multitask quantile regression under the transnormal model abstractwe consider estimating multitask quantile regression under the transnormal model with focus on high dimensional setting we derive a surprisingly simple closed form solution through rank based covariance regularization in particular we propose the rank based l1 penalization with positive definite constraints for estimating sparse covariance matrices and the rank based banded cholesky decomposition regularization for estimating banded precision matrices by taking advantage of the alternating direction method of multipliers nearest correlation matrix projection is introduced that inherits sampling properties of the unprojected one our work combines strengths of quantile regression and rank based covariance regularization to simultaneously deal with nonlinearity and nonnormality for high dimensional regression furthermore the proposed method strikes a good balance between robustness and efficiency achieves the oracle like convergence rate and provides the provable prediction interval und

minimal entropy and collapsing with curvature bounded from below we show that if a closed manifold m admits an structure not necessarily polarized possibly of rank zero then its minimal entropy vanishes in particular this is the case if m admits a non trivial s1 action as a corollary we obtain that the simplicial volume of a manifold admitting an structure is zero we also show that if m admits an structure then it collapses with curvature bounded from below this in turn implies that m collapses with bounded scalar curvature or equivalently its yamabe invariant is non negative we show that structures of rank zero appear rather frequently every compact complex elliptic surface admits one as well as any simply connected closed 5 manifold we use these results to study the minimal entropy problem we show the following two theorems suppose that m is a closed manifold obtained by taking connected sums of copies of s4 p2 overline mathbb c p 2 s2 s2and the k3 surface then m has zero minimal entropy moreover m admits a smooth riemannian metric with zero topological entropy if and only if m is diffeomorphic to s4 p2 s2 s2 p2 overline mathbb c p 2 or p2 p2 finally suppose that m is a closed simply connected 5 manifold then m has zero minimal entropy moreover m admits a smooth riemannian metric with zero topological entropy if and only if m is diffeomorphic to s5 s3 s2 then on trivial s3 bundle over s2 or the wu manifold su 3 so 3

smooth optimization approach for sparse covariance selection in this paper we first study a smooth optimization approach for solving a class of nonsmooth strictly concave maximization problems whose objective functions admit smooth convex minimization reformulations in particular we apply nesterov s smooth optimization technique y e nesterov dokl akad nauk sssr 269 1983 pp 543 547 y e nesterov math programming 103 2005 pp 127 152 to their dual counterparts that are smooth convex problems it is shown that the resulting approach has cal o 1 sqrt epsilon iteration complexity for finding an epsilon optimal solution to both primal and dual problems we then discuss the application of this approach to sparse covariance selection that is approximately solved as an l 1 norm penalized maximum likelihood estimation problem and also propose a variant of this approach which has substantially outperformed the latter one in our computational experiments we finally compare the performance of these approaches with other first order methods namely nesterov s cal o 1 epsilon smooth approximation scheme and block coordinate descent method studied in a d aspremont o banerjee and l el ghaoui siam j matrix anal appl 30 2008 pp 56 66 j friedman t hastie and r tibshirani biostatistics 9 2008 pp 432 441 for sparse covariance selection on a set of randomly generated instances it shows that our smooth optimization approach substantially outperforms the first method above and moreover its variant substantially outperforms both methods above

a nonlinear programming algorithm for solving semidefinite programs via low rank factorization in this paper we present a nonlinear programming algorithm for solving semidefinite programs sdps in standard form the algorithm s distinguishing feature is a change of variables that replaces the symmetric positive semidefinite variable x of the sdp with a rectangular variable r according to the factorization x rr t the rank of the factorization i e the number of columns of r is chosen minimally so as to enhance computational speed while maintaining equivalence with the sdp fundamental results concerning the convergence of the algorithm are derived and encouraging computational results on some large scale test problems are also presented

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

consistency of trace norm minimization regularization by the sum of singular values also referred to as the trace norm is a popular technique for estimating low rank rectangular matrices in this paper we extend some of the consistency results of the lasso to provide necessary and sufficient conditions for rank consistency of trace norm minimization with the square loss we also provide an adaptive version that is rank consistent even when the necessary condition for the non adaptive version is not fulfilled

geometric structures gromov norm and kodaira dimensions abstract we define the kodaira dimension for 3 dimensional manifolds through thurston s eight geometries along with a classification in terms of this kodaira dimension we show this is compatible with other existing kodaira dimensions and the partial order defined by non zero degree maps for higher dimensions we explore the relations of geometric structures and mapping orders with various kodaira dimensions and other invariants especially we show that a closed geometric 4 manifold has nonvanishing gromov norm if and only if it has geometry h 2 h 2 h 2 c or h 4

