joint association graph screening and decomposition for large scale linear dynamical systems this paper studies large scale dynamical networks where the current state of the system is a linear transformation of the previous state contaminated by a multivariate gaussian noise examples include stock markets human brains and gene regulatory networks we introduce a transition matrix to describe the evolution which can be translated to a directed granger transition graph and use the concentration matrix of the gaussian noise to capture the second order relations between nodes which can be translated to an undirected conditional dependence graph we propose regularizing the two graphs jointly in topology identification and dynamics estimation based on the notion of joint association graph jag we develop a joint graphical screening and estimation jgse framework for efficient network learning in big data in particular our method can predetermine and remove unnecessary edges based on the joint graphical structure referred to as jag screening and can decompose a large network into smaller subnetworks in a robust manner referred to as jag decomposition jag screening and decomposition can reduce the problem size and search space for fine estimation at a later stage experiments on both synthetic data and real world applications show the effectiveness of the proposed framework in large scale network topology identification and dynamics estimation

quantum chains of hopf algebras with quantum double cosymmetry given a finite dimensional c hopf algebra h and its dual we construct the infinite crossed product and study its superselection sectors in the framework of algebraic quantum field theory is the observable algebra of a generalized quantum spin chain with h order and disorder symmetries where by a duality transformation the role of order and disorder may also appear interchanged if is a group algebra then becomes an ordinary g spin model we classify all dhr sectors of relative to some haag dual vacuum representation and prove that their symmetry is described by the drinfeld double to achieve this we construct localized coactions and use a certain compressibility property to prove that they are universal amplimorphisms on in this way the double can be recovered from the observable algebra as a universal cosymmetry

heuristics of instability and stabilization in model selection in model selection usually a best predictor is chosen from a collection hat mu cdot s of predictors where hat mu cdot s is the minimum least squares predictor in a collection mathsf u s of predictors here s is a complexity parameter that is the smaller s the lower dimensional smoother the models in mathsf u s if mathsf l is the data used to derive the sequence hat mu cdot s the procedure is called unstable if a small change in mathsf l can cause large changes in hat mu cdot s with a crystal ball one could pick the predictor in hat mu cdot s having minimum prediction error without prescience one uses test sets cross validation and so forth the difference in prediction error between the crystal ball selection and the statistician s choice we call predictive loss for an unstable procedure the predictive loss is large this is shown by some analytics in a simple case and by simulation results in a more complex comparison of four different linear regression methods unstable procedures can be stabilized by perturbing the data getting a new predictor sequence hat mu cdot s and then averaging over many such predictor sequences

learning local dependence in ordered data in many applications data come with a natural ordering this ordering can often induce local dependence among nearby variables however in complex data the width of this dependence may vary making simple assumptions such as a constant neighborhood size unrealistic we propose a framework for learning this local dependence based on estimating the inverse of the cholesky factor of the covariance matrix penalized maximum likelihood estimation of this matrix yields a simple regression interpretation for local dependence in which variables are predicted by their neighbors our proposed method involves solving a convex penalized gaussian likelihood problem with a hierarchical group lasso penalty the problem decomposes into independent subproblems which can be solved efficiently in parallel using first order methods our method yields a sparse symmetric positive definite estimator of the precision matrix encoding a gaussian graphical model we derive theoretical results not found in existing methods attaining this structure in particular our conditions for signed support recovery and estimation consistency rates in multiple norms are as mild as those in a regression problem empirical results show our method performing favorably compared to existing methods we apply our method to genomic data to flexibly model linkage disequilibrium our method is also applied to improve the performance of discriminant analysis in sound recording classification

penalized likelihood methods for estimation of sparse high dimensional directed acyclic graphs directed acyclic graphs are commonly used to represent causal relationships among random variables in graphical models applications of these models arise in the study of physical and biological systems where directed edges between nodes represent the influence of components of the system on each other estimation of directed graphs from observational data is computationally np hard in addition directed graphs with the same structure may be indistinguishable based on observations alone when the nodes exhibit a natural ordering the problem of estimating directed graphs reduces to the problem of estimating the structure of the network in this paper we propose an efficient penalized likelihood method for estimation of the adjacency matrix of directed acyclic graphs when variables inherit a natural ordering we study variable selection consistency of lasso and adaptive lasso penalties in high dimensional sparse settings and propose an error based choice for selecting the tuning parameter we show that although the lasso is only variable selection consistent under stringent conditions the adaptive lasso can consistently estimate the true graph under the usual regularity assumptions

