A Low-Rank Approach to Off-the-Grid Sparse Superresolution

Catala, Paul; Duval, Vincent; Peyré, Gabriel

doi:10.1137/19m124071x

Cited by 11 publications

(11 citation statements)

References 43 publications

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“…However, the overall complexity of the latter is polynomial in O( f 2d c ), where d is the dimension of the domain X, which restricts its application to small dimensional problems. This limitation has led to recent developments [16] where the authors proposed a relaxed low rank SDP formulation of the BLASSO in order to use a Frank-Wolfetype method (see below). The resulting method enjoys the better overall complexity of…”

Section: Fixed Spectral Discretization and Semidefinite Programming (...mentioning

confidence: 99%

The sliding Frank–Wolfe algorithm and its application to super-resolution microscopy

Denoyelle¹,

Duval²,

Peyré³

et al. 2019

Inverse Problems

167

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This paper showcases the theoretical and numerical performance of the Sliding Frank-Wolfe, which is a novel optimization algorithm to solve the BLASSO sparse spikes super-resolution problem.The BLASSO is a continuous (i.e. off-the-grid or grid-less) counterpart to the well-known 1 sparse regularisation method (also known as LASSO or Basis Pursuit). Our algorithm is a variation on the classical Frank-Wolfe (also known as conditional gradient) which follows a recent trend of interleaving convex optimization updates (corresponding to adding new spikes) with non-convex optimization steps (corresponding to moving the spikes). Our main theoretical result is that this algorithm terminates in a finite number of steps under a mild non-degeneracy hypothesis. We then target applications of this method to several instances of single molecule fluorescence imaging modalities, among which certain approaches rely heavily on the inversion of a Laplace transform. Our second theoretical contribution is the proof of the exact support recovery property of the BLASSO to invert the 1-D Laplace transform in the case of positive spikes. On the numerical side, we conclude this paper with an extensive study of the practical performance of the Sliding Frank-Wolfe on different instantiations of single molecule fluorescence imaging, including convolutive and non-convolutive (Laplace-like) operators. This shows the versatility and superiority of this method with respect to alternative sparse recovery technics.

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Section: Fixed Spectral Discretization and Semidefinite Programming (...mentioning

confidence: 99%

The sliding Frank–Wolfe algorithm and its application to super-resolution microscopy

Denoyelle¹,

Duval²,

Peyré³

et al. 2019

Inverse Problems

167

View full text Add to dashboard Cite

show abstract

“…One can expect to recover the positions of the Dirac masses exactly since no discretization is performed. However, these techniques are limited to the case where the dual problem involves trigonometrical polynomials, and usually to the one-dimensional setting (although some extensions exist in higher dimensions [38,39]). A last line of approaches tackles the BLASSO problem directly over the space of Radon measures.…”

Section: Related Workmentioning

confidence: 99%

On the uniqueness of solutions for the basis pursuit in the continuum

Debarre¹,

Denoyelle²,

Fageot³

2022

Inverse Problems

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This paper studies the continuous-domain inverse problem of recovering Radon measures on the one-dimensional torus from low-frequency Fourier coeﬃcients, where Kc is the cutoﬀ frequency. Our approach consists in minimizing the total-variation norm among all Radon measures that are consistent with the observations. We call this problem the basis pursuit in the continuum (BPC). We characterize the solution set of (BPC) in terms of uniqueness and describe its sparse solutions which are sums of few signed Dirac masses. The characterization is determined by the spectrum of a Toeplitz and Hermitian-symmetric matrix that solely depends on the observations. More precisely, we prove that (BPC) has a unique solution if and only if this matrix is neither positive deﬁnite nor negative deﬁnite. If it has both a positive and negative eigenvalue, then the unique solution is the sum of at most 2Kc Dirac masses, with at least one positive and one negative weight. If this matrix is positive (respectively negative) semi-deﬁnite and rank deﬁcient, then the unique solution is composed of a number of Dirac masses equal to the rank of the matrix, all of which have nonnegative (respectively nonpositive) weights. Finally, in cases where (BPC) has multiple solutions, we demonstrate that there are inﬁnitely many solutions composed of Kc + 1 Dirac masses, with nonnegative (respectively nonpositive) weights if the matrix is positive (respectively negative) deﬁnite.

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“…These methods have been further extended to the multidimensional case by considering SDP approximations of the problem [32]. The conditional gradient method (CGM) has also proven to be applicable to address the BLasso problem [29] and further enhanced with nonconvex local optimization extra steps [33][34][35]. Interestingly, the CGM has been shown to be equivalent to the so-called exchange method in [36,37].…”

Section: Related Work and State Of The Artmentioning

confidence: 99%