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month = {feb},  url = {http://arxiv.org/abs/1502.06932}  }  % Encoding: UTF8  @article{azais_spike,  title = {Spike detection from inaccurate samplings},  issn = {1063-5203},  url = {http://www.sciencedirect.com/science/article/pii/S106352031400044X},  doi = {10.1016/j.acha.2014.03.004},  abstract = {This article investigates the support detection problem using the {LASSO} estimator in the space of measures. More precisely, we study the recovery of a discrete measure (spike train) from few noisy observations (Fourier samples, moments, etc.) using an l 1 -regularization procedure. In particular, we provide an explicit quantitative localization of the spikes.},  urldate = {2014-11-01},  journal = {Applied and Computational Harmonic Analysis},  year = {in press},  author = {Aza{\"i}s, Jean-Marc and de Castro, Yohann and Gamboa, Fabrice},  keywords = {compressed sensing, {LASSO}, Semidefinite programming, Signed measure, super-resolution},  annote = {Extracted Annotations (03/11/2014 17:22:29)"theoretical guarantees of source detection are of crucial importance in practice." (Aza{\"i}s et al :2)"The aforementioned results suggest that the recovered spike locations should be close to the input support. This is investigated, for the first time, in this paper for an unconstrained l1-minimization problem under a general sampling scheme." (Aza{\"i}s et al :2)"One knows that the extreme points of the unit ball of the {TV}-norm are the atoms $\delta$x where x ?T. Therefore, l1-minimization compels the solutions to be discrete measures. Nevertheless, the {TV}-norm is inappropriated in measuring the distance between spikes, and it seems difficult to localize {BLASSO}. Two questions immediately arise: {\textbullet} How close is the recovered spike train {\textasciicircum}? from the target ?? {\textbullet} How accurate is the localization of {BLASSO} in terms of the noise and the amplitude of the recovered/original spike? To the best of our knowledge, this paper is the first to address these questions in a general frame. In particular, it is the first paper to provide quantitative localization guarantees from the output amplitudes." (Aza{\"i}s et al :4)"The present paper is concerned with the problem of the localization of the target spikes. In the general frame, we show that the large recovered spikes are very close to the target spikes, and that the mass of all the recovered spikes far from the original support is small. In the Fourier case and the moment case, we specify that under mild assumptions on the support of the target, the mass of the reconstructed measure is concentrated around large spikes, i.e. spikes above the noise level. In particular, we derive error bounds in terms of the recovered measure and the original measure." (Aza{\"i}s et al :4)"All solution to l1-regularization satisfies some optimality condition based on the fact that the sub-gradient of the regularization function vanishes at the solution point. Then a sufficient condition for exact recovery is that the target measure satisfies this optimality condition. This analysis has led to the notion of dual certificate, see [7,6] for instance, and the notion of source condition, see [4]." (Aza{\"i}s et al :4)"Our results show that if the spikes are sufficiently separated, at least 2.5/fc apart, then one can detect some point sources with a known precision solving a simple convex optimization program. (See Fig. 1.)" (Aza{\"i}s et al :8)}  }  @article{batenkov_numerical_2014,  title = {Numerical stability bounds for algebraic systems of {Prony} type and their accurate solution by decimation},  url = {http://arxiv.org/abs/1409.3137},  urldate = {2014-12-25},  journal = {{arXiv} preprint {arXiv}:1409.3137},  author = {Batenkov, Dmitry},  year = {2014}  }  @article{batenkov_accurate_2014,  title = {Accurate solution of near-colliding {Prony} systems via decimation and homotopy continuation},  url = {http://arxiv.org/abs/1501.00160},  abstract = {We consider polynomial systems of Prony type, appearing in many areas of mathematics. Their robust numerical solution is considered to be difficult, especially in "near-colliding" situations. We transform the nonlinear part of the Prony system into a Hankel-type polynomial system. Combining this representation with a recently discovered "decimation" technique, we present an algorithm which applies homotopy continuation to an appropriately chosen Hankel-type system as above. In this way, we are able to solve for the nonlinear variables of the original system with high accuracy when the data is perturbed.