how to project onto an isotone projection cone abstract the solution of the complementarity problem defined by a mapping f r n r n and a cone k r n consists of finding the fixed points of the operator p k i f where p k is the projection onto the cone k and i stands for the identity mapping for the class of isotone projection cones cones admitting projections isotone with respect to the order relation they generate and f satisfying certain monotonicity properties the solution can be obtained by iterative processes see g isac a b nemeth projection methods isotone projection cones and the complementarity problem j math anal appl 153 1 1990 258 275 and s z nemeth iterative methods for nonlinear complementarity problems on isotone projection cones j math anal appl 350 1 2009 340 347 these algorithms require computing at each step the projection onto the cone k in general computing the projection mapping onto a cone k is a difficult and computationally expensive problem in this note it is shown that the projection of an arbitrary point onto an isotone projection cone in r n can be obtained by projecting recursively at most n 1 times into subspaces of decreasing dimension this emphasizes the efficiency of the algorithms mentioned above and furnishes a handy tool for some problems involving special isotone projection cones as for example the non negative monotone cones occurring in reconstruction problems see e g section 5 13 in j dattorro convex optimization and euclidean distance geometry meboo 2005 v2009 04 11

a convergence analysis of dykstra s algorithm for polyhedral sets let h be a nonempty closed convex set in a hilbert space x determined by the intersection of a finite number of closed half spaces it is well known that given an x 0 in x dykstra s algorithm applied to this collection of closed half spaces generates a sequence of iterates that converge to ph x0 the orthogonal projection of x0 onto h the iterates however do not necessarily lie in h we propose a combined dykstra conjugate gradient method such that given an varepsilon 0 the algorithm computes an x in h with x p h x 0 varepsilon moreover for each iterate xm of dykstra s algorithm we calculate a bound for x m p h x 0 that approaches zero as m tends to infinity applications are made to computing bounds for x m p h x 0 where h is a polyhedral cone numerical results are presented from a sample isotone regression problem

efficient rank reduction of correlation matrices geometric optimisation algorithms are developed that efficiently find the nearest low rank correlation matrix we show in numerical tests that our methods compare favourably with the existing methods in the literature the connection with the lagrange multiplier method is established along with an identification of whether a local minimum is a global minimum an additional benefit of the geometric approach is that any weighted norm can be applied the problem of finding the nearest low rank correlation matrix occurs as part of the calibration of multi factor interest rate market models to correlation

global risk bounds and adaptation in univariate convex regression we consider the problem of nonparametric estimation of a convex regression function phi 0 we study the risk of the least squares estimator lse under the natural squared error loss we show that the risk is always bounded from above by n 4 5 modulo logarithmic factors while being much smaller when phi 0 is well approximable by a piecewise affine convex function with not too many affine pieces in which case the risk is at most 1 n up to logarithmic factors on the other hand when phi 0 has curvature we show that no estimator can have risk smaller than a constant multiple of n 4 5 in a very strong sense by proving a local minimax lower bound we also study the case of model misspecification where we show that the lse exhibits the same global behavior provided the loss is measured from the closest convex projection of the true regression function in the process of deriving our risk bounds we prove new results for the metric entropy of local neighborhoods of the space of univariate convex functions these results which may be of independent interest demonstrate the non uniform nature of the space of univariate convex functions in sharp contrast to classical function spaces based on smoothness constraints

additive conjoint isotonic probabilistic models adisop the isop model or model of twodimensional or bi isotonicity scheiblechner 1995 postulates that the probabilities of ordered response categories increase isotonically in the order of subject ability and item easiness adding a conventional cancellation axiom for the factors of subjects and items gives the adisop model where the c d f s of response categories are functions of an additive item and subject parameter and an ordinal category parameter extending cancellation to the interactions of subjects and categories as well as of items and categories independence axiom of the category factor from the subject and item factor gives the cadisop model completely additive model in which the parallel c d f s are functions of the sum of subject item and category parameters the cadisop model is very close to the unidimensional version of the polytomous rasch model with the logistic item category characteristic s replaced by nonparametric axioms and statistics the axioms representation theorems and algorithms for model fitting of the additive models are presented

existence of solutions for generalized differential equations with unbounded right hand side abstract this paper is concerned with the existence of solutions to the initial value problem for generalized differential equations orientor fields when the right hand side f t x may be unbounded two global and one local existence theorems are established when f satisfies caratheodory type conditions involving weak integral boundedness conditions the multiple valued function f t is assumed to have closed graph and to be lower semicontinuous at each point x where f t x is not convex

