Curse of Dimensionality in Approximation of Random Fields Mikhail ...
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Curse of Dimensionality in Approximation of Random Fields Mikhail ...
Curse of Dimensionality in Approximation of Random Fields. Mikhail Lifshits and Ekaterina Tulyakova. Consider a random f
Curse of Dimensionality in Approximation of Random Fields Mikhail Lifshits and Ekaterina Tulyakova Consider a random field of tensor product type X(t), t ∈ [0, 1]d , given by d d X Y Y ξk X(t) = λ(kl ) ϕkl (tl ), k∈Nd
l=1
l=1
where (λ(i))i>0 ∈ `2 , (ϕi )i>0 is an orthonormal system in L2 [0, 1] and (ξk )k∈Nd are noncorrelated random variables with zero mean and unit variance. We investigate the quality of approximation (both in the average and in the probabilistic sense) to X by the n-term partial sums Xn minimizing the quadratic error EkX − Xn k2 . In the first part of the article we consider the case of fixed dimension d. In the second part, following the suggestion of H.Wo´zniakowski, we consider the same problem for d → ∞. We show that, for any fixed level of relative error, approximation complexity increases exponentially and find the explosion coefficient. We also show that the behavior of the probabilistic and average complexity is essentially the same in the large domain of parameters.