decomposable markov decision processes a fluid optimization approach decomposable markov decision processes mdps are problems where the stochastic system can be decomposed into multiple individual components although such mdps arise naturally in many practical applications they are often difficult to solve exactly due to the enormous size of the state space of the complete system which grows exponentially with the number of components in this paper we propose an approximate solution approach to decomposable mdps that is based on re solving a fluid linear optimization formulation of the problem at each decision epoch this formulation tractably approximates the problem by modeling transition behavior at the level of the individual components rather than the complete system we prove that our fluid formulation provides a tighter bound on the optimal value function than three state of the art formulations the approximate linear optimization formulation the classical lagrangian relaxation formulation and a novel alternate lagrangian relaxation that is based on relax

collapsing manifolds obtained by kummer type constructions we construct f structures on a bott manifold and on some other manifolds obtained by kummer type constructions we also prove that if m e x where e is a fiber bundle with structure group g and a fiber admitting a g invariant metric of non negative sectional curvature and x admits an f structure with one trivial covering then one can construct on m a sequence of metrics with sectional curvature uniformly bounded from below and volume tending to zero i e vol k m 0 as a corollary we prove that all the elements in the spin cobordism ring can be represented by manifolds m with vol k m 0

lusternik schnirelmann category and systolic category of low dimensional manifolds we show that the geometry of a riemannian manifold m is sensitive to the apparently purely homotopy theoretic invariant of m known as the lusternik schnirelmann category denoted catls m here we introduce a riemannian analogue of catls m called the systolic category of m it is denoted catsys m and defined in terms of the existence of systolic inequalities satisfied by every metric as initiated by c loewner and later developed by m gromov we compare the two categories in all our examples the inequality catsysm catlsm is satisfied which typically turns out to be an equality e g in dimension 3 we show that a number of existing systolic inequalities can be reinterpreted as special cases of such equality and that both categories are sensitive to massey products the comparison with the value of catls m leads us to prove or conjecture new systolic inequalities on m 2006 wiley periodicals inc

a new inexact alternating directions method for monotone variational inequalities the alternating directions method adm is an effective method for solving a class of variational inequalities vi when the proximal and penalty parameters in sub vi problems are properly selected in this paper we propose a new adm method which needs to solve two strongly monotone sub vi problems in each iteration approximately and allows the parameters to vary from iteration to iteration the convergence of the proposed adm method is proved under quite mild assumptions and flexible parameter conditions

robust subspace discovery via relaxed rank minimization this letter examines the problem of robust subspace discovery from input data samples instances in the presence of overwhelming outliers and corruptions a typical example is the case where we are given a set of images each image contains for example a face at an unknown location of an unknown size our goal is to identify or detect the face in the image and simultaneously learn its model we employ a simple generative subspace model and propose a new formulation to simultaneously infer the label information and learn the model using low rank optimization solving this problem enables us to simultaneously identify the ownership of instances to the subspace and learn the corresponding subspace model we give an efficient and effective algorithm based on the alternating direction method of multipliers and provide extensive simulations and experiments to verify the effectiveness of our method the proposed scheme can also be used to tackle many related high dimensional combinatorial selection problems

entropy and collapsing of compact complex surfaces we study the problem of existence of mathcal f structures in the sense of cheeger and gromov but not necessarily polarized on compact complex surfaces we give a complete classification of compact complex surfaces of kahler type admitting mathcal f structures in the non kahler case we give a complete classification modulo the gap in the classification of surfaces of class vii we then use these results to study the minimal entropy problem for compact complex surfaces we prove that modulo the gap in the classification of surfaces of class vii all compact complex surfaces of kodaira dimension less than or equal to 1 have minimal entropy zero and such a surface admits a smooth metric g with zero topological entropy if and only if it is mathbb c p 2 a ruled surface of genus 0 or 1 a hopf surface a complex torus a kodaira surface a hyperelliptic surface or a kodaira surface modulo a finite group the key result we use to prove this is a new topological obstruction to the existence of metrics with vanishing topological entropy finally we show that these results fit perfectly into wall s study of geometric structures on compact complex surfaces for instance we show that the minimal entropy problem can be solved for a minimal compact kahler surface s of kodaira dimension infty 0 or 1 if and only if s admits a geometric structure modelled on mathbb c p 2 s 2 times s 2 s 2 times mathbb e 2 or mathbb e 4

