Super-resolution of generalized spikes and spectra of confluent Vandermonde matrices

TL;DR

This paper addresses super-resolution of Dirac distributions and their derivatives on a circle from noisy Fourier data, providing sharp bounds on matrix singular values and error estimates under partial clustering conditions.

Contribution

It introduces new asymptotic bounds for the singular values of confluent Vandermonde matrices and derives matching error bounds for super-resolution with grid-constrained nodes.

Findings

Sharp asymptotic bounds for singular values of confluent Vandermonde matrices.

Matching lower and upper min-max error bounds for super-resolution.

Results applicable under partial clustering and grid assumptions.

Abstract

We study the problem of super-resolution of a linear combination of Dirac distributions and their derivatives on a one-dimensional circle from noisy Fourier measurements. Following numerous recent works on the subject, we consider the geometric setting of "partial clustering", when some Diracs can be separated much below the Rayleigh limit. Under this assumption, we prove sharp asymptotic bounds for the smallest singular value of a corresponding rectangular confluent Vandermonde matrix with nodes on the unit circle. As a consequence, we derive matching lower and upper min-max error bounds for the above super-resolution problem, under the additional assumption of nodes belonging to a fixed grid.