,
)
I
(
s
1
s
1
I
s
1
s
2
I
s
1
s
n
(
,
)
(
,
)
...
(
,
)
I
s
2
s
1
I
s
2
s
2
I
s
2
s
n
V
=
.
.
.
.
.
.
I
(
s
n
,
s
1
)
I
(
s
n
,
s
2
)
...
I
(
s
n
,
s
n
)
is symmetric and nonnegative definite.
The proof is given in the appendix. Through the work of Moore [17] and due to
the Moore-Aronszajn theorem [1], the following two properties result as corollaries
of Property 1.2.
Property 1.3.
The mCI kernel is a symmetric and positive definite kernel. Thus,
by definition, for any set of
n
≥
1 point processes and corresponding
n
scalars
a
1
,
a
2
,...,
a
n
∈
R
,
n
i
=
1
n
j
=
1
a
i
a
j
I
(
s
i
,
s
j
)
≥
0
.
(1.15)
Property 1.4.
There exists a Hilbert space for which the mCI kernel is a reproducing
kernel.
Actually, Property 1.3 can be obtained explicitly by verifying that the inequality
of Equation (1.15) is implied by Equations (1.44) and (1.45) in the proof of Prop-
erty 1.2 (in the appendix).
Properties 1.2, 1.3, and 1.4 are equivalent in the sense that any of these properties
implies the other two. The most important consequence of these properties, explic-
itly stated through Property 1.4, is that the
mCI kernel induces an unique RKHS
,
henceforth denoted by
H
I
.
Property 1.5.
The mCI kernel verifies the Cauchy-Schwarz inequality,
I
2
(
s
i
,
s
j
)
≤
I
(
s
i
,
s
i
)
I
(
s
j
,
s
j
)
∀
s
i
,
s
j
∈S
(
T
)
.
(1.16)
The proof is given in the appendix. The Cauchy-Schwarz inequality is important
since the triangle inequality results as an immediate consequence and it induces a
correlation coefficient-like measure very useful for matching spike trains. Indeed,
the Cauchy-Schwarz inequality is the concept behind the spike train measure pro-
posed by Schreiber et al. [32]. However, our proof in appendix verifies that all it
is required is a spike train kernel inducing an RKHS, and, therefore, the idea by
Schreiber and colleagues is easily extendible.