fused multiple graphical lasso in this paper we consider the problem of estimating multiple graphical models simultaneously using the fused lasso penalty which encourages adjacent graphs to share similar structures a motivating example is the analysis of brain networks of alzheimer s disease using neuroimaging data specifically we may wish to estimate a brain network for the normal controls nc a brain network for the patients with mild cognitive impairment mci and a brain network for alzheimer s patients ad we expect the two brain networks for nc and mci to share common structures but not to be identical to each other similarly for the two brain networks for mci and ad the proposed formulation can be solved using a second order method our key technical contribution is to establish the necessary and sufficient condition for the graphs to be decomposable based on this key property a simple screening rule is presented which decomposes the large graphs into small subgraphs and allows an efficient estimation of multiple independent small subgraphs dramatically reducing the computational cost we perform experiments on both synthetic and real data our results demonstrate the effectiveness and efficiency of the proposed approach

the generally covariant locality principle a new paradigm for local quantum field theory a new approach to the model independent description of quantum field theories will be introduced in the present work the main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity thus giving rise to the concept of a locally covariant quantum field theory such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories on one side of globally hyperbolic spacetimes with isometric embeddings as morphisms and on the other side of algebras with unital injective monomorphisms as morphisms moreover locally covariant quantum fields can be described in this framework as natural transformations between certain functors the usual haag kastler framework of nets of operator algebras over a fixed spacetime background manifold together with covariant automorphic actions of the isometry group of the background spacetime can be re gained from this new approach as a special case examples of this new approach are also outlined in case that a locally covariant quantum field theory obeys the time slice axiom one can naturally associate to it certain automorphic actions called relative cauchy evolutions which describe the dynamical reaction of the quantum field theory to a local change of spacetime background metrics the functional derivative of a relative cauchy evolution with respect to the spacetime metric is found to be a divergence free quantity which has as will be demonstrated in an example the significance of an energy momentum tensor up to addition of scalar functions for the locally covariant quantum field theory furthermore we discuss the functorial properties of state spaces of locally covariant quantum field theories that entail the validity of the principle of local definiteness

block diagonal covariance selection for high dimensional gaussian graphical models abstractgaussian graphical models are widely used to infer and visualize networks of dependencies between continuous variables however inferring the graph is difficult when the sample size is small compared to the number of variables to reduce the number of parameters to estimate in the model we propose a nonasymptotic model selection procedure supported by strong theoretical guarantees based on an oracle type inequality and a minimax lower bound the covariance matrix of the model is approximated by a block diagonal matrix the structure of this matrix is detected by thresholding the sample covariance matrix where the threshold is selected using the slope heuristic based on the block diagonal structure of the covariance matrix the estimation problem is divided into several independent problems subsequently the network of dependencies between variables is inferred using the graphical lasso algorithm in each block the performance of the procedure is illustrated on simulated data an application t

sparse estimation of a covariance matrix we suggest a method for estimating a covariance matrix on the basis of a sample of vectors drawn from a multivariate normal distribution in particular we penalize the likelihood with a lasso penalty on the entries of the covariance matrix this penalty plays two important roles it reduces the effective number of parameters which is important even when the dimension of the vectors is smaller than the sample size since the number of parameters grows quadratically in the number of variables and it produces an estimate which is sparse in contrast to sparse inverse covariance estimation our method s close relative the sparsity attained here is in the covariance matrix itself rather than in the inverse matrix zeros in the covariance matrix correspond to marginal independencies thus our method performs model selection while providing a positive definite estimate of the covariance the proposed penalized maximum likelihood problem is not convex so we use a majorize minimize approach in which we iteratively solve convex approximations to the original nonconvex problem we discuss tuning parameter selection and demonstrate on a flow cytometry dataset how our method produces an interpretable graphical display of the relationship between variables we perform simulations that suggest that simple elementwise thresholding of the empirical covariance matrix is competitive with our method for identifying the sparsity structure additionally we show how our method can be used to solve a previously studied special case in which a desired sparsity pattern is prespecified

