An isomorphic Dvoretzky-Milman Theorem using general random ensembles

TL;DR

This paper introduces a broad class of non-Gaussian, non-spherical random ensembles that achieve optimal isomorphic estimates in the Dvoretzky-Milman Theorem, expanding the types of ensembles known to exhibit such behavior.

Contribution

It constructs the first non-Gaussian, non-spherical ensembles with heavy tails that attain the optimal isomorphic estimates in the Dvoretzky-Milman Theorem.

Findings

Constructed general random ensembles with optimal estimates

Ensembles do not require rotation invariance

Ensembles can be heavy-tailed

Abstract

We construct rather general random ensembles that yield the optimal (isomorphic) estimate in the Dvoretzky-Milman Theorem. This is the first construction of non gaussian/spherical ensembles that exhibit the optimal behaviour. The ensembles constructed here need not satisfy any rotation invariance and can be rather heavy-tailed.