further relaxations of the semidefinite programming approach to sensor network localization recently a semidefinite programming sdp relaxation approach has been proposed to solve the sensor network localization problem although it achieves high accuracy in estimating the sensor locations the speed of the sdp approach is not satisfactory for practical applications in this paper we propose methods to further relax the sdp relaxation more precisely to relax the single semidefinite matrix cone into a set of small size semidefinite submatrix cones which we call a sub sdp ssdp approach we present two such relaxations although they are weaker than the original sdp relaxation they retain the key theoretical property and numerical experiments show that they are both efficient and accurate the speed of the ssdp is even faster than that of other approaches based on weaker relaxations the ssdp approach may also pave a way to efficiently solving general sdp problems without sacrificing the solution quality

graph theoretic methods for matrix completion problems abstract a pattern is a list of positions in an n n real matrix a matrix completion problem for the class of matrices asks whether every partial matrix whose specified entries are exactly the positions of the pattern can be completed to a matrix we survey the current state of research on matrix completion problems for many subclasses of p 0 matrices including positive definite matrices m matrices inverse m matrices p matrices and matrices defined by various sign symmetry and positivity conditions on p 0 and p matrices graph theoretic techniques used to study completion problems are discussed several new results are also presented including the solution to the m 0 matrix completion problem and the sign symmetric p 0 matrix completion problem

a comparison of permutation and mixed model regression methods for the analysis of simulated data in the context of a group randomized trial our first purpose was to determine whether in the context of a group randomized trial grt with gaussian errors permutation or mixed model regression methods fare better in the presence of measurable confounding in terms of their monte carlo type i error rates and power our results indicate that given a proper randomization the type i error rate is similar for both methods whether unadjusted or adjusted even in small studies however our results also show that should the investigator face the unfortunate circumstance in which modest confounding exists in the only realization available the unadjusted analysis risks a type i error in this regard there was little to distinguish the two methods finally our results show that power is similar for the two methods and not surprisingly better for the adjusted tests our second purpose was to examine the relative performance of permutation and mixed model regression methods in the context of a grt when the normality assumptions underlying the mixed model are violated published studies have examined the impact of violation of this assumption at the member level only our findings indicate that both methods perform well when the assumption is violated so long as the icc is very small and the design is balanced at the group level however at icc 0 01 the permutation test carries the nominal type i error rate while the model based test is conservative and so less powerful binomial group and member level errors did not otherwise change the relative performance of the two methods with regard to confounding copyright 2005 john wiley sons ltd

exploiting sparsity in sdp relaxation for sensor network localization a sensor network localization problem can be formulated as a quadratic optimization problem qop for qops semidefinite programming sdp relaxation by lasserre with relaxation order 1 for general polynomial optimization problems pops is known to be equivalent to the sparse sdp relaxation by waki et al with relaxation order 1 except for the size and sparsity of the resulting sdp relaxation problems we show that the sparse sdp relaxation applied to the qop is at least as strong as the biswas ye sdp relaxation for the sensor network localization problem a sparse variant of the biswas ye sdp relaxation which is equivalent to the original biswas ye sdp relaxation is also derived we compare numerically the sparse sdp relaxation applied to the qop the biswas ye sdp relaxation its sparse variant and the edge based sdp relaxation by wang et al to confirm the effectiveness of the proposed techniques for exploiting the sparsity in sdp relaxation for sensor network localization problems the sparse sdp relaxation applied to the qop is much faster than the biswas ye sdp relaxation and the sparse variant of the biswas ye sdp relaxation outperforms all other sdp relaxations in speed

on the behavior of randomization tests without a group invariance assumption abstract fisher s randomization construction of hypothesis tests is a powerful tool to yield tests that are nonparametric in nature in that their level is exactly equal to the nominal level in finite samples over a wide range of distributional assumptions for example the usual permutation t test to test equality of means is valid without a normality assumption of the underlying populations on the other hand fisher s randomization construction is not applicable in this example unless the underlying populations differ only in location in general the basis for the randomization construction is invariance of the probability distribution of the data under a transformation group it is the goal of this article to understand the robustness properties of randomization tests by studying their asymptotic validity in situations where the basis for their construction breaks down here asymptotic validity refers to whether the probability of a type i error tends asymptotically to the nominal level in particula