general projective splitting methods for sums of maximal monotone operators we describe a general projective framework for finding a zero of the sum of n maximal monotone operators over a real hilbert space unlike prior methods for this problem we neither assume n 2 nor first reduce the problem to the case n 2 our analysis defines a closed convex extended solution set for which we can construct a separating hyperplane by individually evaluating the resolvent of each operator at the cost of a single computationally simple projection step this framework gives rise to a family of splitting methods of unprecedented flexibility numerous parameters including the proximal stepsize may vary by iteration and by operator the order of operator evaluation may vary by iteration and may be either serial or parallel the analysis essentially generalizes our prior results for the case n 2 we also include a relative error criterion for approximately evaluating resolvents which was not present in our earlier work

latent variable graphical model selection via convex optimization suppose we observe samples of a subset of a collection of random variables no additional information is provided about the number of latent variables nor of the relationship between the latent and observed variables is it possible to discover the number of latent components and to learn a statistical model over the entire collection of variables we address this question in the setting in which the latent and observed variables are jointly gaussian with the conditional statistics of the observed variables conditioned on the latent variables being specified by a graphical model as a first step we give natural conditions under which such latent variable gaussian graphical models are identifiable given marginal statistics of only the observed variables essentially these conditions require that the conditional graphical model among the observed variables is sparse while the effect of the latent variables is spread out over most of the observed variables next we propose a tractable convex program based on regularized maximum likelihood for model selection in this latent variable setting the regularizer uses both the ell 1 norm and the nuclear norm our modeling framework can be viewed as a combination of dimensionality reduction to identify latent variables and graphical modeling to capture remaining statistical structure not attributable to the latent variables and it consistently estimates both the number of latent components and the conditional graphical model structure among the observed variables these results are applicable in the high dimensional setting in which the number of latent observed variables grows with the number of samples of the observed variables the geometric properties of the algebraic varieties of sparse matrices and of low rank matrices play an important role in our analysis

recovering low rank and sparse components of matrices from incomplete and noisy observations many problems can be characterized by the task of recovering the low rank and sparse components of a given matrix recently it was discovered that this nondeterministic polynomial time hard np hard task can be well accomplished both theoretically and numerically via heuristically solving a convex relaxation problem where the widely acknowledged nuclear norm and l 1 norm are utilized to induce low rank and sparsity this paper studies the recovery task in the general settings that only a fraction of entries of the matrix can be observed and the observation is corrupted by both impulsive and gaussian noise we show that the resulting model falls into the applicable scope of the classical augmented lagrangian method moreover the separable structure of the new model enables us to solve the involved subproblems more efficiently by splitting the augmented lagrangian function hence some splitting numerical algorithms are developed for solving the new recovery model some preliminary numerical experiments verify that these augmented lagrangian based splitting algorithms are easily implementable and surprisingly efficient for tackling the new recovery model

high dimensional semiparametric gaussian copula graphical models ing the spearman s rho and kendall s tau we prove that the nonparanormal skeptic achieves the optimal parametric rates of convergence for both graph recovery and parameter estimation this result suggests that the nonparanormal graphical models can be used as a safe replacement of the popular gaussian graphical models even when the data are truly gaussian besides theoretical analysis we also conduct thorough numerical simulations to compare the graph recovery performance of dierent estimators under both ideal and noisy settings the proposed methods are then applied on a largescale genomic dataset to illustrate their empirical usefulness the r package huge implementing the proposed methods is available on the comprehensive r archive network http cran r project org

solving log determinant optimization problems by a newton cg primal proximal point algorithm we propose a newton cg primal proximal point algorithm ppa for solving large scale log determinant optimization problems our algorithm employs the essential ideas of ppa the newton method and the preconditioned cg solver when applying the newton method to solve the inner subproblem we find that the log determinant term plays the role of a smoothing term as in the traditional smoothing newton technique focusing on the problem of maximum likelihood sparse estimation of a gaussian graphical model we demonstrate that our algorithm performs favorably compared to existing state of the art algorithms and is much preferred when a high quality solution is required for problems with many equality constraints

alternating direction method with gaussian back substitution for separable convex programming we consider the linearly constrained separable convex minimization problem whose objective function is separable into m individual convex functions with nonoverlapping variables a douglas rachford alternating direction method of multipliers adm has been well studied in the literature for the special case of m 2 but the convergence of extending adm to the general case of m ge 3 is still open in this paper we show that the straightforward extension of adm is valid for the general case of m ge 3 if it is combined with a gaussian back substitution procedure the resulting adm with gaussian back substitution is a novel approach towards the extension of adm from m 2 to m ge 3 and its algorithmic framework is new in the literature for the adm with gaussian back substitution we prove its convergence via the analytic framework of contractive type methods and we show its numerical efficiency by some application problems

