The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy

Denoyelle, Quentin; Duval, Vincent; Peyré, Gabriel; Soubies, Emmanuel

doi:10.48550/arxiv.1811.06416

Cited by 3 publications

(8 citation statements)

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“…In addition, the instabilities mentioned earlier are significantly reduced for these methods. This observation was already made in [3,11] and proved in [28] for specific problems.…”

Section: Continuous Methodssupporting

confidence: 61%

“…The proposed algorithmic scheme somehow captures the best of all methods. Let us notice that it is very similar in spirit to the sliding Frank-Wolfe algorithm proposed in [11], apart from the fact that we suggest adding all the points X k violating the constraints, while the single most violating point is added in [11]. We believe that the proposed analysis sheds some light on the good numerical performance of this method.…”

Section: Description Of the Hybrid Approachmentioning

confidence: 67%

“…, x p ). This principle is similar to continuous methods in semi-infinite programming [23] and was proposed specifically for total variation minimization in [3,11,28,8]. By Proposition 3.9, we know that after a finite number of iterations, X k will contain exactly s points located in a neighborhood of ξ.…”

Section: Convergence Of Continuous Methodsmentioning

confidence: 88%

“…The authors gave sufficient conditions for a simple gradient descent on the product-space (α, x) to converge. In [3] and [11], this method was used alternatively with a Frank-Wolfe algorithm, the idea being to first add Dirac masses roughly at the right locations and then to optimize their locations and position continuously, leading to promising numerical results. Surprisingly enough, it seems that the connection with the mature field of semi-infinite programming has been ignored (or not explicitly stated) in all the mentioned references.…”

Section: Numerical Approaches In Signal Processingmentioning

confidence: 99%

“…It can be shown to be equivalent to a Frank-Wolfe (a.k.a. conditional gradient) method up to an epigraphical lift [11]. These methods were introduced in the field of signal processing in [5] and the connection with exchange algorithms was proposed in [14].…”

Section: Algorithm 1 a Generic Exchange Algorithmmentioning

confidence: 99%

See 4 more Smart Citations

On the linear convergence rates of exchange and continuous methods for total variation minimization

Flinth¹,

Gournay²,

Weiss³

2019

Preprint

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We analyze an exchange algorithm for the numerical solution total-variation regularized inverse problems over the space M(Ω) of Radon measures on a subset Ω of R d . Our main result states that under some regularity conditions, the method eventually converges linearly. Additionally, we prove that continuously optimizing the amplitudes of positions of the target measure will succeed at a linear rate with a good initialization. Finally, we propose to combine the two approaches into an alternating method and discuss the comparative advantages of this approach.

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“…In addition, the instabilities mentioned earlier are significantly reduced for these methods. This observation was already made in [3,11] and proved in [28] for specific problems.…”

Section: Continuous Methodssupporting

confidence: 61%

Section: Description Of the Hybrid Approachmentioning

confidence: 67%

Section: Convergence Of Continuous Methodsmentioning

confidence: 88%

Section: Numerical Approaches In Signal Processingmentioning

confidence: 99%

Section: Algorithm 1 a Generic Exchange Algorithmmentioning

confidence: 99%

See 3 more Smart Citations

On the linear convergence rates of exchange and continuous methods for total variation minimization

Flinth¹,

Gournay²,

Weiss³

2019

Preprint

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Linear convergence of accelerated conditional gradient algorithms in spaces of measures

Pieper¹

2019

Preprint

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A class of generalized conditional gradient algorithms for the solution of optimization problem in spaces of Radon measures is presented. The method iteratively inserts additional Dirac-delta functions and optimizes the corresponding coefficients. Under general assumptions, a sub-linear O(1/k) rate in the objective functional is obtained, which is sharp in most cases. To improve efficiency, one can fully resolve the finite-dimensional subproblems occurring in each iteration of the method. We provide an analysis for the resulting procedure: under a structural assumption on the optimal solution, a linear O(ζ k ) convergence rate is obtained locally.

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Sparse Optimization on Measures with Over-parameterized Gradient Descent

Chizat

2019

Preprint

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Minimizing a convex function of a measure with a sparsity-inducing penalty is a typical problem arising, e.g., in sparse spikes deconvolution or two-layer neural networks training. We show that this problem can be solved by discretizing the measure and running nonconvex gradient descent on the positions and weights of the particles. For measures on a d-dimensional manifold and under some non-degeneracy assumptions, this leads to a global optimization algorithm with a complexity scaling as log(1/ ) in the desired accuracy , instead of −d for convex methods. The key theoretical tools are a local convergence analysis in Wasserstein space and an analysis of a perturbed mirror descent in the space of measures. Our bounds involve quantities that are exponential in d which is unavoidable under our assumptions.

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The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy

Cited by 3 publications

References 0 publications

On the linear convergence rates of exchange and continuous methods for total variation minimization

On the linear convergence rates of exchange and continuous methods for total variation minimization

Linear convergence of accelerated conditional gradient algorithms in spaces of measures

Sparse Optimization on Measures with Over-parameterized Gradient Descent

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