computing maximum likelihood estimators of convex density functions we consider the problem of estimating a density function that is known in advance to be convex the maximum likelihood estimator is then the solution of a linearly constrained convex minimization problem this problem turns out to be numerically difficult we show that interior point algorithms perform well on this class of optimization problem though for large samples numerical difficulties are still encountered to eliminate those difficulties we propose a clustering scheme that is reasonable from a statistical point of view we display results for problems with up to 40000 observations we also give a typical picture of the estimated density a piecewise linear function with only very few pieces

1 f α noise in the fluctuations of the spectra of tridiagonal random matrices from the β hermite ensemble the 1 f ensuremath alpha noise displayed by the fluctuation of the n th unfolded eigenvalue where n plays the role of a discrete time was recently characterized for the classical gaussian ensembles of n ifmmode times else texttimes fi n random matrices it is investigated here for the ensuremath beta hermite ensemble by wavelet analysis of monte carlo simulated series both as a function of ensuremath beta and of n when ensuremath beta decreases from 1 to 0 for a given and large enough n the evolution from a 1 f noise at ensuremath beta 1 gaussian orthogonal ensemble goe to a 1 f 2 noise at ensuremath beta 0 gaussian diagonal ensemble gde is heterogeneous with a ensuremath sim 1 f 2 noise at the finest scales and a ensuremath sim 1 f noise at the coarsest ones

multiview stereo and silhouette consistency via convex functionals over convex domains we propose a convex formulation for silhouette and stereo fusion in 3d reconstruction from multiple images the key idea is to show that the reconstruction problem can be cast as one of minimizing a convex functional where the exact silhouette consistency is imposed as convex constraints that restrict the domain of feasible functions as a consequence we can retain the original stereo weighted surface area as a cost functional without heuristic modifications of this energy by balloon terms or other strategies yet still obtain meaningful nonempty reconstructions which are guaranteed to be silhouette consistent we prove that the proposed convex relaxation approach provides solutions that lie within a bound of the optimal solution compared to existing alternatives the proposed method does not depend on initialization and leads to a simpler and more robust numerical scheme for imposing silhouette consistency obtained by projection onto convex sets we show that this projection can be solved exactly using an efficient algorithm we propose a parallel implementation of the resulting convex optimization problem on a graphics card given a photoconsistency map and a set of image silhouettes we are able to compute highly accurate and silhouette consistent reconstructions for challenging real world data sets in particular experimental results demonstrate that the proposed silhouette constraints help to preserve fine scale details of the reconstructed shape computation times depend on the resolution of the input imagery and vary between a few seconds and a couple of minutes for all experiments in this paper

hubert l arabie p meulman j 2006 the structural representation of proximity matrices with matlab philadelphia siam xvi 214 pp us 79 00 isbn 0898716071

dykstra s algorithm with strategies for projecting onto certain polyhedral cones we consider dykstra s alternating projection method when it is used to find the projection onto polyhedral cones of the form i 1 n email protected h 0 i j 1 n based on some properties of the projection we propose strategies with the aim to reduce the number of cycles and the execution time these strategies consist in previous discarding and arrangement and in projecting cyclically onto the intersection of two halfspaces encouraging preliminary numerical results with cut semimetrics as vectors v i are presented

the approximation of two mode proximity matrices by sums of order constrained matrices a least squares strategy is proposed for representing a two mode proximity matrix as an approximate sum of a small number of matrices that satisfy certain simple order constraints on their entries the primary class of constraints considered define q forms or anti q forms for a two mode matrix where after suitable and separate row and column reorderings the entries within each row and within each column are nondecreasing or nonincreasing to a maximum or minimum and thereafter nonincreasing or nondecreasing several other types of order constraints are also mentioned to show how alternative structures can be considered using the same computational strategy

signal recovery by best feasible approximation the objective of set theoretical signal recovery is to find a feasible signal in the form of a point in the intersection of s of sets modeling the information available about the problem for problems in which the true signal is known to lie near a reference signal r the solution should not be any feasible point but one which best approximates r i e a projection of r onto s such a solution cannot be obtained by the feasibility algorithms currently in use e g the method of projections onto convex sets pocs and its offsprings methods for projecting a point onto the intersection of closed and convex sets in a hilbert space are introduced and applied to signal recovery by best feasible approximation of a reference signal these algorithms are closely related to the above projection methods to which they add little computational complexity