google matrix and ulam networks of intermittency maps we study the properties of the google matrix of an ulam network generated by intermittency maps this network is created by the ulam method which gives a matrix approximant for the perron frobenius operator of dynamical map the spectral properties of eigenvalues and eigenvectors of this matrix are analyzed we show that the pagerank of the system is characterized by a power law decay with the exponent ensuremath beta dependent on map parameters and the google damping factor ensuremath alpha under certain conditions the pagerank is completely delocalized so that the google search in such a situation becomes inefficient

convergence behavior of interior point algorithms we show that most interior point algorithms for linear programming generate a solution sequence in which every limit point satisfies the strict complementarity condition these algorithms include all path following algorithms and some potential reduction algorithms the result also holds for the monotone complementarity problem if a strict complementarity solution exists in general the limit point is a solution that maximizes the number of its nonzero components among all solutions

robust edge based semidefinite programming relaxation of sensor network localization recently wang zheng boyd and ye siam j optim 19 655 673 2008 proposed a further relaxation of the semidefinite programming sdp relaxation of the sensor network localization problem named edge based sdp esdp in simulation the esdp is solved much faster by interior point method than sdp relaxation and the solutions found are comparable or better in approximation accuracy we study some key properties of the esdp relaxation showing that when distances are exact zero individual trace is not only sufficient but also necessary for a sensor to be correctly positioned by an interior solution we also show via an example that when distances are inexact zero individual trace is insufficient for a sensor to be accurately positioned by an interior solution we then propose a noise aware robust version of esdp relaxation for which small individual trace is necessary and sufficient for a sensor to be accurately positioned by a certain analytic center solution assuming the noise level is sufficiently small for this analytic center solution the position error for each sensor is shown to be in the order of the square root of its trace lastly we propose a log barrier penalty coordinate gradient descent method to find such an analytic center solution in simulation this method is much faster than interior point method for solving esdp and the solutions found are comparable in approximation accuracy moreover the method can distribute its computation over the sensors via local communication making it practical for positioning and tracking in real time

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

iterative greedy algorithm for solving the fir paraunitary approximation problem in this paper a method for approximating a multi input multi output mimo transfer function by a causal finite impulse response fir paraunitary pu system in a weighted least squares sense is presented using a complete parameterization of fir pu systems in terms of householder like building blocks an iterative algorithm is proposed that is greedy in the sense that the observed mean squared error at each iteration is guaranteed to not increase for certain design problems in which there is a phase type ambiguity in the desired response which is formally defined in the paper a phase feedback modification is proposed in which the phase of the fir approximant is fed back to the desired response with this modification in effect it is shown that the resulting iterative algorithm not only still remains greedy but also offers a better magnitude type fit to the desired response simulation results show the usefulness and versatility of the proposed algorithm with respect to the design of principal component filter bank pcfb like filter banks and the fir pu interpolation problem concerning the pcfb design problem it is shown that as the mcmillan degree of the fir pu approximant increases the resulting filter bank behaves more and more like the infinite order pcfb consistent with intuition in particular this pcfb like behavior is shown in terms of filter response shape multiresolution coding gain noise reduction with zeroth order wiener filtering in the subbands and power minimization for discrete multitone dmt type transmultiplexers

on the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues we derive some basic results on the geometry of semidefinite programming sdp and eigenvalue optimization i e the minimization of the sum of the k largest eigenvalues of a smooth matrix valued function we provide upper bounds on the rank of extreme matrices in sdps and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue optimization in the spectrum of an optimal matrix the kth and k 1 st largest eigenvalues tend to be equal and frequently have multiplicity greater than two this clustering is intuitively plausible and has been observed as early as 1975 when the matrix valued function is affine we prove that clustering must occur at extreme points of the set of optimal solutions if the number of variables is sufficiently large we also give a lower bound on the multiplicity of the critical eigenvalue these results generalize to the case of a general matrix valued function under appropriate conditions

