Gaussian Graphs Research Articles

Frequentist model averaging for undirected Gaussian graphical models.

Advances in information technologies have made network data increasingly frequent in a spectrum of big data applications, which is often explored by probabilistic graphical models. To precisely estimate the precision matrix, we propose an optimal model averaging estimator for Gaussian graphs. We prove that the proposed estimator is asymptotically optimal when candidate models are misspecified. The consistency and the asymptotic distribution of model averaging estimator, and the weight convergence are also studied when at least one correct model is included in the candidate set. Furthermore, numerical simulations and a real data analysis on yeast genetic data are conducted to illustrate that the proposed method ispromising.

On Symmetry of Complete Graphs over Quadratic and Cubic Residues

In this study, we investigate two graphs, one of which has units of a ring Z n as vertices (or nodes) and an edge will be built between two vertices u and v if and only if u 3 ≡ v 3 mod n . This graph will be termed as cubic residue graph. While the other is called Gaussian quadratic residue graph whose vertices are the elements of a Gaussian ring Z n i of the form α = a + i b , β = c + i d , where a , b , c , d are the units of Z n . Two vertices α and β are adjacent to each other if and only if α 2 ≡ β 2 mod n . In this piece of work, we characterize cubic and Gaussian quadratic residue graphs for each positive integer n in terms of complete graphs.

Near-Optimal Graph Signal Sampling by Pareto Optimization.

In this paper, we focus on the bandlimited graph signal sampling problem. To sample graph signals, we need to find small-sized subset of nodes with the minimal optimal reconstruction error. We formulate this problem as a subset selection problem, and propose an efficient Pareto Optimization for Graph Signal Sampling (POGSS) algorithm. Since the evaluation of the objective function is very time-consuming, a novel acceleration algorithm is proposed in this paper as well, which accelerates the evaluation of any solution. Theoretical analysis shows that POGSS finds the desired solution in quadratic time while guaranteeing nearly the best known approximation bound. Empirical studies on both Erdos-Renyi graphs and Gaussian graphs demonstrate that our method outperforms the state-of-the-art greedy algorithms.

Convergence rate analysis of Gaussian belief propagation for Markov networks

Gaussian belief propagation algorithm ( GaBP ) is one of the most important distributed algorithms in signal processing and statistical learning involving Markov networks. It is well known that the algorithm correctly computes marginal density functions from a high dimensional joint density function over a Markov network in a finite number of iterations when the underlying Gaussian graph is acyclic. It is also known more recently that the algorithm produces correct marginal means asymptotically for cyclic Gaussian graphs under the condition of walk summability ( or generalised diagonal dominance ) . This paper extends this convergence result further by showing that the convergence is exponential under the generalised diagonal dominance condition, and provides a simple bound for the convergence rate. Our results are derived by combining the known walk summability approach for asymptotic convergence analysis with the control systems approach for stability analysis.

Benchmarking the contention aware nature inspired metaheuristic task scheduling algorithms

In this paper, we consider the contention aware task scheduling problem on a grid topology of processors. By contention awareness, we mean that simultaneous communication on a link has to be serialized. To solve this problem, we propose several nature inspired metaheuristic algorithms: Simulated Annealing (SA), Genetic Algorithm (GA), Differential Evolution (DE), Particle Swarm Optimization (PSO), Bat Algorithm (BA), Cuckoo Search (CS), and Firefly Algorithm (FA). We perform benchmark evaluation of these algorithms for the Normalized Schedule Length (NSL) parameter. The benchmark task graphs that we consider are: random task graphs, peer set task graphs, systolic array task graphs, Gaussian elimination task graphs, divide and conquer task graphs, and fast Fourier transform task graphs.

Tuning-Free Heterogeneous Inference in Massive Networks

Heterogeneity is often natural in many contemporary applications involving massive data. While posing new challenges to effective learning, it can play a crucial role in powering meaningful scientific discoveries through the integration of information among subpopulations of interest. In this article, we exploit multiple networks with Gaussian graphs to encode the connectivity patterns of a large number of features on the subpopulations. To uncover the underlying sparsity structures across subpopulations, we suggest a framework of large-scale tuning-free heterogeneous inference, where the number of networks is allowed to diverge. In particular, two new tests, the chi-based and the linear functional-based tests, are introduced and their asymptotic null distributions are established. Under mild regularity conditions, we establish that both tests are optimal in achieving the testable region boundary and the sample size requirement for the latter test is minimal. Both theoretical guarantees and the tuning-free property stem from efficient multiple-network estimation by our newly suggested heterogeneous group square-root Lasso for high-dimensional multi-response regression with heterogeneous noises. To solve this convex program, we further introduce a scalable algorithm that enjoys provable convergence to the global optimum. Both computational and theoretical advantages are elucidated through simulation and real data examples. Supplementary materials for this article are available online.