fast alternating linearization methods for minimizing the sum of two convex functions we present in this paper alternating linearization algorithms based on an alternating direction augmented lagrangian approach for minimizing the sum of two convex functions our basic methods require at most o 1 epsilon iterations to obtain an epsilon optimal solution while our accelerated i e fast versions of them require at most o 1 sqrt epsilon iterations with little change in the computational effort required at each iteration for both types of methods we present one algorithm that requires both functions to be smooth with lipschitz continuous gradients and one algorithm that needs only one of the functions to be so algorithms in this paper are gauss seidel type methods in contrast to the ones proposed by goldfarb and ma in fast multiple splitting algorithms for convex optimization columbia university 2009 where the algorithms are jacobi type methods numerical results are reported to support our theoretical conclusions and demonstrate the practical potential of our algorithms

fixed point continuation for ell_1 minimization methodology and convergence we present a framework for solving the large scale ell 1 regularized convex minimization problem min x 1 mu f x our approach is based on two powerful algorithmic ideas operator splitting and continuation operator splitting results in a fixed point algorithm for any given scalar mu continuation refers to approximately following the path traced by the optimal value of x as mu increases in this paper we study the structure of optimal solution sets prove finite convergence for important quantities and establish q linear convergence rates for the fixed point algorithm applied to problems with f x convex but not necessarily strictly convex the continuation framework motivated by our convergence results is demonstrated to facilitate the construction of practical algorithms

classification and characterization of rationally elliptic manifolds in low dimensions we give a characterization of closed simply connected rationally elliptic 6 manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings we classify rational real and complex homotopy types of closed simply connected rationally elliptic 7 manifolds we give partial results in dimensions 8 and 9

a fast algorithm for sparse reconstruction based on shrinkage subspace optimization and continuation we propose a fast algorithm for solving the ell 1 regularized minimization problem min x in mathbb r n mu x 1 ax b 2 2 for recovering sparse solutions to an undetermined system of linear equations ax b the algorithm is divided into two stages that are performed repeatedly in the first stage a first order iterative shrinkage method yields an estimate of the subset of components of x likely to be nonzero in an optimal solution restricting the decision variables x to this subset and fixing their signs at their current values reduces the ell 1 norm x 1 to a linear function of x the resulting subspace problem which involves the minimization of a smaller and smooth quadratic function is solved in the second phase our code fpc as embeds this basic two stage algorithm in a continuation homotopy approach by assigning a decreasing sequence of values to mu this code exhibits state of the art performance in terms of both its speed and its ability to recover sparse signals

positive definite l1 penalized estimation of large covariance matrices the thresholding covariance estimator has nice asymptotic properties for estimating sparse large covariance matrices but it often has negative eigenvalues when used in real data analysis to fix this drawback of thresholding estimation we develop a positive definite l1 penalized covariance estimator for estimating sparse large covariance matrices we derive an efficient alternating direction method to solve the challenging optimization problem and establish its convergence properties under weak regularity conditions nonasymptotic statistical theory is also established for the proposed estimator the competitive finite sample performance of our proposal is demonstrated by both simulation and real applications

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

solving semidefinite quadratic linear programs using sdpt3 this paper discusses computational experiments with linear optimization problems involving semidefinite quadratic and linear cone constraints sqlps many test problems of this type are solved using a new release of sdpt3 a matlab implementation of infeasible primal dual path following algorithms the software developed by the authors uses mehrotra type predictor corrector variants of interior point methods and two types of search directions the hkm and nt directions a discussion of implementation details is provided and computational results on problems from the sdplib and dimacs challenge collections are reported

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

graphical model selection for gaussian conditional random fields in the presence of latent variables we consider the problem of learning a conditional gaussian graphical model in the presence of latent variables building on recent advances in this field we suggest a method that decomposes the pa

regularization methods for semidefinite programming we introduce a new class of algorithms for solving linear semidefinite programming sdp problems our approach is based on classical tools from convex optimization such as quadratic regularization and augmented lagrangian techniques we study the theoretical properties and we show that practical implementations behave very well on some instances of sdp having a large number of constraints we also show that the boundary point method from povh rendl and wiegele computing 78 2006 pp 277 286 is an instance of this class

continuity properties of entropy on donne une borne superieure sur le defaut en semicontinuite superieure des entropies topologiques et metriques d une auto application c r d une variete c r r 1

