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The principle of penalized empirical risk in severely ill-posed problems

Abstract.

We study a standard method of regularization by projections of the linear inverse problem Y=Ax+ε, where ε 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 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.

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Correspondence to Yu Golubev.

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Mathematics Subject Classification (2000):Primary 62G05, 62G20; secondary 62C20

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Golubev, Y. The principle of penalized empirical risk in severely ill-posed problems. Probab. Theory Relat. Fields 130, 18–38 (2004). https://doi.org/10.1007/s00440-004-0362-y

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Keywords

  • Ill-posed problem
  • Partial differential equation
  • Singular value decomposition
  • Projection estimator
  • Minimax risk
  • Penalization
  • Empirical risk