Chernoff information between Gaussian trees

In this paper, we deal with Gaussian graphical classifiaction problem. We aim to provide a systematic study of the relationship between Chernoff information and topological, as well as algebraic properties of the corresponding Gaussian graphs for the underlying graphical testing problems. We first show the relationship between Chernoff information and generalized eigenvalues of the associated covariance matrices. It is then proved that Chernoff information between two Gaussian trees sharing certain local subtree structures can be transformed into that of two smaller trees. In this way, we provide a sequence of equivalent Gaussian tree pairs in terms of Chernoff Information. Under our proposed grafting operations, bottleneck Gaussian trees, namely, Gaussian trees connected by one such operation, can thus be simplified into two 3-node Gaussian trees. Thereafter, we provide a thorough study about how Chernoff information changes when small differences are accumulated into bigger ones via concatenated grafting operations, as well as partial ordering. In the end, we propose an optimal linear dimension reduction method based on generalized eigenvalues for the purpose of classification, which is proved to achieve maximum Chernoff information among all linear transformations.

Tuning-Free Heterogeneity Pursuit in Massive Networks

Heterogeneity is often natural in many contemporary applications involving massive data. While posing new challenges to effective learning, it can play a crucial role in powering meaningful scientific discoveries through the understanding of important differences among subpopulations of interest. In this paper, we exploit multiple networks with Gaussian graphs to encode the connectivity patterns of a large number of features on the subpopulations. To uncover the heterogeneity of these structures across subpopulations, we suggest a new framework of tuning-free heterogeneity pursuit (THP) via large-scale inference, where the number of networks is allowed to diverge. In particular, two new tests, the chi-based test and the linear functional-based test, are introduced and their asymptotic null distributions are established. Under mild regularity conditions, we establish that both tests are optimal in achieving the testable region boundary and the sample size requirement for the latter test is minimal. Both theoretical guarantees and the tuning-free feature stem from efficient multiple-network estimation by our newly suggested approach of heterogeneous group square-root Lasso (HGSL) for high-dimensional multi-response regression with heterogeneous noises. To solve this convex program, we further introduce a tuning-free algorithm that is scalable and enjoys provable convergence to the global optimum. Both computational and theoretical advantages of our procedure are elucidated through simulation and real data examples.

Nonparametric spatial models for clustered ordered periodontal data.

Clinical attachment level (CAL) is regarded as the most popular measure to assess periodontal disease (PD). These probed tooth-site level measures are usually rounded and recorded as whole numbers (in mm) producing clustered (site measures within a mouth) error-prone ordinal responses representing some ordering of the underlying PD progression. In addition, it is hypothesized that PD progression can be spatially-referenced, i.e., proximal tooth-sites share similar PD status in comparison to sites that are distantly located. In this paper, we develop a Bayesian multivariate probit framework for these ordinal responses where the cut-point parameters linking the observed ordinal CAL levels to the latent underlying disease process can be fixed in advance. The latent spatial association characterizing conditional independence under Gaussian graphs is introduced via a nonparametric Bayesian approach motivated by the probit stick-breaking process, where the components of the stick-breaking weights follows a multivariate Gaussian density with the precision matrix distributed as G-Wishart. This yields a computationally simple, yet robust and flexible framework to capture the latent disease status leading to a natural clustering of tooth-sites and subjects with similar PD status (beyond spatial clustering), and improved parameter estimation through sharing of information. Both simulation studies and application to a motivating PD dataset reveal the advantages of considering this flexible nonparametric ordinal framework over other alternatives.

Multiple Gaussian graphical estimation with jointly sparse penalty

In this paper, we consider estimating multiple Gaussian graphs with a similar sparsity structure. Most related solving methods, such as GGL (Group graphical lasso) and FMGL (Fused multiple graphical lasso), focus on the information of the edge values, and pay few attention to the estimation based on structure information. We construct a jointly sparse penalty to encourage graphs to share a similar sparsity structure by utilizing information of the common structure across the graphs. The new objective function is neither convex nor differentiable. Combining block coordinate descent and majorization–minimization strategies, we derive a new re-weighed algorithm to solve the problem by transforming the subproblems in every iteration into convex ones. Experimental results show that the proposed algorithm outperforms FMGL and GGL when the sparsity structure is similar but the edge values are not.

