Related papers: Complex best $r$-term approximations almost always…

This paper investigates functions from $\mathbb{R}^d$ to $\mathbb{R} \cup \{\pm \infty\}$ that satisfy axioms of linearity wherever allowed by extended-value arithmetic. They have a nontrivial structure defined inductively on $d$, and…

The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest…

In this paper, we consider the rank-one tensor completion problem. We address the question of existence and uniqueness of the rank-one solution. In particular we show that the global uniqueness over the field of real numbers can be verified…

We use deep sparsely connected neural networks to measure the complexity of a function class in $L^2(\mathbb R^d)$ by restricting connectivity and memory requirement for storing the neural networks. We also introduce representation system -…

We address the problem of the best uniform approximation by linear combinations of a finite system of functions. If the system is Chebyshev and the problem is unconstrained, then the classical Remez algorithm provides a fast and precise…

In this work we study different notions of ranks and approximation of tensors. We consider the tensor rank, the nuclear rank and we introduce the notion of symmetric decomposable rank, a notion of rank defined only on symmetric tensors. We…

We investigate the existence of approximation algorithms for maximization of submodular functions, that run in fixed parameter tractable (FPT) time. Given a non-decreasing submodular set function $v: 2^X \to \mathbb{R}$ the goal is to…

Low-rank tensor approximation error bounds are proposed for the case of noisy input data that depend on low-rank representation type, rank and the dimensionality of the tensor. The bounds show that high-dimensional low-rank structured…

Low rank tensor learning, such as tensor completion and multilinear multitask learning, has received much attention in recent years. In this paper, we propose higher order matching pursuit for low rank tensor learning problems with a convex…

Submodular continuous functions are a category of (generally) non-convex/non-concave functions with a wide spectrum of applications. We characterize these functions and demonstrate that they can be maximized efficiently with approximation…

The extremum value theorem for function spaces plays the central role in optimal control. It is known that computation of optimal control actions and policies is often prone to numerical errors which may be related to computability issues.…

We study the consistency of the $k$-nearest neighbor regressor under complex survey designs. While consistency results for this algorithm are well established for independent and identically distributed data, corresponding results for…

We consider the problem of under and over-approximating the image of general vector-valued functions over bounded sets, and apply the proposed solution to the estimation of reachable sets of uncertain non-linear discrete-time dynamical…

We introduce and study the problem of consistent low-rank approximation, in which rows of an input matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ arrive sequentially and the goal is to provide a sequence of subspaces that well-approximate the…

The sparse polynomial approximation of continuous functions has emerged as a prominent area of interest in function approximation theory in recent years. A key challenge within this domain is the accurate estimation of approximation errors.…

By universal formulas we understand parameterized analytic expressions that have a fixed complexity, but nevertheless can approximate any continuous function on a compact set. There exist various examples of such formulas, including some in…

The main contribution of this note is to establish a framework to extend results of tensor functions over specific field to general field. As a consequence of this framework, we extend the existing work to more general settings: \emph{(1)}…

Low rank tensor representation underpins much of recent progress in tensor completion. In real applications, however, this approach is confronted with two challenging problems, namely (1) tensor rank determination; (2) handling real tensor…

Upper semicontinuous (usc) functions arise in the analysis of maximization problems, distributionally robust optimization, and function identification, which includes many problems of nonparametric statistics. We establish that every usc…

Given sufficiently many components, it is often cited that finite mixture models can approximate any other probability density function (pdf) to an arbitrary degree of accuracy. Unfortunately, the nature of this approximation result is…