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Operator $\ell^\infty \to \ell^\infty$ norm of products of random matrices

Jean-Christophe Mourrat

TL;DR

This work analyzes the $\ell^ty \to \ell^ty$ operator norm of products of i.i.d. random matrices in the high-dimensional limit. By rewriting the norm as a maximum over row sums and employing a quantitative Gaussian approximation together with concentration inequalities, the authors show a universal scaling $n_p^{-1}(n_1\cdots n_{p-1})^{-1/2}\|P\|_{\ell^ty\to\ell^ty} \to \sqrt{2/\pi}$ for centered square cases and extend the results to non-square and non-centered matrices via decomposition into centered and $\mathbf{1}$-type components. The presence and position of $\mathbf{1}$-type factors lead to distinct regimes: with ends both being $\mathbf{1}$-types, a Gaussian limit governs the sum of entries; with a left end $\mathbf{1}$-type factor the same leading-order is preserved, while a right end $\mathbf{1}$-type factor introduces a logarithmic correction reflecting maximal-sum effects. The results have implications for neural networks initialized with random weights, providing explicit asymptotics for Lipschitz-type norms and offering a broader understanding of operator norms of random matrix products in high dimensions.

Abstract

We study the $\ell^\infty \to \ell^\infty$ operator norm of products of independent random matrices with independent and identically distributed entries. For $n$-by-$n$ matrices whose entries are centered, have unit variance, and have a finite moment of order $4α$ for some $α> 1$, we find that the operator norm of the product of $p$ matrices behaves asymptotically like $n^{\frac {p+1}{2}}\sqrt{2/π}$. The case of products of possibly non-square matrices with possibly non-centered entries is also covered.

Operator $\ell^\infty \to \ell^\infty$ norm of products of random matrices

TL;DR

This work analyzes the operator norm of products of i.i.d. random matrices in the high-dimensional limit. By rewriting the norm as a maximum over row sums and employing a quantitative Gaussian approximation together with concentration inequalities, the authors show a universal scaling for centered square cases and extend the results to non-square and non-centered matrices via decomposition into centered and -type components. The presence and position of -type factors lead to distinct regimes: with ends both being -types, a Gaussian limit governs the sum of entries; with a left end -type factor the same leading-order is preserved, while a right end -type factor introduces a logarithmic correction reflecting maximal-sum effects. The results have implications for neural networks initialized with random weights, providing explicit asymptotics for Lipschitz-type norms and offering a broader understanding of operator norms of random matrix products in high dimensions.

Abstract

We study the operator norm of products of independent random matrices with independent and identically distributed entries. For -by- matrices whose entries are centered, have unit variance, and have a finite moment of order for some , we find that the operator norm of the product of matrices behaves asymptotically like . The case of products of possibly non-square matrices with possibly non-centered entries is also covered.

Paper Structure

This paper contains 6 sections, 17 theorems, 162 equations.

Table of Contents

  1. Introduction and main results
  2. The case of one matrix
  3. Products of centered matrices
  4. Products that start and end with 1
  5. Products starting with 1
  6. Products ending with 1

Key Result

Proposition 1.1

Let $\alpha > 1$ and let $A = (A_{ij})_{1 \leqslant i \leqslant m, 1 \leqslant j \leqslant n}$ be a family of i.i.d. random variables with finite moment of order $2\alpha$. As $m$ and $n$ tend to infinity with $m = O(n)$, we have

Theorems & Definitions (41)

  • Proposition 1.1: single matrix
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: centered entries
  • Remark 1.5
  • Theorem 1.6: Products that start with $\mathbf{1}$
  • Remark 1.7
  • Theorem 1.8: the case of $\mathbf{1} P\mathbf{1}$
  • Theorem 1.9: Products that end with $\mathbf{1}$
  • Remark 1.10
  • ...and 31 more