Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses

We investigate the relationship between the structure of a discrete graphical model and the support of the inverse of a generalized covariance matrix. We show that for certain graph structures, the support of the inverse covariance matrix of indicator variables on the vertices of a graph reflects the conditional independence structure of the graph. Our work extends results that have previously been established only in the context of multivariate Gaussian graphical models, thereby addressing an open question about the significance of the inverse covariance matrix of a non-Gaussian distribution. The proof exploits a combination of ideas from the geometry of exponential families, junction tree theory and convex analysis. These population-level results have various consequences for graph selection methods, both known and novel, including a novel method for structure estimation for missing or corrupted observations. We provide nonasymptotic guarantees for such methods and illustrate the sharpness of these predictions via simulations.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1162 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

High-dimensional regression with noisy and missing data: Provable guarantees with nonconvexity

Although the standard formulations of prediction problems involve fully-observed and noiseless data drawn in an i.i.d. manner, many applications involve noisy and/or missing data, possibly involving dependence, as well. We study these issues in the context of high-dimensional sparse linear regression, and propose novel estimators for the cases of noisy, missing and/or dependent data. Many standard approaches to noisy or missing data, such as those using the EM algorithm, lead to optimization problems that are inherently nonconvex, and it is difficult to establish theoretical guarantees on practical algorithms. While our approach also involves optimizing nonconvex programs, we are able to both analyze the statistical error associated with any global optimum, and more surprisingly, to prove that a simple algorithm based on projected gradient descent will converge in polynomial time to a small neighborhood of the set of all global minimizers. On the statistical side, we provide nonasymptotic bounds that hold with high probability for the cases of noisy, missing and/or dependent data. On the computational side, we prove that under the same types of conditions required for statistical consistency, the projected gradient descent algorithm is guaranteed to converge at a geometric rate to a near-global minimizer. We illustrate these theoretical predictions with simulations, showing close agreement with the predicted scalings.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1018 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Efficacy of a Dissolvable Strip with Calcium Sodium Phosphosilicate (NovaMin®) in Providing Rapid Dentine Hypersensitivity Relief

Objective To evaluate the efficacy of a dissolvable strip containing 15% w/w calcium sodium phosphosilicate (CSPS) (Novamin®) in providing rapid relief from dentine hypersensitivity (DH). Methods In this examiner-blind, proof-of-principle study, 120 healthy adults with DH were randomized 1:1 to the Test strip, professionally applied to facial surfaces of two selected teeth, or to No treatment. Sensitivity was assessed at baseline and 10 min, 2 h and 4 h post-application in response to evaporative (air) and tactile stimuli (measured by Schiff sensitivity scale/a numeric rating scale and tactile threshold, respectively). Change from baseline was analyzed by ANCOVA. Results At 10 min post-application, mean Schiff score change from baseline (primary endpoint) was statistically significant with the Test strip (−0.46; 95% confidence intervals [CI]: −0.563, −0.356; p \u3c 0.0001) but not with No treatment (−0.02; 95% CI: −0.119, 0.088; p = 0.7664). The between-treatment group difference favored the Test strip (difference: −0.44; 95% CI: −0.591, −0.297; p \u3c 0.0001). Similar improvements with the Test strip were reported for all other evaporative (air) and tactile sensitivity endpoints (p \u3c 0.0001 vs no-treatment) at all timepoints (10 min, 2 h, 4 h). Test strips were considered by most staff and participants slightly/moderately easy to apply (98%). Many participants rated the overall usage experience as “like moderately” (40%) or “like extremely” (20%). There were no treatment-related adverse events. Conclusion This new CSPS-based technology may provide a novel treatment option for rapid relief from DH (Clinicaltrials.gov NCT02937623). Clinical significance A dissolvable strip containing 15% w/w calcium sodium phosphosilicate (CSPS) demonstrated significantly greater dentine hypersensitivity reductions following a single application compared with no treatment. Strips were well-liked by participants and generally well tolerated. A strip containing CSPS, which dissolves within 10 min, may provide rapid relief from dentine hypersensitivity

