The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its @math consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is @math and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than @math apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT (Estimation of Signal Parameters via Rotation Invariance Techniques) is an efficient method that does not depend on the sign of the measure and this paper provides a : we prove an explicit upper bound on the recovery error of ESPRIT in terms of the minimum singular value of Vandermonde matrices. Using prior estimates for the minimum singular value explains its resolution limit -- when the support of @math consists of multiple well-separated clumps, the noise level that ESPRIT can tolerate scales like @math , where the super-resolution factor @math governs the difficulty of the problem and @math is the cardinality of the largest clump. Our theory is validated by numerical experiments.
@cite_37: 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 (Δ). If m > 1 Δ + 1, our estimator converges to the true values at an inverse polynomial rate in terms of the magnitude of the noise. And when m
@cite_26: Abstract This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data are turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the adjoint of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent smaller than an upper bound established by Donoho.
@cite_8: Performance analyses of subspace algorithms for cisoid parameter estimation available in the literature are predominantly of statistical nature with a focus on asymptotic—either in the sample size or the SNR—statements. This paper presents a deterministic, finite sample size, and finite-SNR performance analysis of the ESPRIT algorithm and the matrix pencil method. Our results are based, inter alia, on a new upper bound on the condition number of Vandermonde matrices with nodes inside the unit disk. This bound is obtained through a generalization of Hilbert's inequality frequently used in large sieve theory.
@cite_2: In this paper Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) is developed for spectral estimation with single-snapshot measurement. Stability and resolution analysis with performance guarantee for Single-Snapshot ESPRIT (SS-ESPRIT) is the main focus. In the noise-free case, exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise and under the assumption that the true frequencies are separated by at least two times Rayleigh's Resolution Length, an explicit error bound for frequency reconstruction is given in terms of the dynamic range and the separation of the frequencies. The separation and sparsity constraint compares favorably with those of the leading approaches to compressed sensing in the continuum.
@cite_13: Super-resolution refers to the process of recovering the locations and amplitudes of a collection of point sources, represented as a discrete measure, given @math of its noisy low-frequency Fourier coefficients. The recovery process is highly sensitive to noise whenever the distance @math between the two closest point sources is less than @math . This paper studies the fundamental difficulty of super-resolution and the performance guarantees of a subspace method called MUSIC in the regime that @math . 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 @math 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 @math point sources located on a grid with spacing @math , 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 @math .
Regarding the stability analysis of subspace methods, there have been works on bounding the error in terms of the minimum singular value of Vandermonde matrices. Such inequalities can be found in @cite_26 @cite_13 for MUSIC and in @cite_2 @cite_8 for ESPRIT, as well as in @cite_37 for the matrix pencil method. One major roadblock for this approach is that accurate bounds for the smallest singular value in the @math regime were not readily available. This difficulty was addressed in @cite_13 , which provided the first accurate analysis of MUSIC in the @math regime. As for ESPRIT, the bounds in @cite_2 @cite_8 do not capture the exact dependence of the error on the minimum singular value, and consequently, are inaccurate when @math (see and and the discussion there).