shrinking characteristics of precision matrix estimators we propose a framework to shrink a user specified characteristic of a precision matrix estimator that is needed to fit a predictive model estimators in our framework minimize the gaussian negative loglikelihood plus an l 1 penalty on a linear or affine function evaluated at the optimization variable corresponding to the precision matrix we establish convergence rate bounds for these estimators and propose an alternating direction method of multipliers algorithm for their computation our simulation studies show that our estimators can perform better than competitors when they are used to fit predictive models in particular we illustrate cases where our precision matrix estimators perform worse at estimating the population precision matrix but better at prediction

bayesian reconstructions from emission tomography data using a modified em algorithm a novel method of reconstruction from single photon emission computerized tomography data is proposed this method builds on the expectation maximization em approach to maximum likelihood reconstruction from emission tomography data but aims instead at maximum posterior probability estimation which takes account of prior belief about smoothness in the isotope concentration a novel modification to the em algorithm yields a practical method the method is illustrated by an application to data from brain scans

nonuniversality of front fluctuations for compact colonies of nonmotile bacteria the front of a compact bacterial colony growing on a petri dish is a paradigmatic instance of non equilibrium fluctuations in the celebrated eden or kardar parisi zhang kpz universality class while in many experiments the scaling exponents crucially differ from the expected kpz values the source of this disagreement has remained poorly understood we have performed growth experiments with b subtilis 168 and e coli atcc 25922 under conditions leading to compact colonies in the classically alleged eden regime where individual motility is suppressed non kpz scaling is indeed observed for all accessible times kpz asymptotics being ruled out for our experiments due to the monotonic increase of front branching with time simulations of an effective model suggest the occurrence of transient nonuniversal scaling due to diffusive morphological instabilities agreeing with expectations from detailed models of the relevant biological reaction diffusion processes

computable elastic distances between shapes we dene distances between geometric curves by the square root of the minimal energy required to transform one curve into the other the energy is formally dened from a left invariant riemannian distance on an innite dimensional group acting on the curves which can be explicitly computed the obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed an analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem which can eciently be implemented as illustrated by experiments 1 1 generalities the problem of matching two objects together is very impor tant in computer vision and shape recognition in many cases recognition is based on shapes outlines with the help of some suitably designed distance a general principle is to associate with any pair o1 o 2 of objects to be compared a measure of discrepancy d o1 o 2 the recognition of an observed object o may be done by nding from a dictionary of templates the previously recorded object otemp for which d o otemp is minimal clearly the denition of the distance is the crucial step of the method and much research has been done in this direction we shall not try here to provide a review of the huge literature existing on the subject see for example 17 but rather focus on methods related to deformable templates with which we are directly concerned instead of basing recognition on a nite collection of points of interest primitives taken from the outline of an object corners inflexion points etc which is a popular way of handling the problem our purpose is to base the comparison on the whole outline considered as a plane curve the distance we shall dene incorporates some deformation energy between the curves the approach as we will see turns out to be intrinsic and robust to usual euclidean transformations the method is related to the wide literature on snakes 14 7 21 etc in the way that our distance corresponds to some continuous process of deformation of one curve into another it is also related to papers on elastic matching such as 8 however we provide an elastic matching algorithm which is based on a true distance between intrinsic properties of the shapes taking into account possible invariance to scaling or euclidean transformations in the case they are required from this point of view our results are indebted to the seminal work of grenander on group theory applied to pattern recognition cf 10 and 11 in particular see also 1 2 12 another source of inspiration may come from mathematical physics since we are going to look to the path process of lowest energy which deforms one object into

bayesian estimation of density via multiple sequential inversions of two dimensional images with application to electron microscopy we present a new bayesian methodology to learn the unknown material density of a given sample by inverting its two dimensional images that are taken with a scanning electron microscope an image results from a sequence of projections of the convolution of the density function with the unknown microscopy correction function that we also learn from the data thus learning of the unknowns demands multiple inversions we invoke a novel design of experiment involving imaging at multiple values of the parameter that controls the subsurface depth from which information about the density structure is carried to result in the image real life material density functions are characterized by high density contrasts and are highly discontinuous implying that they exhibit correlation structures that do not vary smoothly in the absence of training data modeling such correlation structures of real material density functions is not possible so we discretize the material sample and treat values of the density functi

