Signal Recovery by Proximal Forward-Backward Splitting

By Patrick L. Combettes, Valérie R. Wajs. 2005

3,028 citations recorded by the source corpus.

Subfields: Biomedical Engineering, Computational Mechanics, Mathematical Physics

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Citations in the corpus

Spectral Regularization Algorithms for Learning Large Incomplete Matrices (2010, methodology): “All these algorithms [CCS08, CW05 , MGC09] require the specification of a step size, and can be quite sensitive to the chosen value.”
Incremental Majorization-Minimization Optimization with Application to Large-Scale Machine Learning (2014, background): “For a review of proximal operators and their computations, we refer the reader to [2, 15].”
A Singular Value Thresholding Algorithm for Matrix Completion (2008, methodology): “for some �� > 0. Theorem 2.1 asserts that �� is the proximity operator of �� ∥�� ∥∗ and Proposition 3.1(iii) in [24] gives that the solution to this unconstrained problem is characterized by the fixed point equation �� = �� (�� + �� Ω(�� − �� )) for each �� > 0. One can then apply a simpli”
Optimization with Sparsity-Inducing Penalties (2011, background): “For a complete study of the properties of the proximal operator, we refer the interested reader to [39].”
Sparse coding for multitask and transfer learning (2012, methodology): “Brieﬂy, assuming that the loss function ℓ is convex and has Lipschitz continuous gradient, either minimization problem is convex and can be solved efﬁciently by proximal gradient methods, see e.g. (Beck & Teboulle, 2009; Combettes & Wajs, 2006).”
A Unified Convergence Analysis of Block Successive Minimization Methods for Nonsmooth Optimization (2012, neutral): “12) [13], [14]: x = proxγf1 } {{ } backward step (x − γ∇f2(x)) } {{ } forward step , (8.”
Image deblurring with Poisson data: from cells to galaxies (2009, methodology): “Since all conditions required for applying the forward–backward splitting proximal algorithm [41] are satisfied, this optimization method is used for the minimization of the resulting functional.”
The Convex Geometry of Linear Inverse Problems (2010, background): “sociated with the atomic norm, ΠA(x;µ) := argmin z 1 2kz −xk2 +µkzkA . (18) Here µis some positive parameter. Proximity operators have already been harnessed for fast algorithms involving the ℓ1 norm [34, 20, 21, 39, 71] and the nuclear norm [50, 12, 69] where these maps can be quickly computed in c”
Split Bregman Methods and Frame Based Image Restoration (2009, methodology): “This is essentially the approach in [31, 40]; see also [24, 35, 46, 49].”
Applied Iterative Methods (2007, methodology): “The second term in f(x) is differentiable, allowing us to apply the forwardbackward splitting method of Combettes and Wajs [68], to be discussed in a subsequent chapter.”
Linearized Bregman iterations for compressed sensing (2009, methodology): “The following convergence theorem is an immediate consequence of the main results in [ 15 ].”
A variable splitting augmented Lagrangian approach to linear spectral unmixing (2009, background): “The minimization with respect to z, in line 5, is, by definition the proximity operator of of the convex function II zllh [22], which is similar to the soft threshold function but applied just to the negative part of its argument: where F b (JLI + TA A) JLqt - g + 7A + dj ).”
Inverse problems in spaces of measures (2013, background): “This algorithm is known to converge in the sequence-space setting, in many cases at a q-linear rate [6, 12, 14].”