Problem One-Dimensional Problems Multiple Dimensions References

Product Kernels with Weights

To avoid the curse of dimensionality in the exponent of ε−1 or the coeﬃcient, we typically must

appeal to weighted spaces, i.e., spaces where not all coordinates are equally important, e.g,

Kd(x, t) =

u⊂1:d

γu

j∈u

xj + tj − xj − tj , γu → 0 as max j : j ∈ u or |u| → ∞

f, g H

=

f(0)g(0)

γ∅

+

1

2γ{1}

1

−1

∂f

∂x1

(x1, 0, · · · )

∂g

∂x1

(x1, 0, · · · ) dx1

+

1

2γ{2}

1

−1

∂f

∂x2

(0, x2, 0, · · · )

∂g

∂x2

(0, x1, 0, · · · ) dx2 + · · ·

+

1

4γ{1,2}

1

−1

1

−1

∂2f

∂x1∂x2

(x1, x2, 0, · · · )

∂2g

∂x1∂x2

(x1, x2, 0, · · · ) dx1dx2 + · · ·

The discrepancy, ηn H

, tends to zero as n → ∞ for your favorite data sites if γu → 0 fast

enough. For f H

to be of reasonable size as γu → 0, the importance of the coordinates in u

must vanish. Problem: How do you know which coordinates are most important a priori?

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