Related papers: A remark on approximation with polynomials and gre…

We show that a (semi-normalized) basis in a Banach space is quasi-greedy with quasi-greedy constant equal to 1 if and only if it is unconditional with suppression-unconditional constant equal to 1.

In this paper we continue the study of Lebesgue-type inequalities for greedy algorithms. We introduce the notion of strong partially greedy Markushevich bases and study the Lebesgue-type parameters associated with them. We prove that this…

This paper is a follow up to the previous author's paper on convex optimization. In that paper we began the process of adjusting greedy-type algorithms from nonlinear approximation for finding sparse solutions of convex optimization…

We prove thatthe Banach space $(\oplus_{n=1}^\infty \ell_p^n)_{\ell_q}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever $1\leq p \leq\infty$ and $1<q<\infty$. Furthermore, the Banach spaces $(\oplus_{n=1}^\infty…

Let $X$ be a Banach holomorphic function space on the unit disk. A linear polynomial approximation scheme for $X$ is a sequence of bounded linear operators $T_n:X\to X$ with the property that, for each $f\in X$, the functions $T_n(f)$ are…

We show that a very simple modification of the Pure Greedy Algorithm for approximating functions by sparse sums from a dictionary in a Hilbert or more generally a Banach space has optimal convergence rates on the class of convex…

Let K be a closed bounded convex subset of $\Bbb R^n$; then by a result of the first author, which extends a classical theorem of Whitney there is a constant $w_m(K)$ so that for every continuous function f on K there is a polynomial $\phi$…

We propose a new concept named adaptive submodularity ratio to study the greedy policy for sequential decision making. While the greedy policy is known to perform well for a wide variety of adaptive stochastic optimization problems in…

Let $X$ be a Banach space and $\mathcal{K}$ be a compact subset in $X$. We consider a greedy algorithm for finding an $n$-dimensional subspace $V_n\subset X$ which can be used to approximate the elements of $\mathcal{K}$. We are interested…

It is known that a basis is almost greedy if and only if the thresholding greedy algorithm gives essentially the smallest error term compared to errors from projections onto intervals or in other words, consecutive terms of $\mathbb{N}$. In…

The aim of this article is to use Banach lattice techniques to study coordinate systems in function spaces. We begin by proving that the greedy algorithm of a basis is order convergent if and only if a certain maximal inequality is…

Kernel methods are versatile tools for function approximation and surrogate modeling. In particular, greedy techniques offer computational efficiency and reliability through inherent sparsity and provable convergence. Inspired by the…

The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of…

Submodularity is a key property in discrete optimization. Submodularity has been widely used for analyzing the greedy algorithm to give performance bounds and providing insight into the construction of valid inequalities for mixed-integer…

The problem of column subset selection has recently attracted a large body of research, with feature selection serving as one obvious and important application. Among the techniques that have been applied to solve this problem, the greedy…

We investigate, in the setting of UMD Banach spaces E, the continuous dependence on the data A, F, G and X_0 of mild solutions of semilinear stochastic evolution equations with multiplicative noise of the form dX(t) = [AX(t) + F(t,X(t))]dt…

A flaw in the greedy approximation algorithm proposed by Zhang et al. for minimum connected set cover problem is corrected, and a stronger result on the approximation ratio of the modified greedy algorithm is established. The results are…

Relatively recently it was proved that if $\Gamma$ is an arbitrary set, then any equivalent norm on $c_0(\Gamma)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^\infty$ smooth norms, with arbitrary precision. We…

We generalize the Gr\"obner basis method for free D-modules to the case of several term orderings induced by a partition of the set of basic variables. Using this generalized Gr\"obner basis technique we prove the existence and give a…

Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on…