joint estimation of multiple graphical models from high dimensional time series in this manuscript we consider the problem of jointly estimating multiple graphical models in high dimensions we assume that the data are collected from n subjects each of which consists of t possibly dependent observations the graphical models of subjects vary but are assumed to change smoothly corresponding to a measure of closeness between subjects we propose a kernel based method for jointly estimating all graphical models theoretically under a double asymptotic framework where both t n and the dimension d can increase we provide the explicit rate of convergence in parameter estimation it characterizes the strength one can borrow across different individuals and the impact of data dependence on parameter estimation empirically experiments on both synthetic and real resting state functional magnetic resonance imaging rs fmri data illustrate the effectiveness of the proposed method

high dimensional analysis of semidefinite relaxations for sparse principal components principal component analysis pca is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix however pca is well known to behave poorly in the large p small n setting in which the problem dimension p is comparable to or larger than the sample size n this paper studies pca in this high dimensional regime but under the additional assumption that the maximal eigenvector is sparse say with at most k nonzero components we consider a spiked covariance model in which a base matrix is perturbed by adding a k sparse maximal eigenvector and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector as follows a a simple diagonal thresholding method which transitions from success to failure as a function of the rescaled sample size dia n p k n k 2 log p k and b a more sophisticated semidefinite programming sdp relaxation which succeeds once the rescaled sample size sdp n p k n klog p k is larger than a critical threshold in addition we prove that no method including the best method which has exponential time complexity can succeed in recovering the support if the order parameter sdp n p k is below a threshold our results thus highlight an interesting trade off between computational and statistical efficiency in high dimensional inference

better subset regression using the nonnegative garrote a new method called the nonnegative nn garrote is proposed for doing subset regression it both shrinks and zeroes coefficients in tests on real and simulated data it produces lower prediction error than ordinary subset selection it is also compared to ridge regression if the regression equations generated by a procedure do not change drastically with small changes in the data the procedure is called stable subset selection is unstable ridge is very stable and the nn garrote is intermediate simulation results illustrate the effects of instability on prediction error

event triggered fuzzy adaptive quantized control for nonlinear multi agent systems in nonaffine pure feedback form abstract in this paper we address the problem of event triggered fuzzy adaptive quantized control for stochastic nonlinear non affine pure feedback multi agent systems the fuzzy logic system is used to estimate the stochastic disturbance term and unknown nonlinear functions a nonlinearity decomposition method of asymmetric hysteresis quantizer is proposed by applying sector bound property moreover to reduce the communication burden an adaptive event triggered protocol with a varying threshold is constructed based on the backstepping technique and stochastic lyapunov function method a novel adaptive event triggered fuzzy control protocol and adaptive laws are constructed by using stochastic lyapunov stability theory it is demonstrated that all signals are bounded in the closed loop systems in probability and all the outputs of followers converge to the neighborhood of the leader output simulation results illustrate the effectiveness of our proposed scheme

an imputation regularized optimization algorithm for high dimensional missing data problems and beyond missing data are frequently encountered in high dimensional problems but they are usually difficult to deal with by using standard algorithms such as the expectation maximization algorithm and its variants to tackle this difficulty some problem specific algorithms have been developed in the literature but there still lacks a general algorithm this work is to fill the gap we propose a general algorithm for high dimensional missing data problems the algorithm works by iterating between an imputation step and a regularized optimization step at the imputation step the missing data are imputed conditionally on the observed data and the current estimates of parameters and at the regularized optimization step a consistent estimate is found via the regularization approach for the minimizer of a kullback leibler divergence defined on the pseudocomplete data for high dimensional problems the consistent estimate can be found under sparsity constraints the consistency of the averaged estimate for the true parameter can be established under quite general conditions the algorithm is illustrated by using high dimensional gaussian graphical models high dimensional variable selection and a random coefficient model

blessing of massive scale spatial graphical model estimation with a total cardinality constraint approach we consider the problem of estimating high dimensional spatial graphical models with a total cardinality constraint i e the ell 0 constraint though this problem is highly nonconvex we show that its primal dual gap diminishes linearly with the dimensionality and provide a convex geometry justification of this blessing of massive scale phenomenon motivated by this result we propose an efficient algorithm to solve the dual problem which is concave and prove that the solution achieves optimal statistical properties extensive numerical results are also provided

