Slide 6
Slide 6 text
Background Rep Ker & Riesz Rep Thm Kernel Ex Assoc Measures Error Bds References
Reproducing Kernels for Functions on {1, . . . , d}, aka Vectors
Let F := all functions on {1, . . . , d} “=” Rd
Pick a symmetric, positive definite (positive eigenvalues) matrix W ∈ Rd×d to define an inner product
⟨f, h⟩ := fTWh, ∀f, h ∈ F, where f = f(t) d
t=1
Reproducing kernel, K, is defined by K(t, x) d
t,x=1
= K := W−1, and has the properties
Symmetry K(t, x) = K(x, t) because W is symmetric and thus so is K
Positive Definiteness K(xi
, xj
) n
i,j=1
is positive definite for any distinct x1
, . . . , xn ∈ {1, . . . , d}
Belonging K(·, x) = xth column of K =: Kx ∈ F
Reproduction ⟨K(·, x), f⟩ = KT
x
Wf = ex
f = f(x) since K := W−1; ex
:= (0, . . . , 0, 1
xth position
, 0, . . .)T
Riesz Representation Theorem says that for any linear function, LINEAR, there is a representer g such
that LINEAR(f) = ⟨g, f⟩ = gTWf. Note
g(1)
.
.
.
g(d)
= g = KWg =
KT
1
Wg
.
.
.
KT
d
Wg
=
⟨K(·, 1), g⟩
.
.
.
⟨K(·, d), g⟩
=
LINEAR(K(·, 1))
.
.
.
LINEAR(K(·, d))
3/20