Related papers: On the approximation of vector-valued functions by…
We develop sampling formulas for high-dimensional functions in reproducing kernel Hilbert spaces, where we rely on irregular samples that are taken at determining sequences of data points. We place particular emphasis on sampling formulas…
A lower semi-definite self-adjoint linear operator in a Hilbert space is taken whose discrete spectrum is not empty and comprises at least several eigenvalues $\lambda_{min}=\lambda_1\leqslant\ldots\leqslant\lambda_m<\sigma_{ess}$. The…
Linear fixed point equations in Hilbert spaces arise in a variety of settings, including reinforcement learning, and computational methods for solving differential and integral equations. We study methods that use a collection of random…
We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864--885], obtained by searching for an optimal probability…
We study the problem of robustly learning multi-dimensional histograms. A $d$-dimensional function $h: D \rightarrow \mathbb{R}$ is called a $k$-histogram if there exists a partition of the domain $D \subseteq \mathbb{R}^d$ into $k$…
This paper addresses the problem of regression to reconstruct functions, which are observed with superimposed errors at random locations. We address the problem in reproducing kernel Hilbert spaces. It is demonstrated that the estimator,…
We introduce the notion of a pseudomultiplier of a Hilbert space $\mathcal H$ of functions on a set $\Omega$. Roughly, a pseudomultiplier of $\mathcal H$ is a function which multiplies a finite-codimensional subspace of $\mathcal H$ into…
We show that any decoherence functional $D$ can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural…
Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to…
Halfspace (or Tukey) depth is a fundamental and robust measure of centrality of data points in multivariate datasets. Computing the depth of a point with respect to the uniform distribution on an open convex body in $\mathbb{R}^d$ is a…
It is known that for a $\rho$-weighted $L_q$-approximation of single variable functions $f$ with the $r$th derivatives in a $\psi$-weighted $L_p$ space, the minimal error of approximations that use $n$ samples of $f$ is proportional to…
The holographic principle suggests that the Hilbert space of quantum gravity is locally finite-dimensional. Motivated by this point-of-view, and its application to the observable Universe, we introduce a set of numerical and conceptual…
In this paper, we develop and analyze sub-sampled trust-region methods for solving finite-sum optimization problems. These methods employ subsampling strategies to approximate the gradient and Hessian of the objective function,…
We study the min-max optimization problem where each function contributing to the max operation is strongly-convex and smooth with bounded gradient in the search domain. By smoothing the max operator, we show the ability to achieve an…
In this paper, both semidiscrete and completely discrete finite volume element methods (FVEMs) are analyzed for approximating solutions of a class of linear hyperbolic integro- differential equations in a two-dimensional convex polygonal…
Most current sampling algorithms for high-dimensional distributions are based on MCMC techniques and are approximate in the sense that they are valid only asymptotically. Rejection sampling, on the other hand, produces valid samples, but is…
Low-rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high-dimensional applications. In this work, we study the low-rank approximability of solutions to linear systems…
Finite volume methods for problems involving second order operators with full diffusion matrix can be used thanks to the definition of a discrete gradient for piecewise constant functions on unstructured meshes satisfying an orthogonality…
The purpose of this paper is twofold. First, we provide an optimal $\Omega(\sqrt{n})$ bits lower bound for any two-way protocol for the Vector in Subspace Communication Problem which is of bounded total rank. This result complements Raz's…
We describe a new MCMC method optimized for the sampling of probability measures on Hilbert space which have a density with respect to a Gaussian; such measures arise in the Bayesian approach to inverse problems, and in conditioned…