On the Convergence Rate of Decomposable Submodular Function Minimization

Submodular functions describe a variety of discrete problems in machine learning, signal processing, and computer vision. However, minimizing submodular functions poses a number of algorithmic challenges. Recent work introduced an easy-to-use, parallelizable algorithm for minimizing submodular functions that decompose as the sum of "simple" submodular functions. Empirically, this algorithm performs extremely well, but no theoretical analysis was given. In this paper, we show that the algorithm converges linearly, and we provide upper and lower bounds on the rate of convergence. Our proof relies on the geometry of submodular polyhedra and draws on results from spectral graph theory.Comment: 17 pages, 3 figure

A General Analysis of the Convergence of ADMM

We provide a new proof of the linear convergence of the alternating direction method of multipliers (ADMM) when one of the objective terms is strongly convex. Our proof is based on a framework for analyzing optimization algorithms introduced in Lessard et al. (2014), reducing algorithm convergence to verifying the stability of a dynamical system. This approach generalizes a number of existing results and obviates any assumptions about specific choices of algorithm parameters. On a numerical example, we demonstrate that minimizing the derived bound on the convergence rate provides a practical approach to selecting algorithm parameters for particular ADMM instances. We complement our upper bound by constructing a nearly-matching lower bound on the worst-case rate of convergence.Comment: 10 pages, 6 figure

Atomistic Dynamics of the Richtmyer-Meshkov Instability in Cylindrical and Planar Geometries

We apply molecular dynamics (MD) simulations to study the evolution of the shock-driven Richtmyer-Meshkov instability (RMI) in the cylindrical and planar geometries. Compared to traditional hydrodynamic simulations, MD has a number of fundamental advantages: it accounts for strong gradients of the pressure and temperature, and captures accurately the heat and mass transfers at the early stage (shock passage) as well as the late stage (perturbation growth) of the instability evolution. MD has no hydrodynamic limitations for spatial resolution and thermodynamic quasi-equilibrium at atomic scale. We study the instability evolution for different perturbation modes and analyze the role of the vorticity production for RMI dynamics

The alternatively spliced fibronectin CS1 isoform regulates IL-17A levels and mechanical allodynia after peripheral nerve injury.

BackgroundMechanical pain hypersensitivity associated with physical trauma to peripheral nerve depends on T-helper (Th) cells expressing the algesic cytokine, interleukin (IL)-17A. Fibronectin (FN) isoform alternatively spliced within the IIICS region encoding the 25-residue-long connecting segment 1 (CS1) regulates T cell recruitment to the sites of inflammation. Herein, we analyzed the role of CS1-containing FN (FN-CS1) in IL-17A expression and pain after peripheral nerve damage.MethodsMass spectrometry, immunoblotting, and FN-CS1-specific immunofluorescence analyses were employed to examine FN expression after chronic constriction injury (CCI) in rat sciatic nerves. The acute intra-sciatic nerve injection of the synthetic CS1 peptide (a competitive inhibitor of the FN-CS1/α4 integrin binding) was used to elucidate the functional significance of FN-CS1 in mechanical and thermal pain hypersensitivity and IL-17A expression (by quantitative Taqman RT-PCR) after CCI. The CS1 peptide effects were analyzed in cultured primary Schwann cells, the major source of FN-CS1 in CCI nerves.ResultsFollowing CCI, FN expression in sciatic nerve increased with the dominant FN-CS1 deposition in endothelial cells, Schwann cells, and macrophages. Acute CS1 therapy attenuated mechanical allodynia (pain from innocuous stimulation) but not thermal hyperalgesia and reduced the levels of IL-17A expression in the injured nerve. CS1 peptide inhibited the LPS- or starvation-stimulated activation of the stress ERK/MAPK pathway in cultured Schwann cells.ConclusionsAfter physical trauma to the peripheral nerve, FN-CS1 contributes to mechanical pain hypersensitivity by increasing the number of IL-17A-expressing (presumably, Th17) cells. CS1 peptide therapy can be developed for pharmacological control of neuropathic pain

A system of relational syllogistic incorporating full Boolean reasoning

We present a system of relational syllogistic, based on classical propositional logic, having primitives of the following form: Some A are R-related to some B; Some A are R-related to all B; All A are R-related to some B; All A are R-related to all B. Such primitives formalize sentences from natural language like `All students read some textbooks'. Here A and B denote arbitrary sets (of objects), and R denotes an arbitrary binary relation between objects. The language of the logic contains only variables denoting sets, determining the class of set terms, and variables denoting binary relations between objects, determining the class of relational terms. Both classes of terms are closed under the standard Boolean operations. The set of relational terms is also closed under taking the converse of a relation. The results of the paper are the completeness theorem with respect to the intended semantics and the computational complexity of the satisfiability problem.Comment: Available at http://link.springer.com/article/10.1007/s10849-012-9165-