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))

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