Proximal Splitting Methods in Signal Processing

Abstract

The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of inverse problems and, especially, in signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.

Blind image deblurring with unknown boundaries using the alternating direction method of multipliers

“…has a closed form expression for several choices of f [12]. Concerning line 4, it was show in [1,2] that the required inversion can be efficiently obtained in several cases of interest, namely using the FFT and/or fast wavelet/frame transforms.…”

Tractable Optimization in Machine Learning

“…This operator enjoys several other useful properties, and we encourage the reader to read the excellent survey [Combettes and Pesquet, 2010] for more information.…”

Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems

“…Now one can choose a primal-dual splitting algorithm as those proposed in [4,6,11,12,23] to solve this problem. One step in all these algorithms consists of the orthogonal projections onto the epigraphs of ϕ i for all i ∈ {1, .…”

“…We can apply the projection onto the epigraph of ϕ in combination with any primal-dual algorithm proposed in [4,6,11,12,23] or an alternating direction method of multipliers. For example, we use here the primal-dual hybrid gradient algorithm from [6,19] with an extrapolation (modification) of the dual variable which will be designated by PDHGMp.…”

“…As can be seen, the estimates τ I = n/2 and τ = n are good approximations of the true constraints D(f, Hu) and T (Hu) − T (f ) 2 2 . We computed a minimizer of our functional (3) with C = [0, +∞) n , the ℓ 2,1 -norm for Φ, and the discrete gradient operator L (q = 2n) by using Algorithm 1 with τ = n. We compared the result with the I-divergence constrained approach (11) and Algorithm 2 with τ I = n/2. The parameters σ and ρ appearing in PDHGMp (in this setting convergence is theoretically guaranteed for σρ < 1/9) are fitted such that the algorithms give (up to two digits after the comma) the same PSNR, MAE and TV-norm (times 10 5 or 10 6 ) after 1000 iterations as after 100000 iterations.…”