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Small Ball Probabilities for Simple Random Tensors

Xuehan Hu, Grigoris Paouris

Abstract

We study the small ball probability of an order-$\ell$ simple random tensor $X=X^{(1)}\otimes\cdots\otimes X^{(\ell)}$ where $X^{(i)}, 1\leq i\leq\ell$ are independent random vectors in $\mathbb{R}^n$ that are log-concave or have independent coordinates with bounded densities. We show that the probability that the projection of $X$ onto an $m$-dimensional subspace $F$ falls within an Euclidean ball of length $\varepsilon$ is upper bounded by $\frac{\varepsilon}{(\ell-1)!}\left(C\log\left(\frac{e}{\varepsilon}\right)\right)^{\ell}$ and also this upper bound is sharp when $m$ is small. We also established that a much better estimate holds true for a random subspace.

Small Ball Probabilities for Simple Random Tensors

Abstract

We study the small ball probability of an order- simple random tensor where are independent random vectors in that are log-concave or have independent coordinates with bounded densities. We show that the probability that the projection of onto an -dimensional subspace falls within an Euclidean ball of length is upper bounded by and also this upper bound is sharp when is small. We also established that a much better estimate holds true for a random subspace.
Paper Structure (11 sections, 34 theorems, 284 equations)

This paper contains 11 sections, 34 theorems, 284 equations.

Table of Contents

  1. Introduction
  2. Background and Related Results
  3. Motivation
  4. Preliminaries
  5. Stochastic Dominance
  6. Related Results
  7. Stochastic Dominance
  8. Anti-concentration of "log-concave" simple random tensors
  9. Proof of Theorem \ref{['1.5']}
  10. Proof of Theorem \ref{['1.1']} and \ref{['1.2']}
  11. Application

Key Result

Theorem 1.1

Let $X^{(j)}\in\mathbb{R}^{n_j}, 1\leq j\leq\ell$ be independent random vectors with independent coordinates whose densities have uniform norms bounded by some constant $M>0$. Suppose $F$ is a subspace in $\mathbb{R}^{n_1\otimes\cdots\otimes n_{\ell}}$ with dimension $m$ and suppose $z_j\in\mathbb{R

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • ...and 44 more