on random tomography with unobservable projection angles we formulate and investigate a statistical inverse problem of a random tomographic nature where a probability density function on mathbb r 3 is to be recovered from observation of finitely many of its two dimensional projections in random and unobservable directions such a problem is distinct from the classic problem of tomography where both the projections and the unit vectors normal to the projection plane are observable the problem arises in single particle electron microscopy a powerful method that biophysicists employ to learn the structure of biological macromolecules strictly speaking the problem is unidentifiable and an appropriate reformulation is suggested hinging on ideas from kendall s theory of shape within this setup we demonstrate that a consistent solution to the problem may be derived without attempting to estimate the unknown angles if the density is assumed to admit a mixture representation

parametric deconvolution of positive spike trains this paper describes a parametric deconvolution method pdps appropriate for a particular class of signals which we call spike convolution models these models arise when a sparse spike train dirac deltas according to our mathematical treatment is convolved with a fixed point spread function and additive noise or measurement error is superimposed we view deconvolution as an estimation problem regarding the locations and heights of the underlying spikes as well as the baseline and the measurement error variance as unknown parameters our estimation scheme consists of two parts model fitting and model selection to fit a spike convolution model of a specific order we estimate peak locations by trigonometric moments and heights and the baseline by least squares the model selection procedure has two stages its first stage is so designed that we expect a model of a somewhat larger order than the truth to be selected in the second stage the final model is obtained using backwards deletion this results in not only an estimate of the model order but also an estimate of peak locations and heights with much smaller bias and variation than that found in a direct trigonometric moment estimate a more efficient maximum likelihood estimate can be calculated from these estimates using a gauss newton algorithm we also present some relevant results concerning the spectral structure of toeplitz matrices which play a key role in the estimation finally we illustrate the behavior of these estimates using simulated and real dna sequencing data

a study of least squares and maximum likelihood for image reconstruction in positron emission tomography positron emission tomography pet is a radiologic tool offering a unique capability for measuring tissue metabolism in vivo a number of biological and physical factors limit the resolution of pet so often the statistical aspects of image reconstruction have an appreciable effect on the quality of information obtained from a study to a first approximation reconstruction involves the solution of a linear inverse problem with a line integral radon type transform standard filtered back projection reconstruction is based on the method of least squares although computationally efficient the method does not enforce positivity constraints leading to undesirable negative artifacts in the results maximum likelihood based approaches to reconstruction do not suffer from this problem but their computational complexity has limited the ability to determine quantitatively the improvements in image quality in this paper asymptotic approximations and numerical simulations are used to examine the least squares and maximum likelihood methods in some detail the studies are carried out for idealized representations of conventional and time of flight tomographs the asymptotic analysis indicates that for a range of sobolev norms the rates of estimation of least squares and maximum likelihood reconstructions are of the same order this is borne out by numerical studies however in these studies maximum likelihood is found to be more efficient than least squares on a conventional distance angle tomograph the root mean square error is on the order of 10 20 smaller for maximum likelihood reconstructions the corresponding reduction in the root mean square error on a time of flight tomograph is in the 5 range similar results are found using more complex region of interest based measures of reconstruction accuracy in addition it is found that much of the effect of maximum likelihood can apparently be realized by postprocessing least squares solutions in an ad hoc manner to reduce negativity artifacts since the postprocessing scheme is computationally fast this suggests that in pet it may be possible to realize the benefits of maximum likelihood without the substantial computational overhead of the em algorithm

