Abstract
In this paper, we present selected old and new results on the optimal solution of linear problems based on noisy information, where the noise is bounded or random. This is done in the framework of information-based complexity (IBC), and the main focus is on the following questions:
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(i)
What is an optimal algorithm for given noisy information?
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(ii)
What is the \(\varepsilon\)-complexity of a problem with noisy information?
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(iii)
When is a multivariate problem with noisy information tractable?
The answers are given for the worst case, average case, and randomized (Monte Carlo) settings. For (ii) and (iii) we present a computational model in which the cost of information depends on the noise level. For instance, for integrating a function \(f: D \rightarrow \mathbb{R}\), available information may be given as
with \(x_{j}\mathop{ \sim }\limits^{\mathrm{ i.i.d.}}\mathcal{N}(0,\sigma _{j}^{2})\). For this information one pays \(\sum _{j=1}^{n}c(\sigma _{j})\) where c:[0, ∞) → [0, ∞] is a given cost function. We will see how the complexity and tractability of linear multivariate problems depend on the cost function, and compare the obtained results with noiseless case, in which c ≡ 1.
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References
Bakhvalov, N.S.: On the optimality of linear methods for operator approximation in convex classes. Comput. Math. Math. Phys. 11, 244–249 (1971)
Casella, G., Strawderman, W.E.: Estimating bounded normal mean. Ann. Statist. 9, 870–878 (1981)
Donoho, D.L.: Statistical estimation and optimal recovery. Ann. Statist. 22, 238–270 (1994)
Donoho, D.L., Johnstone, I.M.: Minimax estimation via wavelet shrinkage. Ann. Statist. 26, 879–921 (1998)
Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D.: Wavelet shrinkage: asymptopia? J. Roy. Stat. Soc. ser. B 57, 301–369 (1995)
Donoho, D.L., Liu, R.C., MacGibbon, K.B.: Minimax risk over hyperrectangles, and implications. Ann. Statist. 18, 1416–1437 (1990)
Heinrich, S.: Random approximation in numerical analysis. In: Berstadt et al. (ed.) Proceedings of the Functional Analysis Conference, Essen 1991, pp. 123–171. Marcel Dekker, New York (1993)
Hinrichs, A., Novak, E., Ulrich, M., Woźniakowski, H.: The curse of dimensionalty for numerical integration of smooth functions. Submitted
Ibragimov I.A., Hasminski, R.Z.: On the nonparametric estimation of the value of a linear functional in Gaussian white noise. Theory Probab. Appl. 29, 19–32 (1984). In Russian
Kacewicz, B.Z., Plaskota, L.: Noisy information for linear problems in the asymptotic setting. J. Complexity 7, 35–57 (1991)
Kuo, H.H.: Gaussian measures in banach spaces. In: Lecture Notes in Math, vol. 463. Springer, Berlin (1975)
Magaril-Il’yaev G.G., Osipenko, K.Yu.: On optimal recovery of functionals from inaccurate data. Matem. Zametki 50, 85–93 (1991). In Russian
Melkman, A.A., Micchelli, C.A.: Optimal estimation of linear operators in Hilbert spaces from inaccurate data. SIAM J. Numer. Anal. 16, 87–105 (1979)
Micchelli, C.A., Rivlin, T.J.: A survey of optimal recovery. In: Estimation in Approximation Theory, pp. 1–54. Plenum, New York (1977)
Novak, E.: Deterministic and stochastic error bounds in numerical analysis. In: Lecture Notes in Mathematics, vol. 1349. Springer, Berlin (1988)
Novak, E.: On the power of adaption. J. Complexity 12, 199–237 (1996)
Novak, E., Woźniakowski, H.: Tractability of multivariate problems. Volume I: linear information. In: EMS Tracts in Mathematics, vol. 6. European Mathematical Society, Zürich (2008)
Novak, E., Woźniakowski, H.: Tractability of multivariate problems. Volume II: standard information for functionals. In: EMS Tracts in Mathematics, vol. 6. European Mathematical Society, Zürich (2010)
Novak, E., Woźniakowski, H.: Tractability of multivariate problems. Volume III: standard information for operators. In: EMS Tracts in Mathematics, vol. 6. European Mathematical Society, Zürich (2012)
Nussbaum, M.: Spline smoothing in regression model and asymptotic efficiency in l 2. Ann. Stat. 13, 984–997 (1985)
Papageorgiou, A., Wasilkowski, G.: On the average complexity of multivariate problems. J. Complexity 6, 1–23 (1990)
Pinsker, M.S.: Optimal filtering of square integrable signals in Gaussian white noise. Probl. Inform. Transm. 16, 52–68 (1980). In Russian
Plaskota, L.: On average case complexity of linear problems with noisy information. J. Complexity 6, 199–230 (1990)
Plaskota, L.: A note on varying cardinality in the average case setting. J. Complexity 9, 458–470 (1993)
Plaskota, L.: Average case approximation of linear functionals based on information with deterministic noise. J. Comput. Inform. 4, 21–39 (1994)
Plaskota, L.: Average complexity for linear problems in a model with varying noise of information. J. Complexity 11, 240–264 (1995)
Plaskota, L.: How to benefit from noise. J. Complexity 12, 175–184 (1996)
Plaskota, L.: Worst case complexity of problems with random information noise. J. Complexity 12, 416–439 (1996)
Plaskota, L.: Noisy Information and Computational Complexity. Cambridge University Press, Cambridge (1996)
Plaskota, L.: Average case uniform approximation in the presence of Gaussian noise. J. Approx. Theory 93, 501–515 (1998)
Smolyak, S.A.: On optimal recovery of functions and functionals of them. Ph.D. thesis, Moscow State University (1965). In Russian
Sukharev, A.G.: On the existence of optimal affine methods for approximating linear functionals. J. Complexity 2, 317–322 (1986)
Tikhonov, A.N., Goncharsky A.V., Stepanov, V.V., Yagola, A.G.: Numerical Methods for the Solution of Ill-Posed Problems. Kluwer, Dordrecht (1995)
Traub, J.F., Wasilkowski, G.W., Woźniakowski, H.: Information-Based Complexity. Academic, New York (1980)
Vakhania, N.N., Tarieladze, V.I., Chobanyan, S.A.: Probability Distributions on Banach Spaces. Reidel, Dordrecht (1987)
Wahba, G.: Spline models for observational data. In: CBMS-NSF Series in Applied Mathematics, vol. 59. SIAM, Philadelphia (1990)
Wasilkowski, G.W.: Information of varying cardinality. J. Complexity 2, 204–228 (1986)
Werschulz, A.G., Woźniakowski, H.: Are linear algorithms always good for linear problems? Aequations Math. 30, 202–212 (1986)
Woźniakowski, H.: Tractability and strong tractability of linear mulivariate problems. J. Complexity 10, 96–128 (1994)
Acknowledgements
This research was supported by the Ministry of Science and Higher Education of Poland under the research grant N N201 547738. The author highly appreciates valuable comments from Henryk Woźniakowski and two anonymous referees.
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Plaskota, L. (2013). Noisy Information: Optimality, Complexity, Tractability. In: Dick, J., Kuo, F., Peters, G., Sloan, I. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics & Statistics, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41095-6_7
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