selective alternating projections to find the nearest sdd matrix we extend and improve recently proposed algorithms to solve the problem of minimizing the distance from a given matrix to the cone of symmetric and diagonally dominant matrices with positive diagonal sdd we present a variety of criteria to select a subset of the supporting hyperplanes of the faces of sdd and also of the polar cone sdd 0 to then apply dykstra s alternating projection method these selections reduce the number of projections and therefore reduce the required computational work in all our new algorithms the symmetry and the diagonal dominance of the obtained matrix are guaranteed preliminary numerical experiments indicate that some of the selection criteria produce a significant reduction in cpu time

misleading signatures of quantum chaos the main signature of chaos in a quantum system is provided by spectral statistical analysis of the nearest neighbor spacing distribution p s and the spectral rigidity given by the delta 3 l statistic it is shown that some standard unfolding procedures such as local unfolding and gaussian broadening lead to a spurious saturation of delta 3 l that spoils the relationship of this statistic with the regular or chaotic motion of the system this effect can also be misinterpreted as berry s saturation

global existence of solutions of differential inclusions abstract in this paper we consider sufficient conditions for global existence of solutions to a differential inclusion dx dt f t x assuming existence of local solutions to it by means of lyapunov s second method our theorem includes as special cases s w seah s theorems 1982 j math anal appl 89 648 663

normal mode analysis of spectra of random networks abstract several spectral fluctuation measures of random matrix theory rmt have been applied in the study of spectral properties of networks however the calculation of those statistics requires performing an unfolding procedure which may not be an easy task in this work network spectra are interpreted as time series and we show how their short and long range correlations can be characterized without implementing any previous unfolding in particular we consider three different representations of erd s renyi er random networks standard er networks er networks with random weighted self edges and fully random weighted er networks in each case we apply singular value decomposition svd such that the spectra are decomposed in trend and fluctuation normal modes we obtain that the fluctuation modes exhibit a clear crossover between the poisson and the gaussian orthogonal ensemble statistics when the average degree of er networks changes moreover by using the trend modes we perform a data adaptive unfolding to calculate for comparison purposes traditional fluctuation measures such as the nearest neighbor spacing distribution number variance 2 as well as 3 and n statistics the thorough comparison of rmt short and long range correlation measures make us identify the svd method as a robust tool for characterizing random network spectra

projection onto simplicial cones by picard s method abstract by using moreau s decomposition theorem for projecting onto cones the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations it is shown that picard s method applied to the system of equations associated with the problem of projecting onto a simplicial cone generates a sequence that converges linearly to the solution of the system numerical experiments are presented making the comparison between picard s and semi smooth newton s methods to solve the nonsmooth system associated with the problem of projecting a point onto a simplicial cone

norm product belief propagation primal dual message passing for approximate inference inference problems in graphical models can be represented as a constrained optimization of a free energy function in this paper we treat both forms of probabilistic inference estimating marginal probabilities of the joint distribution and finding the most probable assignment through a unified message passing algorithm architecture in particular we generalize the belief propagation bp algorithms of sum product and max product and tree reweighted trw sum and max product algorithms trbp and introduce a new set of convergent algorithms based on convex free energy and linear programming lp relaxation as a zero temperature of a convex free energy the main idea of this work arises from taking a general perspective on the existing bp and trbp algorithms while observing that they all are reductions from the basic optimization formula of f ihi where the function f is an extended valued strictly convex but nonsmooth and the functions hi are extended valued functions not necessarily convex we use tools from convex duality to present the primal dual ascent algorithm which is an extension of the bregman successive projection scheme and is designed to handle optimization of the general type f ihi we then map the fractional free energy variational principle for approximate inference onto the optimization formula above and introduce the norm product message passing algorithm special cases of the norm product include sum product and max product bp algorithms trbp and nmplp algorithms when the fractional free energy is set to be convex convex free energy the norm product is globally convergent for the estimation of marginal probabilities and for approximating the lp relaxation we also introduce another branch of the norm product which arises as the zero temperature of the convex free energy which we refer to as the convex max product the convex max product is convergent unlike max product and aims at solving the lp relaxation

