Templates for Convex Cone Problems with Applications to Sparse Signal Recovery

This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal first-order method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the total-variation norm, ||Wx||_1 where W is arbitrary, or a combination thereof. In addition, the paper also introduces a number of technical contributions such as a novel continuation scheme, a novel approach for controlling the step size, and some new results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with state-of-the-art methods for solving intensively studied problems such as the LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient large-scale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms.Comment: The TFOCS software is available at http://tfocs.stanford.edu This version has updated reference

LHC Missing-Transverse-Energy Constraints on Models with Universal Extra Dimensions

We consider the performance of the ATLAS and CMS searches for events with missing transverse energy, which were originally motivated by supersymmetry, in constraining extensions of the Standard Model based on extra dimensions, in which the mass differences between recurrences at the same level are generically smaller than the mass hierarchies in typical supersymmetric models. We consider first a toy model with pair-production of a single vector-like quark U1 decaying into a spin-zero stable particle A1 and jet, exploring the sensitivity of the CMS alphaT and ATLAS meff analysis to U1 mass and the U1-A1 mass difference. For this purpose we use versions of the Delphes generic detector simulation with CMS and ATLAS cards, which have been shown to reproduce the published results of CMS and ATLAS searches for supersymmetry. We then explore the sensitivity of these searches to a specific model with two universal extra dimensions, whose signal is dominated by the pair production of quark recurrences, including searches with leptons. We find that the LHC searches have greater sensitivity to this more realistic model, due partly to the contributions of signatures with leptons, and partly to events with large missing transverse energy generated by the decays of higher-level Kaluza-Klein recurrences. We find that the CMS alphaT analysis with ~5/fb of data at 7 TeV excludes a recurrence scale of 600 GeV at a confidence level above 99%, increasing to 99.9% when combined with the CMS single-lepton search, whereas a recurrence scale of 700 GeV is disfavoured at the 72% confidence level.Comment: 29 pages, 11 figures, 5 tables, references adde