We explain some basic theoretical issues regarding nonparametric statistics applied to inverse problems. Simple examples are used to present classical concepts such as the white noise model, risk estimation, minimax risk, model selection and optimal rates of convergence, as well as more recent concepts such as adaptive estimation, oracle inequalities, modern model selection methods, Stein's unbiased risk estimation and the very recent risk hull method.

Nonparametric statistical inverse problems

We consider estimation of the common probability density f of independent identically distributed random variables $X_i$ that are observed with an additive independent identically distributed noise. We assume that the unknown density f belongs to a class ${\cal A}$ of densities whose characteristic function is described by the exponent $\exp(-\alpha |u|^r)$ as $|u|\to\infty$, where $\alpha>0$, $r>0$. The noise density is assumed known and such that its characteristic function decays as $\exp (-\beta|u|^s)$, as $|u|\to\infty$, where $\beta>0$, $s>0$. Assuming that $r<s$, we suggest a kernel-type estimator, whose variance turns out to be asymptotically negligible with respect to its squared bias under both pointwise and ${\mathbb L}_2$ risks. For $r < s/2$ we construct a sharp adaptive estimator of f.

http://www.proba.jussieu.fr/pageperso/tsybakov/TPA1.pdf

Sharp Optimality in Density Deconvolution with Dominating Bias. II

The authors consider the problem of estimating the density g of independent and identically dis tributed variables Xi, from a sample Z1, ... , Z,, such that Zi = Xi + aei for i = 1, . ., n, and e is noise independent of X, with ae having a known distribution. They present a model selection procedure allowing one to construct an adaptive estimator of g and to find nonasymptotic risk bounds. The estimator achieves the minimax rate of convergence, in most cases where lower bounds are available. A simulation study gives an illustration of the good practical performance of the method.

Penalized contrast estimator for adaptive density deconvolution

We study a standard method of regularization by projections of the linear inverse problem Y = Af + ∈, where ∈ is a white Gaussian noise, and A is a known compact operator with singular values converging to zero with polynomial decay. The unknown function f is recovered by a projection method using the singular value decomposition of A. The bandwidth choice of this projection regularization is governed by a data-driven procedure which is based on the principle of risk hull minimization. We provide nonasymptotic upper bounds for the mean square risk of this method and we show, in particular, that in numerical simulations this approach may substantially improve the classical method of unbiased risk estimation.

https://projecteuclid.org/download/pdfview_1/euclid.aos/1162567628

Risk hull method and regularization by projections of ill-posed inverse problems

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.

Inverse Problems in Statistics

We consider the problem of solving linear operator equations from noisy data under the assumptions that the singular values of the operator decrease exponentially fast and that the underlying solution is also exponentially smooth in the Fourier domain. We suggest an estimator of the solution based on a running version of block thresholding in the space of Fourier coefficients. This estimator is shown to be sharp adaptive to the unknown smoothness of the solution.

O. Lepski

Papers

Block Thresholding and Sharp Adaptive Estimation in Severely Ill-Posed Inverse Problems