1,721,307 research outputs found
On convergence rates for iteratively regularized Newton-type methods under a Lipschitz-type nonlinearity condition.
No full textWe investigate a generalization of the well-known iteratively regularized Gauss–Newton method where the Newton equations are regularized variationally using general data delity and penalty terms. To obtain convergence rates, we use a general error assumption which has recently been shown to be useful for impulsive and Poisson noise. We restrict the nonlinearity of the forward operator only by a Lipschitztype condition and compare our results to other convergence rates results proven in the literature. Finally we explicitly state our convergence rates for the aforementioned case of Poisson noise to shed some light on the structure of the posed error assumption
Adaptive minimax optimality in statistical inverse problems via SOLIT - Sharp Optimal Lepskii-Inspired Tuning
No full textWe consider statistical linear inverse problems in separable Hilbert spaces
and filter-based reconstruction methods of the form , where is the available data, the forward
operator, an ordered filter,
and > 0 a regularization parameter. Whenever such a method is used in
practice, has to be chosen appropriately. Typically, the aim is to
find or at least approximate the best possible in the sense that mean
squared error (MSE)
w.r.t.~the true solution is minimized. In this paper, we introduce
the Sharp Optimal Lepski\u{\i}-Inspired Tuning (SOLIT) method, which yields an
a posteriori parameter choice rule ensuring adaptive minimax rates of
convergence. It depends only on and the noise level as well as the
operator and the filter and
does not require any problem-dependent tuning of further parameters. We prove
an oracle inequality for the corresponding MSE in a general setting and derive
the rates of convergence in different scenarios. By a careful analysis we show
that no other a posteriori parameter choice rule can yield a better performance
in terms of the convergence rate of the MSE. In particular, our results reveal
that the typical understanding of Lepskiii-type methods in inverse problems
leading to a loss of a log factor is wrong. In addition, the empirical
performance of SOLIT is examined in simulations
Adaptive minimax optimality in statistical inverse problems via SOLIT -- Sharp Optimal Lepskii-Inspired Tuning
Get PDFWe consider statistical linear inverse problems in separable Hilbert spaces
and filter-based reconstruction methods of the form , where is the available data, the forward
operator, an ordered filter,
and a regularization parameter. Whenever such a method is used in
practice, has to be appropriately chosen. Typically, the aim is to
find or at least approximate the best possible in the sense that mean
squared error (MSE)
w.r.t.~the true solution is minimized. In this paper, we introduce
the Sharp Optimal Lepski\u{\i}-Inspired Tuning (SOLIT) method, which yields an
a posteriori parameter choice rule ensuring adaptive minimax rates of
convergence. It depends only on and the noise level as well as the
operator and the filter and
does not require any problem-dependent tuning of further parameters. We prove
an oracle inequality for the corresponding MSE in a general setting and derive
the rates of convergence in different scenarios. By a careful analysis we show
that no other a posteriori parameter choice rule can yield a better performance
in terms of the order of the convergence rate of the MSE. In particular, our
results reveal that the typical understanding of Lepski\u\i-type methods in
inverse problems leading to a loss of a log factor is wrong. In addition, the
empirical performance of SOLIT is examined in simulations.Comment: Some technical parts are polished, and a comparison with classical
Lepskii is included in the simulation sectio
Convergence analysis of (statistical) inverse problems under conditional stability estimates
No full textMultiscale scanning with nuisance parameters
No full textWe investigate the problem to find anomalies in a -dimensional random field via multiscale scanning in the presence of nuisance parameters. This covers the common situation that either the baseline-level or additional parameters such as the variance are unknown and have to be estimated from the data. We argue that state of the art approaches to determine asymptotically correct critical values for the multiscale scanning statistic will in general fail when naively such parameters are replaced by plug-in estimators. Opposed to this, we suggest to estimate the nuisance parameters on the largest scale and to use the remaining scales for multiscale scanning. We prove a uniform invariance principle for the resulting adjusted multiscale statistic (AMS), which is widely applicable and provides a computationally feasible way to simulate asymptotically correct critical values. We illustrate the implications of our theoretical results in a simulation study and in a real data example from super-resolution STED microscopy. This allows us to identify interesting regions inside a specimen in a pre-scan with controlled family-wise error rate
Inverse problems with Poisson data: Statistical regularization theory, applications and algorithms.
No full textInverse problems with Poisson data arise in many photonic imaging modalities in medicine, engineering and astronomy. The design of regularization methods and estimators for such problems has been studied intensively over the last two decades. In this review we give an overview of statistical regularization theory for such problems, the most important applications, and the most widely used algorithms. The focus is on variational regularization methods in the form of penalized maximum likelihood estimators, which can be analyzed in a general setup. Complementing a number of recent convergence rate results we will establish consistency results. Moreover, we discuss estimators based on a wavelet-vaguelette decomposition of the (necessarily linear) forward operator. As most prominent applications we briefly introduce Positron emission tomography, inverse problems in fluorescence microscopy, and phase retrieval problems. The computation of a penalized maximum likelihood estimator involves the solution of a (typically convex) minimization problem. We also review several efficient algorithms which have been proposed for such problems over the last five years
What is resolution? A statistical minimax testing perspective on super-resolution microscopy
No full textAs a general rule of thumb the resolution of a light microscope (i.e. the ability to discern objects) is predominantly described by the full width at half maximum (FWHM) of its point spread function (psf)---the diameter of the blurring density at half of its maximum. Classical wave optics suggests a linear relationship between FWHM and resolution also manifested in the well known Abbe and Rayleigh criteria, dating back to the end of 19th century. However, during the last two decades conventional light microscopy has undergone a shift from microscopic scales to nanoscales. This increase in resolution comes with the need to incorporate the random nature of observations (light photons) and challenges the classical view of discernability, as we argue in this paper. Instead, we suggest a statistical description of resolution obtained from such random data. Our notion of discernability is based on statistical testing whether one or two objects with the same total intensity are present. For Poisson measurements we get linear dependence of the (minimax) detection boundary on the FWHM, whereas for a homogeneous Gaussian model the dependence of resolution is nonlinear. Hence, at small physical scales modeling by homogeneous gaussians is inadequate, although often implicitly assumed in many reconstruction algorithms. In contrast, the Poisson model and its variance stabilized Gaussian approximation seem to provide a statistically sound description of resolution at the nanoscale. Our theory is also applicable to other imaging setups, such as telescopes
Multiscale scanning with nuisance parameters
No full textAbstract We develop a multiscale scanning method to find anomalies in a d-dimensional random field in the presence of nuisance parameters. This covers the common situation that either the baseline-level or additional parameters such as the variance are unknown and have to be estimated from the data. We argue that state of the art approaches to determine asymptotically correct critical values for multiscale scanning statistics will in general fail when such parameters are naively replaced by plug-in estimators. Instead, we suggest to estimate the nuisance parameters on the largest scale and to use (only) smaller scales for multiscale scanning. We prove a uniform invariance principle for the resulting adjusted multiscale statistic, which is widely applicable and provides a computationally feasible way to simulate asymptotically correct critical values. We illustrate the implications of our theoretical results in a simulation study and in a real data example from super-resolution STED microscopy. This allows us to identify interesting regions inside a specimen in a pre-scan with controlled family-wise error rate
Error estimates for variational regularization of inverse problems with general noise models for data and operator
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