the statistical analysis of dynamic curves and sections abstract by a curve we shall understand a one dimensional smooth path lying in r 2 or r 3 which can be naturally parametrized by a real coordinate the coordinate could represent physical time or any other variable which can be interpreted dynamically in some cases the curve will arise as the linear section of a higher dimensional structure such as the planar section of a surface in r 3 in this paper we develop a model for the shape of planar curves based on their curvatures that is reasonably robust to the location of landmarks or knots used to approximate the contours of the curve the measurement for the shape difference between two curves that we propose is also based on the curvatures of the curves and directly inherits the simple euclidean property for averages

multidimensional density estimation by tomography the paper explores the application of the filtered backprojection algorithm of computerized tomography to statistical smoothing of multidimensional densities the method makes use of automated one dimensional density smoothers to obtain an automated smoothed multidimensional density no structural assumptions such as additivity or log additivity are used an important computational aspect of this method is its inherently parallel structure we demonstrate linear speed up on a parallel computer architecture various practical details are discussed together with some illustrative two dimensional examples

covariance enhanced discriminant analysis linear discriminant analysis has been widely used to characterize or separate multiple classes via linear combinations of features however the high dimensionality of features from modern biological experiments defies traditional discriminant analysis techniques possible interfeature correlations present additional challenges and are often underused in modelling in this paper by incorporating possible interfeature correlations we propose a covariance enhanced discriminant analysis method that simultaneously and consistently selects informative features and identifies the corresponding discriminable classes under mild regularity conditions we show that the method can achieve consistent parameter estimation and model selection and can attain an asymptotically optimal misclassification rate extensive simulations have verified the utility of the method which we apply to a renal transplantation trial

a smoothed em approach to indirect estimation problems with particular reference to stereology and emission tomography there are many practical problems where the observed data are not drawn directly from the density g of real interest but rather from another distribution derived from g by the application of an integral operator the estimation of g then entails both statistical and numerical difficulties a natural statistical approach is by maximum likelihood conveniently implemented using the em algorithm but this provides unsatisfactory reconstructions of g in this paper we modify the maximum likelihood em approach by introducing a simple smoothing step at each em iteration in our experience this algorithm converges in relatively few iterations to good estimates of g that do not depend on the choice of starting configuration some theoretical background is given that relates this smoothed em algorithm to a maximum penalized likelihood approach two applications are considered in detail the first is the classical stereology problem of determining particle size distributions from data collected on a plane section through a composite medium the second concerns the recovery of the structure of a section of the human body from external observations obtained by positron emission tomography for this problem we also suggest several technical improvements on existing methodology

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

power accretion in social systems we consider a model of power distribution in a social system where a set of agents plays a simple game on a graph the probability of winning each round is proportional to the agent s current power and the winner gets more power as a result we show that when the agents are distributed on simple one dimensional and two dimensional networks inequality grows naturally up to a certain stationary value characterized by a clear division between a higher and a lower class of agents high class agents are separated by one or several lower class agents which serve as a geometrical barrier preventing further flow of power between them moreover we consider the effect of redistributive mechanisms such as proportional nonprogressive taxation sufficient taxation will induce a sharp transition towards a more equal society and we argue that the critical taxation level is uniquely determined by the system geometry interestingly we find that the roughness and shannon entropy of the power distributions are a very useful complement to the standard measures of inequality such as the gini index and the lorenz curve

covariance regularized regression and classification for high dimensional problems in recent years many methods have been developed for regression in high dimensional settings we propose covariance regularized regression a family of methods that use a shrunken estimate of the inverse covariance matrix of the features in order to achieve superior prediction an estimate of the inverse covariance matrix is obtained by maximizing its log likelihood under a multivariate normal model subject to a constraint on its elements this estimate is then used to estimate coefficients for the regression of the response onto the features we show that ridge regression the lasso and the elastic net are special cases of covariance regularized regression and we demonstrate that certain previously unexplored forms of covariance regularized regression can outperform existing methods in a range of situations the covariance regularized regression framework is extended to generalized linear models and linear discriminant analysis and is used to analyze gene expression data sets with multiple class and survival outcomes

on nonparametric estimation of intercept and slope distributions in random coefficient regression an experiment records stimulus and response for a random sample of cases the relationship between response and stimulus is thought to be linear the values of the slope and intercept varying by case from such data we construct a consistent asymptotically normal nonparametric estimator for the joint density of the slope and intercept our methodology incorporates the radial projection slice theorem for the radon transform a technique for locally linear nonparametric regression and a tapered fourier inversion computationally the new density estimator is more feasible than competing nonparametric estimators one of which is based on moments and the other on minimum distance considerations

