Super-resolution, Extremal Functions and the Condition Number of Vandermonde Matrices

Super-resolution is a fundamental task in imaging, where the goal is to extract fine-grained structure from coarse-grained measurements. Here we are interested in a popular mathematical abstraction of this problem that has been widely studied in the statistics, signal processing and machine learning communities. We exactly resolve the threshold at which noisy super-resolution is possible. In particular, we establish a sharp phase transition for the relationship between the cutoff frequency ($m$) and the separation ($\Delta$). If $m > 1/\Delta + 1$, our estimator converges to the true values at an inverse polynomial rate in terms of the magnitude of the noise. And when $m < (1-\epsilon) /\Delta$ no estimator can distinguish between a particular pair of $\Delta$-separated signals even if the magnitude of the noise is exponentially small. Our results involve making novel connections between {\em extremal functions} and the spectral properties of Vandermonde matrices. We establish a sharp phase transition for their condition number which in turn allows us to give the first noise tolerance bounds for the matrix pencil method. Moreover we show that our methods can be interpreted as giving preconditioners for Vandermonde matrices, and we use this observation to design faster algorithms for super-resolution. We believe that these ideas may have other applications in designing faster algorithms for other basic tasks in signal processing.Comment: 19 page

Stable super-resolution limit and smallest singular value of restricted Fourier matrices

Super-resolution refers to the process of recovering the locations and amplitudes of a collection of point sources, represented as a discrete measure, given $M+1$ of its noisy low-frequency Fourier coefficients. The recovery process is highly sensitive to noise whenever the distance $\Delta$ between the two closest point sources is less than $1/M$. This paper studies the {\it fundamental difficulty of super-resolution} and the {\it performance guarantees of a subspace method called MUSIC} in the regime that $\Delta<1/M$. The most important quantity in our theory is the minimum singular value of the Vandermonde matrix whose nodes are specified by the source locations. Under the assumption that the nodes are closely spaced within several well-separated clumps, we derive a sharp and non-asymptotic lower bound for this quantity. Our estimate is given as a weighted $\ell^2$ sum, where each term only depends on the configuration of each individual clump. This implies that, as the noise increases, the super-resolution capability of MUSIC degrades according to a power law where the exponent depends on the cardinality of the largest clump. Numerical experiments validate our theoretical bounds for the minimum singular value and the resolution limit of MUSIC. When there are $S$ point sources located on a grid with spacing $1/N$, the fundamental difficulty of super-resolution can be quantitatively characterized by a min-max error, which is the reconstruction error incurred by the best possible algorithm in the worst-case scenario. We show that the min-max error is closely related to the minimum singular value of Vandermonde matrices, and we provide a non-asymptotic and sharp estimate for the min-max error, where the dominant term is $(N/M)^{2S-1}$.Comment: 47 pages, 8 figure

Beurling-Selberg Extremization for Dual-Blind Deconvolution Recovery in Joint Radar-Communications

Recent interest in integrated sensing and communications has led to the design of novel signal processing techniques to recover information from an overlaid radar-communications signal. Here, we focus on a spectral coexistence scenario, wherein the channels and transmit signals of both radar and communications systems are unknown to the common receiver. In this dual-blind deconvolution (DBD) problem, the receiver admits a multi-carrier wireless communications signal that is overlaid with the radar signal reflected off multiple targets. The communications and radar channels are represented by continuous-valued range-times or delays corresponding to multiple transmission paths and targets, respectively. Prior works addressed recovery of unknown channels and signals in this ill-posed DBD problem through atomic norm minimization but contingent on individual minimum separation conditions for radar and communications channels. In this paper, we provide an optimal joint separation condition using extremal functions from the Beurling-Selberg interpolation theory. Thereafter, we formulate DBD as a low-rank modified Hankel matrix retrieval and solve it via nuclear norm minimization. We estimate the unknown target and communications parameters from the recovered low-rank matrix using multiple signal classification (MUSIC) method. We show that the joint separation condition also guarantees that the underlying Vandermonde matrix for MUSIC is well-conditioned. Numerical experiments validate our theoretical findings.Comment: 5 pages, 3 figure