Nonparametric instrumental regression with non-convex constraints

This paper considers the nonparametric regression model with an additive error that is dependent on the explanatory variables. As is common in empirical studies in epidemiology and economics, it also supposes that valid instrumental variables are observed. A classical example in microeconomics considers the consumer demand function as a function of the price of goods and the income, both variables often considered as endogenous. In this framework, the economic theory also imposes shape restrictions on the demand function, like integrability conditions. Motivated by this illustration in microeconomics, we study an estimator of a nonparametric constrained regression function using instrumental variables by means of Tikhonov regularization. We derive rates of convergence for the regularized model both in a deterministic and stochastic setting under the assumption that the true regression function satisfies a projected source condition including, because of the non-convexity of the imposed constraints, an additional smallness condition

Local Analysis of Inverse Problems: H\"{o}lder Stability and Iterative Reconstruction

We consider a class of inverse problems defined by a nonlinear map from parameter or model functions to the data. We assume that solutions exist. The space of model functions is a Banach space which is smooth and uniformly convex; however, the data space can be an arbitrary Banach space. We study sequences of parameter functions generated by a nonlinear Landweber iteration and conditions under which these strongly converge, locally, to the solutions within an appropriate distance. We express the conditions for convergence in terms of H\"{o}lder stability of the inverse maps, which ties naturally to the analysis of inverse problems

Fast parallel algorithms for a broad class of nonlinear variational diffusion approaches

Variational segmentation and nonlinear diffusion approaches have been very active research areas in the fields of image processing and computer vision during the last years. In the present paper, we review recent advances in the development of efficient numerical algorithms for these approaches. The performance of parallel implement at ions of these algorithms on general-purpose hardware is assessed. A mathematically clear connection between variational models and nonlinear diffusion filters is presented that allows to interpret one approach as an approximation of the other, and vice versa. Numerical results confirm that, depending on the parametrization, this approximation can be made quite accurate. Our results provide a perspective for uniform implement at ions of both nonlinear variational models and diffusion filters on parallel architectures

Older adults' perspectives on key domains of childhood social and economic experiences and opportunities: a first step to creating a multidimensional measure

ObjectivesAlthough research has found that childhood socioeconomic status (SES) is associated with physical and mental health in mid- and later life, most of these studies used conventional, single dimension SES measures for the childhood period such as household income or educational attainment of parents. Life course and health disparities research would benefit from identification and measurement of a variety of childhood social and economic experiences and opportunities that might affect health in later life.DesignThis study utilized qualitative research methods to identify key dimensions of childhood experiences related to SES. We conducted in-depth interviews with 25 adults age 55 to 80 years from diverse economic and ethnic backgrounds. Topics included home, neighborhood, school, and work experiences during early childhood and adolescence. Interviews were audio-taped and transcripts were coded to identify thematic domains.ResultsWe identified eight thematic domains, many of which had clear subdomains: home and family circumstances, neighborhood, work and money, potential for advancement through schooling, school quality and content, discrimination, influence and support of adults, and leisure activities. These domains highlight individual characteristics and experiences and also economic and educational opportunities.ConclusionThese domains of childhood social and economic circumstances add breadth and depth to conventional conceptualization of childhood SES. When the domains are translated into a measurement tool, it will allow for the possibility of classifying people along multiple dimensions, such as from a low economic circumstance with high levels of adult support

Discretization of variational regularization in Banach spaces

Consider a nonlinear ill-posed operator equation $F(u)=y$ where $F$ is defined on a Banach space $X$. In general, for solving this equation numerically, a finite dimensional approximation of $X$ and an approximation of $F$ are required. Moreover, in general the given data \yd of $y$ are noisy. In this paper we analyze finite dimensional variational regularization, which takes into account operator approximations and noisy data: We show (semi-)convergence of the regularized solution of the finite dimensional problems and establish convergence rates in terms of Bregman distances under appropriate sourcewise representation of a solution of the equation. The more involved case of regularization in nonseparable Banach spaces is discussed in detail. In particular we consider the space of finite total variation functions, the space of functions of finite bounded deformation, and the $L^\infty$--space

Computing Topology Preservation of RBF Transformations for Landmark-Based Image Registration

