Incremental proximal methods for large scale convex optimization

Laboratory for Information and Decision Systems Report LIDS-P-2847We consider the minimization of a sum∑m [over]i=1 fi (x) consisting of a large number of convex component functions fi . For this problem, incremental methods consisting of gradient or subgradient iterations applied to single components have proved very effective. We propose new incremental methods, consisting of proximal iterations applied to single components, as well as combinations of gradient, subgradient, and proximal iterations. We provide a convergence and rate of convergence analysis of a variety of such methods, including some that involve randomization in the selection of components.We also discuss applications in a few contexts, including signal processing and inference/machine learning.United States. Air Force Office of Scientific Research (grant FA9550-10-1-0412

Fractal Image Coding as Projections Onto Convex Sets

Abstract. We show how fractal image coding can be viewed and gen-eralized in terms of the method of projections onto convex sets (POCS). In this approach, the fractal code denes a set of spatial domain sim-ilarity constraints. We also show how such a reformulation in terms of POCS allows additional contraints to be imposed during fractal image decoding. Two applications are presented: image construction with an incomplete fractal code and image denoising.

Alternating Projections on Nontangential Manifolds

We consider sequences of points obtained by projecting a given point B=B (0) back and forth between two manifolds and , and give conditions guaranteeing that the sequence converges to a limit . Our motivation is the study of algorithms based on finding the limit of such sequences, which have proved useful in a number of areas. The intersection is typically a set with desirable properties but for which there is no efficient method for finding the closest point B (opt) in . Under appropriate conditions, we prove not only that the sequence of alternating projections converges, but that the limit point is fairly close to B (opt) , in a manner relative to the distance ayenB (0)-B (opt) ayen, thereby significantly improving earlier results in the field

Structure of metal-rich (001) surfaces of III-V compound semiconductors

The atomic structure of the group-III-rich surface of III-V semiconductor compounds has been under intense debate for many years, yet none of the models agrees with the experimental data available. Here we present a model for the three-dimensional structure of the (001)-c(8x2) reconstruction on InSb, InAs, and GaAs surfaces based on surface x-ray diffraction data that was analyzed by direct methods and subsequent least squares refinement. Contrary to common belief the main building blocks of the structure are not dimers on the surface but subsurface dimers in the second bilayer. This essential feature of the structure is accompanied by linear arrays of atoms on nonbulklike sites at the surface which, depending on the compounds, exhibit a certain degree of disorder. A tendency to group-III-dimer formation within these chains increases when descending the periodic table. We propose that all the c(8 x 2) reconstructions of III-V semiconductor surfaces contain the same essential building blocks