global continuation for distance geometry problems distance geometry problems arise in the determination of protein structure we consider the case where only a subset of the distances between atoms is given and formulate this distance geometry problem as a global minimization problem with special structure we show that global smoothing techniques and a continuation approach for global optimization can be used to determine global solutions of this problem reliably and efficiently the global continuation approach determines a global solution with less computational effort than is required by a standard multistart algorithm moreover the continuation approach usually finds the global solution from any given starting point while the multistart algorithm tends to fail

regression analysis of multivariate incomplete failure time data by modeling marginal distributions abstract many survival studies record the times to two or more distinct failures on each subject the failures may be events of different natures or may be repetitions of the same kind of event in this article we consider the regression analysis of such multivariate failure time observations each marginal distribution of the failure times is formulated by a cox proportional hazards model no specific structure of dependence among the distinct failure times on each subject is imposed the regression parameters in the cox models are estimated by maximizing the failure specific partial likelihoods the resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated simultaneous inferential procedures are then proposed extensive monte carlo studies indicate that the normal approximation is adequate for practical use the new methods allow time dependent covariates missing observations and arbitrary patterns of censorship they are illustrat

detecting treatment covariate interactions using permutation methods the primary objective of a randomized clinical trial usually is to investigate whether one treatment is better than its alternatives on average however treatment effects may vary across different patient subpopulations in contrast to demonstrating one treatment is superior to another on the average sense one is often more concerned with the question that for a particular patient or a group of patients with similar characteristics which treatment strategy is most appropriate to achieve a desired outcome various interaction tests have been proposed to detect treatment effect heterogeneity however they typically examine covariates one at a time do not offer an integrated approach that incorporates all available information and can greatly increase the chance of a false positive finding when the number of covariates is large we propose a new permutation test for the null hypothesis of no interaction effects for any covariate the proposed test allows us to consider the interaction effects of many covariates simultaneously without having to group subjects into subsets based on pre specified criteria and applies generally to randomized clinical trials of multiple treatments the test provides an attractive alternative to the standard likelihood ratio test especially when the number of covariates is large we illustrate the proposed methods using a dataset from the treatment of adolescents with depression study

graph rigidity via euclidean distance matrices abstract let g v e be an incomplete graph with node set v edge set e and nonnegative weights ij s on the edges let each edge v i v j be viewed as a rigid bar of length ij which can rotate freely around its end nodes a realization of a graph g is an assignment of coordinates in some euclidean space to each node of g in this paper we consider the problem of determining whether or not a given realization of a graph g is rigid we show that each realization of g can be epresented as a point in a compact convex set r m and that a generic realization of g is rigid if and only if its corresponding point is a vertex of i e an extreme point with full dimensional normal cone

explicit sensor network localization using semidefinite representations and facial reductions the sensor network localization snl problem in embedding dimension r consists of locating the positions of wireless sensors given only the distances between sensors that are within radio range and the positions of a subset of the sensors called anchors current solution techniques relax this problem to a weighted nearest positive semidefinite programming sdp completion problem by using the linear mapping between euclidean distance matrices edm and semidefinite matrices the resulting sdp is solved using primal dual interior point solvers yielding an expensive and inexact solution this relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the sdp cone implying that the slater constraint qualification fails cliques in the graph of the snl problem give rise to this degeneracy in the sdp relaxation in this paper we take advantage of the absence of the slater constraint qualification and derive a technique for the snl problem with exact data that explicitly solves the corresponding rank restricted sdp problem no sdp solvers are used for randomly generated instances we are able to efficiently solve many huge instances of this np hard problem to high accuracy by finding a representation of the minimal face of the sdp cone that contains the sdp matrix representation of the edm the main work of our algorithm consists in repeatedly finding the intersection of subspaces that represent the faces of the sdp cone that correspond to cliques of the snl problem

cooperative wireless sensor network positioning via implicit convex feasibility we propose a distributed positioning algorithm to estimate the unknown positions of a number of target nodes given distance measurements between target nodes and between target nodes and a number of reference nodes at known positions based on a geometric interpretation we formulate the positioning problem as an implicit convex feasibility problem in which some of the sets depend on the unknown target positions and apply a parallel projection onto convex sets approach to estimate the unknown target node positions the proposed technique is suitable for parallel implementation in which every target node in parallel can update its position and share the estimate of its location with other targets we mathematically prove convergence of the proposed algorithm simulation results reveal enhanced performance for the proposed approach compared to available techniques based on projections especially for sparse networks

