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In this paper, we characterize the rectifiability (both uniform and not) of an Ahlfors regular set, E, of arbitrary co-dimension by the behavior of a regularized distance function in the complement of that set. In particular, we establish a…
Estimation of the covariance structure of spatial processes is of fundamental importance in spatial statistics. In the literature, several non-parametric and semi-parametric methods have been developed to estimate the covariance structure…
We present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence…
Sufficient dimension reduction [J. Amer. Statist. Assoc. 86 (1991) 316-342] has long been a prominent issue in multivariate nonparametric regression analysis. To uncover the central dimension reduction space, we propose in this paper an…
This study debuts a new spline dimensional decomposition (SDD) for uncertainty quantification analysis of high-dimensional functions, including those endowed with high nonlinearity and nonsmoothness, if they exist, in a proficient manner.…
Automatic algorithms attempt to provide approximate solutions that differ from exact solutions by no more than a user-specified error tolerance. This paper describes an automatic, adaptive algorithm for approximating the solution to a…
The article considers the problem of estimating a high-dimensional sparse parameter in the presence of side information that encodes the sparsity structure. We develop a general framework that involves first using an auxiliary sequence to…
We propose a new space-variant anisotropic regularisation term for variational image restoration, based on the statistical assumption that the gradients of the target image distribute locally according to a bivariate generalised Gaussian…
We propose and study the framework of dissipative statistical solutions for the incompressible Euler equations. Statistical solutions are time-parameterized probability measures on the space of square-integrable functions, whose…
Line spectral estimation theory aims to estimate the off-the-grid spectral components of a time signal with optimal precision. Recent results have shown that it is possible to recover signals having sparse line spectra from few temporal…
An important problem in space-time adaptive detection is the estimation of the large p-by-p interference covariance matrix from training signals. When the number of training signals n is greater than 2p, existing estimators are generally…
A simple technique is described, that provides improved nearside-farside (NF) decompositions of elastic scattering amplitudes. The technique, involving the resummation of a Legendre partial wave series, reduces the importance of unphysical…
Regularization methods have been recently developed to construct stable approximate solutions to classical partial differential equations considered as final value problems. In this paper, we investigate the backward parabolic problem with…
Signal decomposition is a classical problem in signal processing, which aims to separate an observed signal into two or more components each with its own property. Usually each component is described by its own subspace or dictionary.…
State estimation that combines observational data with mathematical models is central to many applications and is commonly addressed through filtering methods, such as ensemble Kalman filters. In this article, we examine the signal-tracking…
Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori…
The goal of unsupervised anomaly segmentation (UAS) is to detect the pixel-level anomalies unseen during training. It is a promising field in the medical imaging community, e.g, we can use the model trained with only healthy data to segment…
This paper addresses the problem of estimating a convex regression function under both the sup-norm risk and the pointwise risk using B-splines. The presence of the convex constraint complicates various issues in asymptotic analysis,…
We propose to approximate the conditional expectation of a spatial random variable given its nearest-neighbour observations by an additive function. The setting is meaningful in practice and requires no unilateral ordering. It is capable of…
This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is both motivated by…