Abstract
We propose several variants of the primal–dual method due to Chambolle and Pock. Without requiring full strong convexity of the objective functions, our methods are accelerated on subspaces with strong convexity. This yields mixed rates, O(1{/}N^2) with respect to initialisation and O(1 / N) with respect to the dual sequence, and the residual part of the primal sequence. We demonstrate the efficacy of the proposed methods on image processing problems lacking strong convexity, such as total generalised variation denoising and total variation deblurring.
Highlights
Let G : X → R and F : Y → R be convex, proper, and lower semicontinuous functionals on Hilbert spaces X and Y, possibly infinite dimensional
G(x) = G0(Px) for a projection operator P to a subspace X0 ⊂ X, and strongly convex G0 : X0 → R? This kind of structure is common in many applications in image processing and data science, as we will more closely review in Sect
We demonstrate our algorithms on TGV2 denoising and total variation (TV) deblurring
Summary
Let G : X → R and F : Y → R be convex, proper, and lower semicontinuous functionals on Hilbert spaces X and Y , possibly infinite dimensional. This can under mild conditions on F (see, for example, [1,2]) be written with the help of the convex conjugate F∗ in the minimax form min max G(x) + K x, y − F∗(y). If either G or F∗ is strongly convex, the method can be accelerated to O(1/N 2) convergence rates of the iterates and an ergodic duality gap [3]. Under such partial strong convexity, can we obtain a method that would give an accelerated rate of convergence at least for Px?. Let us be given convex, proper, lower semicontinuous functionals G : X → R and F∗ : Y → R on Hilbert spaces X and Y , as well as a bounded linear operator K ∈ L(X ; Y ). If G is strongly convex with factor γ , we may use the acceleration scheme [3]
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