the molecule problem exploiting structure in global optimization the molecule problem is that of determining the relative locations of a set of objects in euclidean space relying only upon a sparse set of pairwise distance measurements this np hard problem has applications in the determination of molecular conformation the molecule problem can be naturally expressed as a continuous global optimization problem but it also has a rich combinatorial structure this paper investigates how that structure can be exploited to simplify the optimization problem in particular we present a novel divide and conquer algorithm in which a large global optimization problem is replaced by a sequence of smaller ones since the cost of the optimization can grow exponentially with problem size this approach holds the promise of a substantial improvement in performance our algorithmic development relies upon some recently published results in graph theory we describe an implementation of this algorithm and report some results of its performance on a sample molecule

on equivalence of semidefinite relaxations for quadratic matrix programming we analyze two popular semidefinite programming relaxations for quadratically constrained quadratic programs with matrix variables these relaxations are based on vector lifting and on matrix lifting they are of different size and expense we prove under mild assumptions that these two relaxations provide equivalent bounds thus our results provide a theoretical guideline for how to choose a less expensive semidefinite programming relaxation and still obtain a strong bound the main technique used to show the equivalence and that allows for the simplified constraints is the recognition of a class of nonchordal sparse patterns that admit a smaller representation of the positive semidefinite constraint

fast computation of low rank matrix approximations given a matrix a it is often desirable to find a good approximation to a that has low rank we introduce a simple technique for accelerating the computation of such approximations when a has strong spectral features that is when the singular values of interest are significantly greater than those of a random matrix with size and entries similar to a our technique amounts to independently sampling and or quantizing the entries of a thus speeding up computation by reducing the number of nonzero entries and or the length of their representation our analysis is based on observing that the acts of sampling and quantization can be viewed as adding a random matrix n to a whose entries are independent random variables with zero mean and bounded variance since with high probability n has very weak spectral features we can prove that the effect of sampling and quantization nearly vanishes when a low rank approximation to a p n is computed we give high probability bounds on the quality of our approximation both in the frobenius and the 2 norm

on bar frameworks stress matrices and semidefinite programming a bar framework g p in r dimensional euclidean space is a graph g on the vertices 1 2 n where each vertex i is located at point p i in mathbb r r given a framework g p in mathbb r r a problem of great interest is that of determining whether or not there exists another framework g q not obtained from g p by a rigid motion such that q i q j 2 p i p j 2 for each edge i j of g this problem is known as either the global rigidity problem or the universal rigidity problem depending on whether such a framework g q is restricted to be in the same r dimensional space or not the stress matrix s of a bar framework g p plays a key role in these and other related problems in this paper semidefinite programming sdp theory is used to address in a unified manner several problems concerning universal rigidity new results are presented as well as new proofs of previously known theorems in particular we use the notion of sdp non degeneracy to obtain a sufficient condition for universal rigidity and we show that this condition yields the previously known sufficient condition for generic universal rigidity we present new results concerning positive semidefinite stress matrices and we use a semidefinite version of farkas lemma to characterize bar frameworks that admit a nonzero positive semidefinite stress matrix s

interior point trajectories in semidefinite programming in this paper we study interior point trajectories in semidefinite programming sdp including the central path of an sdp this work was inspired by the seminal work of megiddo on linear programming trajectories progress in math programming interior point algorithms and related methods n megiddo ed springer verlag berlin 1989 pp 131 158 under an assumption of primal and dual strict feasibility we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of respectively the primal and the dual problems we consider a class of trajectories that are similar to the central path but can be constructed to pass through any given interior feasible or infeasible point and study their convergence finally we study the derivatives of these trajectories and their convergence

instability statistics and mixing rates we claim that looking at probability distributions of finite time largest lyapunov exponents and more precisely studying their large deviation properties yields an extremely powerful technique to get quantitative estimates of polynomial decay rates of time correlations and poincare recurrences in the quite delicate case of dynamical systems with weak chaotic properties

