We consider the problem of stable recovery of sparse signals of the form @math from their spectral measurements, known in a bandwidth @math with absolute error not exceeding @math . We consider the case when at most @math nodes @math of @math form a cluster of size @math , while the rest of the nodes are well separated. Provided that @math , we show that the minimax error rate for reconstruction of the cluster nodes is of order @math , while for recovering the corresponding amplitudes @math the rate is of the order @math . Moreover, the corresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are @math and @math , respectively. Our numerical experiments show that the well-known Matrix Pencil method achieves the above accuracy bounds. These results suggest that stable super-resolution is possible in much more general situations than previously thought, and have implications for analyzing stability of super-resolution algorithms in this regime.
Stable super-resolution in the on-grid'' setting of @cite_9 @cite_1 @cite_17 is closely related to the smallest singular value of a certain class of Fourier-type matrices. Using the decimation technique, we have shown in a recent paper @cite_24 that the asymptotic scaling of the condition number for on-grid super-resolution is @math , matching the off-grid setting of the present paper. This result extends and generalizes previously known bounds @cite_36 @cite_10 @cite_43 @cite_39 , as well as recent works @cite_0 @cite_17 .
@cite_36 @cite_9 @cite_1 @cite_39 @cite_24 @cite_43 @cite_0 @cite_10 @cite_17
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Abstract We derive bounds on the extremal singular values and the condition number of N × K , with N ⩾ K , Vandermonde matrices with nodes in the unit disk. The mathematical techniques we develop to prove our main results are inspired by a link—first established by Selberg [1] and later extended by Moitra [2] —between the extremal singular values of Vandermonde matrices with nodes on the unit circle and large sieve inequalities. Our main conceptual contribution lies in establishing a connection between the extremal singular values of Vandermonde matrices with nodes in the unit disk and a novel large sieve inequality involving polynomials in z ∈ C with | z | ⩽ 1 . Compared to Bazan's upper bound on the condition number [3] , which, to the best of our knowledge, constitutes the only analytical result—available in the literature—on the condition number of Vandermonde matrices with nodes in the unit disk, our bound not only takes a much simpler form, but is also sharper for certain node configurations. Moreover, the bound we obtain can be evaluated consistently in a numerically stable fashion, whereas the evaluation of Bazan's bound requires the solution of a linear system of equations which has the same condition number as the Vandermonde matrix under consideration and can therefore lead to numerical instability in practice. As a byproduct, our result—when particularized to the case of nodes on the unit circle—slightly improves upon the Selberg–Moitra bound. We consider the problem of robustly recovering a @math -sparse coefficient vector from the Fourier series that it generates, restricted to the interval @math . The difficulty of this problem is linked to the superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse of @math ) by the spacing of the grid supporting the sparse vector. In the presence of additive deterministic noise of norm @math , we show upper and lower bounds on the minimax error rate that both scale like @math , providing a partial answer to a question posed by Donoho in 1992. The scaling arises from comparing the noise level to a restricted isometry constant at sparsity @math , or equivalently from comparing @math to the so-called @math -spark of the Fourier system. The proof involves new bounds on the singular values of restricted Fourier matrices, obtained in part from old techniques in complex analysis. Consider the problem of recovering a measure @math supported on a lattice of span @math , when measurements are only available concerning the Fourier Transform @math at frequencies @math . If @math is much smaller than the Nyquist frequency @math and the measurements are noisy, then, in general, stable recovery of @math is impossible. In this paper it is shown that if, in addition, we know that the measure @math satisfies certain sparsity constraints, then stable recovery is possible. Say that a set has Rayleigh index less than or equal to R if in any interval of length @math there are at most R elements. Indeed, if the (unknown) support of @math is known, a priori, to have Rayleigh index at most R, then stable recovery is possible with a stability coefficient that grows at most like @math as @math . This result validates certain practical efforts, in spectroscopy, seismic prospecting, and astrono... The purpose of this paper is to study the conditioning of complex Vandermonde matrices, in reference to applications such as superresolution and the problem of recovering missing samples in band-limited signals. The results include bounds for the singular values of Vandermonde matrices whose nodes are complex numbers on the unit circle. It is shown that, under certain conditions, such matrices can be quite well-conditioned, contrarily to what happens in the real case. 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. This generalizes previously known results for the extreme cases when all of the nodes either form a single cluster, or are completely separated. We briefly discuss possible implications for the theory and practice of super-resolution under sparsity constraints. Let WN=WN(z1,z2, . . . z1) be a rectangular Vandermonde matrix of order n × N, @math with distinct nodes zj in the unit disk and @math as its (j,k) entry. Matrices of this type often arise in frequency estimation and system identification problems. In this paper, the conditioning of WN is analyzed and bounds for the spectral condition number @math are derived. The bounds depend on n, N, and the separation of the nodes. By analyzing the behavior of the bounds as functions of N, we conclude that these matrices may become well conditioned, provided the nodes are close to the unit circle but not extremely close to each other and provided the number of columns of WN is large enough. The asymptotic behavior of both the conditioning itself and the bounds is analyzed and the theoretical results arising from this analysis verified by numerical examples. We prove upper and lower bounds for the spectral condition number of rectangular Vandermonde matrices with nodes on the complex unit circle. The nodes are "off the grid", pairs of nodes nearly collide, and the studied condition number grows linearly with the inverse separation distance. We provide reasonable sharp constants that are independent from the number of nodes as long as non-colliding nodes are well-separated. 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 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 .