How exponentially ill-conditioned are contiguous submatrices of the Fourier matrix?

TL;DR

This paper establishes exponential lower bounds on the condition number of contiguous submatrices of the Fourier matrix, revealing their severe ill-conditioning in various applications like super-resolution and diffraction imaging.

Contribution

It provides explicit, non-asymptotic lower bounds on the condition number of Fourier submatrices, using novel analytical techniques including the Kaiser-Bessel transform and sinc function estimates.

Findings

Condition number grows exponentially with N for fixed shape ratios.

Bounds are within a factor of two of empirical rates, sharp in certain regions.

Results are explicit and apply to all relevant matrix sizes, not just asymptotic cases.

Abstract

We show that the condition number of any cyclically contiguous $p\times q$ submatrix of the $N\times N$ discrete Fourier transform (DFT) matrix is at least $$ \exp \left( \frac{\pi}{2} \left[\min(p,q)- \frac{pq}{N}\right] \right)~, $$ up to algebraic prefactors. That is, fixing any shape parameters $(\alpha,\beta):=(p/N,q/N)\in(0,1)^2$, the growth is $e^{\rho N}$ as $N\to\infty$ with rate $\rho = \frac{\pi}{2}[\min(\alpha,\beta)- \alpha\beta]$. Such Vandermonde system matrices arise in many applications, such as Fourier continuation, super-resolution, and diffraction imaging. Our proof uses the Kaiser-Bessel transform pair (of which we give a self-contained proof), and estimates on sums over distorted sinc functions, to construct a localized trial vector whose DFT is also localized. We warm up with an elementary proof of the above but with half the rate, via a periodized Gaussian trial vector. Using low-rank approximation of the kernel $e^{ixt}$, we also prove another lower bound $(4/e\pi \alpha)^q$, up to algebraic prefactors, which is stronger than the above for small $\alpha, \beta$. When combined, the bounds are within a factor of two of the numerically-measured empirical asymptotic rate, uniformly over $(0,1)^2$, and they become sharp in certain regions. However, the results are not asymptotic: they apply to essentially all $N$, $p$, and $q$, and with all constants explicit.