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?
11/17