Related papers: $(n D+1)$ Pad\'e approximation
Optimization problems, generalized equations, and the multitude of other variational problems invariably lead to the analysis of sets and set-valued mappings as well as their approximations. We review the central concept of set-convergence…
Approximating complex probability densities is a core problem in modern statistics. In this paper, we introduce the concept of Variational Inference (VI), a popular method in machine learning that uses optimization techniques to estimate…
Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincar\'e inequality, have been handy for estimating the…
Dimension reduction is often the first step in statistical modeling or prediction of multivariate spatial data. However, most existing dimension reduction techniques do not account for the spatial correlation between observations and do not…
A framework to boost the efficiency of Bayesian inference in probabilistic programs is introduced by embedding a sampler inside a variational posterior approximation. We call it the refined variational approximation. Its strength lies both…
Two aspects of neural networks that have been extensively studied in the recent literature are their function approximation properties and their training by gradient descent methods. The approximation problem seeks accurate approximations…
Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications the…
Fixpoints are ubiquitous in computer science and when dealing with quantitative semantics and verification one often considers least fixpoints of (higher-dimensional) functions over the non-negative reals. We show how to approximate the…
Value functions arise as a component of algorithms as well as performance metrics in statistics and engineering applications. Computation of the associated Bellman equations is numerically challenging in all but a few special cases. A…
When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given…
We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms…
This paper presents an identity between the multivariate and univariate saddlepoint approximations applied to sample path probabilities for a certain class of stochastic processes. This class, which we term the recursively compounded…
Models based on approximation capabilities have recently been studied in the context of Optimal Recovery. These models, however, are not compatible with overparametrization, since model- and data-consistent functions could then be…
Equational unification of two terms consists of finding a substitution that, when applied to both terms, makes them equal modulo some equational properties. Equational unification is of special relevance to automated deduction, theorem…
In this paper we combine two existing approaches for approximating attractors. One of them approximates the attractors arbitrarily well by sublevel sets related to solutions of infinite dimensional linear programming problems. A downside…
A simple sparse coding mechanism appears in the sensory systems of several organisms: to a coarse approximation, an input $x \in \R^d$ is mapped to much higher dimension $m \gg d$ by a random linear transformation, and is then sparsified by…
We consider approximating analytic functions on the interval $[-1,1]$ from their values at a set of $m+1$ equispaced nodes. A result of Platte, Trefethen \& Kuijlaars states that fast and stable approximation from equispaced samples is…
The paper presents new and known results on estimates of important linear and nonlinear approximation characteristics of generalized Wiener classes of functions of several variables in different metrics.
Submodular functions are a fundamental object of study in combinatorial optimization, economics, machine learning, etc. and exhibit a rich combinatorial structure. Many subclasses of submodular functions have also been well studied and…
We study adaptive approximation algorithms for general multivariate linear problems where the sets of input functions are non-convex cones. While it is known that adaptive algorithms perform essentially no better than non-adaptive…