Related papers: Randomized least-squares with minimal oversampling…
We introduce an interpolation--regression operator for polynomial approximation on the unit sphere $\mathbb{S}^2$ from discrete samples. The approximant is a spherical polynomial of degree $r$ which interpolates the data on a prescribed…
The purpose of the paper is to provide a characterization of the error of the best polynomial approximation of composite functions in weighted spaces. Such a characterization is essential for the convergence analysis of numerical methods…
While the theory of operator approximation with any given accuracy is well elaborated, the theory of {best constrained} constructive operator approximation is still not so well developed. Despite increasing demands from applications this…
We investigate iterative methods with randomized preconditioners for solving overdetermined least-squares problems, where the preconditioners are based on a random embedding of the data matrix. We consider two distinct approaches: the…
Stochastic approximation techniques play an important role in solving many problems encountered in machine learning or adaptive signal processing. In these contexts, the statistics of the data are often unknown a priori or their direct…
In this work we present a novel optimization strategy for image reconstruction tasks under analysis-based image regularization, which promotes sparse and/or low-rank solutions in some learned transform domain. We parameterize such…
We give a new probabilistic algorithm for interpolating a "sparse" polynomial f given by a straight-line program. Our algorithm constructs an approximation f* of f, such that their difference probably has at most half the number of terms of…
We exploit the idea to use the maximal-entropy method, successfully tested in information theory and statistical thermodynamics, to determine approximating function's coefficients and squared errors' weights simultaneously as output of one…
The standard estimator for the two-point function of a homogeneous and isotropic random field is a special case of a larger class of least squares estimators that interpolate the function values. Using a different interpolation scheme,…
The goal of this paper is to design compact support basis spline functions that best approximate a given filter (e.g., an ideal Lowpass filter). The optimum function is found by minimizing the least square problem ($\ell$2 norm of the…
We describe stochastic Newton and stochastic quasi-Newton approaches to efficiently solve large linear least-squares problems where the very large data sets present a significant computational burden (e.g., the size may exceed computer…
Linear Least Squares is a very well known technique for parameter estimation, which is used even when sub-optimal, because of its very low computational requirements and the fact that exact knowledge of the noise statistics is not required.…
We study the following basic machine learning task: Given a fixed set of $d$-dimensional input points for a linear regression problem, we wish to predict a hidden response value for each of the points. We can only afford to attain the…
Over the course of the past decade, a variety of randomized algorithms have been proposed for computing approximate least-squares (LS) solutions in large-scale settings. A longstanding practical issue is that, for any given input, the user…
The so-called constrained least mean-square algorithm is one of the most commonly used linear-equality-constrained adaptive filtering algorithms. Its main advantages are adaptability and relative simplicity. In order to gain analytical…
Estimating the values of unknown parameters from corrupted measured data faces a lot of challenges in ill-posed problems. In such problems, many fundamental estimation methods fail to provide a meaningful stabilized solution. In this work,…
The paper deals with two fundamental types of trigonometric polynomials and splines on uniform grids, which allow us to construct interpolation approximations that depend linearly on the values of the interpolated function. Fundamental on…
Kernel-based schemes are state-of-the-art techniques for learning by data. In this work we extend some ideas about kernel-based greedy algorithms to exponential-polynomial splines, whose main drawback consists in possible overfitting and…
In this paper, we study a class of approximation problems, appearing in data approximation and signal processing. The approximations are constructed as combinations of polynomial splines (piecewise polynomials), whose parameters are subject…
The algorithm of modified wavelet analysis is discussed. It is based on the weighted least squares approximation. Contrary to the Gaussian as a weight function, we propose to use a compact weight function. The accuracy estimates using the…