Related papers: Adaptive sampling recovery of functions with highe…
This paper demonstrates that the space of piecewise smooth functions can be well approximated by the space of functions defined by a set of simple (non-linear) operations on smooth uniform splines. The examples include bivariate functions…
In this paper, we present a new multiscale model reduction technique for the Stokes flows in heterogeneous perforated domains. The challenge in the numerical simulations of this problem lies in the fact that the solution contains many…
We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with $\alpha$-H\"older derivatives (for some $0<\alpha\leq 1$). The smooth approximation is given by means of an…
In a previous paper, the author constructed frames and oversampling formulas for band-limited functions, in the framework of the theory of shift-invariant spaces. In this article we study the problem of recovering missing samples. We find a…
In the framework of scalar-on-function regression models, in which several functional variables are employed to predict a scalar response, we propose a methodology for selecting relevant functional predictors while simultaneously providing…
We introduce a method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we…
We develop fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications…
Successive differences on a sequence of data help to discover some smoothness features of this data. This was one of the main reasons for rewriting the classical interpolation formula in terms of such data differences. The aim of this paper…
We consider the problem of approximating smoothing spline estimators in a nonparametric regression model. When applied to a sample of size $n$, the smoothing spline estimator can be expressed as a linear combination of $n$ basis functions,…
Many data-fitting applications require the solution of an optimization problem involving a sum of large number of functions of high dimensional parameter. Here, we consider the problem of minimizing a sum of $n$ functions over a convex…
In this work, we propose an adaptive sparse learning algorithm that can be applied to learn the physical processes and obtain a sparse representation of the solution given a large snapshot space. Assume that there is a rich class of…
In this paper we consider the approximation of a function by its interpolating multilinear spline and the approximation of its derivatives by the derivatives of the corresponding spline. We derive formulas for the uniform approximation…
Sampling recovery on some function classes is studied in this paper. Typically, function classes are defined by imposing smoothness conditions. It was understood in nonlinear approximation that structural conditions in the form of control…
A basis expansion with regularization methods is much appealing to the flexible or robust nonlinear regression models for data with complex structures. When the underlying function has inhomogeneous smoothness, it is well known that…
Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by…
We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…
We extend deconvolution in a periodic setting to deal with functional data. The resulting functional deconvolution model can be viewed as a generalization of a multitude of inverse problems in mathematical physics where one needs to recover…
A new representation is proposed for functions in a Sobolev space with dominating mixed smoothness on an $N$-dimensional hyperrectangle. In particular, it is shown that these functions can be expressed in terms of their highest-order mixed…
Adaptive sparse coding methods learn a possibly overcomplete set of basis functions, such that natural image patches can be reconstructed by linearly combining a small subset of these bases. The applicability of these methods to visual…
The estimation of functions with varying degrees of smoothness is a challenging problem in the nonparametric function estimation. In this paper, we propose the LABS (L\'{e}vy Adaptive B-Spline regression) model, an extension of the LARK…