on rigidity and realizability of weighted graphs abstract recently a new characterization of rigid graphs was introduced using euclidean distance matrices edms a y alfakih linear algebra appl 310 2000 149 in this paper we address the computational aspects of this characterization also we present a characterization of graphs which are realizable in r r for some 1 r n 2 but not realizable in r n 1 where n is the number of nodes

lyapunov statistics and mixing rates for intermittent systems we consider here a recent conjecture stating that correlation functions and tail probabilities of finite time lyapunov exponents would have the same power law decay in weakly chaotic systems we demonstrate that this conjecture fails for a generic class of maps of the pomeau manneville type we show further that typically the decay properties of such tail probabilities do not provide significant information on key aspects of weakly chaotic dynamics such as ergodicity and instability regimes our approaches are firmly based on rigorous results particularly the aaronson darling kac theorem and are also confirmed by exhaustive numerical simulations

extensive numerical investigations on the ergodic properties of two coupled pomeau manneville maps abstract we present extensive numerical investigations on the ergodic properties of two identical pomeau manneville maps interacting on the unit square through a diffusive linear coupling the system exhibits anomalous statistics as expected but with strong deviations from the single intermittent map such differences are characterized by numerical experiments with densities which do not have singularities in the marginal fixed point escape and poincare recurrence time statistics that share a power law decay exponent modified by a clear dimensional scaling while the rate of phase space filling and the convergence of ensembles of lyapunov exponents show a stretched instead of pure exponential behavior in spite of the lack of rigorous results about this system the dependence on both the intermittency and the coupling parameters appears to be smooth paving the way for further analytical development we remark that dynamical exponents appear to be independent of the nonzero coupling strength

euclidean distance geometry and applications euclidean distance geometry is the study of euclidean geometry based on the concept of distance this is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in euclidean space realizing those given distances we survey the theory of euclidean distance geometry and its most important applications with special emphasis on molecular conformation problems

chaotic hamiltonian systems survival probability we consider the dynamical system described by the area preserving standard mapping it is known for this system that p t the normalized number of recurrences staying in some given domain of the phase space at time t so called survival probability has the power law asymptotics p t ensuremath sim t ensuremath ensuremath nu we present new semiphenomenological arguments which enable us to map the dynamical system near the chaos border onto the effective ultrametric diffusion on the boundary of a treelike space with hierarchically organized transition rates in the framework of our approach we have estimated the exponent ensuremath nu as ensuremath nu text ln text 2 text ln 1 r g ensuremath approx 1 44 where r g sqrt 5 ensuremath 1 2 is the critical rotation number

exploiting sparsity in semidefinite programming via matrix completion i general framework a critical disadvantage of primal dual interior point methods compared to dual interior point methods for large scale semidefinite programs sdps has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse based on some fundamental results about positive semidefinite matrix completion this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage our method is used in two ways one is a conversion of a sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior point method for sdps employing a standard block diagonal matrix data structure the other way is an incorporation of our method into primal dual interior point methods which we can apply directly to a given sdp in part ii of this article we will investigate an implementation of such a primal dual interior point method based on positive definite matrix completion and report some numerical results

solving large scale sparse semidefinite programs for combinatorial optimization we present a dual scaling interior point algorithm and show how it exploits the structure and sparsity of some large scale problems we solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints we report the first computational results of interior point algorithms for approximating maximum cut semidefinite programs with dimension up to 3 000

on a registration based approach to sensor network localization we consider a registration based approach for localizing sensor networks from range measurements this is based on the assumption that one can find overlapping cliques spanning the network that is for each sensor one can identify geometric neighbors for which all inter sensor ranges are known such cliques can be efficiently localized using multidimensional scaling however since each clique is localized in some local coordinate system we are required to register them in a global coordinate system in other words our approach is based on transforming the localization problem into a problem of registration in this context the main contributions are as follows first we describe an efficient method for partitioning the network into overlapping cliques second we study the problem of registering the localized cliques and formulate a necessary rigidity condition for uniquely recovering the global sensor coordinates in particular we present a method for efficiently testing rigidity and a proposal for augmenting the partitioned network to enforce rigidity a recently proposed semidefinite relaxation of global registration is used for registering the cliques we present simulation results on random and structured sensor networks to demonstrate that the proposed method compares favourably with state of the art methods in terms of run time accuracy and scalability