퍼지신경망을 사용한 네이브 베이지안 분류기의 분산 그래프 학습

Abstract Naive Bayesian classifiers are a powerful and well-known type of classifiers that can be easily induced from a dataset of sample cases. However, the strong conditional independence assumptions can sometimes lead to weak classification performance. Normally, naive Bayesian classifiers use Gaussian distributions to handle continuous attributes and to represent the likelihood of the features conditioned on the classes. The probability density of attributes, however, is not always well fitted by a Gaussian distribution. Another eminent type of classifier is the neuro-fuzzy classifier, which can learn fuzzy rules and fuzzy sets using supervised learning. Since there are specific structural similarities between a neuro-fuzzy classifier and a naive Bayesian classifier, the purpose of this study is to apply learning distribution graphs constructed by a neuro-fuzzy network to naive Bayesian classifiers. We compare the Gaussian distribution graphs with the fuzzy distribution graphs for the naive Bayesian classifier. We applied these two types of distribution graphs to classify leukemia and colon DNA microarray data sets. The results demonstrate that a naive Bayesian classifier with fuzzy distribution graphs is more reliable than that with Gaussian distribution graphs.

Maximum Likelihood Estimation Over Directed Acyclic Gaussian Graphs.

Estimation of multiple directed graphs becomes challenging in the presence of inhomogeneous data, where directed acyclic graphs (DAGs) are used to represent causal relations among random variables. To infer causal relations among variables, we estimate multiple DAGs given a known ordering in Gaussian graphical models. In particular, we propose a constrained maximum likelihood method with nonconvex constraints over elements and element-wise differences of adjacency matrices, for identifying the sparseness structure as well as detecting structural changes over adjacency matrices of the graphs. Computationally, we develop an efficient algorithm based on augmented Lagrange multipliers, the difference convex method, and a novel fast algorithm for solving convex relaxation subproblems. Numerical results suggest that the proposed method performs well against its alternatives for simulated and real data.

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.

Estimation of Gaussian graphs by model selection

We investigate in this paper the estimation of Gaussian graphs by model selection from a non-asymptotic point of view. We start from an n-sample of a Gaussian law ℙC in ℝp and focus on the disadvantageous case where n is smaller than p. To estimate the graph of conditional dependences of ℙC, we introduce a collection of candidate graphs and then select one of them by minimizing a penalized empirical risk. Our main result assesses the performance of the procedure in a non-asymptotic setting. We pay special attention to the maximal degree D of the graphs that we can handle, which turns to be roughly n/(2logp).

Sand drawings and Gaussian graphs§

Sand drawings form a part of many cultural traditions. Depending on the part of the world in which they occur, such drawings have different names such as sona, kolam, and nitus drawings. In this paper, we show connections between a special class of sand drawings and mathematical objects studied in the disciplines of graph theory and topology called Gaussian graphs. Motivated by this connection, we further our study to include analysis of some properties of sand drawings. In particular, we study the number of different drawings, show how to generate them, and show connections to the well-known Traveling Salesman Problem in computer science. §A preliminary version of this paper appeared at BRIDGES 2006 1.

Gradient directed regularization for sparse Gaussian concentration graphs, with applications to inference of genetic networks

Large-scale microarray gene expression data provide the possibility of constructing genetic networks or biological pathways. Gaussian graphical models have been suggested to provide an effective method for constructing such genetic networks. However, most of the available methods for constructing Gaussian graphs do not account for the sparsity of the networks and are computationally more demanding or infeasible, especially in the settings of high dimension and low sample size. We introduce a threshold gradient descent (TGD) regularization procedure for estimating the sparse precision matrix in the setting of Gaussian graphical models and demonstrate its application to identifying genetic networks. Such a procedure is computationally feasible and can easily incorporate prior biological knowledge about the network structure. Simulation results indicate that the proposed method yields a better estimate of the precision matrix than the procedures that fail to account for the sparsity of the graphs. We also present the results on inference of a gene network for isoprenoid biosynthesis in Arabidopsis thaliana. These results demonstrate that the proposed procedure can indeed identify biologically meaningful genetic networks based on microarray gene expression data.

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%.

Perfect Codes in the Lee Metric and the Packing of Polyominoes

Perfect Codes in the Lee Metric and the Packing of Polyominoes