inference for eigenvalues and eigenvectors of gaussian symmetric matrices this article presents maximum likelihood estimators mles and log likelihood ratio llr tests for the eigenvalues and eigenvectors of gaussian random symmetric matrices of arbitrary dimension where the observations are independent repeated samples from one or two populations these inference problems are relevant in the analysis of diffusion tensor imaging data and polarized cosmic background radiation data where the observations are respectively 3 3 and 2 2 symmetric positive definite matrices the parameter sets involved in the inference problems for eigenvalues and eigenvectors are subsets of euclidean space that are either affine subspaces embedded submanifolds that are invariant under orthogonal transformations or polyhedral convex cones we show that for a class of sets that includes the ones considered in this paper the mles of the mean parameter do not depend on the covariance parameters if and only if the covariance structure is orthogonally invariant closed form expressions for the mles and the associated llrs are derived for this covariance structure

ordinary differential equations and systems with time dependent discontinuity sets in this paper we prove new existence results for non autonomous systems of first order ordinary differential equations under weak conditions on the nonlinear part discontinuities with respect to the unknown are allowed to occur over general classes of time dependent sets which are assumed to satisfy a kind of inverse viability condition

dykstra s algorithm for constrained least squares rectangular matrix problems abstract in a recent paper the authors applied dykstra s alternating projection algorithm to solve constrained least squares n n matrix problems we extend these results in two different directions first we make use of the singular value decomposition to solve now constrained least squares rectangular m n matrix problems that arise in several applications second we propose a new and improved implementation of the projection algorithm onto the e positive definite set of matrices this implementation does not require the computation of all eigenvalues and eigenvectors of a matrix per iteration and still guarantees convergence finally encouraging preliminary numerical results are discussed

spectral statistics of the two body random ensemble revisited using longer spectra we reanalyze spectral properties of the two body random ensemble studied 30 years ago at the center of the spectra the old results are largely confirmed and we show that the nonergodicity is essentially due to the variance of the lowest moments of the spectra the longer spectra allow us to test and reach the limits of validity of french s correction for the number variance at the edge of the spectra we discuss the problems of unfolding in more detail with a gaussian unfolding of each spectrum the nearest neighbor spacing distribution between ground state and first exited state is shown to be stable using such an unfolding the distribution tends toward a semi poisson distribution for longer spectra for comparison with the nuclear table ensemble we could use such unfolding obtaining similar results as in the early papers but an ensemble with realistic splitting gives reasonable results if we just normalize the spacings in accordance with the procedure used for the data

accelerating the alternating projection algorithm for the case of affine subspaces using supporting hyperplanes abstract the von neumann halperin method of alternating projections converges strongly to the projection of a given point onto the intersection of finitely many closed affine subspaces we propose acceleration schemes making use of two ideas firstly each projection onto an affine subspace identifies a hyperplane of codimension 1 containing the intersection and secondly it is easy to project onto a finite intersection of such hyperplanes we give conditions for which our accelerations converge strongly finally we perform numerical experiments to show that these accelerations perform well for a matrix model updating problem

distribution of the ratio of consecutive level spacings in random matrix ensembles we derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices this ratio distribution was recently introduced to study spectral properties of many body problems as contrary to the standard level spacing distributions it does not depend on the local density of states our wigner like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit quantitative improvements are found through a polynomial expansion examples from a quantum many body lattice model and from zeros of the riemann zeta function are presented

universality in the spectral and eigenfunction properties of random networks by the use of extensive numerical simulations we show that the nearest neighbor energy level spacing distribution p s and the entropic eigenfunction localization length of the adjacency matrices of erd s renyi er fully random networks are universal for fixed average degree n and n being the average network connectivity and the network size respectively we also demonstrate that the brody distribution characterizes well p s in the transition from 0 when the vertices in the network are isolated to 1 when the network is fully connected moreover we explore the validity of our findings when relaxing the randomness of our network model and show that in contrast to standard er networks er networks with diagonal disorder also show universality finally we also discuss the spectral and eigenfunction properties of small world networks

determination of scale invariance in random matrix spectral fluctuations without unfolding we apply the singular value decomposition svd method based on normal mode analysis to decompose the spectra of finite random matrices of standard gaussian ensembles in trend and fluctuation modes we use the fact that the fluctuation modes are scale invariant and follow a power law to characterize the transition between the extreme regular and chaotic cases thereby we quantify the quantum chaos in systems described by random matrix theory in a direct way without performing any previous unfolding procedure and therefore avoiding possible artifacts

