Inverse Problems in Statistics

There exist many fields where inverse problems appear. Some examples are: astronomy (blurred images of the Hubble satellite), econometrics (instrumental variables), financial mathematics (model calibration of the volatility), medical image processing (X-ray tomography), and quantum physics (quantum homodyne tomography). These are problems where we have indirect observations of an object (a function) that we want to reconstruct, through a linear operator A. Due to its indirect nature, solving an inverse problem is usually rather difficult. For this reason, one needs regularization methods in order to get a stable and accurate reconstruction. We present the framework of statistical inverse problems where the data are corrupted by some stochastic error. This white noise model may be discretized in the spectral domain using Singular Value Decomposition (SVD), when the operator A is compact. Several examples of inverse problems where the SVD is known are presented (circular deconvolution, heat equation, tomography). We explain some basic issues regarding nonparametric statistics applied to inverse problems. Standard regularization methods and their counterpart as estimation procedures by use of SVD are discussed (projection, Landweber, Tikhonov, . . . ). Several classical statistical approaches like minimax risk and optimal rates of convergence, are presented. This notion of optimality leads to some optimal choice of the tuning parameter. However these optimal parameters are unachievable since they depend on the unknown smoothness of the function. This leads to more recent concepts like adaptive estimation and oracle inequalities. Several data-driven selection procedures of the regularization parameter are discussed in details, among these: model selection methods, Stein’s unbiased risk estimation and the recent risk hull method.