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In-Depth Information
PCA finds an orthonormal transformation providing a compact description of the
data. Determining the principal components of spike trains in the RKHS can be
formulated as the problem of finding the set of orthonormal vectors in the RKHS
such that the projection of the centered transformed spike trains
˜
has the
maximum variance . This means that the principal components can be obtained by
solving
{
Λ s i }
an
optimization
problem
in
the
RKHS.
A
function
ξ ∈H I
(i.e.,
ξ
:
S ( T ) −→ R
)
is
a
principal
component
if
it
maximizes
the
cost
function
i = 1 Proj ξ ( Λ s i ) 2
1
N
2
( ξ )=
ρ
ξ
,
(1.30)
J
˜
where Proj
ξ (
Λ
)
denotes the projection of the i th centered transformed spike train
s i
1 imposing that
the principal components have unit norm. To evaluate this cost function one needs
to be able to compute the projection and the norm of the principal components.
However, in an RKHS, an inner product is the projection operator and the norm
is naturally defined (see Equation (1.13)). Thus, the above cost function can be
expressed as
is the Lagrange multiplier to the constraint
2
onto
ξ
, and
ρ
ξ
1
N
i = 1 Λ s i , ξ 2
J
( ξ )=
H I ρ
ξ , ξ H I
.
(1.31)
Because in practice we always have a finite number of spike trains,
ξ
is restricted
˜
{
}
to the subspace spanned by the centered transformed spike trains
Λ
. Conse-
s i
,...,
R
quently, there exist coefficients b 1
b N
such that
N
j = 1 b j
˜
T
˜
ξ =
Λ s j = b
Λ
(1.32)
)= ˜
) T . Substituting in Equa-
˜
˜
T
where
b
=[
b 1 ,...,
b N ]
and
Λ (
t
Λ s 1 (
t
) ,...,
Λ s N (
t
tion (1.31) yields
N
N
N
i = 1
j = 1 b j Λ s i ,
Λ s j
k = 1 b k Λ s i ,
Λ s k
˜
˜
J
( ξ )=
1
N
j = 1
k = 1 b j b k Λ s i ,
N
Λ s k
(1.33)
˜
+ ρ
b + ρ 1
Ib ,
T
˜
2
T
˜
= b
I
b
where ˜
I
is the Gram matrix of the centered spike trains; that is, the N
×
N matrix
with elements
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