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A tutorial on MM algorithms
 Amer. Statist
, 2004
"... Most problems in frequentist statistics involve optimization of a function such as a likelihood or a sum of squares. EM algorithms are among the most effective algorithms for maximum likelihood estimation because they consistently drive the likelihood uphill by maximizing a simple surrogate function ..."
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Cited by 154 (6 self)
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Most problems in frequentist statistics involve optimization of a function such as a likelihood or a sum of squares. EM algorithms are among the most effective algorithms for maximum likelihood estimation because they consistently drive the likelihood uphill by maximizing a simple surrogate function for the loglikelihood. Iterative optimization of a surrogate function as exemplified by an EM algorithm does not necessarily require missing data. Indeed, every EM algorithm is a special case of the more general class of MM optimization algorithms, which typically exploit convexity rather than missing data in majorizing or minorizing an objective function. In our opinion, MM algorithms deserve to part of the standard toolkit of professional statisticians. The current article explains the principle behind MM algorithms, suggests some methods for constructing them, and discusses some of their attractive features. We include numerous examples throughout the article to illustrate the concepts described. In addition to surveying previous work on MM algorithms, this article introduces some new material on constrained optimization and standard error estimation. Key words and phrases: constrained optimization, EM algorithm, majorization, minorization, NewtonRaphson 1 1
K.L.: Fast modelbased estimation of ancestry in unrelated individuals. Genome Res. 19, 1655– 1664
 Information Systems and Data Analysis
, 1997
"... Population stratification has long been recognized as a confounding factor in genetic association studies. Estimated ancestries, derived from multilocus genotype data, can be used as covariates to correct for population stratification. One popular technique for estimation of ancestry is the modelb ..."
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Cited by 127 (4 self)
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Population stratification has long been recognized as a confounding factor in genetic association studies. Estimated ancestries, derived from multilocus genotype data, can be used as covariates to correct for population stratification. One popular technique for estimation of ancestry is the modelbased approach embodied by the widelyapplied program structure. Another approach, implemented in the program eigenstrat, relies on principal component analysis rather than modelbased estimation and does not directly deliver admixture fractions. eigenstrat has gained in popularity in part due to its remarkable speed in comparison to structure. We present a new algorithm and a program, admixture, for modelbased estimation of ancestry in unrelated individuals. admixture adopts the likelihood model embedded in structure. However, admixture runs considerably faster, solving problems in minutes that take structure hours. In many of our experiments we have found that admixture is almost as fast as eigenstrat. The runtime improvements of admixture rely on a fast block relaxation scheme using sequential quadratic programming for block updates, coupled with a novel quasiNewton acceleration of convergence. Our algorithm also runs faster and with greater accuracy than the implementation of an ExpectationMaximization (EM) algorithm incorporated in the program frappe. Our simulations show that admixture’s maximum likelihood estimates of the underlying admixture coefficients and ancestral allele frequencies are as accurate as structure’s Bayesian estimates. On real world datasets, admixture’s estimates are directly comparable to those from structure and eigenstrat. Taken together, our results show that admixture’s computational speed opens up the possibility of using a much larger setof markers in modelbased ancestry estimation and that its estimates are suitable for use in correcting for population stratification in association studies. 2 1
Applications of Multidimensional Scaling to Molecular Conformation
, 1997
"... Multidimensional scaling (MDS) is a collection of data analytic techniques for constructing configurations of points from information about interpoint distances. Such constructions arise in computational chemistry when one endeavors to infer the conformation (3dimensional structure) of a molecule fr ..."
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Cited by 24 (6 self)
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Multidimensional scaling (MDS) is a collection of data analytic techniques for constructing configurations of points from information about interpoint distances. Such constructions arise in computational chemistry when one endeavors to infer the conformation (3dimensional structure) of a molecule from information about its interatomic distances. For a number of reasons, this application of MDS poses computational challenges not encountered in more traditional applications. In this report we sketch the mathematical formulation of MDS for molecular conformation problems and describe two approaches that can be employed for their solution. 1 Molecular Conformation Consider a molecule with n atoms. We can represent its conformation, or 3dimensional structure, by specifying the coordinates of each atom with respect to a Euclidean coordinate system for ! 3 . We store these coordinates in an n \Theta 3 configuration matrix X. Given X, we can easily compute the matrix of interatomic distan...
Discovering Graphical Granger Causality Using the Truncating Lasso Penalty
, 2010
"... Components of biological systems interact with each other in order to carry out vital cell functions. Such information can be used to improve estimation and inference, and to obtain better insights into the underlying cellular mechanisms. Discovering regulatory interactions among genes is therefore ..."
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Cited by 22 (6 self)
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Components of biological systems interact with each other in order to carry out vital cell functions. Such information can be used to improve estimation and inference, and to obtain better insights into the underlying cellular mechanisms. Discovering regulatory interactions among genes is therefore an important problem in systems biology. Wholegenome expression data over time provides an opportunity to determine how the expression levels of genes are affected by changes in transcription levels of other genes, and can therefore be used to discover regulatory interactions among genes. In this paper, we propose a novel penalization method, called truncating lasso, for estimation of causal relationships from timecourse gene expression data. The proposed penalty can correctly determine the order of the underlying time series, and improves the performance of the lassotype estimators. Moreover, the resulting estimate provides information on the time lag between activation of transcription factors and their effects on regulated genes. We provide an efficient algorithm for estimation of model parameters, and show that the proposed method can consistently discover causal relationships in the large p, small n setting. The performance of the proposed model is evaluated favorably in simulated, as well as real, data examples. The proposed truncating lasso method is implemented in the Rpackage grangerTlasso and is available at www.stat.lsa.umich.edu/∼shojaie. 1
Multidimensional Scaling Using Majorization: Smacof In R
"... In this paper we present the methodology of multidimensional scaling problems (MDS) solved by means of the majorization algorithm. The objective function to be minimized is known as stress and functions which majorize stress are elaborated. This strategy to solve MDS problems is called SMACOF and it ..."
