Conditioning of Partial Nonuniform Fourier Matrices with Clustered Nodes

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.

Conditioning of restricted Fourier matrices and super-resolution of MUSIC

“…These results are also consistent with our intuition that σ min (Φ M ) is smallest when Ω consists of S closely spaced points; more details about this can be found in [13]. In [16], a lower bound of σ min (Φ M ) is derived for a model called clustered nodes; a detail comparison between Theorem 1 and results in [16] can be found in [13].…”

On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

“…For nodes on a grid, the results in [7,9] imply that the condition number grows like the super-resolution factor raised to the power of 1 if all nodes nearly collide. More recently, the practically relevant situation of groups of nearly colliding nodes was studied in [1,4,16,19]. In different setups and oversimplifying a bit, all of these refinements are able to replace the exponent 1 by the smaller number 1, where denotes the number of nodes that are in the largest group of nearly colliding nodes.…”

Super-Resolution Limit of the ESPRIT Algorithm

“…One is a rectangular Vandermonde matrix with nodes on the unit disk and the other one arises particularly from ESPRIT. The minimum singular value of the rectangular Vandermonde matrix with nodes closely spaced on the unit disk was not addressed till recent works in [3,25,21]. To study the second matrix, one needs to exploit its structure, but little research was done in this direction.…”

“…The best available bound for the case ∆ ≥ C/M was provided in [30], which relied on the Beurling-Selberg machinery, see [43]. Recently there are several independent works which provide estimates for ∆ ≤ 1/M by incorporating additional geometric information about the support set, see [3,25,21]. Accurate lower bounds under a clumps model can be found in [25].…”