geometric interpretation of vertex operator algebras abstract in this paper vafa s approach to the formulation of conformal field theories is combined with the formal calculus developed in frenkel lepowsky and meurman s work on the vertex operator construction of the monster to give a geometric definition of vertex operator algebras the main result announced is the equivalence between this definition and the algebraic one in the sense that the categories determined by these definitions are isomorphic

joint estimation of multiple precision matrices with common structures estimation of inverse covariance matrices known as precision matrices is important in various areas of statistical analysis in this article we consider estimation of multiple precision matrices sharing some common structures in this setting estimating each precision matrix separately can be suboptimal as it ignores potential common structures this article proposes a new approach to parameterize each precision matrix as a sum of common and unique components and estimate multiple precision matrices in a constrained l1 minimization framework we establish both estimation and selection consistency of the proposed estimator in the high dimensional setting the proposed estimator achieves a faster convergence rate for the common structure in certain cases our numerical examples demonstrate that our new estimator can perform better than several existing methods in terms of the entropy loss and frobenius loss an application to a glioblastoma cancer data set reveals some interesting gene networks across multiple cancer subtypes

robust gaussian graphical modeling via l1 penalization gaussian graphical models have been widely used as an effective method for studying the conditional independency structure among genes and for constructing genetic networks however gene expression data typically have heavier tails or more outlying observations than the standard gaussian distribution such outliers in gene expression data can lead to wrong inference on the dependency structure among the genes we propose a l1 penalized estimation procedure for the sparse gaussian graphical models that is robustified against possible outliers the likelihood function is weighted according to how the observation is deviated where the deviation of the observation is measured based on its own likelihood an efficient computational algorithm based on the coordinate gradient descent method is developed to obtain the minimizer of the negative penalized robustified likelihood where nonzero elements of the concentration matrix represents the graphical links among the genes after the graphical structure is obtained we re estimate the positive definite concentration matrix using an iterative proportional fitting algorithm through simulations we demonstrate that the proposed robust method performs much better than the graphical lasso for the gaussian graphical models in terms of both graph structure selection and estimation when outliers are present we apply the robust estimation procedure to an analysis of yeast gene expression data and show that the resulting graph has better biological interpretation than that obtained from the graphical lasso

a new approach to cholesky based covariance regularization in high dimensions summary in this paper we propose a new regression interpretation of the cholesky factor of the covariance matrix as opposed to the well known regression interpretation of the cholesky factor of the inverse covariance which leads to a new class of regularized covariance estimators suitable for high dimensional problems regularizing the cholesky factor of the covariance via this regression interpretation always results in a positive definite estimator in particular one can obtain a positive definite banded estimator of the covariance matrix at the same computational cost as the popular banded estimator of bickel levina 2008b which is not guaranteed to be positive definite we also establish theoretical connections between banding cholesky factors of the covariance matrix and its inverse and constrained maximum likelihood estimation under the banding constraint and compare the numerical performance of several methods in simulations and on a sonar data example

on l _ q optimization and sparse inverse covariance selection graphical models are well established in providing meaningful conditional probability descriptions of complex multivariable interactions in the gaussian case the conditional independencies between different variables correspond to zero entries in the precision inverse covariance matrix hence there has been much recent interest in sparse precision matrix estimation in areas such as statistics machine learning computer vision pattern recognition and signal processing a popular estimation method involves optimizing a penalized log likelihood problem the penalty is responsible for inducing sparsity and a common choice is the convex l1 norm even though the l0 penalty is the natural choice guaranteeing maximum sparsity it has been avoided due to lack of convexity as a result in this paper we bridge the gap between these two penalties and propose the non concave lq penalized log likelihood problem for sparse precision matrix estimation where 0 q 1 a novel algorithm is developed for the optimization and we provide some of its theoretic properties that are useful in sparse linear regression we illustrate on synthetic and real data showing reconstruction quality comparisons of sparsity inducing penalties l0 lq with 0 q 1 l1 and scad

sparse estimation of conditional graphical models with application to gene networks in many applications the graph structure in a network arises from two sources intrinsic connections and connections due to external effects we introduce a sparse estimation procedure for graphical models that is capable of isolating the intrinsic connections by removing the external effects technically this is formulated as a conditional graphical model in which the external effects are modeled as predictors and the graph is determined by the conditional precision matrix we introduce two sparse estimators of this matrix using the reproduced kernel hilbert space combined with lasso and adaptive lasso we establish the sparsity variable selection consistency oracle property and the asymptotic distributions of the proposed estimators we also develop their convergence rate when the dimension of the conditional precision matrix goes to infinity the methods are compared with sparse estimators for unconditional graphical models and with the constrained maximum likelihood estimate that assumes a know