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

the riemannian structure of euclidean shape spaces a novel environment for statistics the riemannian metric structure of the shape space sum k m for k labelled points in mathbb r m was given by kendall for the atypically simple situations in which m 1 or 2 and k geq 2 here we deal with the general case m geq 1 k geq 2 by using the properties of riemannian submersions and warped products as studied by o neill the approach is via the associated size and shape space that is the warped product of the shape space and the half line mathbb r carrying size the warping function being equal to the square of the size when combined with parallel studies by le of the corresponding global geodesic geometry the results obtained here determine the environment in which shape statistical calculations have to be acted out finally three different applications are discussed that illustrate the theory and its use in practice

a direct estimation approach to sparse linear discriminant analysis this article considers sparse linear discriminant analysis of high dimensional data in contrast to the existing methods which are based on separate estimation of the precision matrix and the difference of the mean vectors we introduce a simple and effective classifier by estimating the product directly through constrained l1 minimization the estimator can be implemented efficiently using linear programming and the resulting classifier is called the linear programming discriminant lpd rule the lpd rule is shown to have desirable theoretical and numerical properties it exploits the approximate sparsity of and as a consequence allows cases where it can still perform well even when and or cannot be estimated consistently asymptotic properties of the lpd rule are investigated and consistency and rate of convergence results are given the lpd classifier has superior finite sample performance and significant computational advantages over the existing methods that require separate estimation

speed of estimation in positron emission tomography and related inverse problems we study a continuous idealization of the image reconstruction problem in positron emission tomography considered as an example of bivariate density estimation based on indirect observations given a large sample of indirect observations we consider the size of the equivalent sample of observations whose original exact positions would allow equally accurate estimation of the image of interest both for indirect and for direct observations we establish exact minimax rates of convergence of estimation for all possible estimators over suitable smoothness classes of functions the approach of the paper is applicable to a wide class of linear inverse problems

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

generalized thresholding of large covariance matrices we propose a new class of generalized thresholding operators that combine thresholding with shrinkage and study generalized thresholding of the sample covariance matrix in high dimensions generalized thresholding of the covariance matrix has good theoretical properties and carries almost no computational burden we obtain an explicit convergence rate in the operator norm that shows the tradeoff between the sparsity of the true model dimension and the sample size and shows that generalized thresholding is consistent over a large class of models as long as the dimension p and the sample size n satisfy log p n 0 in addition we show that generalized thresholding has the sparsistency property meaning it estimates true zeros as zeros with probability tending to 1 and under an additional mild condition is sign consistent for nonzero elements we show that generalized thresholding covers as special cases hard and soft thresholding smoothly clipped absolute deviation and adaptive lasso and comp

penalized classification using fisher s linear discriminant we consider the supervised classification setting in which the data consist of p features measured on n observations each of which belongs to one of k classes linear discriminant analysis lda is a classical method for this problem however in the high dimensional setting where p n lda is not appropriate for two reasons first the standard estimate for the within class covariance matrix is singular and so the usual discriminant rule cannot be applied second when p is large it is difficult to interpret the classification rule obtained from lda since it involves all p features we propose penalized lda a general approach for penalizing the discriminant vectors in fisher s discriminant problem in a way that leads to greater interpretability the discriminant problem is not convex so we use a minorization maximization approach in order to efficiently optimize it when convex penalties are applied to the discriminant vectors in particular we consider the use of l 1 and fused lasso penalties our proposal is equivalent to recasting fisher s discriminant problem as a biconvex problem we evaluate the performances of the resulting methods on a simulation study and on three gene expression data sets we also survey past methods for extending lda to the high dimensional setting and explore their relationships with our proposal

