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