Tight embedding of subspaces of $L_p$ in $\ell_p^n$ for even $p$

TL;DR

This paper provides a precise estimate on the minimal dimension of ^n needed to nearly preserve all k-dimensional subspaces of L_p for even p, leveraging recent advances in matrix sparsification.

Contribution

It introduces a tight bound on embedding dimensions for subspaces of L_p into ^n, extending previous work with new optimal estimates for even p.

Findings

Derived a tight estimate for embedding dimension n

Applied recent matrix sparsification results to L_p spaces

Achieved near-isometric embeddings for all k-dimensional subspaces

Abstract

Using a recent result of Batson, Spielman and Srivastava, We obtain a tight estimate on the dimension of $\ell_p^n$, $p$ an even integer, needed to almost isometrically contain all $k$-dimensional subspaces of $L_p$.