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

a kpz cocktail shaken not stirred the stochastic partial differential equation proposed nearly three decades ago by kardar parisi and zhang kpz continues to inspire intrigue and confound its many admirers here we i pay debts to heroic predecessors ii highlight additional experimentally relevant aspects of the recently solved 1 1 kpz problem iii use an expanding substrates formalism to gain access to the 3d radial kpz equation and lastly iv examining extremal paths on disordered hierarchical lattices set our gaze upon the fate of d infty kpz clearly there remains ample unexplored territory within the realm of kpz and for the hearty much work to be done especially in higher dimensions where numerical and renormalization group methods are providing a deeper understanding of this iconic equation

a road to classification in high dimensional space the regularized optimal affine discriminant for high dimensional classification it is well known that naively performing the fisher discriminant rule leads to poor results due to diverging spectra and noise accumulation therefore researchers proposed independence rules to circumvent the diverging spectra and sparse independence rules to mitigate the issue of noise accumulation however in biological applications there are often a group of correlated genes responsible for clinical outcomes and the use of the covariance information can significantly reduce misclassification rates in theory the extent of such error rate reductions is unveiled by comparing the misclassification rates of the fisher discriminant rule and the independence rule to materialize the gain based on finite samples a regularized optimal affine discriminant road is proposed road selects an increasing number of features as the regularization relaxes further benefits can be achieved when a screening method is employed to narrow the feature pool before hitting the road an efficient constrained coordinate descent algorithm ccd is also developed to solve the associated optimization problems sampling properties of oracle type are established simulation studies and real data analysis support our theoretical results and demonstrate the advantages of the new classification procedure under a variety of correlation structures a delicate result on continuous piecewise linear solution path for the road optimization problem at the population level justifies the linear interpolation of the ccd algorithm

circular kardar parisi zhang equation as an inflating self avoiding ring polymer we consider the kardar parisi zhang equation for a circular interface in two dimensions unconstrained by the standard small slope and no overhang approximations numerical simulations using an adaptive scheme allow us to elucidate the complete time evolution as a crossover between a short time regime with the interface fluctuations of a self avoiding ring or two dimensional vesicle and a long time regime governed by the tracy widom distribution expected for this geometry for small noise amplitudes scaling behavior is only of the latter type large noise is also seen to renormalize the bare physical parameters of the ring akin to analogous parameter renormalization for equilibrium three dimensional membranes our results bear particular importance on the relation between relevant universality classes of scale invariant systems in two dimensions

consistent estimation of mixture complexity the consistent estimation of mixture complexity is of fundamental importance in many applications of finite mixture models an enormous body of literature exists regarding the application computational issues and theoretical aspects of mixture models when the number of components is known but estimating the unknown number of components remains an area of intense research effort this article presents a semiparametric methodology yielding almost sure convergence of the estimated number of components to the true but unknown number of components the scope of application is vast as mixture models are routinely employed across the entire diverse application range of statistics including nearly all of the social and experimental sciences

multiparticle biased diffusion limited aggregation with surface diffusion a comprehensive model of electrodeposition we present a complete study of the multiparticle biased diffusion limited aggregation mbdla model supplemented with surface diffusion sd focusing on the relevance and effects of the latter transport mechanism by comparing different algorithms we show that mbdla 1 sd is a very good qualitative model for electrodeposition in essentially the whole range of current intensities provided one introduces sd in the model in the proper fashion we have found that the correct procedure involves simultaneous bulk diffusion and sd introducing a time scale arising from the ratio of the rates of the two processes we discuss in detail the different morphologies obtained and compare them to the available experimental data with very satisfactory results we also characterize the aggregates thus obtained by means of the dynamic scaling exponents of the interface height allowing us to distinguish several regimes in the mentioned interface growth our asymptotic scaling exponents are again in good agreement with recent experiments we conclude by discussing a global picture of the influence and consequences of sd in electrodeposition