Greedy Algorithms and Kolmogorov Widths in Banach Spaces

TL;DR

This paper analyzes the effectiveness of greedy algorithms in approximating compact sets in Banach spaces, relating their performance to Kolmogorov widths and geometric properties of subspaces.

Contribution

It establishes bounds on greedy algorithm performance in Banach spaces based on Kolmogorov widths and subspace geometry, extending previous results.

Findings

Greedy algorithms achieve approximation errors bounded by Kolmogorov widths with logarithmic factors.

When additional set information is available, the logarithmic factor can be eliminated.

The results connect approximation quality to subspace geometry via Banach-Mazur distances.

Abstract

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 in how well the space $V_n$ approximates the elements of $\mathcal{K}$. For this purpose we compare the performance of greedy algorithm measured by $\sigma_n(\mathcal{K})_X:=\text{dist}(\mathcal{K},V_n)_X$ with the Kolmogorov width $d_n(\mathcal{K})_X$ which is the best possible error one can achieve when approximating $\mathcal{K}$ by $n$-dimensional subspaces. Various results in this direction have been given, e.g., in Binev et al. (SIAM J. Math. Anal. (2011)), DeVore et al. (Constr. Approx. (2013)) and Wojtaszczyk (J. Math. Anal. Appl. (2015)). The purpose of the present paper is to continue this line. We shall show that there exists a constant $C>0$ such that $$ \sigma_n(\mathcal{K})_X\leq C n^{-s+\mu}\big(\log(n+2)\big)^{\min(s,1/2)}, \quad \ n\geq 1\,, $$ if Kolmogorov widths $d_n(\mathcal{K})_X$ decay as $n^{-s}$ and the Banach-Mazur distance between an arbitrary $n$-dimensional subspace $V_n \subset X$ and $\ell_2^n$ satisfies $d(V_n,\ell_2^n)\leq C_1 n^\mu$. In particular, when some additional information about the set $\mathcal{K}$ is given then there is no logarithmic factor in this estimate.