General regularization schemes for signal detection in inverse problems

Clément Marteau1, Peter Mathé2
1Inst. Math. de Toulouse, INSA de Toulouse, Univ. de Toulouse, Toulouse, France
2Weierstrass Inst. for Appl. Anal. and Stoch., Berlin, Germany

Tóm tắt

Từ khóa


Tài liệu tham khảo

Ya. Baraud, “Non-Asymptotic Minimax Rates of Testing in Signal Detection”, Bernoulli 8(5), 577–606 (2002).

Ya. Baraud, S. Huet, and B. Laurent, “Adaptive Tests of Linear Hypotheses by Model Selection”, Ann. Statist. 31, 225–251 (2003).

N. Bissantz, Th. Hohage, A. Munk, and F. H. Ruymgaart, “Convergence Rates of General Regularization Methods for Statistical Inverse Problems and Applications”, SIAM J. Numer. Anal. 45(6), 2610–2636 (2007).

G. Blanchard and P. Mathé, “Discrepancy Principle for Statistical Inverse Problems with Application to Conjugate Gradient Regularization”, InverseProblems 28(11), 115011 (2012).

R. Bojanic and E. Seneta, “AUnified Theory of Regularly Varying Sequences”, Math. Z. 134, 91–106 (1973).

A. Caponnetto, Optimal Rates for Regularization Operators in Learning Theory, Techn. Rep. CSAILTR 2006-062 (Massachusetts Inst. of Technology, 2006).

L. Cavalier, “Inverse Problems in Statistics”, in Lect. Notes Statist. Proc., Vol. 203: Inverse Problems and High-Dimensional Estimation (Springer, Heidelberg, 2011), pp. 3–96.

C. de Boor, “Bounding the Error in Spline Interpolation”, SIAMRev. 16, 531–544 (1974).

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, in Mathematics and its Applications (Kluwer Academic Publishers Group, Dordrecht, 1996), Vol. 375.

M. Fromont and B. Laurent, “Adaptive Goodness-of-Fit Tests in a Density Model”, Ann. Statist. 34, 1–45 (2006).

B. Hofmann and P. Mathé, “Analysis of Profile Functions for General Linear Regularization Methods”, SIAM J. Numer. Anal. 45(3), 1122–1141 (electronic) (2007).

Yu. I. Ingster, “Asymptotically Minimax Hypothesis Testing for Nonparametric Alternatives. I”, Math. Methods Statist. 2(2), 85–114 (1993).

Yu. I. Ingster, Asymptotically Minimax Hypothesis Testing for Nonparametric Alternatives. II”, Math. Methods Statist. 2(3), 171–189 (1993).

Yu. I. Ingster, “Asymptotically Minimax Hypothesis Testing for Nonparametric Alternatives. III”, Math. Methods Statist. 2(4), 249–268 (1993).

Yu. I. Ingster, T. Sapatinas, and I. A. Suslina, “Minimax Signal Detection in Ill-Posed Inverse Problems”, Ann. Statist. 40, 1524–1549 (2012).

Q. Jin and P. Mathé, “Oracle Inequality for a Statistical Raus-Gfrerer-Type Rule”, SIAM/ASA J. Uncertainty Quantification 1(1), 386–407 (2013).

B. Laurent, J.-M. Loubes, and C. Marteau, “Testing Inverse Problems: A Direct or an Indirect Problem?”, J. Statist. Plann. Inference 141(5), 1849–1861 (2011).

B. Laurent, J.-M. Loubes, and C. Marteau, “Non-Asymptotic Minimax Rates of Testing in Signal Detection with Heterogeneous Variances”, Electron. J. Statist. 6, 91–122 (2012).

M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes (Springer-Verlag, Berlin, 1991).

P. Mathé and B. Hofmann, “How General are General Source Conditions?”, Inverse Problems 24(1), 015009, 5 (2008).

P. Mathé and N. Schöne, “Regularization by Projection in Variable Hilbert Scales”, Appl. Anal. 87(2), 201–219 (2008).

P. Mathé and U. Tautenhahn, “Interpolation in Variable Hilbert Scales with Application to Inverse Problems”, Inverse Problems 22(6), 2271–2297 (2006).

F. Natterer, “Regularisierung schlecht gestellter Probleme durch Projektionsverfahren”, Numer. Math. 28(3), 329–341 (1977).

V. Spokoiny, “Adaptive Hypothesis Testing Using Wavelets”, Ann. Statist. 24, 2477–2498 (1996).

G.M. Vaĭnikko and U. A. Khyamarik. “Projection Methods and Self-Regularization in Ill-Posed Problems”, Izv. Vyssh. Uchebn. Zaved. Mat. 84(10), 3–17 (1985).

Tong Zhang. “Learning Bounds for Kernel Regression Using Effective Data Dimensionality”, Neural Comput. 17(9), 2077–2098 (2005).