Circular law for random discrete matrices of given row sum

Let $M_n$ be a random matrix of size $n\times n$ and let $\lambda_1,...,\lambda_n$ be the eigenvalues of $M_n$. The empirical spectral distribution $\mu_{M_n}$ of $M_n$ is defined as \mu_{M_n}(s,t)=\frac{1}{n}# \{k\le n, \Re(\lambda_k)\le s; \Im(\lambda_k)\le t\}. The circular law theorem in random matrix theory asserts that if the entries of $M_n$ are i.i.d. copies of a random variable with mean zero and variance $\sigma^2$, then the empirical spectral distribution of the normalized matrix $\frac{1}{\sigma\sqrt{n}}M_n$ of $M_n$ converges almost surely to the uniform distribution \mu_\cir over the unit disk as $n$ tends to infinity. In this paper we show that the empirical spectral distribution of the normalized matrix of $M_n$, a random matrix whose rows are independent random $(-1,1)$ vectors of given row-sum $s$ with some fixed integer $s$ satisfying $|s|\le (1-o(1))n$, also obeys the circular law. The key ingredient is a new polynomial estimate on the least singular value of $M_n$

Random matrices: Law of the determinant

Let $A_n$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\det A_n|$ satisfies a central limit theorem. More precisely, \begin{eqnarray*}\sup_{x\in {\mathbf {R}}}\biggl|{\mathbf {P}}\biggl(\frac{\log(|\det A_n|)-({1}/{2})\log (n-1)!}{\sqrt{({1}/{2})\log n}}\le x\biggr)-{\mathbf {P}}\bigl(\mathbf {N}(0,1)\le x\bigr)\biggr|\\\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}Comment: Published in at http://dx.doi.org/10.1214/12-AOP791 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The development and effectiveness of an osteoporosis prevention education intervention

Title from PDF of title page (University of Missouri--Columbia, viewed on November 6, 2012).The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file.Dissertation advisor: Dr. Alexander C. WaigandtVita.Ph. D. University of Missouri-Columbia 2011."May 2011"Osteoporosis prevention education interventions intended to increase the osteoporosis preventive behaviors of weight-bearing physical activity and calcium consumption in young individuals have been found to be ineffective. An osteoporosis prevention education intervention was developed and modeled after an effective health threat prevention education intervention based on the health belief model, which emphasized the health threat's visible severity and proximal time of onset. To test its effectiveness, it was experimentally researched in a sample of 109 college women who were students in an undergraduate health education course, and were randomly assigned to either the treatment or a control group to receive the osteoporosis prevention education intervention or a stress management intervention, respectively. The treatment group did not positively alter their osteoporosis health beliefs or increase self-reported weight-bearing physical activity and calcium consumption compared to the control group. And the control group who received the stress management intervention showed a significant increase in health motivation while the treatment group who received the osteoporosis prevention education intervention did not. A probable reason is that due to the distal time of onset of osteoporosis, young individuals may not be concerned with modifying their behaviors to prevent the disease. Recommendations for future research and effective ways to promote weight-bearing physical activity and calcium consumption are provided.Includes bibliographical reference

Bayesian Nonparametric Multilevel Clustering with Group-Level Contexts

We present a Bayesian nonparametric framework for multilevel clustering which utilizes group-level context information to simultaneously discover low-dimensional structures of the group contents and partitions groups into clusters. Using the Dirichlet process as the building block, our model constructs a product base-measure with a nested structure to accommodate content and context observations at multiple levels. The proposed model possesses properties that link the nested Dirichlet processes (nDP) and the Dirichlet process mixture models (DPM) in an interesting way: integrating out all contents results in the DPM over contexts, whereas integrating out group-specific contexts results in the nDP mixture over content variables. We provide a Polya-urn view of the model and an efficient collapsed Gibbs inference procedure. Extensive experiments on real-world datasets demonstrate the advantage of utilizing context information via our model in both text and image domains.Comment: Full version of ICML 201

System Energy-Efficient Hybrid Beamforming for mmWave Multi-user Systems

This paper develops energy-efficient hybrid beamforming designs for mmWave multi-user systems where analog precoding is realized by switches and phase shifters such that radio frequency (RF) chain to transmit antenna connections can be switched off for energy saving. By explicitly considering the effect of each connection on the required power for baseband and RF signal processing, we describe the total power consumption in a sparsity form of the analog precoding matrix. However, these sparsity terms and sparsity-modulus constraints of the analog precoding make the system energy-efficiency maximization problem non-convex and challenging to solve. To tackle this problem, we first transform it into a subtractive-form weighted sum rate and power problem. A compressed sensing-based re-weighted quadratic-form relaxation method is employed to deal with the sparsity parts and the sparsity-modulus constraints. We then exploit alternating minimization of the mean-squared error to solve the equivalent problem where the digital precoding vectors and the analog precoding matrix are updated sequentially. The energy efficiency upper bound and a heuristic algorithm are also examined for comparison purposes. Numerical results confirm the superior performances of the proposed algorithm over benchmark energy-efficiency hybrid precoding algorithms and heuristic ones.Comment: submitted to TGC