Related papers: On the approximation by weighted ridge functions
Given data $y$ and $k$ covariates $x$ the problem is to decide which covariates to include when approximating $y$ by a linear function of the covariates. The decision is based on replacing subsets of the covariates by i.i.d. normal random…
We provide a quantitative two weight estimate for the dyadic paraproduct $\pi_b$ under certain conditions on a pair of weights $(u;v)$ and $b$ in $Carl_{u,v}$, a new class of functions that we show coincides with BMO when $u = v \in A^d_2$.…
We propose and analyse numerical algorithms based on weighted least squares for the approximation of a real-valued function on a general bounded domain $\Omega \subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n…
We consider the problem of classification of functional data into two groups by linear classifiers based on one-dimensional projections of functions. We reformulate the task to find the best classifier as an optimization problem and solve…
As with classic statistics, functional regression models are invaluable in the analysis of functional data. While there are now extensive tools with accompanying theory available for linear models, there is still a great deal of work to be…
A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations…
This paper considers the problem of approximating a Boolean function $f$ using another Boolean function from a specified class. Two classes of approximating functions are considered: $k$-juntas, and linear Boolean functions. The $n$ input…
Computing the probability of a formula given the probabilities or weights associated with other formulas is a natural extension of logical inference to the probabilistic setting. Surprisingly, this problem has received little attention in…
In this paper we develop an optimisation based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalised rational approximation. In the…
We construct explicit easily implementable polynomial approximations of sufficiently high accuracy for locally constant functions on the union of disjoint segments. This problem has important applications in several areas of numerical…
We consider the problem of estimating a function $s$ on $[-1,1]^{k}$ for large values of $k$ by looking for some best approximation by composite functions of the form $g\circ u$. Our solution is based on model selection and leads to a very…
Due to their flexibility, frames of Hilbert spaces are attractive alternatives to bases in approximation schemes for problems where identifying a basis is not straightforward or even feasible. Computing a best approximation using frames,…
We study the problem of estimating the joint probability mass function (pmf) over two random variables. In particular, the estimation is based on the observation of $m$ samples containing both variables and $n$ samples missing one fixed…
We propose a general approach to construct weighted likelihood estimating equations with the aim of obtain robust estimates. The weight, attached to each score contribution, is evaluated by comparing the statistical data depth at the model…
We deal with a problem of the reconstruction of any holomorphic function $f$ on the unit ball of $\mathbb{C}^2$ from its restricions on a union of complex lines. We give an explicit formula of Lagrange interpolation's type that is…
Approximations of non-smooth multivariate functions return low-order approximations in the vicinities of the singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating…
In projection pursuit regression (PPR), an unknown response function is approximated by the sum of M "ridge functions," which are flexible functions of one-dimensional projections of a multivariate input space. Traditionally, optimization…
If $f\in L^2(R^d)$ and if the function $f(x)f(y)$ is close in $L^2(R^{2d})$ norm to a radially symmetric function of $(x,y)$ then $f$ is close in $L^2$ norm to a centered Gaussian function. This is proved in a quantitative form with the…
Adaptive estimation of a quadratic functional over both Besov and $L_p$ balls is considered. A collection of nonquadratic estimators are developed which have useful bias and variance properties over individual Besov and $L_p$ balls. An…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…