Inexact first-order primal–dual algorithms

Rasch, Julian; Chambolle, Antonin

doi:10.1007/s10589-020-00186-y

Cited by 45 publications

(39 citation statements)

References 58 publications

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“…for ω = (1 + θ)/(2 + µ) < 1 (precisely, ω = θ in case θ = 1/(1 + µ), ω = 1/(1 + µ/2) in case θ = 1). Hence, one should expect X k , Y k to converge to the solution of (31) if g, f * and are smooth enough, following the analysis for inexact primal-dual algorithms (see for instance [22]). We can sketch a convergence analysis, following [6,7,22], as follows.…”

Section: A3 Convergence Analysismentioning

confidence: 99%

Learning Consistent Discretizations of the Total Variation

Chambolle¹,

Pock²

2021

SIAM J. Imaging Sci.

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In this work, we study a general framework of discrete approximations of the total variation for image reconstruction problems. The framework, for which we can show consistency in the sense of Γ-convergence, unifies and extends several existing discretization schemes. In addition, we propose algorithms for learning discretizations of the total variation in order to achieve the best possible reconstruction quality for particular image reconstruction tasks. Interestingly, the learned discretizations significantly differ between the tasks, illustrating that there is no universal best discretization of the total variation.

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Section: A3 Convergence Analysismentioning

confidence: 99%

Learning Consistent Discretizations of the Total Variation

Chambolle¹,

Pock²

2021

SIAM J. Imaging Sci.

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“…Sincew (n) approximates the minimum of the proximal function with precision C E n , the calculated w (n) is a so-called type-one approximation of the proximal point w (n) := S τ (w) with precision C E n , see [50, p. 385]. Since the precisions (C E n ) 1 /2 are summable, [50,Thm. 2] guarantees the convergence of the inexact primal-dual method to a saddlepoint (w † , y † ).…”

Section: Remark 12mentioning

confidence: 99%

Tensor-Free Proximal Methods for Lifted Bilinear/Quadratic Inverse Problems with Applications to Phase Retrieval

Beinert

Bredies

2021

Found Comput Math

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We propose and study a class of novel algorithms that aim at solving bilinear and quadratic inverse problems. Using a convex relaxation based on tensorial lifting, and applying first-order proximal algorithms, these problems could be solved numerically by singular value thresholding methods. However, a direct realization of these algorithms for, e.g., image recovery problems is often impracticable, since computations have to be performed on the tensor-product space, whose dimension is usually tremendous. To overcome this limitation, we derive tensor-free versions of common singular value thresholding methods by exploiting low-rank representations and incorporating an augmented Lanczos process. Using a novel reweighting technique, we further improve the convergence behavior and rank evolution of the iterative algorithms. Applying the method to the two-dimensional masked Fourier phase retrieval problem, we obtain an efficient recovery method. Moreover, the tensor-free algorithms are flexible enough to incorporate a priori smoothness constraints that greatly improve the recovery results.

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“…Generalizations of [15] involving inexactness already exist in the form of [45] and [14], however, [45] only considers determinstic inexactness and proximal operators computed in the euclidean sense, i.e., with entropy equal to the euclidean energy, and requires Lipschitz-smoothness. It's worth noting that the inexactness considered in their paper allows for the inexact computation of the proximal operators, in contrast to our work.…”

Section: Contribution and Prior Workmentioning

confidence: 99%

A Stochastic Bregman Primal-Dual Splitting Algorithm for Composite Optimization

Silveti-Falls¹,

Molinari²,

Fadili³

2021

Preprint

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We study a stochastic first order primal-dual method for solving convex-concave saddle point problems over real reflexive Banach spaces using Bregman divergences and relative smoothness assumptions, in which we allow for stochastic error in the computation of gradient terms within the algorithm. We show ergodic convergence in expectation of the Lagrangian optimality gap with a rate of O (1/k) and that every almost sure weak cluster point of the ergodic sequence is a saddle point in expectation under mild assumptions. Under slightly stricter assumptions, we show almost sure weak convergence of the pointwise iterates to a saddle point. Under a relative strong convexity assumption on the objective functions and a total convexity assumption on the entropies of the Bregman divergences, we establish almost sure strong convergence of the pointwise iterates to a saddle point. Our framework is general and does not need strong convexity of the entropies inducing the Bregman divergences in the algorithm. Numerical applications are considered including entropically regularized Wasserstein barycenter problems and regularized inverse problems on the simplex.

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Inexact first-order primal–dual algorithms

Cited by 45 publications

References 58 publications

Learning Consistent Discretizations of the Total Variation

Learning Consistent Discretizations of the Total Variation

Tensor-Free Proximal Methods for Lifted Bilinear/Quadratic Inverse Problems with Applications to Phase Retrieval

A Stochastic Bregman Primal-Dual Splitting Algorithm for Composite Optimization

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