partial correlation estimation by joint sparse regression models this article features online supplementary material in this article we propose a computationally efficient approach space sparse partial correlation estimation for selecting nonzero partial correlations under the high dimension low sample size setting this method assumes the overall sparsity of the partial correlation matrix and employs sparse regression techniques for model fitting we illustrate the performance of space by extensive simulation studies it is shown that space performs well in both nonzero partial correlation selection and the identification of hub variables and also outperforms two existing methods we then apply space to a microarray breast cancer dataset and identify a set of hub genes that may provide important insights on genetic regulatory networks finally we prove that under a set of suitable assumptions the proposed procedure is asymptotically consistent in terms of model selection and parameter estimation

fast multiple splitting algorithms for convex optimization we present in this paper two different classes of general multiple splitting algorithms for solving finite dimensional convex optimization problems under the assumption that the function being minimized can be written as the sum of k convex functions each of which has a lipschitz continuous gradient we prove that the number of iterations needed by the first class of algorithms to obtain an epsilon optimal solution is o k 1 l epsilon where l is an upper bound on all of the lipschitz constants the algorithms in the second class are accelerated versions of those in the first class where the complexity result is improved to o sqrt k 1 l epsilon while the computational effort required at each iteration is almost unchanged to the best of our knowledge the complexity results presented in this paper are the first ones of this type that have been given for splitting and alternating direction type methods moreover all algorithms proposed in this paper are parallelizable which makes the

alternating direction methods for latent variable gaussian graphical model selection chandrasekaran parrilo and willsky 2012 proposed a convex optimization problem for graphical model selection in the presence of unobserved variables this convex optimization problem aims to estimate an inverse covariance matrix that can be decomposed into a sparse matrix minus a low rank matrix from sample data solving this convex optimization problem is very challenging especially for large problems in this letter we propose two alternating direction methods for solving this problem the first method is to apply the classic alternating direction method of multipliers to solve the problem as a consensus problem the second method is a proximal gradient based alternating direction method of multipliers our methods take advantage of the special structure of the problem and thus can solve large problems very efficiently a global convergence result is established for the proposed methods numerical results on both synthetic data and gene expression data show that our methods usually solve problems with 1 million variables in 1 to 2 minutes and are usually 5 to 35i times faster than a state of the art newton cg proximal point algorithm

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

proximal decomposition via alternating linearization a new approximate proximal point method for minimizing the sum of two convex functions is introduced it replaces the original problem by a sequence of regularized subproblems in which the functions are alternately represented by linear models the method updates the linear models and the prox center as well as the prox coefficient it is monotone in terms of the objective values and converges to a solution of the problem if any a dual version of the method is derived and analyzed applications of the methods to multistage stochastic programming problems are discussed and preliminary numerical experience is presented

a proximal based decomposition method for convex minimization problems this paper presents a decomposition method for solving convex minimization problems at each iteration the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables we derive this algorithm from rockafellar s proximal method of multipliers which involves an augmented lagrangian with an additional quadratic proximal term the algorithm preserves the good features of the proximal method of multipliers with the additional advantage that it leads to a decoupling of the constraints and is thus suitable for parallel implementation we allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem s data the method is globally convergent and at a linear rate the method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets

on the linear convergence of the alternating direction method of multipliers we analyze the convergence rate of the alternating direction method of multipliers admm for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints previous analysis of the admm typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks moreover there has been no rate of convergence analysis for the admm without strong convexity in the objective function in this paper we establish the global r linear convergence of the admm for minimizing the sum of any number of convex separable functions assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping and a nonsmooth ell 1 l1 regularizer this result implies the linear convergence of the admm for contemporary applications such as lasso without assuming strong convexity of the objective function

ricci curvature minimal volumes and seiberg witten theory we derive new sharp lower bounds for certain curvature functionals on the space of riemannian metrics of a smooth compact 4 manifold with non trivial seiberg witten invariants these allow one for example to exactly compute the infimum of the l 2 norm of ricci curvature for any complex surface of general type we are also able to show that the standard metric on any complex hyperbolic 4 manifold minimizes volume among all metrics satisfying a point wise lower bound on sectional curvature plus suitable multiples of the scalar curvature these estimates also imply new non existence results for einstein metrics

model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data we consider the problem of estimating the parameters of a gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse our approach is to solve a maximum likelihood problem with an added l1 norm penalty term the problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes we present two new algorithms for solving problems with at least a thousand nodes in the gaussian case our first algorithm uses block coordinate descent and can be interpreted as recursive l1 norm penalized regression our second algorithm based on nesterov s first order method yields a complexity estimate with a better dependence on problem size than existing interior point methods using a log determinant relaxation of the log partition function wainwright and jordan 2006 we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case we test our algorithms on synthetic data as well as on gene expression and senate voting records data