Randomized Complexity of Vector-Valued Approximation

06/23/2023
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by   Stefan Heinrich, et al.
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We study the randomized n-th minimal errors (and hence the complexity) of vector valued approximation. In a recent paper by the author [Randomized complexity of parametric integration and the role of adaption I. Finite dimensional case (preprint)] a long-standing problem of Information-Based Complexity was solved: Is there a constant c>0 such that for all linear problems 𝒫 the randomized non-adaptive and adaptive n-th minimal errors can deviate at most by a factor of c? That is, does the following hold for all linear 𝒫 and nβˆˆβ„• e_n^ ran-non (𝒫)≀ ce_n^ ran (𝒫) ? The analysis of vector-valued mean computation showed that the answer is negative. More precisely, there are instances of this problem where the gap between non-adaptive and adaptive randomized minimal errors can be (up to log factors) of the order n^1/8. This raises the question about the maximal possible deviation. In this paper we show that for certain instances of vector valued approximation the gap is n^1/2 (again, up to log factors).

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