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The principle of penalized empirical risk in severely ill-posed problems. (English) Zbl 1064.62011

Summary: We study a standard method of regularization by projections of the linear inverse problem \(Y=Ax+\varepsilon\), where \(\varepsilon\) is a white Gaussian noise and \(A\) is a known compact operator. It is assumed that the eigenvalues of \(AA^*\) converge to zero with exponential decay. Such behavior of the spectrum is typical for inverse problems related to elliptic differential operators. As a model example we consider recovering of unknown boundary conditions in the Dirichlet problem for the Laplace equation on the unit disk. By using the singular value decomposition of \(A\), we construct a projection estimator of \(x\). The bandwidth of this estimator is chosen by a data-driven procedure based on the principle of minimization of penalized empirical risk. We provide non-asymptotic upper bounds for the mean square risk of this method and we show, in particular, that this approach gives asymptotically minimax estimators in our model example.

MSC:

62C20 Minimax procedures in statistical decision theory
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
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[1] Akaike, H.: Information theory and an extension of the maximum likelihood principle. Proc. 2nd Intern. Symp. Inf. Theory, Petrov P.N. and Csaki F. (eds.), Budapest, 1973, pp. 267-281 · Zbl 0283.62006
[2] Barron, Probab. Theory Relat. Fields, 113, 301 (1999) · Zbl 0946.62036 · doi:10.1007/s004400050210
[3] Belitser, Math. Meth. Statist., 4, 259 (1995) · Zbl 0836.62070
[4] Cavalier, L., Golubev, G., Lepski, O., Tsybakov, A: (2003) Block thresholding and sharp adaptive estimation in severely ill-posed inverse problems. Probab. Theory Appl. 3 to appear. · Zbl 1130.62313
[5] Cavalier, Probab. Theory Relat. Fields, 123, 323 (2002) · Zbl 1039.62031 · doi:10.1007/s004400100169
[6] Cavalier, Ann. Stat., 30, 843 (3) · Zbl 1029.62032
[7] Engl, H., Hanke, M., Nuebauer, A.: Regularization of inverse problems. Kluwer academic publishers, 2000
[8] Golubev, Probl. Inform. Transm., 35, 51 (2) · Zbl 0947.35174
[9] Efromovich, IEEE Trans. Inform. Theory, 43, 1184 (1997) · Zbl 0881.93081 · doi:10.1109/18.605581
[10] Ermakov, Inverse Probl., 6, 863 (5) · Zbl 0729.62089 · doi:10.1088/0266-5611/6/5/012
[11] Johnstone, Statistica Sinica, 9, 51 (1999) · Zbl 1065.62519
[12] Johnstone, J. Royal Stat. Soc. Ser. B., 59, 319 (1997) · Zbl 0886.62044 · doi:10.1111/1467-9868.00071
[13] Kneip, Ann. Stat., 22, 835 (1994) · Zbl 0815.62022
[14] Landau, Bell System Tech. J., 40, 65 (1961) · Zbl 0184.08602
[15] Landau, Bell System Tech. J., 41, 1295 (1962) · Zbl 0184.08603
[16] Lavrentiev, M.M.: Some improperly posed problems of mathematical physics. Springer Verlag, Berlin Heidelberg New-York, 1967 · Zbl 0149.41902
[17] Mair, SIAM J. Appl. Math., 56, 1424 (1996) · Zbl 0864.62020
[18] Mallows, Technometrics, 15, 661 (1973) · Zbl 0269.62061
[19] Pinsker, Probl. Inform. Transm., 16, 120 (1980) · Zbl 0452.94003
[20] Shibata, Boimetrika, 68, 45 (1981) · Zbl 0464.62054
[21] Slepjan, Bell System Tech. J., 40, 43 (1961) · Zbl 0184.08601
[22] Slepjan, J. Math. Rhys., 44, 99 (1965) · Zbl 0128.29601
[23] Stein, Ann. Stat., 9, 1135 (1981) · Zbl 0476.62035
[24] Sudakov, Soviet Mathematics Doklady, 157, 1094 (1964) · Zbl 0132.33302
[25] Sullivan, Stat. Sci., 1, 502 (1996) · Zbl 0955.65500
[26] Tsybakov, A. B., On the best rate of adaptive estimation in some inverse problems, C.R. Acad. Sci. Paris, 330, 835-840 (2000) · Zbl 1163.62316
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