A statistical view of iterative methods for linear inverse problems
Tóm tắt
In this article, we study the problem of recovering the unknown solution of a linear ill-posed problem, via iterative regularization methods, from a statistical point of view. The basic purpose of the paper is to develop adaptive model selection techniques for determining the regularization parameters, i.e., the iteration index. We assume observations are taken over a fixed grid and we consider solutions over a sequence of finite-dimensional subspaces. Based on concentration inequalities techniques, we derive non-asymptotic optimal upper bounds for the mean square error of the proposed estimator.
Tài liệu tham khảo
Baraud Y (2000) Model selection for regression on a fixed design. Probab Theory Relat Fields 117:467–493
Birgé L, Massart P (1998) Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4:329–395
Bousquet O (2003) Concentration inequalities for sub-additive functions using of entropy method. Stoch Inequal Appl 56:213–247
Calveltti D, Hansen P, Reichel L (2002) L-curve curvature bounds via Lanczos bidiagonalization. Electron Trans Numer Anal 14:20–35
Cavalier L, Golubev G, Picard D, Tsybakov A (2002) Oracle inequalities for inverse problem. Ann Stat 30(3):843–874
Dey AK (1996) Cross-validation for parameter selection in inverse estimation problems. Scand J Stat 23(4):609–620
Egger H (2005) Accelerated Newton–Landweber iterations for regularizing nonlinear inverse problems. Technical Report SBF F013, RICAM, Austria
Engl H, Grever W (1994) Using the L-curve for determining optimal regularization parameters. Num Math 69:25–31
Engl HW, Hanke M, Neubauer A (1996) Reguralization of inverse problems. Mathematics and its applications, vol 375. Kluwer Academic, Dordrecht
Gilyazov S, Gol’dman N (2000) Reguralization of ill-posed problems by iteration methods. Mathematics and its applications, vol 499. Kluwer Academic, Dordrecht
Hansen PC, O’Leary DP (1993) The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J Sci Comput 14:1487–1503
Kilmer ME, O’Leary DP (2001) Choosing regularization parameters in iterative methods for ill-posed problems. SIAM J Matrix Anal Appl 22(4):1204–1221
Loubes JM, Ludeña C (2004) Estimators for nonlinear inverse problems. Preprints. Available at http://arxiv.org/pdf/math.ST/0502033
Morozov VA (1966) On the solution of functional equations by the method of regularization. Sov Math Dokl 7:414–417
O’Sullivan F (1986) A statistical perspective on ill-posed inverse problems. Stat Sci 1(4):502–527
Tikhonov A, Arsenin V (1977) Solutions of ill-posed problems. Wiley, New York