Discrete norms of a matrix and the converse to the Expander Mixing Lemma

TL;DR

This paper introduces discrete matrix norms based on binary vectors and establishes bounds relating these norms to traditional operator norms, providing new proofs and generalizations of the converse to the expander mixing lemma.

Contribution

The paper defines discrete and Rayleigh matrix norms, derives bounds relating them to standard norms, and offers a new proof of the converse to the expander mixing lemma.

Findings

Bounds for discrete norm in terms of standard operator norm.

Bounds for discrete Rayleigh norm in terms of standard operator norm.

New proof and generalization of the converse to the expander mixing lemma.

Abstract

We define the discrete norm of a complex $m\times n$ matrix $A$ by $$ \|A\|_\Delta := \max_{0\ne\xi\in\{0,1\}^n} \frac{\|A\xi\|}{\|\xi\|}, $$ and show that $$ \frac c{\sqrt{\log h(A)+1}}\,\|A\| \le \|A\|_\Delta \le \|A\|, $$ where $c>0$ is an explicitly indicated absolute constant, $h(A)=\sqrt{\|A\|_1\|A\|_\infty}/\|A\|$, and $\|A\|_1,\|A\|_\infty$, and $\|A\|=\|A\|_2$ are the induced operator norms of $A$. Similarly, for the \emph{discrete Rayleigh norm} $$ \|A\|_P := \max_{\substack{0\ne\xi\in\{0,1\}^m \\ 0\ne\eta\in\{0,1\}^n}} \frac{|\xi^tA\eta|}{\|\xi\|\|\eta\|} $$ we prove the estimate $$ \frac c{\log h(A)+1}\,\|A\| \le \|A\|_P \le \|A\|. $$ These estimates are shown to be essentially best possible. As a consequence, we obtain another proof of the (slightly sharpened and generalized version of the) converse to the expander mixing lemma by Bollobas-Nikiforov and Bilu-Linial.