sparse precision matrix estimation via lasso penalized d trace loss we introduce a constrained empirical loss minimization framework for estimating high dimensional sparse precision matrices and propose a new loss function called the d trace loss for that purpose a novel sparse precision matrix estimator is defined as the minimizer of the lasso penalized d trace loss under a positive definiteness constraint under a new irrepresentability condition the lasso penalized d trace estimator is shown to have the sparse recovery property examples demonstrate that the new condition can hold in situations where the irrepresentability condition for the lasso penalized gaussian likelihood estimator fails we establish rates of convergence for the new estimator in the elementwise maximum frobenius and operator norms we develop a very efficient algorithm based on alternating direction methods for computing the proposed estimator simulated and real data are used to demonstrate the computational efficiency of our algorithm and the finite sample performance of the new estimator the lasso penalized d trace estimator is found to compare favourably with the lasso penalized gaussian likelihood estimator

aqft from n functorial qft there are essentially two different approaches to the axiomatization of quantum field theory qft algebraic qft going back to haag and kastler and functorial qft going back to atiyah and segal more recently based on ideas by baez and dolan the latter is being refined to extended functorial qft by freed hopkins lurie and others the first approach uses local nets of operator algebras which assign to each patch an algebra of observables the latter uses n functors which assign to each patch a propagator of states

model selection and estimation in the gaussian graphical model summary we propose penalized likelihood methods for estimating the concentration matrix in the gaussian graphical model the methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite and thus conduct model selection and estimation simultaneously the implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization we propose a bic type criterion for the selection of the tuning parameter in the penalized likelihood methods the connection between our methods and existing methods is illustrated simulations and real examples demonstrate the competitive performance of the new methods

joint mean and covariance estimation with unreplicated matrix variate data abstractit has been proposed that complex populations such as those that arise in genomics studies may exhibit dependencies among observations as well as among variables this gives rise to the challenging problem of analyzing unreplicated high dimensional data with unknown mean and dependence structures matrix variate approaches that impose various forms of inverse covariance sparsity allow flexible dependence structures to be estimated but cannot directly be applied when the mean and covariance matrices are estimated jointly we present a practical method utilizing generalized least squares and penalized inverse covariance estimation to address this challenge we establish consistency and obtain rates of convergence for estimating the mean parameters and covariance matrices the advantages of our approaches are i dependence graphs and covariance structures can be estimated in the presence of unknown mean structure ii the mean structure becomes more efficiently estimated when accounting for

a constrained l1 minimization approach to sparse precision matrix estimation this article proposes a constrained l1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p variate random variables the resulting estimator is shown to have a number of desirable properties in particular the rate of convergence between the estimator and the true s sparse precision matrix under the spectral norm is when the population distribution has either exponential type tails or polynomial type tails we present convergence rates under the elementwise l norm and frobenius norm in addition we consider graphical model selection the procedure is easily implemented by linear programming numerical performance of the estimator is investigated using both simulated and real data in particular the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods

functional graphical models graphical models have attracted increasing attention in recent years especially in settings involving high dimensional data in particular gaussian graphical models are used to model the conditio

higher algebraic structures and quantization we derive quasi quantum groups in 2 1 dimensional topological field theory directly from the classical action and the path integral detailed computations are carried out for the chern simons theory with finite gauge group the principles behind our computations are presumably more general we extend the classical action in ad 1 dimensional topological theory to manifolds of dimension less thand 1 we then construct a generalized path integral which ind 1 dimensions reduces to the standard one and ind dimensions reproduces the quantum hilbert space in a 2 1 dimensional topological theory the path integral over the circle is the category of representations of a quasi quantum group in this paper we only consider finite theories in which the generalized path integral reduces to a finite sum new ideas are needed to extend beyond the finite theories treated here

