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Regularization plays a pivotal role when facing the challenge of solving ill-posed inverse problems, where the number of observations is smaller than the ambient dimension of the object to be estimated. A line of recent work has studied regularization models with various types of low-dimensional structures. In such settings, the general approach is to solve a regularized optimization problem, which combines a data fidelity term and some regularization penalty that promotes the assumed low-dimensional/simple structure. This paper provides a general framework to capture this low-dimensional structure using what we call partly smooth functions relative to a linear manifold. These are convex, non-negative, closed and finite-valued functions that will promote objects living on low-dimensional subspaces. This class of regularizers encompasses many popular examples such as the ℓ¹-norm, ℓ¹ - ℓ²-norm (group sparsity), as well as several others including the ℓ∞ norm. We also show that the set of partly smooth functions relative to a linear manifold is closed under addition and pre-composition by a linear operator, which allows us to cover mixed regularization, and the so-called analysis-type priors (e.g. total variation, fused Lasso, finite-valued polyhedral gauges). Our main result presents a unified sharp analysis of exact and robust recovery of the low-dimensional subspace model associated to the object to recover from partial measurements. This analysis is illustrated on a number of special and previously studied cases, and on an analysis of the performance of ℓ∞ regularization in a compressed sensing scenario. [ABSTRACT FROM AUTHOR]

We investigate the estimation of a weighted density taking the formg=w(F)f, wherefdenotes an unknown density,Fthe associated distribution function andwis a known non-negative weight. Such a class encompasses many examples, including those arising in order statistics or whengis related to the maximum or the minimum ofN(random or fixed) independent and identically distributed (i.i.d.) random variables. We here construct a new adaptive non-parametric estimator forgbased on a plug-in approach and the wavelets methodology. For a wide class of models, we show that it attains fast rates of convergence under therisk withp⩾ 2 over Besov balls. Our estimator is also simple to implement and fast. We also report an extensive simulation study to support our findings. [ABSTRACT FROM PUBLISHER]

This special issue is in honor of our outstanding colleague Mila Nikolova (1962-2018), Research Director at Centre National de la Recherche Scientifique (CNRS), Microsoft Fellow, Simons Fellow, and IEEE Senior Member. Mila was awarded with the Michel Monpetit Prize from the French Academy of Science in 2010 "for the originality and depth of her research in mathematical image processing and in solving certain inverse problems." She gave a systematic convergence rate analysis of half-quadratic minimization algorithms and achieved tightness results on the two-phase, piecewise constant Mumford Shah model for image segmentation, to mention only a few of her results. [Extracted from the article]

In this paper, we propose in a Hilbertian setting a second-order time-continuous dynamic system with fast convergence guarantees to solve structured convex minimization problems with an affine constraint. The system is associated with the augmented Lagrangian formulation of the minimization problem. The corresponding dynamics brings into play three general time-varying parameters, each with specific properties, and which are, respectively, associated with viscous damping, extrapolation and temporal scaling. By appropriately adjusting these parameters, we develop a Lyapunov analysis which provides fast convergence properties of the values and of the feasibility gap. These results will naturally pave the way for developing corresponding accelerated ADMM algorithms, obtained by temporal discretization. [ABSTRACT FROM AUTHOR]

In a Hilbert space setting, for convex optimization, we analyze the convergence rate of a class of first-order algorithms involving inertial features. They can be interpreted as discrete time versions of inertial dynamics involving both viscous and Hessian-driven dampings. The geometrical damping driven by the Hessian intervenes in the dynamics in the form ∇ 2 f (x (t)) x ˙ (t) . By treating this term as the time derivative of ∇ f (x (t)) , this gives, in discretized form, first-order algorithms in time and space. In addition to the convergence properties attached to Nesterov-type accelerated gradient methods, the algorithms thus obtained are new and show a rapid convergence towards zero of the gradients. On the basis of a regularization technique using the Moreau envelope, we extend these methods to non-smooth convex functions with extended real values. The introduction of time scale factors makes it possible to further accelerate these algorithms. We also report numerical results on structured problems to support our theoretical findings. [ABSTRACT FROM AUTHOR]

An introduction is presented on the recent developments in mathematical image analysis in which it notes the application of image processing, computer vision and computer graphics. It highlights several papers related to image analysis which include Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis, "Matrix Recipes for Hard Thresholding Methods" and "Poisson Noise Reduction with Non-Local PCA". The author also discusses the contribution of mathematician Vicent Caselles.

Over the past decades, operator splitting methods have become ubiquitous for non-smooth optimization owing to their simplicity and efficiency. In this paper, we consider the Forward–Douglas–Rachford splitting method and study both global and local convergence rates of this method. For the global rate, we establish a sublinear convergence rate in terms of a Bregman divergence suitably designed for the objective function. Moreover, when specializing to the Forward–Backward splitting, we prove a stronger convergence rate result for the objective function value. Then locally, based on the assumption that the non-smooth part of the optimization problem is partly smooth, we establish local linear convergence of the method. More precisely, we show that the sequence generated by Forward–Douglas–Rachford first (i) identifies a smooth manifold in a finite number of iteration and then (ii) enters a local linear convergence regime, which is for instance characterized in terms of the structure of the underlying active smooth manifold. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from applicative fields including, for instance, signal/image processing, inverse problems and machine learning. [ABSTRACT FROM AUTHOR]