A Geometrical Study of Matching Pursuit Parametrization

Jacques, Laurent; Vleeschouwer, Christophe De

doi:10.1109/tsp.2008.917379

Cited by 26 publications

(26 citation statements)

References 45 publications

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“…These greedy pursuit algorithms can be applied without grid discretization [56] which enables the use of local optimizations over the spikes' positions [37]. Finally, let us mention the class of nonconvex optimization methods which include the well known Iterative Hard Thresholding (IHT) [7,8]…”

Section: Other Methods For Super-resolutionmentioning

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: Other Methods For Super-resolutionmentioning

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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“…A global invariance measure is then derived by averaging over a sufficiently large sample set. Equipped with our generic definition of invariance, we leverage the techniques used in the analysis of manifolds of transformed visual patterns [9,14,38] and design the Manitest method built on the efficient Fast Marching algorithm [15,35] to compute the invariance of classifiers.…”

Section: Arxiv:150706535v1 [Cscv] 23 Jul 2015mentioning

confidence: 99%

“…Equipped with the L 2 metric, M(I) defines a metric space and a continuous submanifold of L 2 . Following the works of [14,38] that considered similar manifolds in different contexts, we call M(I) an Image Appearance Manifold (IAM), and we follow here their approach. Assuming that γ : [0, 1] → T is a C 1 curve in T , and that I γ(t) is differentiable with respect to t, we define the length L(γ) of γ as…”

Section: Transformation Metricmentioning

confidence: 99%

Manitest: Are classifiers really invariant?

Fawzi¹,

Frossard²

2015

Procedings of the British Machine Vision Conference 2015

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Invariance to geometric transformations is a highly desirable property of automatic classifiers in many image recognition tasks. Nevertheless, it is unclear to which extent state-of-the-art classifiers are invariant to basic transformations such as rotations and translations. This is mainly due to the lack of general methods that properly measure such an invariance. In this paper, we propose a rigorous and systematic approach for quantifying the invariance to geometric transformations of any classifier. Our key idea is to cast the problem of assessing a classifier's invariance as the computation of geodesics along the manifold of transformed images. We propose the Manitest method, built on the efficient Fast Marching algorithm to compute the invariance of classifiers. Our new method quantifies in particular the importance of data augmentation for learning invariance from data, and the increased invariance of convolutional neural networks with depth. We foresee that the proposed generic tool for measuring invariance to a large class of geometric transformations and arbitrary classifiers will have many applications for evaluating and comparing classifiers based on their invariance, and help improving the invariance of existing classifiers.

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“…With the advent of CS, many variants of OMP have been applied to recovery including methods called MOMP, ROMP, CoSaMP, etc. (Needell and Tropp, 2008;Needell and Vershynin, 2009;Huang and Zhu, 2011) but with one exception (Jacques and De Vleeschouwer, 2008) discussed below, all of these methods recover frequencies (or other parameters) from discrete grids. The basic idea of all matching pursuit algorithms is to minimize a cost function to obtain frequencies of sinusoids present in the signal.…”

Section: Orthogonal Matching Pursuit Methodsmentioning

confidence: 99%