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Cited by 15 (0 self)
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In this paper we present the methodology of multidimensional scaling problems (MDS) solved by means of the majorization algorithm. The objective function to be minimized is known as stress and functions which majorize stress are elaborated. This strategy to solve MDS problems is called SMACOF and it is implemented in an R package of the same name which is presented in this article. We extend the basic SMACOF theory in terms of configuration constraints, threeway data, unfolding models, and projection of the resulting configurations onto spheres and other quadratic surfaces. Various examples are presented to show the possibilities of the SMACOF approach offered by the corresponding package.
Computing Distances Between Convex Sets and Subsets of the Positive Semidefinite Matrices
, 1997
"... We describe an important class of semidefinite programming problems that has received scant attention in the optimization community. These problems are derived from considerations in distance geometry and multidimensional scaling and therefore arise in a variety of disciplines, e.g. computational ch ..."
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Cited by 14 (8 self)
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We describe an important class of semidefinite programming problems that has received scant attention in the optimization community. These problems are derived from considerations in distance geometry and multidimensional scaling and therefore arise in a variety of disciplines, e.g. computational chemistry and psychometrics. In most applications, the feasible positive semidefinite matrices are restricted in rank, so that recent interiorpoint methods for semidefinite programming do not apply. We establish some theory for these problems and discuss what remains to be accomplished. Key words: Distance geometry, multidimensional scaling, semidefinite programming. Contents 1 Introduction 2 2 Projection into Subsets of\Omega n 4 3 Reducible Programming Formulations 5 3.1 Variable Alternation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 3.2 Variable Reduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 4 Optimization by Variab...
Multidimensional Scaling Using Majorization
 SMACOF in R.” Journal of Statistical Software
, 2009
"... In this paper we present the methodology of multidimensional scaling problems (MDS) solved by means of the majorization algorithm. The objective function to be minimized is known as stress and functions which majorize stress are elaborated. This strategy to solve MDS problems is called SMACOF and it ..."
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Cited by 14 (2 self)
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In this paper we present the methodology of multidimensional scaling problems (MDS) solved by means of the majorization algorithm. The objective function to be minimized is known as stress and functions which majorize stress are elaborated. This strategy to solve MDS problems is called SMACOF and it is implemented in an R package of the same name which is presented in this article. We extend the basic SMACOF theory in terms of configuration constraints, threeway data, unfolding models, and projection of the resulting configurations onto spheres and other quadratic surfaces. Various examples are presented to show the possibilities of the SMACOF approach offered by the corresponding package. Keywords:˜SMACOF, multidimensional scaling, majorization, R. 1.
Surrogate Maximization/Minimization Algorithms for AdaBoost and the Logistic Regression Model
 In The 21th International Conference on Machine Learning
, 2004
"... Surrogate maximization (or minimization) (SM) algorithms are a family of algorithms that can be regarded as a generalization of expectationmaximization (EM) algorithms. ..."
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Cited by 9 (1 self)
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Surrogate maximization (or minimization) (SM) algorithms are a family of algorithms that can be regarded as a generalization of expectationmaximization (EM) algorithms.
Tensor regression with applications in neuroimaging data analysis
, 2012
"... Classical regression methods treat covariates as a vector and estimate a corresponding vector of regression coefficients. Modern applications in medical imaging generate covariates of more complex form such as multidimensional arrays (tensors). Traditional statistical and computational methods are ..."
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Cited by 8 (2 self)
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Classical regression methods treat covariates as a vector and estimate a corresponding vector of regression coefficients. Modern applications in medical imaging generate covariates of more complex form such as multidimensional arrays (tensors). Traditional statistical and computational methods are proving insufficient for analysis of these highthroughput data due to their ultrahigh dimensionality as well as complex structure. In this article, we propose a new family of tensor regression models that efficiently exploit the special structure of tensor covariates. Under this framework, ultrahigh dimensionality is reduced to a manageable level, resulting in efficient estimation and prediction. A fast and highly scalable estimation algorithm is proposed for maximum likelihood estimation and its associated asymptotic properties are studied. Effectiveness of the new methods is demonstrated on both synthetic and real MRI imaging data. Key Words: Brain imaging; dimension reduction; generalized linear model (GLM); magnetic resonance imaging (MRI); multidimensional array; tensor regression. 1
Solving the PnP Problem with Anisotropic Orthogonal Procrustes Analysis
 2012 SECOND JOINT 3DIM/3DPVT CONFERENCE: 3D IMAGING, MODELING, PROCESSING, VISUALIZATION & TRANSMISSION
, 2012
"... In this paper we formulate the PerspectivenPoint (a.k.a. exterior orientation) problem in terms of an instance of the anisotropic orthogonal Procrustes problem, and derive its solution. Experiments with synthetic and real data demonstrate that our method reaches the best tradeoff between speed an ..."
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Cited by 8 (2 self)
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In this paper we formulate the PerspectivenPoint (a.k.a. exterior orientation) problem in terms of an instance of the anisotropic orthogonal Procrustes problem, and derive its solution. Experiments with synthetic and real data demonstrate that our method reaches the best tradeoff between speed and accuracy. The MATLAB code reported in the paper testifies that it is also exceedingly simple to implement.