Average best $m$-term approximation

TL;DR

This paper introduces average best m-term approximation widths in high-dimensional spaces, estimates these for embeddings between ℓp and ℓq spaces, and shows typical vectors are nearly sparse under certain measures.

Contribution

It defines and estimates average best m-term approximation widths for ℓp^n to ℓq^n embeddings and demonstrates the near sparsity of typical vectors under specific measures.

Findings

Estimated approximation widths for embeddings with p ≤ q.

Typical vectors are nearly sparse under certain measures.

Provided bounds for average best m-term approximation in high dimensions.

Abstract

We introduce the concept of average best $m$-term approximation widths with respect to a probability measure on the unit ball of $\ell_p^n$. We estimate these quantities for the embedding $id:\ell_p^n\to\ell_q^n$ with $0<p\le q\le \infty$ for the normalized cone and surface measure. Furthermore, we consider certain tensor product weights and show that a typical vector with respect to such a measure exhibits a strong compressible (i.e. nearly sparse) structure.