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The condition number of a random banded Toeplitz matrix is typically large

Paulo Manrique

TL;DR

The paper analyzes when random banded Toeplitz matrices $T_n(r,s)$ are well-conditioned by tying conditioning to the zeros of the associated Laurent polynomial $P_{r,s}(z)=\sum_{j=-r}^{s} \xi_j z^j$ and its Wiener–Hopf factors. It proves a sharp dichotomy: if the bandwidth is symmetric $(r=s)$, the matrices remain well-conditioned with high probability, while if the bandwidth is asymmetric $(r\neq s)$, the inverse norm grows at least as $e^{\alpha_m n}$, indicating ill-conditioning. Central to the results is showing that asymptotically half of the zeros of the related random polynomial lie inside the unit disk, together with bounds on Mahler measure and small-ball probabilities. These findings reveal that symmetry in the bandwidth dramatically affects numerical stability in random structured matrices and have implications for the analysis of Toeplitz-type discretizations and random polynomials.

Abstract

It is well known that square matrices with independent and identically distributed (iid) random entries are typically well conditioned. A natural question is whether this favorable behavior persists for random matrices whose entries obey additional structure, i.e., their position inside of the matrix. A prominent class of structured matrices is given by {\it Toeplitz matrices}, characterized by constant diagonals. A particular tractable subclass is that of circulant matrices, whose additional characteristic (its entries {\it circulate} row by row) allows one to express their conditioning in terms of the localization of the zeros of a associated polynomial. When the entries of a circulant matrix are iid, the matrix is well conditioned precisely when the corresponding random polynomial has no zeros on the unit circle. This connection is especially relevant because, as the degree of a random polynomial increases, its zeros tend to concentrate near the unit circle, making it a delicate problem to quantify how close the closest zeros lie to the unit circle. Another notable family within the Toeplitz class is that of {\it banded Toeplitz matrices}, namely matrices for which only finitely many diagonals around the main diagonal may be nonzero. These matrices have been extensively studied in Operator Theory, nevertheless, despite their apparent simplicity, they raise subtle questions regarding the behavior of their condition numbers. In present work we show that the bandwidth asymmetry plays a decisive role: if the band contains $r$ diagonals below and $s$ diagonal above the main diagonal, then if $r=s$ the banded Toeplitz matriz is well conditioned with high probability, whereas if $r\neq s$ it is typically ill conditioned. This highlights that structural constraints can have a impact on the numerical behavior of random matrices.

The condition number of a random banded Toeplitz matrix is typically large

TL;DR

The paper analyzes when random banded Toeplitz matrices are well-conditioned by tying conditioning to the zeros of the associated Laurent polynomial and its Wiener–Hopf factors. It proves a sharp dichotomy: if the bandwidth is symmetric , the matrices remain well-conditioned with high probability, while if the bandwidth is asymmetric , the inverse norm grows at least as , indicating ill-conditioning. Central to the results is showing that asymptotically half of the zeros of the related random polynomial lie inside the unit disk, together with bounds on Mahler measure and small-ball probabilities. These findings reveal that symmetry in the bandwidth dramatically affects numerical stability in random structured matrices and have implications for the analysis of Toeplitz-type discretizations and random polynomials.

Abstract

It is well known that square matrices with independent and identically distributed (iid) random entries are typically well conditioned. A natural question is whether this favorable behavior persists for random matrices whose entries obey additional structure, i.e., their position inside of the matrix. A prominent class of structured matrices is given by {\it Toeplitz matrices}, characterized by constant diagonals. A particular tractable subclass is that of circulant matrices, whose additional characteristic (its entries {\it circulate} row by row) allows one to express their conditioning in terms of the localization of the zeros of a associated polynomial. When the entries of a circulant matrix are iid, the matrix is well conditioned precisely when the corresponding random polynomial has no zeros on the unit circle. This connection is especially relevant because, as the degree of a random polynomial increases, its zeros tend to concentrate near the unit circle, making it a delicate problem to quantify how close the closest zeros lie to the unit circle. Another notable family within the Toeplitz class is that of {\it banded Toeplitz matrices}, namely matrices for which only finitely many diagonals around the main diagonal may be nonzero. These matrices have been extensively studied in Operator Theory, nevertheless, despite their apparent simplicity, they raise subtle questions regarding the behavior of their condition numbers. In present work we show that the bandwidth asymmetry plays a decisive role: if the band contains diagonals below and diagonal above the main diagonal, then if the banded Toeplitz matriz is well conditioned with high probability, whereas if it is typically ill conditioned. This highlights that structural constraints can have a impact on the numerical behavior of random matrices.
Paper Structure (5 sections, 7 theorems, 138 equations)

This paper contains 5 sections, 7 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Number of zeros of a random polynomial inside the unit circle
  4. Bounding expectation of Mahler measure
  5. Small ball probability

Key Result

Theorem 2.1

Let $r,s$ be nonnegative integers and set $m:=r+s$. Let We assume that $P_{r,s}$ is a random Laurent polynomial whose random coefficients $\xi_j$ satisfy assumptions $\mathcal{H}$. For each realization of $G_m$, let $\left\{T_n : n\in\mathds{N}\right\}$ denote the associated sequence of $n\times n$ banded Toeplitz matrices with symbol $P_{r,s}$. We have:

Theorems & Definitions (14)

  • Theorem 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['thm:main']}
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['thm:main01']}
  • Lemma 3.2
  • Lemma 3.3
  • proof : Proof of Lemma \ref{['lem:expectation']}
  • proof : Proof of Lemma \ref{['lem:dist_approx']}
  • Lemma 4.1
  • ...and 4 more