estimation of multivariate distributions under stochastic ordering abstract let f and g be the cdf s of two p dimensional multivariate distributions such that f is stochastically larger than g a straightforward derivation is given of the generalized maximum likelihood estimators of f and g based on random samples from each population an algorithmic approach to computing these estimators is described and motivating numerical examples are discussed the special case when f and g correspond to multivariate ordinal contingency tables is also presented the relationship of these results to those of robertson and wright 1974 is considered

projection on the intersection of convex sets abstract in this paper we give a solution of the problem of projecting a point onto the intersection of several closed convex sets when a projection on each individual convex set is known the existing solution methods for this problem are sequential in nature here we propose a highly parallelizable method the key idea in our approach is the reformulation of the original problem as a system of semi smooth equations the benefits of the proposed reformulation are twofold a a fast semi smooth newton iterative technique based on clarke s generalized gradients becomes applicable and b the mechanics of the iterative technique is such that an almost decentralized solution method emerges we proved that the corresponding semi smooth newton algorithm converges near the optimal point quadratically these features make the overall method attractive for distributed computing platforms e g sensor networks

least squares estimation of covariance matrices in balanced multivariate variance components models abstract the problem of estimating covariance matrices in balanced multivariate variance components models is discussed as with univariate models it is possible for the traditional estimators based on differences of the mean square matrices to produce estimates that are outside the parameter space in fact in many cases it is extremely likely that traditional estimates of the covariance matrices will not be nonnegative definite nnd in this article we develop an iterative estimation procedure satisfying a least squares criterion that is guaranteed to produce nnd estimates of the covariance matrices discuss the speed of convergence and provide an example to show how the estimates change

a simple approach to testing homogeneity of order constrained means abstract the likelihood ratio test for equality of partially ordered means first proposed by bartholomew is known to possess power characteristics that are generally superior to those of competing procedures a limitation of this test however is that the null distribution of the test statistic is heavily dependent on both the specific form of the order restriction and the group sample sizes it is not readily available for most orderings especially when the group sample sizes are unequal furthermore computation of the test statistics often involves intricate algorithms we propose a class of procedures based upon classical methods for combining independent tests these procedures are generally applicable and easily implemented a study of the power functions of the competing tests indicates that the power of the test based on fisher s combination method compares reasonably with that of the likelihood ratio test it is concluded that the procedure based on fisher s combination method is an effective

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

graph theoretic representations for proximity matrices through strongly anti robinson or circular strongly anti robinson matrices there are various optimization strategies for approximating through the minimization of a least squares loss function a given symmetric proximity matrix by a sum of matrices each subject to some collection of order constraints on its entries we extend these approaches to include components in the approximating sum that satisfy what are called the strongly anti robinson sar or circular strongly anti robinson csar restrictions a matrix that is sar or csar is representable by a particular graph theoretic structure where each matrix entry is reproducible from certain minimum path lengths in the graph one published proximity matrix is used extensively to illustrate the types of approximation that ensue when the sar or csar constraints are imposed

two generalizations of dykstra s cyclic projections algorithm dykstra s cyclic projections algorithm allows one to compute best approximations to any pointx in a hilbert space from the intersectionc lrci of a finite number of closed convex setsci by reducing it to a sequence of best approximation problems from theindividual setsci here we present two generalizations of this algorithm first we allow the number of setsci to beinfinite rather than finite secondly we allow arandom rather than cyclic ordering of the setsci

a constrained matrix least squares problem in structural dynamics model updating a constrained matrix least squares problem in structural dynamics model updating is considered in this paper desired matrix properties including the satisfaction of the linear equation symmetric positive semidefiniteness and sparsity are imposed as side constraints alternating projection method is applied to solve the constrained minimization problem the results of the numerical examples show that the proposed method works well all necessary constraints are considered in our model alternating projection method is first used to update the finite element model numerical results are compared with the recent results

computational acceleration of projection algorithms for the linear best approximation problem abstract this is an experimental computational account of projection algorithms for the linear best approximation problem we focus on the sequential and simultaneous versions of dykstra s algorithm and the halpern lions wittmann bauschke algorithm for the best approximation problem from a point to the intersection of closed convex sets in the euclidean space these algorithms employ different iterative approaches to reach the same goal but no mathematical connection has yet been found between their algorithmic schemes we compare these algorithms on linear best approximation test problems that we generate so that the solution will be known a priori and enable us to assess the relative computational merits of these algorithms for the simultaneous versions we present a new component averaging variant that substantially accelerates their initial behavior for sparse systems