},  urldate = {2015-01-06},  journal = {{arXiv}:1501.00160 [cs, math]},  author = {Batenkov, Dmitry},  month = dec,  year = {2014},  note = {{arXiv}: 1501.00160},  keywords = {Computer Science - Numerical Analysis, Computer Science - Symbolic Computation, Mathematics - Numerical Analysis}  }  @Article{Bat.Sar.Yom,  Title = {Accuracy of algebraic {Fourier} reconstruction for shifts of several signals},  Author = {Batenkov, D. and Sarig, N. and Yomdin, Y.},  Journal = {Sampling Theory in Signal and Image Processing},  Year = {2014},  Volume = {13},  Number = {2},  Pages = {151--173}  }  @Article{Bat.Yom2,  Title = {{Geometry} and {Singularities} of the {Prony} mapping},  Author = {Batenkov, D. and Yomdin, Y.},  Journal = {Journal of Singularities},  Year = {2014},  Volume = {10},  Pages = {1--25}  }  @InProceedings{Bat.Yom.Sampta13,  Title = {Algebraic signal sampling, {Gibbs} phenomenon and {Prony}-type systems.},  Author = {Batenkov, D. and Yomdin, Y.},  Booktitle = {Proceedings of the 10th International Conference on Sampling Theory and Applications ({SAMPTA})},  Year = {2013}  }  @Article{Bat.Yom1,  Title = {On the accuracy of solving confluent {Prony} systems},  Author = {Batenkov, D. and Yomdin, Y.},  Journal = {SIAM J.Appl.Math.},  Year = {2013},  Volume = {73},  Number = {1},  Pages = {134--154}  }  @article{candes_towards_2014,  title = {{Towards} a {Mathematical} {Theory} of {Super}-resolution},  volume = {67},  copyright = {{\textcopyright} 2014 Wiley Periodicals, Inc.},  issn = {1097-0312},  url = {http://onlinelibrary.wiley.com/doi/10.1002/cpa.21455/abstract},  doi = {10.1002/cpa.21455},  abstract = {This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object{\textemdash}the high end of its spectrum{\textemdash}from coarse scale information only{\textemdash}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{\textemdash}i.e., recover the exact locations and amplitudes{\textemdash}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. {\textcopyright} 2014 Wiley Periodicals, Inc.},  language = {en},  number = {6},  urldate = {2014-05-18},  journal = {Communications on Pure and Applied Mathematics},  author = {Cand{\`e}s, Emmanuel J. and Fernandez-Granda, Carlos},  month = jun,  year = {2014},  pages = {906--956},  annote = {Extracted Annotations (03/11/2014 19:48:14)  "As to results regarding the robustness of super-resolution in the presence of noise, Donoho [11] studies the modulus of continuity of the recovery of a signed measure on a discrete lattice from its spectrum on the interval {\OE}\&fc; fc{\c c}, a setting that is also equivalent to that of Corollary 1.4 when the size N of the fine grid tends to infinity. More precisely, if the support of the measure is constrained to contain at most ' elements in any interval of length 2=.'fc/, then the modulus of continuity is of order O.{SRF}2'C1/ as {SRF} grows to infinity (note that for 'D 1 the constraint reduces to a minimum distance condition between spikes, which is comparable to the separation condition (1.6)). This means that if the '2 norm of the difference between the measurements generated by two signals satisfying the support constraints is known to be at most {\i}, then the '2 norm of the difference between the signals may be of order O.{SRF}2'C1 {\i}/. This result suggests that, in principle, the super-resolution of spread-out signals is not hopelessly ill conditioned. Having said this, it does not propose any practical recovery algorithm (a brute-force search for sparse measures obeying the low-frequency constraints would be computationally intractable)." (Cand{\`e}s and Fernandez-Granda 2014:916)}  }  @article{candes_super-resolution_2013,  title = {{Super}-{Resolution} from {Noisy} {Data}},  volume = {19},  issn = {1069-5869, 1531-5851},  url = {http://link.springer.com/article/10.1007/s00041-013-9292-3},  doi = {10.1007/s00041-013-9292-3},  abstract = {This paper studies the recovery of a superposition of point sources from noisy bandlimited data. In the fewest possible words, we only have information about the spectrum of an object in the low-frequency band [-f lo,f lo] and seek to obtain a higher resolution estimate by extrapolating the spectrum up to a frequency f hi{\textgreater}f lo. We show that as long as the sources are separated by 2/f lo, solving a simple convex program produces a stable estimate in the sense that the approximation error between the higher-resolution reconstruction and the truth is proportional to the noise level times the square of the super-resolution factor ({SRF}) f hi/f lo.