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A large deviation inequality for the rank of a random matrix

M. Rudelson

Abstract

Let $A$ be an $n \times n$ random matrix with independent identically distributed non-constant subgaussian entries. Then for any $k \le c \sqrt{n}$, \[ \text{rank}(A) \ge n-k \] with probability at least $1-\exp(-c'kn)$.

A large deviation inequality for the rank of a random matrix

Abstract

Let be an random matrix with independent identically distributed non-constant subgaussian entries. Then for any , with probability at least .
Paper Structure (12 sections, 18 theorems, 156 equations)

This paper contains 12 sections, 18 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Notation and the outline of the proof
  3. Notation
  4. Outline of the proof
  5. Preliminary results
  6. Almost orthogonal systems of vectors
  7. Concentration and tensorization
  8. Least common denominators and the small ball probability
  9. Integer points inside a ball
  10. Compressible vectors
  11. Incompressible vectors
  12. Rank of a random matrix

Key Result

Theorem 1.1

Let $k,n \in \mathbb{N}$ be numbers such that $k \le c n^{1/2}$. Let $A$ be an $n \times n$ matrix with i.i.d. non-constant subgaussian entries. Then

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3: Almost orthogonal system
  • Remark 3.4
  • proof
  • ...and 28 more