dykstra s algorithm for constrained least squares doubly symmetric matrix problems in this work we apply dykstra s alternating projection algorithm for minimizing email protected where email protected email protected is the frobenius norm and email protected r m x n email protected r m x n and email protected r n x n are doubly symmetric positive definite matrices with entries within prescribed intervals we first solve the constrained least squares matrix problem by using the special structure properties of doubly symmetric matrices and then use the singular value decomposition to transform the original problem into a simpler one that fits nicely with the algorithm originally developed by r escalante m raydan dykstra s algorithm for a constrained least squares matrix problem numer linear algebra appl 3 1996 459 471

consistency of vanishingly smooth fictitious play we discuss consistency of vanishingly smooth fictitious play a strategy in the context of game theory which can be regarded as a smooth fictitious play procedure where the smoothing parameter is time dependent and asymptotically vanishes this answers a question initially raised by drew fudenberg and satoru takahashi

solving hankel matrix approximation problem using semidefinite programming positive semidefinite hankel matrices arise in many important applications some of their properties may be lost due to rounding or truncation errors incurred during evaluation the problem is to find the nearest matrix to a given matrix to retrieve these properties the problem is converted into a semidefinite programming problem as well as a problem comprising a semidefined program and second order cone problem the duality and optimality conditions are obtained and the primal dual algorithm is outlined explicit expressions for a diagonal preconditioned and crossover criteria have been presented computational results are presented a possibility for further improvement is indicated

a unified approach to likelihood inference on stochastic orderings in a nonparametric context abstract for data in a two way contingency table with ordered margins we consider various hypotheses of stochastic orders among the conditional distributions considered by rows and show that each is equivalent to requiring that an invertible transformation of the vectors of conditional row probabilities satisfies an appropriate set of linear inequalities this leads to the construction of a general algorithm for maximum likelihood estimation under multinomial sampling and provides a simple framework for deriving the asymptotic distribution of log likelihood ratio tests the usual stochastic ordering and the so called uniform and likelihood ratio orderings are considered as special cases in particular for each of these three orderings we determine the transformation required to apply the estimation algorithm we then consider testing the hypothesis that the rows are identically distributed against the alternative that they are stochastically ordered as well as testing each stochastic order against an u

hybrid methods for solving the educational testing problem abstract methods for solving the educational testing problem are considered one approach glunt 1995 is to formulate the problem as a linear convex programming problem in which the constraint is the intersection of three convex sets this method is globally convergent but the rate of convergence is slow however the method does have the capability of determining the correct rank of the solution matrix and this can be done in relatively few iterations if the correct rank of the solution matrix is known it is shown how to formulate the problem as a smooth nonlinear minimization problem for which a rapid convergence can be obtained by l1sqp method 6 this paper studies hybrid methods that attempt to combine the best features of both types of method an important feature concerns the interfacing of the component methods thus it has to be decided which method to use first and when to switch between methods difficulties such as these are addressed in the paper comparative numerical results are also reported

on the spectral unfolding of chaotic and mixed systems abstract random matrix theory rmt provides a framework to study the spectral fluctuations in physical systems rmt is capable of making predictions for the fluctuations only after the removal of the secular properties of the spectrum spectral unfolding procedure is used to separate the local level fluctuations from overall energy dependence of the level separation the unfolding procedure is not unique several studies showed that statistics of long term correlation in the spectrum are very sensitive to the choice of the unfolding function in polynomial unfolding this can give misleading results regarding the chaoticity of quantum systems in this letter we consider the spectra of ordered eigenvalues of large random matrices we show that the main cause behind the reported sensitivity to the unfolding polynomial degree is the inclusion of specific extreme eigenvalue s in the unfolding process

crossover in nonstandard random matrix spectral fluctuations without unfolding recently singular value decomposition svd was applied to standard gaussian ensembles of random matrix theory to determine the scale invariance in spectral fluctuations without performing any unfolding procedure here svd is applied directly to the hermite ensemble and to a sparse matrix ensemble decomposing the corresponding spectra in trend and fluctuation modes in correspondence with known results we obtain that fluctuation modes exhibit a crossover between soft and rigid behavior in this way possible artifacts introduced applying unfolding techniques are avoided by using the trend modes we perform data adaptive unfolding and we calculate traditional spectral fluctuation measures additionally ensemble averaged and individual spectrum averaged statistics are calculated consistently within the same basis of normal modes