},  language = {en},  number = {6},  urldate = {2014-01-02},  journal = {Journal of Fourier Analysis and Applications},  author = {Cand{\`e}s, Emmanuel J. and Fernandez-Granda, Carlos},  month = dec,  year = {2013},  keywords = {42A15, 90C22, 90C25, 94A12, Abstract Harmonic Analysis, Approximations and Expansions, Basis mismatch, Deconvolution, Fourier Analysis, Line spectra estimation, Mathematical Methods in Physics, Partial Differential Equations, Signal, Image and Speech Processing, Sparsity, Stable signal recovery, Super-resolution factor},  pages = {1229--1254}  }  @inproceedings{demanet_super-resolution_2013,  title = {Super-resolution via superset selection and pruning},  booktitle = {Proceedings of the 10th International Conference on Sampling Theory and Applications ({SAMPTA})},  author = {Demanet, Laurent and Needell, Deanna and Nguyen, Nam},  year = {2013}  }  @article{donoho_superresolution_1992,  title = {Superresolution via sparsity constraints},  volume = {23},  number = {5},  journal = {{SIAM} Journal on Mathematical Analysis},  author = {Donoho, D.L.},  year = {1992},  pages = {1309--1331}  }  @Article{Don1,  Title = {Uncertainty principles and signal recovery},  Author = {Donoho, D. L. and Stark, P. B.},  Journal = {SIAM J. Appl.Math.},  Year = {1989},  Pages = {906--931},  Volume = {49}  }  @article{duval_exact_2013,  title = {Exact support recovery for sparse spikes deconvolution},  url = {http://arxiv.org/abs/1306.6909},  urldate = {2015-01-06},  journal = {{arXiv} preprint {arXiv}:1306.6909},  author = {Duval, Vincent and Peyr{\'e}, Gabriel},  year = {2013}  }  @inproceedings{fernandez-granda_support_2013,  title = {Support detection in super-resolution},  url = {http://arxiv.org/abs/1302.3921},  urldate = {2014-09-01},  booktitle = {Proc. of 10th Sampling Theory and Applications ({SAMPTA})},  author = {Fernandez-Granda, Carlos},  year = {2013},  pages = {145--148}  }  @article{heckel_super-resolution_2014,  title = {{Super}-{Resolution} {Radar}},  url = {http://arxiv.org/abs/1411.6272},  abstract = {In this paper we study the identification of a time-varying linear system whose response is a weighted superposition of time and frequency shifted versions of the input signal. This problem arises in a multitude of applications such as wireless communications and radar imaging. Due to practical constraints, the input signal has finite bandwidth B, and the received signal is observed over a finite time interval of length T only. This gives rise to a time and frequency resolution of 1/B and 1/T. We show that this resolution limit can be overcome, i.e., we can recover the exact (continuous) time-frequency shifts and the corresponding attenuation factors, by essentially solving a simple convex optimization problem. This result holds provided that the distance between the time-frequency shifts is at least 2.37/B and 2.37/T, in time and frequency. Furthermore, this result allows the total number of time-frequency shifts to be linear (up to a log-factor) in {BT}, the dimensionality of the response of the system. More generally, we show that we can estimate the time-frequency components of a signal that is S-sparse in the continuous dictionary of time-frequency shifts of a random (window) function, from a number of measurements, that is linear (up to a log-factor) in S.},  urldate = {2014-12-06},  journal = {{arXiv}:1411.6272 [cs, math]},  author = {Heckel, Reinhard and Morgenshtern, Veniamin I. and Soltanolkotabi, Mahdi},  month = nov,  year = {2014},  note = {{arXiv}: 1411.6272},  keywords = {Computer Science - Information Theory}  }  @Article{Lev.Ful,  Title = {Reconstruction of a sparse spike train from a portion of its spectrum and application to high-resolution deconvolution},  Author = {Levy, S. and Fullagar, P. K.},  Journal = {Geophysics},  Year = {1981},  Number = {9},  Pages = {1235--1243},  Volume = {46}  }  @article{liao_music_2014,  title = {{MUSIC} for {Single}-{Snapshot} {Spectral} {Estimation}: {Stability} and {Super}-resolution},  shorttitle = {{MUSIC} for Single-Snapshot Spectral Estimation},  url = {http://arxiv.org/abs/1404.1484},  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 is 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 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, 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 4RL, {MUSIC} becomes the best performing one for frequencies separated between 2RL and 3RL. Also, {MUSIC} is more efficient than other methods. {MUSIC} truly shines when the frequency separation drops to one {RL} and below when all other methods fail. Indeed, the resolution of {MUSIC} apparently decreases to zero as noise decreases to zero.