Topological censorship in spacetimes compatible with Λ>0\Lambda > 0

Currently available topological censorship theorems are meant for gravitationally isolated black hole spacetimes with cosmological constant $\Lambda=0$ or $\Lambda<0$. Here, we prove a topological censorship theorem that is compatible with $\Lambda>0$ and which can be applied to whole universes containing possibly multiple collections of black holes. The main assumption in the theorem is that distinct black hole collections eventually become isolated from one another at late times, and the conclusion is that the regions near the various black hole collections have trivial fundamental group, in spite of there possibly being nontrivial topology in the universe.Comment: Comments are welcom

New governance approaches to environmental regulation: an example of the Code for Sustainable Homes (CSH)

Environmental policy in the United Kingdom (UK) is witnessing a shift from command-and-control approaches towards more innovation-orientated environmental governance arrangements. These governance approaches are required which create institutions which support actors within a domain for learning not only about policy options, but also about their own interests and preferences. The need for construction actors to understand, engage and influence this process is critical to establishing policies which support innovation that satisfies each constituent’s needs. This capacity is particularly salient in an era where the expanding raft of environmental regulation is ushering in system-wide innovation in the construction sector. In this paper, the Code for Sustainable Homes (the Code) in the UK is used to demonstrate the emergence and operation of these new governance arrangements. The Code sets out a significant innovation challenge for the house-building sector with, for example, a requirement that all new houses must be zero-carbon by 2016. Drawing upon boundary organisation theory, the journey from the Code as a government aspiration, to the Code as a catalyst for the formation of the Zero Carbon Hub, a new institution, is traced and discussed. The case study reveals that the ZCH has demonstrated boundary organisation properties in its ability to be flexible to the needs and constraints of its constituent actors, yet robust enough to maintain and promote a common identity across regulation and industry boundaries

Simulating Focused Ultrasound Transducers using Discrete Sources on Regular Cartesian Grids

Accurately representing the behaviour of acoustic sources is an important part of ultrasound simulation. This is particularly challenging in ultrasound therapy where multielement arrays are often used. Typically, sources are defined as a boundary condition over a 2D plane within the computational model. However, this approach can become difficult to apply to arrays with multiple elements distributed over a non-planar surface. In this work, a grid-based discrete source model for single and multi-element bowl-shaped transducers is developed to model the source geometry explicitly within a regular Cartesian grid. For each element, the source model is defined as a symmetric, simply-connected surface with a single grid point thickness. Simulations using the source model with the opensource k-Wave toolbox are validated using the Rayleigh integral, O'Neil's solution, and experimental measurements of a focused bowl transducer under both quasi continuous wave and pulsed excitation. Close agreement is shown between the discrete bowl model and the axial pressure predicted by O'Neil's solution for a uniform curved radiator, even at very low grid resolutions. Excellent agreement is also shown between the discrete bowl model and experimental measurements. To accurately reproduce the near-field pressure measured experimentally, it is necessary to derive the drive signal at each grid point of the bowl model directly using holography. However, good agreement is also obtained in the focal region using uniformly radiating monopole sources distributed over the bowl surface. This allows the response of multi-element transducers to be modelled, even where measurement of an input plane is not possible

Regularized M-estimators With Nonconvexity: Statistical and Algorithmic Theory for Local Optima

We establish theoretical results concerning all local optima of various regularized M-estimators, where both loss and penalty functions are allowed to be nonconvex. Our results show that as long as the loss function satisfies restricted strong convexity and the penalty function satisfies suitable regularity conditions, any local optimum of the composite objective function lies within statistical precision of the true parameter vector. Our theory covers a broad class of nonconvex objective functions, including corrected versions of the Lasso for errors-in-variables linear models; regression in generalized linear models using nonconvex regularizers such as SCAD and MCP; and graph and inverse covariance matrix estimation. On the optimization side, we show that a simple adaptation of composite gradient descent may be used to compute a global optimum up to the statistical precision εstat in log(1/εstat) iterations, which is the fastest possible rate of any first-order method. We provide a variety of simulations to illustrate the sharpness of our theoretical predictions