Sparsity and non-Euclidean embeddings

TL;DR

This paper explores the relationship between sparsity and non-Euclidean embeddings, introducing a general isomorphism property to construct embeddings of $\,ell_p^n$ spaces into various Banach or quasi-Banach spaces.

Contribution

It introduces a new restricted isomorphism property and demonstrates how to embed $\,ell_p^n$ into different spaces with optimal bounds, advancing understanding of non-Euclidean embeddings.

Findings

Constructed embeddings of $\,ell_p^n$ into $\,ell_r^{(1+\,eta)n}$ with optimal polynomial bounds

Established a relation between sparsity and non-Euclidean isomorphic embeddings

Introduced a general restricted isomorphism property for embedding constructions

Abstract

We present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables to construct embeddings of $\ell_p^n$, $p > 0$, into various type of Banach or quasi-Banach spaces. In particular, for $0 <r < p<2$ with $r \le 1$, we construct a family of operators that embed $\ell_p^n$ into $\ell_r^{(1+\eta)n}$, with optimal polynomial bounds in $\eta >0$.