},  urldate = {2014-05-18},  journal = {{arXiv}:1404.1484 [cs, math]},  author = {Liao, Wenjing and Fannjiang, Albert},  month = apr,  year = {2014},  keywords = {Computer Science - Information Theory, Mathematics - Numerical Analysis},  annote = {Extracted Annotations (03/11/2014 17:03:21)  "Discretizing [0, 1) as in (4) amounts to rounding frequencies on the continuum to the nearest grid points {inG}, giving rise to a gridding error which is roughly proportional to the grid spacing. On the other hand, as N increases, correlation among adjacent columns of A also increases dramatically [18]." (Liao and Fannjiang 2014:3)  "A key unit of frequency separation is the Rayleigh Length, roughly the minimum resolvable separation of two objects with equal intensities in classical resolution theory [9]. Mathematically, the Rayleigh Length ({RL}) is the distance between the center and the first zero of the Dirichlet kernel D(!) = Hence 1 {RL} = 1/M. The ratio F = N/M between {RL} and the grid spacing is called the refinement factor in [18] and super-resolution factor in [4]. The higher F is, the more coherent the measurement matrix A becomes." (Liao and Fannjiang 2014:3)  "we pursue below a deterministic approach to spectral estimation" (Liao and Fannjiang 2014:3)  "The main contribution of the paper is a stability analysis for the {MUSIC} algorithm with respect to general support {setS} and external noise." (Liao and Fannjiang 2014:6)  "In [4, 3], Cand's and Fernandez-Granda propose total variation minimization and show that, under the assumption of q \% 4RL, that the minimizer yields an L1 reconstruction error linearly proportional to noise level with a magnifying constant proportional to F 2 (F = the refinement or super-resolution factor)." (Liao and Fannjiang 2014:7)  "{MUSIC} is the only algorithm that can resolve two frequencies closely spaced below 1 {RL}. Indeed, the resolution of {MUSIC} can be arbitrarily small for su!ciently small noise." (Liao and Fannjiang 2014:8)  "While we can not at the moment answer this question directly" (Liao and Fannjiang 2014:10)  "How close are the {MUSIC} estimates, namely the s lowest local minimizers of R"(!), to the true frequencies which are the zeros of R(!)? While we can not at the moment answer this question directly, the following asymptotic result says that every true frequency has a near-by strict local minimizer of R" in its vicinity converging to it." (Liao and Fannjiang 2014:10)  "Super-resolution in optics refers to the ability of distinguishing objects separated below 1RL. Its simplest version, two-point resolution, is widely used as metric for resolving power of imaging systems [9]." (Liao and Fannjiang 2014:16)}  }  @Article{McC,  Title = {Superresolution in microscopy and the {Abbe} resolution limit},  Author = {McCutchen, C. W.},  Journal = {J. Opt. Soc. Am.},  Year = {1967},  Number = {10},  Pages = {1190--1190},  Volume = {57}  }  @Article{Min.Kaw.Min,  Title = {Superresolution of {Fourier} {Transform} spectra by autoregressive model fitting  with singular value decomposition},  Author = {Minami, K. and Kawata, S. and Minami, S.},  Journal = {Appl. Optics},  Year = {1985},  Pages = {162--167},  Volume = {24}  }  @article{moitra_threshold_2014,  title = {The {Threshold} for {Super}-resolution via {Extremal} {Functions}},  url = {http://arxiv.org/abs/1408.1681},  abstract = {Super-resolution is a natural mathematical abstraction for the problem of extracting fine-grained structure from coarse-grained measurements, and has received considerable attention following the pioneering works of Donoho and Candes and Fernandez-Granda. Here we introduce new techniques based on extremal functions for studying this and related problems and 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 (\${\textbackslash}Delta\$). If \$m {\textgreater} 1/{\textbackslash}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 {\textless} (1-{\textbackslash}epsilon)/ {\textbackslash}Delta\$ no estimator can distinguish between a particular pair of \${\textbackslash}Delta\$-separated signals even if the magnitude of the noise is exponentially small. Our results involve making novel connections between extremal functions and spectral properties of the Vandermonde matrix, such as bounding its condition number as well as constructing explicit preconditioners for it.