Conditioning of partial nonuniform Fourier matrices with clustered nodes

TL;DR

This paper establishes sharp bounds on the smallest singular value of partial Fourier matrices with clustered nodes, advancing understanding of super-resolution limits in signal processing.

Contribution

It provides new sharp lower bounds for the singular values of Fourier matrices with clustered nodes, extending super-resolution theory to partial and nonuniform cases.

Findings

Sharp lower bounds polynomial in super-resolution factor

Bounds depend on maximal cluster size

Implications for sparse super-resolution on grids

Abstract

We prove sharp lower bounds for the smallest singular value of a partial Fourier matrix with arbitrary "off the grid" nodes (equivalently, a rectangular Vandermonde matrix with the nodes on the unit circle), in the case when some of the nodes are separated by less than the inverse bandwidth. The bound is polynomial in the reciprocal of the so-called "super-resolution factor", while the exponent is controlled by the maximal number of nodes which are clustered together. As a corollary, we obtain sharp minimax bounds for the problem of sparse super-resolution on a grid under the partial clustering assumptions.