The smallest singular value of large random rectangular Toeplitz and circulant matrices

TL;DR

This paper investigates the asymptotic behavior of the smallest singular value of large random Toeplitz and circulant matrices with Gaussian entries, showing it converges to zero and providing bounds on the convergence rate.

Contribution

It proves the convergence of the smallest eigenvalue to zero for large matrices and establishes explicit bounds on the convergence rate, including polynomial bounds for circulant matrices.

Findings

Smallest eigenvalue converges to zero in probability and expectation.

Lower bound on convergence rate is faster than poly-logarithmic.

Polynomial upper bound on convergence rate for circulant matrices.

Abstract

Let $x_i$, $i\in\mathbb{Z}$ be a sequence of i.i.d. standard normal random variables. Consider rectangular Toeplitz $\mathbf{X}=\left(x_{j-i}\right)_{1\leq i\leq p,1\leq j\leq n}$ and circulant $\mathbf{X}=\left(x_{(j-i)\mod n}\right)_{1\leq i\leq p,1\leq j\leq n}$ matrices. Let $p,n\rightarrow\infty$ so that $p/n\rightarrow c\in(0,1]$. We prove that the smallest eigenvalue of $\frac{1}{n}\mathbf{X}\mathbf{X}^\top$ converges to zero in probability and in expectation. We establish a lower bound on the rate of this convergence. The lower bound is faster than any poly-log but slower than any polynomial rate. For the ``rectangular circulant'' matrices, we also establish a polynomial upper bound on the convergence rate, which is a simple explicit function of $c$.