Optimal Estimation of Slope Vector in High-dimensional Linear Transformation Model

In a linear transformation model, there exists an unknown monotone nonlinear transformation function such that the transformed response variable and the predictor variables satisfy a linear regression model. In this paper, we present CENet, a new method for estimating the slope vector and simultaneously performing variable selection in the high-dimensional sparse linear transformation model. CENet is the solution to a convex optimization problem and can be computed efficiently from an algorithm with guaranteed convergence to the global optimum. We show that under a pairwise elliptical distribution assumption on each predictor-transformed-response pair and some regularity conditions, CENet attains the same optimal rate of convergence as the best regression method in the high-dimensional sparse linear regression model. To the best of our limited knowledge, this is the first such result in the literature. We demonstrate the empirical performance of CENet on both simulated and real datasets. We also discuss the connection of CENet with some nonlinear regression/multivariate methods proposed in the literature.Comment: 25 pages, 7 figures, 1 tabl

A New Equation of State for Dark Energy Model

A new parameterization for the dark energy equation of state(EoS) is proposed and some of its cosmological consequences are also investigated. This new parameterization is the modification of Efstathiou' dark energy EoS parameterization. $w (z)$ is a well behaved function for $z\gg1$ and has same behavior in $z$ at low redshifts with Efstathiou' parameterization. In this parameterization there are two free parameter $w_0$ and $w_a$. We discuss the constraints on this model's parameters from current observational data. The best fit values of the cosmological parameters with $1\sigma$ confidence-level regions are: $\Omega_m=0.2735^{+0.0171}_{-0.0163}$, $w_0=-1.0537^{+0.1432}_{-0.1511}$ and $w_a=0.2738^{+0.8018}_{-0.8288}$.Comment: 5 pages, 3 figures.some statement is change

Possible direct measurement of the expansion rate of the universe

A new method is proposed for directly measuring the expansion rate of the universe through very precise measurement of the fluence of extremely stable sources. The method is based on the definition of the luminosity distance and its change along the time due to the cosmic expansion. It is argued that galaxies may be chosen as the targets of the observation to perform the measurement. We show that, by simultaneously increasing the observation time and physically adding the fluences from different galaxies, the requirement on the relative precision of the detector for an observation of 1 second on a single galaxy can be relaxed to $10^{-5}$. Benefiting from the abundance of galaxies in the universe, the method may be quite promising.Comment: 6 pages, 2 figures; added discussion about how to perform the measuremen

Simple and Effective Dynamic Provisioning for Power-Proportional Data Centers

Energy consumption represents a significant cost in data center operation. A large fraction of the energy, however, is used to power idle servers when the workload is low. Dynamic provisioning techniques aim at saving this portion of the energy, by turning off unnecessary servers. In this paper, we explore how much performance gain can knowing future workload information brings to dynamic provisioning. In particular, we study the dynamic provisioning problem under the cost model that a running server consumes a fixed amount energy per unit time, and develop online solutions with and without future workload information available. We first reveal an elegant structure of the off-line dynamic provisioning problem, which allows us to characterize and achieve the optimal solution in a {}"divide-and-conquer" manner. We then exploit this insight to design three online algorithms with competitive ratios $2-\alpha$, $(e-\alpha)/(e-1)\approx1.58-\alpha/(e-1)$ and $e/(e-1+\alpha)$, respectively, where $0\leq\alpha\leq1$ is the fraction of a critical window in which future workload information is available. A fundamental observation is that \emph{future workload information beyond the critical window will not} \emph{improve dynamic provisioning performance}. Our algorithms are decentralized and are simple to implement. We demonstrate their effectiveness in simulations using real-world traces. We also compare their performance with state-of-the-art solutions

Inequalities between the Chern numbers of a singular fiber in a family of algebraic curves

In a family of curves, the Chern numbers of a singular fiber are the local contributions to the Chern numbers of the total space. We will give some inequalities between the Chern numbers of a singular fiber as well as their lower and upper bounds. We introduce the dual fiber of a singular fiber, and prove a duality theorem. As an application, we will classify singular fibers with large or small Chern numbers.Comment: 23 page

High-dimensional robust precision matrix estimation: Cellwise corruption under ϵ\epsilon-contamination