interference between units in randomized experiments in a randomized experiment comparing two treatments there is interference between units if applying the treatment to one unit may affect other units interference implies that treatment effects are not comparisons of two potential responses that a unit may exhibit one under treatment and the other under control but instead are inherently more complex interference is common in social settings where people communicate compete or spread disease in studies that treat one part of an organism using a symmetrical part as control in studies that apply different treatments to the same organism at different times and in many other situations available statistical tools are limited for instance fisher s sharp null hypothesis of no treatment effect implicitly entails no interference and so his randomization test may be used to test no effect but conventional ways of inverting the test to obtain confidence intervals say for an additive effect are not applicable with interference another commonly used

normalized incremental subgradient algorithm and its application the problem of minimizing the sum of a number of component functions is of great importance in the real world in this paper a new incremental optimization algorithm named normalized incremental subgradient nis algorithm is proposed for a class of such problems where the component functions have common local minima the nis algorithm is performed incrementally just as the general incremental subgradient is algorithm and thus can be implemented in a distributed way in the nis algorithm the update of each subiteration is based on a search direction obtained by individually normalizing each component of subgradients of component functions resulting in much better convergence performance as compared to the is algorithm and other traditional optimization methods e g gauss newton method the convergence of the nis algorithm with both diminishing stepsizes and constant stepsizes is proved and analyzed theoretically two important applications are presented one is to solve a class of convex feasibility problems in a distributed way and the other is distributed maximum likelihood estimation numerical examples arising from two important topics in the area of wireless sensor networks source localization and node localization demonstrate the effectiveness and efficiency of the nis algorithm

a diffusion approach to network localization the localization of nodes on a network is a challenging research topic it arises in a variety of applications such as communications and sensor network analysis we propose a computational approach to recovering the positions of network nodes given partial and corrupted distance measurements and the positions of a small subset of anchor nodes first we show how to derive geometrically adaptive diffusion bases defined over the entire network given only partial distance measurements second we propose to utilize several diffusion bases simultaneously to derive multiscale diffusion frames last we utilize the diffusion frames to formulate a l1 regression based extension of the anchor points coordinates to the entire network we experimentally show that under a wide range of conditions our method compares favorably with state of the art approaches

cooperative simultaneous localization and mapping by exploiting multipath propagation an affordable and reliable indoor positioning is a highly needed service moreover maps of the indoor environment are vital to many applications in this paper a method for joint localization and mapping using multipath delay estimates is developed required high resolution estimates of multipath delays may be obtained using radio frequency or acoustic measurements among a set of nodes in a network in this paper the problem is modeled in two dimensional space with arbitrary node configuration and assuming a convex polygonal room shape joint localization and mapping is formulated as an optimization problem it is subdivided and relaxed into two convex subproblems which can be solved in an alternating manner a method for data association and a low complexity mapping algorithm stemming from hough transform are proposed both the estimation performance and identifiability of the indoor localization problem are improved moreover a basic map of the propagation environment is produced

intrinsic stickiness and chaos in open integrable billiards tiny border effects rounding border effects at the escape point of open integrable billiards are analyzed via the escape time statistics and emission angles the model is the rectangular billiard and the shape of the escape point is assumed to have a semicircular form stickiness chaos and self similar structures for the escape times and emission angles are generated inside backgammon like stripes of initial conditions these stripes are born at the boundary between two different emission angles but with the same escape times and when rounding effects increase they start to overlap generating a very rich dynamics tiny rounded borders around 0 1 from the whole billiard size are shown to be sufficient to generate the sticky motion with power law decay esc 1 27 while borders larger than 10 are enough to produce escape times related to the chaotic motion escape exponents in the interval 1 esc 2 are generated due to marginal unstable periodic orbits trapping alternately in time regular and chaotic trajectories

