Linear vs. nonlinear algorithms for linear problems

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. Mathe 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.

[1]  J. Lindenstrauss.  On the extension property for compact operators , 1962 .

[2]  P. Wojtaszczyk.  Banach Spaces For Analysts: Preface , 1991 .

[3]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[4]  G. Pisier.  ASYMPTOTIC THEORY OF FINITE DIMENSIONAL NORMED SPACES (Lecture Notes in Mathematics 1200) , 1987 .

[5]  E. Novak.  Deterministic and Stochastic Error Bounds in Numerical Analysis , 1988 .

[6]  H. Wozniakowski,et al.  Are linear algorithms always good for linear problems? , 1986 .

[7]  E. Packel.  Linear problems (with extended range) have linear optimal algorithms , 1986 .

[8]  J. Bowen,et al.  s -numbers in information-based complexity , 1990 .

[9]  H. Woxniakowski.  Information-Based Complexity , 1988 .

[10]  Y. Gordon,et al.  Relations between some constants associated with finite dimensional Banach spaces , 1971 .

[11]  Charles A. Micchelli,et al.  A Survey of Optimal Recovery , 1977 .

[12]  M. Zippin.  Chapter 40 - Extension of Bounded Linear Operators , 2003 .

[13]  T. J. Rivlin,et al.  Optimal Estimation in Approximation Theory , 1977 .

[14]  A. Pinkus.  n-Widths in Approximation Theory , 1985 .

[15]  B. Carl,et al.  Entropy, Compactness and the Approximation of Operators , 1990 .

[16]  D. R. Lewis.  Finite dimensional subspaces of $L_{p}$ , 1978 .

[17]  J. Lindenstrauss,et al.  Handbook of geometry of Banach spaces , 2001 .

[18]  N. Tomczak-Jaegermann.  Banach-Mazur distances and finite-dimensional operator ideals , 1989 .