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In-Depth Information
n
i
=
1
a
i
λ
s
i
,
j
=
1
a
j
λ
s
j
n
=
L
2
(
T
)
i
=
1
a
i
λ
s
i
L
2
(
T
)
≥
0
,
(1.45)
n
=
since the norm is nonnegative.
Proof (Property 1.5).
Consider the 2
×
2 CI kernel matrix,
I
(
s
i
,
s
i
)
I
(
s
i
,
s
j
)
V
=
.
I
(
s
j
,
s
i
)
I
(
s
j
,
s
j
)
From Property 1.2, this matrix is symmetric and nonnegative definite. Hence, its
determinant is nonnegative [7, p. 245]. Mathematically,
I
2
det
(
V
)=
I
(
s
i
,
s
i
)
I
(
s
j
,
s
j
)
−
(
s
i
,
s
j
)
≥
0
,
which proves the result of Equation (1.16).
Proof (Property 1.6).
Consider two spike trains,
s
i
,
s
j
∈S
(
T
)
. The norm of the sum
of two spike trains is
λ
s
i
+
λ
s
j
=
λ
s
i
+
λ
s
j
,
λ
s
i
+
λ
s
j
2
(1.46a)
2
λ
s
i
,
λ
s
j
+
λ
s
j
,
λ
s
j
=
λ
s
i
,
λ
s
i
+
(1.46b)
λ
s
i
λ
s
j
+
λ
s
j
2
2
≤
λ
s
i
+
2
(1.46c)
=
λ
s
i
+
λ
s
j
2
,
(1.46d)
with the upper bound in step 1.46c established by the Cauchy-Schwarz inequality
(Property 1.5).
References
1. Aronszajn, N. Theory of reproducing kernels. Trans Am Math Soc
68
(3), 337-404 (1950)
2. Berg, C. Christensen, J.P.R., Ressel, P. Harmonic Analysis on Semigroups: Theory of Positive
Definite and Related Functions. Springer-Verlag, New York (1984)
3. Bohte, S.M., Kok, J.N., Poutre, H.L.: Error-backpropagation in temporally encoded networks
of spiking neurons. Neurocomputing
48
(1-4), 17-37 (2002). DOI 10.1016/S0925-2312(01)
00658-0
4. Carnell, A., Richardson, D.: Linear algebra for time series of spikes. In: Proceedings European
Symposium on Artificial Neural Networks, pp. 363-368. Bruges, Belgium (2005)
5. Dayan, P., Abbott, L.F. Theoretical Neuroscience: Computational and Mathematical Modeling
of Neural Systems. MIT Press, Cambridge, MA (2001)
6. Diggle, P., Marron, J.S. Equivalence of smoothing parameter selectors in density and intensity
estimation. J Acoust Soc Am
83
(403), 793-800 (1988)
7. Harville, D.A. Matrix Algebra from a Statistician's Perspective. Springer, New York (1997)
