Conditioning of partial nonuniform Fourier matrices with clustered nodes

Dmitry Batenkov, Gil Goldman, Yosef Yomdin

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 superresolution 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 superresolution on a grid under
the partial clustering assumptions.