Asymptotic theory of finite dimensional normed spaces
Asymptotic theory of finite dimensional normed spaces
Information-based complexity
s-numbers in information-based complexity
Journal of Complexity
Approximation numbers of Sobolev embeddings-Sharp constants and tractability
Journal of Complexity
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We compare linear and non-linear approximations for linear problems. Let X be a linear space and Y a normed space. Let S : X → Y and N : X → Rn be linear mappings and consider the worst-case setting over some balanced convex set F ⊆ X. We compare the minimal error errlin (S, N) achievable by linear algorithms processing N with the minimal error errlin(S, N) achievable by arbitrary algorithms using N. For bounded linear problems, P. Mathé showed that infN errlin(S, N) ≤ (1 + n1/2) ċ infN err(S, N), where the infimum is taken over all bounded linear mappings N : X → Rn. We generalize this result as follows: If the target space Y is complete, then for any linear N : X → Rn. we have errlin(S, N, F) ≤ (1 +n1/2)err(S, N, F). This and some similar results can easily be derived from a general relation of this problem to extension properties of normed spaces, and the manifold and powerful results available for this problem. This allows a unified treatment of the above estimate together with the results of Smolyak and Packel, who showed that linear algorithms are optimal for some Y. The results are also partially extended to noisy information with uniformly bounded noise.