We provide a unified framework for the high-dimensional analysis of“superposition-structured” or “dirty” statistical models: where the model param-eters are a superposition of structurally constrained parameters. We allow for anynumber and types of structures, and any statistical model. We consider the gen-eral class of M -estimators that minimize the sum of any loss function, a...

In this thesis, we have two distinct but related subjects: optimal control and nonlinear programming. In the first part of this thesis, we prove that the value function, propagated from initial or terminal costs, and constraints, in the form of a differential equation, satisfy a subgradient form of the Hamilton-Jacobi equation in which the Hamiltonian is measurable with respect to time. In the ...

Smoothing methods have become part of the standard tool set for the study and solution of nondifferentiable and constrained optimization problems as well as a range of other variational and equilibrium problems. In this note we synthesize and extend recent results due to Beck and Teboulle on infimal convolution smoothing for convex functions with those of X. Chen on gradient consistency for non...

In this paper, a primal-dual algorithm for TV-type image restoration is analyzed and tested. Analytically it turns out that employing a global L-regularization, with s > 1, in the dual problem results in a local smoothing of the TV-regularization term in the primal problem. The local smoothing can alternatively be obtained as the infimal convolution of the `r-norm, with r−1 + s−1 = 1, and a smo...

We consider whether minimizers for total variation regularization of linear inverse problems belong to $L^\infty$ even if the measured data does not. present a simple proof boundedness minimizer fixed parameter, and derive existence uniform bounds sufficiently small noise under source condition adequate priori parameter choices. To show that such result cannot be expected every fidelity term di...

A theory for set-valued functions is developed, which are translative with respect to a linear operator. It shown that such cover wide range of applications, from projections in Hilbert spaces, quantiles vector-valued random variables, scalar or risk measures finance defaultable nondefaultable securities. Primal, dual, and representation results given, among them an infimal convolution represen...

The infimal Heegaard gradient of a 3-manifold was defined and studied by Marc Lackenby in an approach toward the well-known virtually Haken conjecture. As instructive examples, we consider Seifert fibered 3manifolds, and show that a compact orientable Seifert fibered 3-manifold has zero infimal Heegaard gradient if and only if it virtually fibers over the circle or over a surface other than the...

In this paper we study the problem of restoring images affected by mixed Poisson-Square Cauchy noise. A multi-convex variational model is derived integrating infimal convolution likelihood into a process joint maximum posteriori estimation, where modified four-directional fractional-order total variation united with first order to characterize image prior. To solve proposed numerically, block c...

We introduce and study a mathematical framework for broad class of regularization functionals ill-posed inverse problems: Regularization Graphs. graphs allow to construct using as building blocks linear operators convex functionals, assembled by means that can be seen generalizations classical infimal convolution operators. This exhaustively covers existing approaches it is flexible enough craf...