Results 1  10
of
12
SPIKE DETECTION FROM INACCURATE SAMPLINGS
"... Abstract. This article investigates the superresolution phenomenon using the celebrated statistical estimator LASSO in the complex valued measure framework. More precisely, we study the recovery of a discrete measure (spike train) from few noisy observations (Fourier samples, moments, Stieltjes tra ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
Abstract. This article investigates the superresolution phenomenon using the celebrated statistical estimator LASSO in the complex valued measure framework. More precisely, we study the recovery of a discrete measure (spike train) from few noisy observations (Fourier samples, moments, Stieltjes transformation...). In particular, we provide an explicit quantitative localization of the spikes. Moreover, our analysis is based on the Rice method and provide an upper bound on the supremum of white noise perturbation in the measure space. hal00780808, version 1 24 Jan 2013 1.
Model consistency of partly smooth regularizers
, 2014
"... This paper studies leastsquare regression penalized with partly smooth convex regularizers. This class of functions is very large and versatile allowing to promote solutions conforming to some notion of lowcomplexity. Indeed, they force solutions of variational problems to belong to a lowdimensi ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
(Show Context)
This paper studies leastsquare regression penalized with partly smooth convex regularizers. This class of functions is very large and versatile allowing to promote solutions conforming to some notion of lowcomplexity. Indeed, they force solutions of variational problems to belong to a lowdimensional manifold (the socalled model) which is stable under small perturbations of the function. This property is crucial to make the underlying lowcomplexity model robust to small noise. We show that a generalized “irrepresentable condition ” implies stable model selection under small noise perturbations in the observations and the design matrix, when the regularization parameter is tuned proportionally to the noise level. This condition is shown to be almost a necessary condition. We then show that this condition implies model consistency of the regularized estimator. That is, with a probability tending to one as the number of measurements increases, the regularized estimator belongs to the correct lowdimensional model manifold. This work unifies and generalizes several previous ones, where model consistency is known to hold for sparse, group sparse, total variation and lowrank regularizations. Lastly, we also show that this generalized “irrepresentable condition ” implies that the forwardbackward proximal splitting algorithm identifies the model after a finite number of steps.
The recoverability limit for superresolution via sparsity. arXiv:1502.01385,
, 2014
"... Abstract We consider the problem of robustly recovering a ksparse coefficient vector from the Fourier series that it generates, restricted to the interval [−Ω, Ω]. The difficulty of this problem is linked to the superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse of Ω) b ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract We consider the problem of robustly recovering a ksparse coefficient vector from the Fourier series that it generates, restricted to the interval [−Ω, Ω]. The difficulty of this problem is linked to the superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse of Ω) by the spacing of the grid supporting the sparse vector. In the presence of additive deterministic noise of norm σ, we show upper and lower bounds on the minimax error rate that both scale like (SRF ) 2k−1 σ, 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 2k, or equivalently from comparing 2k to the socalled σspark 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. Acknowledgments.
Journal of Machine Learning Research (2014) Submitted; Published Model Consistency of Partly Smooth Regularizers
, 2014
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
Abstract
 Add to MetaCart
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Sparse Spikes Deconvolution on Thin Grids
, 2015
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
Abstract
 Add to MetaCart
(Show Context)
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
A projection method on measures sets
, 2015
"... We consider the problem of projecting a probability measure pi on a setMN of Radon measures. The projection is defined as a solution of the following variational problem: inf µ∈MN ‖h? (µ − pi)‖22, where h ∈ L2(Ω) is a kernel, Ω ⊂ Rd and? denotes the convolution operator. To motivate and illustrate ..."
Abstract
 Add to MetaCart
(Show Context)
We consider the problem of projecting a probability measure pi on a setMN of Radon measures. The projection is defined as a solution of the following variational problem: inf µ∈MN ‖h? (µ − pi)‖22, where h ∈ L2(Ω) is a kernel, Ω ⊂ Rd and? denotes the convolution operator. To motivate and illustrate our study, we show that this problem arises naturally in various practical image rendering problems such as stippling (representing an image with N dots) or continuous line drawing (representing an image with a continuous line). We provide a necessary and sufficient condition on the sequence (MN)N∈N that ensures weak convergence of the projections (µ∗N)N∈N to pi. We then provide a numerical algorithm to solve a discretized version of the problem and show several illustrations related to computerassisted synthesis of artistic paintings/drawings. 1
Exact solutions to Super Resolution on semialgebraic domains in higher dimensions
, 2015
"... We investigate the multidimensional Super Resolution problem on closed semialgebraic domains for various sampling schemes such as Fourier or moments. We present a new semidefinite programming (SDP) formulation of the `1minimization in the space of Radon measures in the multidimensional frame on ..."
Abstract
 Add to MetaCart
(Show Context)
We investigate the multidimensional Super Resolution problem on closed semialgebraic domains for various sampling schemes such as Fourier or moments. We present a new semidefinite programming (SDP) formulation of the `1minimization in the space of Radon measures in the multidimensional frame on semialgebraic sets. While standard approaches have focused on SDP relaxations of the dual program (a popular approach is based on Gram matrix representations), this paper introduces an exact formulation of the primal `1minimization exact recovery problem of Super Resolution that unleashes standard techniques (such as momentsumofsquares hierarchies) to overcome intrinsic limitations of previous works in the literature. Notably, we show that one can exactly solve the Super Resolution problem in dimension greater than 2 and for a large family of domains described by semialgebraic sets.
SUPERRESOLUTION FROM SHORTTIME FOURIER TRANSFORMMEASUREMENTS
"... While spike trains are obviously not bandlimited, the theory of superresolution tells us that perfect recovery of unknown spike locations and weights from lowpass Fourier transform measurements is possible provided that the minimum spacing, ∆, between spikes is not too small. Specifically, for a ..."
Abstract
 Add to MetaCart
(Show Context)
While spike trains are obviously not bandlimited, the theory of superresolution tells us that perfect recovery of unknown spike locations and weights from lowpass Fourier transform measurements is possible provided that the minimum spacing, ∆, between spikes is not too small. Specifically, for a cutoff frequency of fc, Donoho [2] shows that exact recovery is possible if ∆> 1/fc, but does not specify a corresponding recovery method. On the other hand, Candès and FernandezGranda [3] provide a recovery method based on convex optimization, which provably succeeds as long as ∆> 2/fc. In practical applications one often has access to windowed Fourier transform measurements, i.e., shorttime Fourier transform (STFT) measurements, only. In this paper, we develop a theory of superresolution from STFT measurements, and we propose a method that provably succeeds in recovering spike trains from STFT measurements provided that ∆> 1/fc. Index Terms — Superresolution, inverse problems in measure spaces, shorttime Fourier transform. 1.