46,320 research outputs found
Regularization of statistical inverse problems and the Bakushinskii veto
In the deterministic context Bakushinskii's theorem excludes the existence of
purely data driven convergent regularization for ill-posed problems. We will
prove in the present work that in the statistical setting we can either
construct a counter example or develop an equivalent formulation depending on
the considered class of probability distributions. Hence, Bakushinskii's
theorem does not generalize to the statistical context, although this has often
been assumed in the past. To arrive at this conclusion, we will deduce from the
classic theory new concepts for a general study of statistical inverse problems
and perform a systematic clarification of the key ideas of statistical
regularization.Comment: 20 page
Convergence rates of general regularization methods for statistical inverse problems and applications
During the past the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut-off and Tikhonov type estimators. Spectral cut-off estimators achieve minimax rates for a broad range of smoothness classes and operators, but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator. Tikhonov estimators are simpler to compute, but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes. In this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods (spectral methods) including the aforementioned estimators as well as many iterative methods, such as í-methods and the Landweber iteration. The latter estimators converge at the same rate as spectral cut-off, but only require matrixvector products. Our results are applied to various problems, in particular we obtain precise convergence rates for satellite gradiometry, L2-boosting, and errors in variable problems. --Statistical inverse problems,iterative regularization methods,Tikhonov regularization,nonparametric regression,minimax convergence rates,satellite gradiometry,Hilbert scales,boosting,errors in variable
Pinsker estimators for local helioseismology
A major goal of helioseismology is the three-dimensional reconstruction of
the three velocity components of convective flows in the solar interior from
sets of wave travel-time measurements. For small amplitude flows, the forward
problem is described in good approximation by a large system of convolution
equations. The input observations are highly noisy random vectors with a known
dense covariance matrix. This leads to a large statistical linear inverse
problem.
Whereas for deterministic linear inverse problems several computationally
efficient minimax optimal regularization methods exist, only one
minimax-optimal linear estimator exists for statistical linear inverse
problems: the Pinsker estimator. However, it is often computationally
inefficient because it requires a singular value decomposition of the forward
operator or it is not applicable because of an unknown noise covariance matrix,
so it is rarely used for real-world problems. These limitations do not apply in
helioseismology. We present a simplified proof of the optimality properties of
the Pinsker estimator and show that it yields significantly better
reconstructions than traditional inversion methods used in helioseismology,
i.e.\ Regularized Least Squares (Tikhonov regularization) and SOLA (approximate
inverse) methods.
Moreover, we discuss the incorporation of the mass conservation constraint in
the Pinsker scheme using staggered grids. With this improvement we can
reconstruct not only horizontal, but also vertical velocity components that are
much smaller in amplitude
Convergence rates for variational regularization of inverse problems in exponential families
We consider statistical inverse problems with statistical noise. By using regularization methods one can approximate the true solution of the inverse problem by a regularized solution. The previous investigation of convergence rates for variational regularization with Poisson and empirical process data is shown to be suboptimal. In this thesis we obtain improved convergence rates for variational regularization methods of nonlinear ill-posed inverse problems with certain stochastic noise models described by exponential families and derive better reconstruction error bounds by applying deviation inequalities for stochastic process in some function spaces. Furthermore, we also consider iteratively regularized Newton-method as an alternative while the operator is non-linear. Due to the difficulty of deriving suitable deviation inequalities for stochastic processes in some function spaces, we are currently not able to obtain optimal convergence rates for variational regularization such that we state our desired result as a conjecture. If our conjecture holds true, then we can immediately obtain our desired results
IDENTIFICATION AND ESTIMATION OF NONPARAMETRIC STRUCTURAL
This paper concerns a new statistical approach to instrumental variables (IV) method for nonparametric structural models with additive errors. A general identifying condition of the model is proposed, based on richness of the space generated by marginal discretizations of joint density functions. For consistent estimation, we develop statistical regularization theory to solve a random Fredholm integral equation of the first kind. A\ minimal set of conditions are given for consistency of a general regularization method. Using an abstract smoothness condition, we derive some optimal bounds, given the accuracies of preliminary estimates, and show the convergence rates of various regularization methods, including (the ordinary/iterated/generalized) Tikhonov and Showalter's methods. An application of the general regularization theory is discussed with a focus on a kernel smoothing method. We show an exact closed form, as well as the optimal convergence rate, of the kernel IV estimates of various regularization methods. The finite sample properties of the estimates are investigated via a small-scale Monte Carlo experimentNonparametric Strucutral Models, IV estimation, Statistical inverse problems
Variational Downscaling, Fusion and Assimilation of Hydrometeorological States via Regularized Estimation
Improved estimation of hydrometeorological states from down-sampled
observations and background model forecasts in a noisy environment, has been a
subject of growing research in the past decades. Here, we introduce a unified
framework that ties together the problems of downscaling, data fusion and data
assimilation as ill-posed inverse problems. This framework seeks solutions
beyond the classic least squares estimation paradigms by imposing proper
regularization, which are constraints consistent with the degree of smoothness
and probabilistic structure of the underlying state. We review relevant
regularization methods in derivative space and extend classic formulations of
the aforementioned problems with particular emphasis on hydrologic and
atmospheric applications. Informed by the statistical characteristics of the
state variable of interest, the central results of the paper suggest that
proper regularization can lead to a more accurate and stable recovery of the
true state and hence more skillful forecasts. In particular, using the Tikhonov
and Huber regularization in the derivative space, the promise of the proposed
framework is demonstrated in static downscaling and fusion of synthetic
multi-sensor precipitation data, while a data assimilation numerical experiment
is presented using the heat equation in a variational setting
Nonlinear estimation for linear inverse problems with error in the operator
We study two nonlinear methods for statistical linear inverse problems when
the operator is not known. The two constructions combine Galerkin
regularization and wavelet thresholding. Their performances depend on the
underlying structure of the operator, quantified by an index of sparsity. We
prove their rate-optimality and adaptivity properties over Besov classes.Comment: Published in at http://dx.doi.org/10.1214/009053607000000721 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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