representation theory of 2 groups on kapranov and voevodsky s 2 vector spaces abstract in this paper the representation theory of 2 groups in 2 categories is considered focusing the attention on the 2 category rep 2 mat k g of representations of a 2 group g in a version of kapranov and voevodsky s 2 category of 2 vector spaces over a field k the set of equivalence classes of such representations is computed in terms of the invariants 0 g 1 g and h 3 0 g 1 g classifying g and the categories of intertwiners are described in terms of categories of vector bundles endowed with a projective action in particular it is shown that the monoidal category of finite dimensional linear representations more generally the category of z projective representations for any given cohomology class z h 2 0 g k of the first homotopy group 0 g as well as its category of representations on finite sets both live in rep 2 mat k g the first as the monoidal category of endomorphisms of the trivial representation more generally as the category of intertwiners between suitable 1 dimensional representations and the second as a non full subcategory of the homotopy category of rep 2 mat k g

optimal sparse volatility matrix estimation for high dimensional ito processes with measurement errors stochastic processes are often used to model complex scientiflc problems in flelds ranging from biology and flnance to engineering and physical science this paper investigates rate optimal estimation of the volatility matrix of a high dimensional it o process observed with measurement errors at discrete time points the minimax rate of convergence is established for estimating sparse volatility matrices by combining the multi scale and threshold approaches we construct a volatility matrix estimator to achieve the optimal convergence rate the minimax lower bound is derived by considering a subclass of it o processes for which the minimax lower bound is obtained through a novel equivalent model of covariance matrix estimation for independent but non identically distributed observations and through a delicate construction of the least favorable parameters in addition a simulation study was conducted to test the flnite sample performance of the optimal estimator and the simulation results were found to support the established asymptotic theory

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

joint estimation of multiple conditional gaussian graphical models in this paper we propose a joint conditional graphical lasso to learn multiple conditional gaussian graphical models also known as gaussian conditional random fields with some similar structures our model builds on the maximum likelihood method with the convex sparse group lasso penalty moreover our model is able to model multiple multivariate linear regressions with unknown noise covariances via a convex formulation in addition we develop an efficient approximated newton s method for optimizing our model theoretically we establish the asymptotic properties of our model on consistency and sparsistency under the high dimensional settings finally extensive numerical results on simulations and real data sets demonstrate that our method outperforms the compared methods on structure recovery and structured output prediction to the best of our knowledge the joint learning of multiple multivariate regressions with unknown covariance is first studied

indirect multivariate response linear regression we propose a class of estimators of the multivariate response linear regression coefficient matrix that exploits the assumption that the response and predictors have a joint multivariate normal distribution this allows us to indirectly estimate the regression coefficient matrix through shrinkage estimation of the parameters of the inverse regression or the conditional distribution of the predictors given the responses we establish a convergence rate bound for estimators in our class and we study two examples which respectively assume that the inverse regression s coefficient matrix is sparse and rank deficient these estimators do not require that the forward regression coefficient matrix is sparse or has small frobenius norm using simulation studies we show that our estimators outperform competitors

representation and character theory in 2 categories we develop a 2 categorical generalization of the theory of group representations and characters we categorify the concept of the trace of a linear transformation associating to any endofunctor of any small category a set called its categorical trace in a linear situation the categorical trace is a vector space and we associate to any two commuting self equivalences a number called their joint trace for a group acting on a linear category we define an analog of the character which is the function on commuting pairs of group elements given by the joint traces of the corresponding functors we call this function the 2 character of such functions of commuting pairs and more generally n tuples appear in the work of hopkins kuhn and ravenel michael j hopkins nicholas j kuhn douglas c ravenel generalized group characters and complex oriented cohomology theories j amer math soc 13 3 2000 553 594 electronic on equivariant morava e theories we define the concept of induced categorical representation and show that the corresponding 2 character is given by the same formula as was obtained in michael j hopkins nicholas j kuhn douglas c ravenel generalized group characters and complex oriented cohomology theories j amer math soc 13 3 2000 553 594 electronic for the transfer map in the second equivariant morava e theory

model selection for gaussian concentration graphs gga multivariate gaussian graphical markov model for an undirected graph g also called a covariance selection model or concentration graph model is defined in terms of the markov properties i e conditional independences associated with g which in turn are equivalent to specified zeros among the set of pairwise partial correlation coefficients by means of fisher s z transformation and sidak s correlation inequality conservative simultaneous confidence intervals for the entire set of partial correlations can be obtained leading to a simple method for model selection that controls the overall error rate for incorrect edge inclusion the simultaneous p values corresponding to the partial correlations are partitioned into three disjoint sets a significant set s an indeterminate set i and a nonsignificant set n our model selection method selects two graphs a graph g si whose edges correspond to the set s i and a more conservative graph g s whose edges correspond to s only similar considerations apply to covariance graph models which are defined in terms of marginal independence rather than conditional independence the method is applied to some well known examples and to simulated data

