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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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 . 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. 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. This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object—the high end of its spectrum—from coarse scale information only—from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0,1] and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up to a frequency cutoff fc. We show that one can super-resolve these point sources with infinite precision—i.e., recover the exact locations and amplitudes—by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2 fc. This result extends to higher dimensions and other models. In one dimension, for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the super-resolution factor vary. © 2014 Wiley Periodicals, Inc.