Gaussian Graph Research Articles

Covariate-Adjusted Precision Matrix Estimation with an Application in Genetical Genomics.

Motivated by analysis of genetical genomics data, we introduce a sparse high dimensional multivariate regression model for studying conditional independence relationships among a set of genes adjusting for possible genetic effects. The precision matrix in the model specifies a covariate-adjusted Gaussian graph, which presents the conditional dependence structure of gene expression after the confounding genetic effects on gene expression are taken into account. We present a covariate-adjusted precision matrix estimation method using a constrained ℓ1 minimization, which can be easily implemented by linear programming. Asymptotic convergence rates in various matrix norms and sign consistency are established for the estimators of the regression coefficients and the precision matrix, allowing both the number of genes and the number of the genetic variants to diverge. Simulation shows that the proposed method results in significant improvements in both precision matrix estimation and graphical structure selection when compared to the standard Gaussian graphical model assuming constant means. The proposed method is also applied to analyze a yeast genetical genomics data for the identification of the gene network among a set of genes in the mitogen-activated protein kinase pathway.

Gapped Two-Body Hamiltonian for Continuous-Variable Quantum Computation

We introduce a family of Hamiltonian systems for measurement-based quantum computation with continuous variables. The Hamiltonians (i) are quadratic, and therefore two body, (ii) are of short range, (iii) are frustration-free, and (iv) possess a constant energy gap proportional to the squared inverse of the squeezing. Their ground states are the celebrated Gaussian graph states, which are universal resources for quantum computation in the limit of infinite squeezing. These Hamiltonians constitute the basic ingredient for the adiabatic preparation of graph states and thus open new venues for the physical realization of continuous-variable quantum computing beyond the standard optical approaches. We characterize the correlations in these systems at thermal equilibrium. In particular, we prove that the correlations across any multipartition are contained exactly in its boundary, automatically yielding a correlation area law.

Quotients of Gaussian graphs and their application to perfect codes

A graph-based model of perfect two-dimensional codes is presented in this work. This model facilitates the study of the metric properties of the codes. Signal spaces are modeled by means of Cayley graphs defined over the Gaussian integers and denoted as Gaussian graphs. Codewords of perfect codes will be represented by vertices of a quotient graph of the Gaussian graph in which the signal space has been defined. It will be shown that any quotient graph of a Gaussian graph is indeed a Gaussian graph. This makes it possible to apply previously known properties of Gaussian graphs to the analysis of perfect codes. To illustrate the modeling power of this graph-based tool, perfect Lee codes will be analyzed in terms of Gaussian graphs and their quotients.

Learning topology of a labeled data set with the supervised generative Gaussian graph

Extracting the topology of a set of a labeled data is expected to provide important information in order to analyze the data or to design a better decision system. In this work, we propose to extend the generative Gaussian graph to supervised learning in order to extract the topology of labeled data sets. The graph obtained learns the intra-class and inter-class connectedness and also the manifold-overlapping of the different classes. We propose a way to vizualize these topological features. We apply it to analyze the well-known Iris database and the three-phase pipe flow database.

First order Gaussian graphs for efficient structure classification

First order random graphs as introduced by Wong are a promising tool for structure-based classification. Their complexity, however, hampers their practical application. We describe an extension to first order random graphs which uses continuous Gaussian distributions to model the densities of all random elements in a random graph. These First Order Gaussian Graphs (FOGGs) are shown to have several nice properties which allow for fast and efficient clustering and classification. Specifically, we show how the entropy of a FOGG may be computed directly from the Gaussian parameters of its random elements. This allows for fast and memoryless computation of the objective function used in the clustering procedure used for learning a graphical model of a class. We give a comparative evaluation between FOGGs and several traditional statistical classifiers. On our example problem, selected from the area of document analysis, our first order Gaussian graph classifier significantly outperforms statistical, feature-based classifiers. The FOGG classifier achieves a classification accuracy of approximately 98%, while the best statistical classifiers only manage approximately 91%.

Determination of creatinine-related urinary uracil excretion in children by high-performance liquid chromatography

Excretion rates of uracil and thymine in children ( n = 140) and circadian rhythms of urinary uracil excretion ( n = 9) were investigated by reversed-phase high-performance liquid chromatography with ultraviolet detection. Excretion values were related to urinary creatinine determined by Jaffé's method. Creatinine-related uracil excretion was not dependent on age or sex. The values seemed to be distributed according to a Gaussian graph in both school children and those in hospital. The intra-individual range was 1.32–23.70 mg uracil per g creatinine over a four-day period in one subject. Uracil excretion seems to be somewhat lower during the night.