Doubly Flexible Estimation under Label Shift

In studies ranging from clinical medicine to policy research, complete data are usually available from a population $\mathscr{P}$, but the quantity of interest is often sought for a related but different population $\mathscr{Q}$ which only has partial data. In this paper, we consider the setting that both outcome $Y$ and covariate ${\bf X}$ are available from $\mathscr{P}$ whereas only ${\bf X}$ is available from $\mathscr{Q}$, under the so-called label shift assumption, i.e., the conditional distribution of ${\bf X}$ given $Y$ remains the same across the two populations. To estimate the parameter of interest in $\mathscr{Q}$ via leveraging the information from $\mathscr{P}$, the following three ingredients are essential: (a) the common conditional distribution of ${\bf X}$ given $Y$, (b) the regression model of $Y$ given ${\bf X}$ in $\mathscr{P}$, and (c) the density ratio of $Y$ between the two populations. We propose an estimation procedure that only needs standard nonparametric technique to approximate the conditional expectations with respect to (a), while by no means needs an estimate or model for (b) or (c); i.e., doubly flexible to the possible model misspecifications of both (b) and (c). This is conceptually different from the well-known doubly robust estimation in that, double robustness allows at most one model to be misspecified whereas our proposal can allow both (b) and (c) to be misspecified. This is of particular interest in our setting because estimating (c) is difficult, if not impossible, by virtue of the absence of the $Y$-data in $\mathscr{Q}$. Furthermore, even though the estimation of (b) is sometimes off-the-shelf, it can face curse of dimensionality or computational challenges. We develop the large sample theory for the proposed estimator, and examine its finite-sample performance through simulation studies as well as an application to the MIMIC-III database

Recent Progress in Image Deblurring

This paper comprehensively reviews the recent development of image deblurring, including non-blind/blind, spatially invariant/variant deblurring techniques. Indeed, these techniques share the same objective of inferring a latent sharp image from one or several corresponding blurry images, while the blind deblurring techniques are also required to derive an accurate blur kernel. Considering the critical role of image restoration in modern imaging systems to provide high-quality images under complex environments such as motion, undesirable lighting conditions, and imperfect system components, image deblurring has attracted growing attention in recent years. From the viewpoint of how to handle the ill-posedness which is a crucial issue in deblurring tasks, existing methods can be grouped into five categories: Bayesian inference framework, variational methods, sparse representation-based methods, homography-based modeling, and region-based methods. In spite of achieving a certain level of development, image deblurring, especially the blind case, is limited in its success by complex application conditions which make the blur kernel hard to obtain and be spatially variant. We provide a holistic understanding and deep insight into image deblurring in this review. An analysis of the empirical evidence for representative methods, practical issues, as well as a discussion of promising future directions are also presented.Comment: 53 pages, 17 figure

Kernel Instrumental Variable Regression

Instrumental variable (IV) regression is a strategy for learning causal relationships in observational data. If measurements of input X and output Y are confounded, the causal relationship can nonetheless be identified if an instrumental variable Z is available that influences X directly, but is conditionally independent of Y given X and the unmeasured confounder. The classic two-stage least squares algorithm (2SLS) simplifies the estimation problem by modeling all relationships as linear functions. We propose kernel instrumental variable regression (KIV), a nonparametric generalization of 2SLS, modeling relations among X, Y, and Z as nonlinear functions in reproducing kernel Hilbert spaces (RKHSs). We prove the consistency of KIV under mild assumptions, and derive conditions under which convergence occurs at the minimax optimal rate for unconfounded, single-stage RKHS regression. In doing so, we obtain an efficient ratio between training sample sizes used in the algorithm's first and second stages. In experiments, KIV outperforms state of the art alternatives for nonparametric IV regression.Comment: 41 pages, 11 figures. Advances in Neural Information Processing Systems. 201

Computation of scattering matrices and resonances for waveguides

Waveguides in Euclidian space are piecewise path connected subsets of R^n that can be written as the union of a compact domain with boundary and their cylindrical ends. The compact and non-compact parts share a common boundary. This boundary is assumed to be Lipschitz, piecewise smooth and piecewise path connected. The ends can be thought of as the cartesian product of the boundary with the positive real half-line. A notable feature of Euclidian waveguides is that the scattering matrix admits a meromorphic continuation to a certain Riemann surface with a countably infinite number of leaves [2], which we will describe in detail and deal with. In order to construct this meromorphic continuation, one usually first constructs a meromorphic continuation of the resolvent for the Laplace operator. In order to do this, we will use a well known glueing construction (see for example [5]), which we adapt to waveguides. The construction makes use of the meromorphic Fredholm theorem and the fact that the resolvent for the Neumann Laplace operator on the ends of the waveguide can be easily computed as an integral kernel. The resolvent can then be used to construct generalised eigenfunctions and, from them, the scattering matrix.Being in possession of the scattering matrix allows us to calculate resonances; poles of the scattering matrix. We are able to do this using a combination of numerical contour integration and Newton s method

Nonlinear Systems

The editors of this book have incorporated contributions from a diverse group of leading researchers in the field of nonlinear systems. To enrich the scope of the content, this book contains a valuable selection of works on fractional differential equations.The book aims to provide an overview of the current knowledge on nonlinear systems and some aspects of fractional calculus. The main subject areas are divided into two theoretical and applied sections. Nonlinear systems are useful for researchers in mathematics, applied mathematics, and physics, as well as graduate students who are studying these systems with reference to their theory and application. This book is also an ideal complement to the specific literature on engineering, biology, health science, and other applied science areas. The opportunity given by IntechOpen to offer this book under the open access system contributes to disseminating the field of nonlinear systems to a wide range of researchers

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

Recent Progress in Image Deblurring

This paper comprehensively reviews the recent development of image deblurring, including non-blind/blind, spatially invariant/variant deblurring techniques. Indeed, these techniques share the same objective of inferring a latent sharp image from one or several corresponding blurry images, while the blind deblurring techniques are also required to derive an accurate blur kernel. Considering the critical role of image restoration in modern imaging systems to provide high-quality images under complex environments such as motion, undesirable lighting conditions, and imperfect system components, image deblurring has attracted growing attention in recent years. From the viewpoint of how to handle the ill-posedness which is a crucial issue in deblurring tasks, existing methods can be grouped into five categories: Bayesian inference framework, variational methods, sparse representation-based methods, homography-based modeling, and region-based methods. In spite of achieving a certain level of development, image deblurring, especially the blind case, is limited in its success by complex application conditions which make the blur kernel hard to obtain and be spatially variant. We provide a holistic understanding and deep insight into image deblurring in this review. An analysis of the empirical evidence for representative methods, practical issues, as well as a discussion of promising future directions are also presented