},  urldate = {2014-08-13},  journal = {{arXiv}:1408.1681 [cs, math, stat]},  author = {Moitra, Ankur},  month = aug,  year = {2014},  note = {{arXiv}: 1408.1681},  keywords = {Computer Science - Data Structures and Algorithms, Computer Science - Information Theory, Mathematics - Statistics Theory},  annote = {Extracted Annotations (03/11/2014 17:22:53)  "Prior to the more modern work on super-resolution, there was a considerable e?ort to develop empirical methods that work when there is some a priori information about the sparsity of the signal (see references in [11]). In a remarkable work, Donoho formulated the notion of the Rayleigh index which can be thought of as a continuous analogue to separation, and studied super-resolution where the fj{\textquoteright}s are restricted to be on a grid [11]. Donoho used Beurling{\textquoteright}s theory of interpolation [2] to prove that any \#-separated signal can be uniquely recovered from from its fourier transform on the continuous range [\#1/\#, 1/\#]. However this result is information theoretic in the sense that there could be two distinct signals that are both \#-separated, but whose fourier transforms restricted to [\#1/\#, 1/\#] di?er in a measure zero set. (Hence they are not noticeably di?erent). Donoho also gave results on the modulus of continuity for the recovery of \#-separated signals, but these results lose an extra factor of two in the range of low-frequency measurements that are needed." (Moitra 2014:3)  "The main question left open by this work was to prove recovery bounds when the signal is not restricted to be on a grid." (Moitra 2014:3)  "Subsequently, Candes and Fernandez-Granda [5, 6] proposed an exciting new method for solving this inverse problem based on convex programming. The authors proved a variety of results in the setting where (a) there is no noise or (b) there is noise, but the fj{\textquoteright}s are restricted to be on a grid. In the noise-free setting, the authors showed their approach recovers the uj{\textquoteright}s and fj{\textquoteright}s exactly provided that m\% 2 and m\% 128 ! where \# is the minimum separation. The authors further improved this bound when the uj{\textquoteright}s are real to a separation condition of m\% 1.87 (unpublished)1. In the grid setting, the authors defined a parameter C ! called the super-resolution factor and proved various bounds using it on the error of their estimator. However the approach used to measure the error is non-standard since it uses the Fejer kernel to {\textquoteleft}mask{\textquoteright} the error on the high-frequency parts of the signal. Finally, Ferndanez-Granda [16] returned to the setting where the fj{\textquoteright}s can be at arbitrary locations, and showed that if the noise is small enough the estimate becomes localized around the true spikes." (Moitra 2014:3)  "Theorem 1.5. If the cut-o? frequency satisfies m {\textgreater}?((1/\#) log 1/?) where\# is the minimum separation to accuracy?, provided that the noise satisfiesk?jk?2/(4k) for each j. ( according to the wrap-around metric) and each uj = 1, there is an {eO}(m2) time a to recover the f0 lgorithm" (Moitra 2014:5)}  }  @Article{Ode.Bar.Pis,  Title = {Two-dimensional superresolution radar imaging using the {MUSIC} algorithm},  Author = {Odendaal, J. and Barnard, E. and Pistorius, C. W. I.},  Journal = {IEEE Transactions on Antennas and Propagation},  Year = {1994},  Number = {10},  Pages = {1386--1391},  Volume = {42}  }  @Article{Sle,  Title = {Prolate spheroidal wave functions},  Author = {Slepian, D.},  Journal = {Fourier Analysis and uncertainty, V. - The discrete case. Bell System Technical Journal},  Year = {1978},  Pages = {1371--1430},  Volume = {57}  }  @Article{Yom2,  Title = {Singularities in algebraic data acquisition},  Author = {Yomdin, Y.},  Journal = {Real and complex singularities, London Math. Soc. Lecture Note Ser.},  Year = {2010},  Pages = {378--396},  Volume = {380}  }  @Article{Yom1,  Title = {Some quantitative results in singularity theory},  Author = {Yomdin, Y.},  Journal = {Ann. Polon. Math.},  Year = {2005},  Pages = {277–-299},  Volume = {87}  }  @Article{Dem.Ngu,  Title = {The recoverability limit for superresolution via sparsity},  Author = {Demanet, L. and Nguyen, N.},  Journal = {Preprint},  Year = {2014}  }  @Article{Mor.Can,  Title = {Stable super-resolution of positive sources: the discrete setup},  Author = {Morgenshtern, V. I. and Candes, E. J.},  Journal = {Preprint},  Year = {2014}  }