positive definite estimators of large covariance matrices using convex optimization we construct a sparse estimator of the covariance matrix that is positive definite and performs well in high dimensional settings a lasso type penalty is used to encourage sparsity and a logarithmic barrier function is used to enforce positive definiteness consistency and convergence rate bounds are established as both the number of variables and sample size diverge an efficient computational algorithm is developed and the merits of the approach are illustrated with simulations and a speech signal classification example copyright 2012 oxford university press

the sparse laplacian shrinkage estimator for high dimensional regression we propose a new penalized method for variable selection and estimation that explicitly incorporates the correlation patterns among predictors this method is based on a combination of the minimax concave penalty and laplacian quadratic associated with a graph as the penalty function we call it the sparse laplacian shrinkage sls method the sls uses the minimax concave penalty for encouraging sparsity and laplacian quadratic penalty for promoting smoothness among coefficients associated with the correlated predictors the sls has a generalized grouping property with respect to the graph represented by the laplacian quadratic we show that the sls possesses an oracle property in the sense that it is selection consistent and equal to the oracle laplacian shrinkage estimator with high probability this result holds in sparse high dimensional settings with p n under reasonable conditions we derive a coordinate descent algorithm for computing the sls estimates simulation studies are conducted to evaluate the performance of the sls method and a real data example is used to illustrate its application

on an additive semigraphoid model for statistical networks with application to pathway analysis we introduce a nonparametric method for estimating non gaussian graphical models based on a new statistical relation called additive conditional independence which is a three way relation among random vectors that resembles the logical structure of conditional independence additive conditional independence allows us to use one dimensional kernel regardless of the dimension of the graph which not only avoids the curse of dimensionality but also simplifies computation it also gives rise to a parallel structure to the gaussian graphical model that replaces the precision matrix by an additive precision operator the estimators derived from additive conditional independence cover the recently introduced nonparanormal graphical model as a special case but outperform it when the gaussian copula assumption is violated we compare the new method with existing ones by simulations and in genetic pathway analysis supplementary materials for this article are available online

higher dimensional algebra and topological quantum field theory the study of topological quantum field theories increasingly relies upon concepts from higher dimensional algebra such as n categories and n vector spaces we review progress towards a definition of n category suited for this purpose and outline a program in which n dimensional topological quantum field theories tqfts are to be described as n category representations first we describe a suspension operation on n categories and hypothesize that the k fold suspension of a weak n category stabilizes for k n 2 we give evidence for this hypothesis and describe its relation to stable homotopy theory we then propose a description of n dimensional unitary extended tqfts as weak n functors from the free stable weak n category with duals on one object to the n category of n hilbert spaces we conclude by describing n categorical generalizations of deformation quantization and the quantum double construction

discussion latent variable graphical model selection via convex optimization since recently there have been an increasing interest in the problem of estimating a high dimensional matrix k that can be decomposed in a sum of a sparse matrix s i e a matrix having only a small number of nonzero entries and a low rank matrix l this is motivated by applications in computer vision video segmentation computational biology semantic indexing etc the main contribution and novelty of chandrasekaran parrilo and willsky paper cpw in what follows is to propose and study a method of inference about such decomposable matrices for a particular setting where k is the precision concentration matrix of a partially observed sparse gaussian graphical model ggm in this case k is the inverse of the covariance matrix of a gaussian vector xo extracted from a larger gaussian vector xo xh with sparse inverse covariance matrix then it is easy to see that k can be represented as a sum of a sparse precision matrix s corresponding to the observed variables xo and a matrix l with rank at most h where h is the dimension of the latent variables xh if h is small which is a typical situation in practice then l has low rank the ggm with latent variables is of major interest for applications in biology or in social networks where one often does not observe all the variables relevant for depicting sparsely the conditional dependencies note that formally this is just one possible motivation and mathematically the problem is dealt with in more generality namely postulating that the precision matrix satisfies