an algorithm for restricted least squares regression abstract a commonly occurring problem in statistics is that of minimizing a least squares expression subject to side constraints here a simple iterative algorithm is presented and shown to converge to the desired solution several examples are presented including finding the closest concave convex function to a set of points and other general quadratic programming problems the dual problem to the basic problem is also discussed and a solution for it is given in terms of the algorithm finally extensions to expressions other than least squares are given

a semi smooth newton method for a special piecewise linear system with application to positively constrained convex quadratic programming in this paper a special piecewise linear system is studied it is shown that under a mild assumption the semi smooth newton method applied to this system is well defined and the method generates a sequence that converges linearly to a solution besides we also show that the generated sequence is bounded for any starting point and a formula for any accumulation point of this sequence is presented as an application we study the convex quadratic programming problem under positive constraints the numerical results suggest that the semi smooth newton method achieves accurate solutions to large scale problems in few iterations

scaling properties of multilayer random networks multilayer networks are widespread in natural and manmade systems key properties of these networks are their spectral and eigenfunction characteristics as they determine the critical properties of many dynamics occurring on top of them here we numerically demonstrate that the normalized localization length ensuremath beta of the eigenfunctions of multilayer random networks follows a simple scaling law given by ensuremath beta x 1 x with x ensuremath gamma b mathrm eff 2 l ensuremath delta ensuremath delta ensuremath sim 1 and b mathrm eff being the effective bandwidth of the adjacency matrix of the network whose size is l the scaling law for ensuremath beta that we validate on real world networks might help to better understand criticality in multilayer networks and to predict the eigenfunction localization properties of them

a geometrical approach to iterative isotone regression in the present paper we propose and analyze a novel method for estimating a univariate regression function of bounded variation the underpinning idea is to combine two classical tools in nonparametric statistics namely isotonic regression and the estimation of additive models a geometrical interpretation enables us to link this iterative method with von neumann s algorithm moreover making a connection with the general property of isotonicity of projection onto convex cones we derive another equivalent algorithm and go further in the analysis

a penalized method for multivariate concave least squares with application to productivity analysis abstract we propose a penalized method for the least squares estimator of a multivariate concave regression function this estimator is formulated as a quadratic programing qp problem with o n2 constraints where n is the number of observations computing such an estimator is a very time consuming task and the computational burden rises dramatically as the number of observations increases by introducing a quadratic penalty function we reformulate the concave least squares estimator as a qp with only non negativity constraints this reformulation can be adapted for estimating variants of shape restricted least squares i e the monotonic concave convex least squares the experimental results and an empirical study show that the reformulated problem and its dual are solved significantly faster than the original problem the matlab and r codes for implementing the penalized problems are provided in the paper

density and spacings for the energy levels of quadratic fermi operators the work presents a proof of convergence of the density of energy levels to a gaussian distribution for a wide class of quadratic forms of fermi operators this general result applies also to quadratic operators with disorder e g containing random coefficients the spacing distribution of the unfolded spectrum is investigated numerically for generic systems the level spacings behave as the spacings in a poisson process level clustering persists in the presence of disorder

order restricted statistical tests on multinomial and poisson parameters the starshaped restriction abstract likelihood ratio statistics for i testing the homogeneity of a collection of multinomial parameters against the alternative which accounts for the restriction that those parameters are starshaped cf shaked ann statist 1979 and for ii testing the null hypothesis that this parameter vector is starshaped are considered for both tests the asymptotic distribution of the test statistic under the null hypothesis is a version of the chi bar square distribution analogous tests on a collection of poisson means are also found to have asymptotic chi bar square distributions author