We analyze the statistical consistency of robust estimators for precision matrices in high dimensions. We focus on a contamination mechanism acting cellwise on the data matrix. The estimators we analyze are formed by plugging appropriately chosen robust covariance matrix estimators into the graphical Lasso and CLIME. Such estimators were recently proposed in the robust statistics literature, but only analyzed mathematically from the point of view of the breakdown point. This paper provides complementary high-dimensional error bounds for the precision matrix estimators that reveal the interplay between the dimensionality of the problem and the degree of contamination permitted in the observed distribution. We also show that although the graphical Lasso and CLIME estimators perform equally well from the point of view of statistical consistency, the breakdown property of the graphical Lasso is superior to that of CLIME. We discuss implications of our work for problems involving graphical model estimation when the uncontaminated data follow a multivariate normal distribution, and the goal is to estimate the support of the population-level precision matrix. Our error bounds do not make any assumptions about the the contaminating distribution and allow for a nonvanishing fraction of cellwise contamination.Comment: 52 pages including appendi

A counterexample to the Nelson-Seiberg theorem

We present a counterexample to the Nelson-Seiberg theorem and its extensions. The model has 4 chiral fields, including one R-charge 2 field and no R-charge 0 filed. Giving generic values of coefficients in the renormalizable superpotential, there is a supersymmetric vacuum with one complex dimensional degeneracy. The superpotential equals zero and the R-symmetry is broken everywhere on the degenerated vacuum. The existence of such a vacuum disagrees with both the original Nelson-Seiberg theorem and its extensions, and can be viewed as the consequence of a non-generic R-charge assignment. Such counterexamples may introduce error to the field counting method for surveying the string landscape, and are worth further investigations.Comment: 7 pages; v2: discussion on non-generic R-charges added, new references updated; v3: minor changes to notation and convention in formulas, JHEP pre-publication versio

Relative Error of Scaled Poisson Approximation via Stein's Method

We study the accuracy of a scaled Poisson approximation to the weighted sum of independent Poisson random variables, focusing on in particular the relative error of the tail distribution. We establish a moderate deviation bound on the approximation error using a modified Stein-Chen method. Numerical experiments are also presented to demonstrate the quality of the approximation

Optimal Estimation of A Quadratic Functional and Detection of Simultaneous Signals

Motivated by applications in genomics, this paper studies the problem of optimal estimation of a quadratic functional of two normal mean vectors, $Q(\mu, \theta) = \frac{1}{n}\sum_{i=1}^n\mu_i^2\theta_i^2$, with a particular focus on the case where both mean vectors are sparse. We propose optimal estimators of $Q(\mu, \theta)$ for different regimes and establish the minimax rates of convergence over a family of parameter spaces. The optimal rates exhibit interesting phase transitions in this family. The simultaneous signal detection problem is also considered under the minimax framework. It is shown that the proposed estimators for $Q(\mu, \theta)$ naturally lead to optimal testing procedures.Comment: 41 pages including appendix, 3 figures, 3 table

Standardization, Distance, Host Galaxy Extinction of Type Ia Supernova and Hubble Diagram from the Flux Ratio Method

In this paper we generalize the flux ratio method Bailey et al. (2009) to the case of two luminosity indicators and search the optimal luminosity-flux ratio relations on a set of spectra whose phases are around not only the date of bright light but also other time. With these relations, a new method is proposed to constrain the host galaxy extinction of SN Ia and its distance. It is first applied to the low redshift supernovas and then to the high redshift ones. The results of the low redshift supernovas indicate that the flux ratio method can indeed give well constraint on the host galaxy extinction parameter E(B-V), but weaker constraints on R_{V}. The high redshift supernova spectra are processed by the same method as the low redshift ones besides some differences due to their high redshift. Among 16 high redshift supernovas, 15 are fitted very well except 03D1gt. Based on these distances, Hubble diagram is drew and the contents of the Universe are analyzed. It supports an acceleration behavior in the late Universe. Therefore, the flux ratio method can give constraints on the host galaxy extinction and supernova distance independently. We believe, through further studies, it may provide a precise tool to probe the acceleration of the Universe than before.Comment: 33 pages, 9 figures and 6 table