In image registration, a proper transformation should be topology preserving. Especially for landmark-based image registration, if the displacement of one landmark is larger enough than those of neighbourhood landmarks, topology violation will be occurred. This paper aim to analyse the topology preservation of some Radial Basis Functions (RBFs) which are used to model deformations in image registration. Mat\'{e}rn functions are quite common in the statistic literature (see, e.g. \cite{Matern86,Stein99}). In this paper, we use them to solve the landmark-based image registration problem. We present the topology preservation properties of RBFs in one landmark and four landmarks model respectively. Numerical results of three kinds of Mat\'{e}rn transformations are compared with results of Gaussian, Wendland's, and Wu's functions

Sparse Regularization with lql^q Penalty Term

We consider the stable approximation of sparse solutions to non-linear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate $O(\sqrt{\delta})$ of the regularized solutions in dependence of the noise level $\delta$. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to $O(\delta)$.Comment: 15 page

CANVAS: case report on a novel repeat expansion disorder with late-onset ataxia

This article presents the case of a 74-year-old female patient who first developed a progressive disease with sensory neuropathy, cerebellar ataxia and bilateral vestibulopathy at the age of 60 years. The family history was unremarkable. Magnetic resonance imaging (MRI) showed atrophy of the cerebellum predominantly in the vermis and atrophy of the spinal cord. The patient was given the syndromic diagnosis of cerebellar ataxia, neuropathy, vestibular areflexia syndrome (CANVAS). In 2019 the underlying genetic cause of CANVAS was discovered to be an intronic repeat expansion in the RFC1 gene with autosomal recessive inheritance. The patient exhibited the full clinical picture of CANVAS and was tested positive for this repeat expansion on both alleles. The CANVAS is a relatively frequent cause of late-onset hereditary ataxia (estimated prevalence 5‑13/100,000). In contrast to the present patient, the full clinical picture is not always present. Therefore, testing for the RFC1 gene expansion is recommended in the work-up of patients with otherwise unexplained late-onset sporadic ataxia. As intronic repeat expansions cannot be identified by next generation sequencing methods, specific testing is necessary

A common polymorphism in SNCA is associated with accelerated motor decline in GBA-Parkinson's disease.

A growing number of genetic susceptibility factors have been identified for Parkinson’s disease (PD). The combination of inherited risk variants is likely to affect not only risk of developing PD but also its clinical course. Variants in the GBA gene are particularly common, being found in approximately 5 to 10% of patients, and they lead to more rapid disease progression1. However, the effect of concomitant genetic risk factors on disease course in GBA-PD is not known.The CamPaIGN study has received financial support from the Wellcome Trust, the Medical Research Council, Parkinson’s UK and the Patrick Berthoud Trust. CHWG is supported by an RCUK/UKRI Innovation Fellowship awarded by the Medical Research Council. RAB is supported by the Wellcome Trust Stem Cell Institute (Cambridge). TBS received financial support from the Cure Parkinson’s Trust. The study is also supported by the National Institute for Health Research (NIHR) Cambridge Biomedical Research Centre Dementia and Neurodegeneration Theme (reference number 146281). The views expressed are those of the author(s) and not necessarily those of the NIHR or the Department of Health and Social Care. CRS' work is supported in part by NIH grants R01AG057331, U01NS100603, R01AG057331, and the American Parkinson Disease Association. Illumina MEGA Chip genotyping was made possible by a philanthropic investment from Dooley LLC (to Brigham & Women's Hospital and CRS)

Necessary conditions for variational regularization schemes

We study variational regularization methods in a general framework, more precisely those methods that use a discrepancy and a regularization functional. While several sets of sufficient conditions are known to obtain a regularization method, we start with an investigation of the converse question: How could necessary conditions for a variational method to provide a regularization method look like? To this end, we formalize the notion of a variational scheme and start with comparison of three different instances of variational methods. Then we focus on the data space model and investigate the role and interplay of the topological structure, the convergence notion and the discrepancy functional. Especially, we deduce necessary conditions for the discrepancy functional to fulfill usual continuity assumptions. The results are applied to discrepancy functionals given by Bregman distances and especially to the Kullback-Leibler divergence.Comment: To appear in Inverse Problem