Related papers: On sparse interpolation and the design of determin…

We construct approximate Fekete point sets for kernel-based interpolation by maximising the determinant of a kernel Gram matrix obtained via truncation of an orthonormal expansion of the kernel. Uniform error estimates are proved for kernel…

Inverse source localization from Helmholtz boundary data collected over a narrow aperture is highly ill-posed and severely undersampled, undermining classical solvers (e.g., the Direct Sampling Method). We present a modular framework that…

One way to investigate the precision of estimates likely to result from planned experiments and planned epidemiological studies is to simulate a large number of possible outcomes and analyse the sets of possible results. This appears to be…

This paper considers the problem of interpolating signals defined on graphs. A major presumption considered by many previous approaches to this problem has been lowpass/ band-limitedness of the underlying graph signal. However, inspired by…

The interpolation of couples of separable Hilbert spaces with a function parameter is studied. The main properties of the classic interpolation are proved. Some applications to the interpolation of isotropic H\"ormander spaces over a closed…

In many practical applications such as direction-of-arrival (DOA) estimation and line spectral estimation, the sparsifying dictionary is usually characterized by a set of unknown parameters in a continuous domain. To apply the conventional…

The Hermite-Birkhoff interpolation problem of a function given on arbitrarily distributed points on the sphere and other manifolds is considered. Each proposed interpolant is expressed as a linear combination of basis functions, the…

A scaled Chebyshev node distribution is studied in this paper. It is proved that the node distribution is optimal for interpolation in $C_M^{s+1}[-1,1]$, the set of $(s+1)$-time differentiable functions whose $(s+1)$-th derivatives are…

Effective verification and validation techniques for modern scientific machine learning workflows are challenging to devise. Statistical methods are abundant and easily deployed, but often rely on speculative assumptions about the data and…

In this paper, we develop a randomized algorithm and theory for learning a sparse model from large-scale and high-dimensional data, which is usually formulated as an empirical risk minimization problem with a sparsity-inducing regularizer.…

We propose a nonparametric factorization approach for sparsely observed tensors. The sparsity does not mean zero-valued entries are massive or dominated. Rather, it implies the observed entries are very few, and even fewer with the growth…

Finding sparse solutions of underdetermined systems of linear equations is a fundamental problem in signal processing and statistics which has become a subject of interest in recent years. In general, these systems have infinitely many…

Subspace clustering methods based on expressing each data point as a linear combination of all other points in a dataset are popular unsupervised learning techniques. However, existing methods incur high computational complexity on…

In this paper I uncover and explain---using contour integrals and residues---a connection between cubic splines and a popular compact finite difference formula. The connection is that on a uniform mesh the simplest Pad\'e scheme for…

Stochastic sampling methods are arguably the most direct and least intrusive means of incorporating parametric uncertainty into numerical simulations of partial differential equations with random inputs. However, to achieve an overall error…

We study the convergence of random function iterations for finding an invariant measure of the corresponding Markov operator. We call the problem of finding such an invariant measure the stochastic fixed point problem. This generalizes…

The nonlinear inverse problem of exponential data fitting is separable since the fitting function is a linear combination of parameterized exponential functions, thus allowing to solve for the linear coefficients separately from the…

In the first part of the series papers, we set out to answer the following question: given specific restrictions on a set of samplers, what kind of signal can be uniquely represented by the corresponding samples attained, as the foundation…

We consider quasi-interpolation with a main application in radial basis function approximations and compression in this article. Constructing and using these quasi-interpolants, we consider wavelet and compression-type approximations from…

In this paper, we give new sparse interpolation algorithms for black box univariate and multivariate rational functions h=f/g whose coefficients are integers with an upper bound. The main idea is as follows: choose a proper integer beta and…