on convergence rates of linearized proximal algorithms for convex composite optimization with applications in the present paper we investigate a linearized proximal algorithm lpa for solving a convex composite optimization problem each iteration of the lpa is a proximal minimization of the convex composite function with the inner function being linearized at the current iterate the lpa has the attractive computational advantage that the solution of each subproblem is a singleton which avoids the difficulty as in the gauss newton method gnm of finding a solution with minimum norm among the set of minima of its subproblem while still maintaining the same local convergence rate as that of the gnm under the assumptions of local weak sharp minima of order p p in 1 2 and a quasi regularity condition we establish a local superlinear convergence rate for the lpa we also propose a globalization strategy for the lpa based on a backtracking line search and an inexact version of the lpa we further apply the lpa to solve a possibly nonconvex feasibility problem as well as a sensor network localiza

visualizing data as objects by dc difference of convex optimization in this paper we address the problem of visualizing in a bounded region a set of individuals which has attached a dissimilarity measure and a statistical value as convex objects this problem which extends the standard multidimensional scaling analysis is written as a global optimization problem whose objective is the difference of two convex functions dc suitable dc decompositions allow us to use the difference of convex algorithm dca in a very efficient way our algorithmic approach is used to visualize two real world datasets

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

sparsity in sums of squares of polynomials representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming sdp relaxation of polynomial optimization problems we discuss effective methods to obtain a simpler representation of a sparse polynomial as a sum of squares of sparse polynomials by eliminating redundancy

spaseloc an adaptive subproblem algorithm for scalable wireless sensor network localization an adaptive rule based algorithm spaseloc is described to solve localization problems for ad hoc wireless sensor networks a large problem is solved as a sequence of very small subproblems each of which is solved by semidefinite programming relaxation of a geometric optimization model the subproblems are generated according to a set of sensor anchor selection rules computational results compared with existing approaches show that the spaseloc algorithm scales well and provides excellent localization accuracy

distributed wireless sensor network localization via sequential greedy optimization algorithm node localization is essential to most applications of wireless sensor networks wsns in this paper we consider both range based node localization and range free node localization with uncertainties in range measurements radio range and anchor positions first a greedy optimization algorithm named sequential greedy optimization sgo algorithm is presented which is more suitable for distributed optimization in networks than the classical nonlinear gauss seidel algorithm then a unified optimization framework is proposed for both range based localization and range free localization and two convex localization formulations are obtained based on semidefinite programming sdp relaxation techniques by applying the sgo algorithm to the edge based sdp relaxation formulation we propose a second order cone programming socp based distributed node localization algorithm two distributed refinement algorithms are also proposed by using the sgo algorithm to nonconvex localization formulations the proposed localization algorithms all can be implemented partially asynchronously in networks finally extensive simulations are conducted to demonstrate the efficiency and accuracy of the proposed distributed localization algorithms

the positive definite completion problem relative to a subspace abstract we consider the problem of completing a partial real symmetric matrix so that its quadratic from is positive semi definite on a subspace presented as the null space of a given matrix under the appropriate definition of partial positive definite the classical result regarding chordal patterns is extended to this setting however unlike the classical setting there are differences between the positive semi definite and positive definite completion problems

universal rigidity and edge sparsification for sensor network localization owing to their high accuracy and ease of formulation there has been great interest in applying convex optimization techniques particularly that of semidefinite programming sdp relaxation to tackle the sensor network localization problem in recent years however a drawback of such techniques is that the resulting convex program is often expensive to solve in order to speed up computation various edge sparsification heuristics have been proposed whose aim is to reduce the number of edges in the input graph although these heuristics do reduce the size of the convex program and hence make it faster to solve they are often ad hoc in nature and do not preserve the localization properties of the input as such one often has to face a tradeoff between solution accuracy and computational effort in this paper we propose a novel edge sparsification heuristic that can provably preserve the localization properties of the original input at the heart of our heuristic is a graph decomposition procedure which allows us to identify certain sparse generically universally rigid subgraphs of the input graph our computational results show that the proposed approach can significantly reduce the computational and memory complexities of sdp based algorithms for solving the sensor network localization problem moreover it compares favorably with existing speedup approaches both in terms of accuracy and solution time

