Many tasks in imaging can be modeled via the minimization of a nonconvex composite function. A popular class of algorithms for solving such problems are majorization-minimization techniques which iteratively approximate the composite nonconvex function by a majorizing function that is easy to minimize. Most techniques, e.g. gradient descent, utilize convex majorizers in order guarantee that the majorizer is easy to minimize. In our work we consider a natural class of nonconvex majorizers for these functions, and show that these majorizers are still sufficient for a provably convergent optimization scheme under quite general assumptions. Numerical results illustrate that by applying this scheme, one can often obtain superior local optima compared to previous majorization-minimization methods, when the nonconvex majorizers are solved to global optimality. Finally, we illustrate the behavior of our algorithm for depth super-resolution from raw time-of-flight data.
The most limiting assumption in these first-order methods is the Lipschitz continuity of the gradient of @math , the first part of the objective function. This class of problems was recently extended in @cite_29 to L-smooth adaptable function @cite_4 , these functions are not necessarily convex or L-smooth, only a smooth and convex 'kernel' function @math must exist, so that @math is convex for some @math . The previously mentioned methods can be extended to a descent relative to these kernel functions. Defining the Bregman distance of @math as @math , @cite_29 's majorizer can be written as They show that the sequence of iterates generated by this type of majorizer converges for appropriate @math and conditions to @math and @math . This is a quite satisfying result as this formulation incorporates logarithmic or quadric data terms and vector-valued step sizes.
@cite_29 @cite_4
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We focus on nonconvex and nonsmooth minimization problems with a composite objective, where the differentiable part of the objective is freed from the usual and restrictive global Lipschitz gradient continuity assumption. This longstanding smoothness restriction is pervasive in first order methods (FOM), and was recently circumvent for convex composite optimization by Bauschke, Bolte and Teboulle, through a simple and elegant framework which captures, all at once, the geometry of the function and of the feasible set. Building on this work, we tackle genuine nonconvex problems. We first complement and extend their approach to derive a full extended descent lemma by introducing the notion of smooth adaptable functions. We then consider a Bregman-based proximal gradient methods for the nonconvex composite model with smooth adaptable functions, which is proven to globally converge to a critical point under natural assumptions on the problem's data. To illustrate the power and potential of our general framework and results, we consider a broad class of quadratic inverse problems with sparsity constraints which arises in many fundamental applications, and we apply our approach to derive new globally convergent schemes for this class. The proximal gradient and its variants is one of the most attractive first-order algorithm for minimizing the sum of two convex functions, with one being nonsmooth. However, it requires the differentiable part of the objective to have a Lipschitz continuous gradient, thus precluding its use in many applications. In this paper we introduce a framework which allows to circumvent the intricate question of Lipschitz continuity of gradients by using an elegant and easy to check convexity condition which captures the geometry of the constraints. This condition translates into a new descent lemma which in turn leads to a natural derivation of the proximal-gradient scheme with Bregman distances. We then identify a new notion of asymmetry measure for Bregman distances, which is central in determining the relevant step-size. These novelties allow to prove a global sublinear rate of convergence, and as a by-product, global pointwise convergence is obtained. This provides a new path to a broad spectrum of problems arising in key applications which were, until now, considered as out of reach via proximal gradient methods. We illustrate this potential by showing how our results can be applied to build new and simple schemes for Poisson inverse problems.