unfolding of the spectrum for chaotic and mixed systems abstract random matrix theory rmt is capable of making predictions for the spectral fluctuations of a physical system only after removing the influence of the level density by unfolding the spectra when the level density is known unfolding is done by using the integrated level density to transform the eigenvalues into dimensionless variables with unit mean spacing when it is not known as in most practical cases one usually approximates the level staircase function by a polynomial we here study the effect of unfolding procedure on the spectral fluctuation of two systems for which the level density is known asymptotically the first is a time reversal invariant chaotic system which is modeled in rmt by a gaussian orthogonal ensemble goe the second is the case of chaotic systems in which m quantum numbers remain almost undistorted in the early stage of the stochastic transition the hamiltonian of a system may be represented by a block diagonal matrix with m blocks of the same size in which each block is a goe unfolding is done once by using the asymptotic level densities for the eigenvalues of the m blocks and once by representing the integrated level density in terms of polynomials of different orders we find that the spacing distribution of the eigenvalues shows a little sensitivity to the unfolding method on the other hand the variance of level number 2 l is sensitive to the choice of the unfolding function unfolding that utilizes low order polynomials enhances 2 l relative to the theoretical value while the use of high order polynomial reduces it the optimal value of the order of the unfolding polynomial depends on the dimension of the corresponding ensemble

convergence of the iterative proportional fitting procedure the iterative proportional fitting procedure ipfp was introduced in 1940 by deming and stephan to estimate cell probabilities in contingency tables subject to certain marginal constraints its convergence and statistical properties have been investigated since then by several authors and by several different methods a natural extension of the ipfp to the case of bivariate densities has been introduced by ireland and kullback it has been conjectured that also in the general case the ipfp converges to the minimum discrimination projection on the class of distributions with given marginals we verify this conjecture under some regularity conditions

computing the nearest euclidean distance matrix with low embedding dimensions euclidean distance embedding appears in many high profile applications including wireless sensor network localization where not all pairwise distances among sensors are known or accurate the classical multi dimensional scaling cmds generally works well when the partial or contaminated euclidean distance matrix edm is close to the true edm but otherwise performs poorly a natural step preceding cmds would be to calculate the nearest edm to the known matrix a crucial condition on the desired nearest edm is for it to have a low embedding dimension and this makes the problem nonconvex there exists a large body of publications that deal with this problem some try to solve the problem directly and some are the type of convex relaxations of it in this paper we propose a numerical method that aims to solve this problem directly our method is strongly motivated by the majorized penalty method of gao and sun for low rank positive semi definite matrix optimization problems the basic geometric object in our study is the set of edms having a low embedding dimension we establish a zero duality gap result between the problem and its lagrangian dual problem which also motivates the majorization approach adopted numerical results show that the method works well for the euclidean embedding of network coordinate systems and for a class of problems in large scale sensor network localization and molecular conformation

the feti family of domain decomposition methods for inequality constrained quadratic programming application to contact problems with conforming and nonconforming interfaces two domain decomposition methods with lagrange multipliers for solving iteratively quadratic programming problems with inequality constraints are presented these methods are based on the feti and feti dp substructuring algorithms in the case of linear constraints they do not perform any newton like iteration instead they solve a constrained problem by an active set strategy and a generalized conjugate gradient based descent method equipped with controls to guarantee convergence monotonicity both methods possess the desirable feature of minimizing numerical oscillations during the iterative solution process performance results and comparisons are reported for several numerical simulations that suggest that both methods are numerically scalable with respect to both the problem size and the number of subdomains their parallel scalability is also illustrated on a linux cluster for a complex 1 4 million degree of freedom multibody problem with frictionless contact and nonconforming discrete interfaces

set intersection problems supporting hyperplanes and quadratic programming we study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem for the problem of finding the closest point in the intersection of closed convex sets we propose an algorithm that like dykstra s algorithm converges strongly in a hilbert space moreover this algorithm converges in finitely many iterations when the closed convex sets are cones in mathbb r n r n satisfying an alignment condition next we propose modifications of the alternating projection algorithm and prove its convergence the algorithm converges superlinearly in mathbb r n r n under some nice conditions under a conical condition the convergence can be finite lastly we discuss the case where the intersection of the sets is empty

constrained maximum likelihood exemplified by isotonic convex logistic regression abstract maximum likelihood estimates in problems in which the likelihood is smooth and the parameter space is defined by linear or smooth nonlinear inequality constraints can be obtained using available nonlinear optimization packages such as npsol this class of models includes generalized linear models with order restrictions convexity or smoothness constraints on the parameters smoothness constraints being in the form of bounds on finite differences of the regression function methods for such models are demonstrated by an analysis of data from four published studies of the incidence of down s syndrome in single year maternal age intervals constrained logistic regression of the incidence of down s syndrome on maternal age shows that the data are fitted well by a nondecreasing convex function p values for the likelihood ratio test of this model against alternatives are obtained by the parametric bootstrap and iterated parametric bootstrap the iterated bootstrap is used as a diagnostic tool it