optimal permutation tests for the analysis of group randomized trials two facts complicate the comparison of interventions in group randomized trials grt s a family of clinical trials in which each member of a particular group receives the same treatment assignment first individual outcomes within each group are often correlated second the number of groups in a grt is often not sufficient to make asymptotic approximations possible therefore the tests used with methods such as generalized estimating equations gee and penalized quasi likelihood pql originally developed for longitudinal studies may not be valid as an alternative a class of permutation tests is derived to maximize the power of testing for an intervention effect in a grt while maintaining a nominal test size the test uses a statistic that is a weighted sum of residuals with the weights based on the group sizes and the variability of each individual outcome through simulation we demonstrate the importance of weights to a permutation test s power and compare the power of permutation tests ge

simple and fast convex relaxation method for cooperative localization in sensor networks using range measurements we address the sensor network localization problem given noisy range measurements between pairs of nodes we approach the nonconvex maximum likelihood formulation via a known simple convex relaxation we exploit its favorable optimization properties to the full to obtain an approach that is completely distributed has a simple implementation at each node and capitalizes on an optimal gradient method to attain fast convergence we offer a parallel but also an asynchronous flavor both with theoretical convergence guarantees and iteration complexity analysis experimental results establish leading performance our algorithms top the accuracy of a comparable state of the art method by one order of magnitude using one order of magnitude fewer communications

the use of permutation tests for the analysis of parallel and stepped wedge cluster randomized trials we investigate the use of permutation tests for the analysis of parallel and stepped wedge cluster randomized trials permutation tests for parallel designs with exponential family endpoints have been extensively studied the optimal permutation tests developed for exponential family alternatives require information on intraclass correlation a quantity not yet defined for time to event endpoints therefore it is unclear how efficient permutation tests can be constructed for cluster randomized trials with such endpoints we consider a class of test statistics formed by a weighted average of pair specific treatment effect estimates and offer practical guidance on the choice of weights to improve efficiency we apply the permutation tests to a cluster randomized trial evaluating the effect of an intervention to reduce the incidence of hospital acquired infection in some settings outcomes from different clusters may be correlated and we evaluate the validity and efficiency of permutation test in such settings lastly we propose a permutation test for stepped wedge designs and compare its performance with mixed effect modeling and illustrate its superiority when sample sizes are small the underlying distribution is skewed or there is correlation across clusters copyright 2017 john wiley sons ltd

distributed maximum likelihood sensor network localization we propose a class of convex relaxations to solve the sensor network localization problem based on a maximum likelihood ml formulation this class as well as the tightness of the relaxations depends on the noise probability density function pdf of the collected measurements we derive a computational efficient edge based version of this ml convex relaxation class and we design a distributed algorithm that enables the sensor nodes to solve these edge based convex programs locally by communicating only with their close neighbors this algorithm relies on the alternating direction method of multipliers admm it converges to the centralized solution it can run asynchronously and it is computation error resilient finally we compare our proposed distributed scheme with other available methods both analytically and numerically and we argue the added value of admm especially for large scale networks

second order cone programming relaxation of sensor network localization the sensor network localization problem has been much studied recently biswas and ye proposed a semidefinite programming sdp relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed here we study a second order cone programming socp relaxation of this problem motivated by its simpler structure and its potential to be solved faster than sdp we show that the socp relaxation though weaker than the sdp relaxation has nice properties that make it useful as a problem preprocessor in particular sensors that are uniquely positioned among interior solutions of the socp relaxation are accurate up to the square root of the distance error thus these sensors which are easily identified are accurately positioned in our numerical simulation the interior solution found can accurately position up to 80 90p of the sensors we also propose a smoothing coordinate gradient descent method for finding an interior solution that is faster than an interior point method

sensor network localization by augmented dual embedding in this work we propose an anchor based sensor networks localization scheme that utilizes a dual spectral embedding the input noisy distance measurements are first embedded by diffusion embedding and then by isomap this allows to better estimate the intrinsic network geometry and derive improved adaptive bases that are used to estimate the global localization via l 1 regression we then introduce the augmented dual embedding by computationally augmenting the set of measured distances and computing the dual embedding this significantly improves the scheme s robustness and accuracy we also propose a straightforward approach to preprocessing the noisy distances via the triangle inequality the proposed scheme is experimentally shown to outperform contemporary state of the art localization schemes