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Quantitative estimates of the spectral norm of random matrices with independent columns

Guozheng Dai, Zhonggen Su, Hanchao Wang

TL;DR

The paper addresses nonasymptotic control of the spectral norm for random matrices $W=BA$ where $A$ has independent mean-zero subexponential entries and $B$ is deterministic. It develops a decomposition-based proof that combines almost-square analysis, truncation, and a Gaussian-comparison principle to bound $(\mathbb{E}\|W\|^{p})^{1/p}$ by $\lesssim p(\sqrt{m}+\sqrt{n})$, with the crucial feature that the bound is independent of the ambient dimension $N$. The main contributions include a $N$-independent moment bound for general $B$ and an accompanying tail-analysis via a comparison theorem, plus results for the almost-square regime. Overall, the work provides sharp, finite-sample spectral-norm control for random matrices with independent columns, with implications for high-dimensional statistics and related applications where $N$ can be large.

Abstract

This paper investigates the nonasymptotic properties of the spectral norm of some random matrices with independent columns. In particular, we consider an $m\times n$ random matrix $BA$, where $A$ is an $N\times n$ random matrix with independent mean-zero subexponential entries, and $B$ is an $m\times N$ deterministic matrix. We prove that the $L_{p}$ norm of the spectral norm of $BA$ is upper bounded by $(\sqrt{m}+\sqrt{n})p$. It is remarkable that this result is independent of the dimension $N$.

Quantitative estimates of the spectral norm of random matrices with independent columns

TL;DR

The paper addresses nonasymptotic control of the spectral norm for random matrices where has independent mean-zero subexponential entries and is deterministic. It develops a decomposition-based proof that combines almost-square analysis, truncation, and a Gaussian-comparison principle to bound by , with the crucial feature that the bound is independent of the ambient dimension . The main contributions include a -independent moment bound for general and an accompanying tail-analysis via a comparison theorem, plus results for the almost-square regime. Overall, the work provides sharp, finite-sample spectral-norm control for random matrices with independent columns, with implications for high-dimensional statistics and related applications where can be large.

Abstract

This paper investigates the nonasymptotic properties of the spectral norm of some random matrices with independent columns. In particular, we consider an random matrix , where is an random matrix with independent mean-zero subexponential entries, and is an deterministic matrix. We prove that the norm of the spectral norm of is upper bounded by . It is remarkable that this result is independent of the dimension .
Paper Structure (13 sections, 16 theorems, 116 equations)

This paper contains 13 sections, 16 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Matrices with independent entries
  3. Matrices with independent columns
  4. Preliminaries
  5. Notations
  6. Concentration of subgaussian variables
  7. Subexponentional random variables
  8. Moments and Tails
  9. Nets
  10. Proof of the main result
  11. Proof of Theorem \ref{['Theo_main_weak']}
  12. Tail bounds for matrices with small columns
  13. Almost square matrices

Key Result

Theorem 1.1

Let $W=BA$ be an $m\times n$ random matrix, where $A=(a_{ij})$ is an $N\times n$ random matrix whose entries are independent subexponential random variables with mean zero, and $B$ is an $m\times N$ non-random matrix such that $\Vert B\Vert\le 1$. Then for $p\ge 1$ where $C$ is a universal constant.

Theorems & Definitions (24)

  • Theorem 1.1
  • Lemma 2.1: Gaussian concentration
  • Lemma 2.2
  • Lemma 2.3: Proposition 2.7.1 in highdimension
  • Lemma 2.4: Theorem 2.8.1 in highdimension
  • Lemma 2.5
  • Remark 2.1
  • proof
  • Lemma 2.6: Lemma 4.6 in inventions
  • Lemma 2.7: Lemma A.1 in ejp
  • ...and 14 more