Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the structure of marginals in high dimensions

Daniel Bartl, Shahar Mendelson

Abstract

Let $G, G_1,\dots,G_N$ be independent copies of a standard gaussian random vector in $\mathbb{R}^d$ and denote by $Γ= \sum_{i=1}^N \langle G_i,\cdot\rangle e_i$ the standard gaussian ensemble. We show that, for any set $A\subset S^{d-1}$, with exponentially high probability, \[ \sup_{x\in A} \frac{1}{N}\sum_{i=1}^N \big| (Γx)^\sharp_i - q_i\big| \le c \frac{ \mathbb{E} \sup_{x\in A} \langle G,x\rangle + \log^2N }{\sqrt N }. \] Here each $q_i$ is the $\frac{i}{N+1}$-quantile of the standard normal distribution and $(Γx)^\sharp $ denotes the monotone increasing rearrangement of the vector $Γx$. The estimate is sharp up to a possible logarithmic factor and significantly extends previously known bounds. Moreover, we show that similar estimates hold in much greater generality: after replacing the gaussian quantiles by the appropriate ones, the same phenomenon persists for a broad class of random vectors.

On the structure of marginals in high dimensions

Abstract

Let be independent copies of a standard gaussian random vector in and denote by the standard gaussian ensemble. We show that, for any set , with exponentially high probability, Here each is the -quantile of the standard normal distribution and denotes the monotone increasing rearrangement of the vector . The estimate is sharp up to a possible logarithmic factor and significantly extends previously known bounds. Moreover, we show that similar estimates hold in much greater generality: after replacing the gaussian quantiles by the appropriate ones, the same phenomenon persists for a broad class of random vectors.
Paper Structure (18 sections, 29 theorems, 160 equations)

This paper contains 18 sections, 29 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. A uniform estimate on the Wasserstein distance
  3. On Assumption \ref{['ass:main']}
  4. Proofs of Theorem \ref{['thm:main']} and Lemma \ref{['lem:optimality']}
  5. Setting up the chaining argument
  6. The heart of the proof
  7. Estimating $\mathcal{E}_{1}(f)$
  8. Controlling $\mathbb{E} Z$
  9. Concentration around $\mathbb{E} Z$
  10. Controlling $\mathcal{E}_{2}(f)$
  11. Putting it all together: proof of Theorem \ref{['thm:main']}
  12. Proof of Lemma \ref{['lem:optimality']}
  13. Proof of Theorems \ref{['thm:main.intro']} and \ref{['thm:main.structure']} and optimality
  14. An application of Theorem \ref{['thm:main']}
  15. On Assumption \ref{['ass:main']}
  16. ...and 3 more sections

Key Result

Theorem 1.2

There exist absolute constants $c_1,c_2>0$ such that the following holds. For every set $A\subset S^{d-1}$, if then with probability at least $1-\exp(-c_2\Delta N)$,

Theorems & Definitions (51)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Lemma 2.7
  • Corollary 2.8
  • Definition 3.1
  • ...and 41 more