Optimal non-gaussian Dvoretzky-Milman embeddings

TL;DR

This paper introduces a novel non-Gaussian ensemble that achieves optimal Euclidean embedding estimates in arbitrary normed spaces, challenging the traditional reliance on Gaussian ensembles despite being heavy-tailed.

Contribution

It constructs the first non-Gaussian ensemble that matches Gaussian optimal estimates in the Dvoretzky-Milman Theorem, expanding the understanding of embeddings beyond Gaussian distributions.

Findings

Non-Gaussian ensemble achieves optimal Euclidean sections

Ensemble is heavy-tailed and far from Gaussian

Matches Gaussian embedding estimates in arbitrary normed spaces

Abstract

We construct the first non-gaussian ensemble that yields the optimal estimate in the Dvoretzky-Milman Theorem: the ensemble exhibits almost Euclidean sections in arbitrary normed spaces of the same dimension as the gaussian embedding -- despite being very far from gaussian (in fact, it happens to be heavy-tailed).