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kernel satisfying the above requirements is theoretically valid and understood under
the framework proposed here although, obviously, the practical results may vary.
An example of a family of kernels that can be used (but not limited to) is the
radial basis functions [2],
p
κ
(
t
m
,
t
n
)=
exp
(
−|
t
m
−
t
n
|
)
,
t
m
,
t
n
∈T,
(1.6)
for any 0
2. Some well-known kernels, such as the widely used Gaussian and
Laplacian kernels, are special cases of this family for
p
<
p
≤
1, respectively.
It is interesting to notice that shift-invariant kernels result in a natural norm in-
duced by the inner product with the following property:
=
2 and
p
=
Φ
m
=
κ
(
0
)
,
∀
Φ
m
∈H
κ
.
(1.7)
Since the norm of the transformed spike times in
H
κ
is constant, all the spike times
are mapped to the surface of a hypersphere in
H
κ
. The set of transformed spike
times is called the manifold of
. Moreover, this shows in a different perspective
why the kernel used needs to be nonnegative. Furthermore, the
geodesic distance
corresponding to the length of the smallest path contained within this manifold (in
this case, the hypersphere) between two functions in this manifold,
S
(
T
)
Φ
m
and
Φ
n
,is
given by
arccos
Φ
m
,
Φ
n
Φ
m
Φ
n
d
(
Φ
m
,
Φ
n
)=
Φ
m
arccos
κ
(
(1.8)
t
m
,
t
n
)
=
κ
(
0
)
.
κ
(
0
)
Put differently, from the geometry of the transformed spike times, the kernel func-
tion is proportional to the cosine of the angle between two transformed spike times
in
2, which re-
stricts the manifold of transformed spike times to a small area of the hypersphere.
With the kernel inducing the above metric, the manifold of the transformed points
forms a
Riemannian space
. However, this space is
not
a linear space. Fortunately,
its span is obviously a linear space. In fact, it equals the RKHS associated with the
kernel. Although this is not a major problem, computing with the transformed points
will almost surely yield points outside of the manifold of transformed spike times.
This means that such points cannot be mapped back to the input space directly. De-
pending on the aim of the application this may not be necessary, but if required, it
may be solvable through a projection to the manifold of transformed input points.
H
κ
. Because the kernel is nonnegative, the maximum angle is
π
/
1.4 Inner Product for Spike Trains
Although any kernel verifying the conditions discussed in the previous section in-
duces an RKHS and therefore is of interest on itself, the fact that it only operates
