max norm optimization for robust matrix recovery this paper studies the matrix completion problem under arbitrary sampling schemes we propose a new estimator incorporating both max norm and nuclear norm regularization based on which we can conduct efficient low rank matrix recovery using a random subset of entries observed with additive noise under general non uniform and unknown sampling distributions this method significantly relaxes the uniform sampling assumption imposed for the widely used nuclear norm penalized approach and makes low rank matrix recovery feasible in more practical settings theoretically we prove that the proposed estimator achieves fast rates of convergence under different settings computationally we propose an alternating direction method of multipliers algorithm to efficiently compute the estimator which bridges a gap between theory and practice of machine learning methods with max norm regularization further we provide thorough numerical studies to evaluate the proposed method using both simulated and real datasets

algebraic tensegrity form finding abstract this paper concerns the form finding problem for general and symmetric tensegrity structures with shape constraints a number of different geometries are treated and several fundamental properties of tensegrity structures are identified that simplify the form finding problem the concept of a tensegrity invariance similarity transformation is defined and it is shown that tensegrity equilibrium is preserved under affine node position transformations this result provides the basis for a new tensegrity form finding tool the generality of the problem formulation makes it suitable for the automated generation of the equations and their derivatives state of the art numerical algorithms are applied to solve several example problems examples are given for tensegrity plates shell class symmetric tensegrity structures and structures generated by applying similarity transformation

learning multiple tasks with kernel methods we study the problem of learning many related tasks simultaneously using kernel methods and regularization the standard single task kernel methods such as support vector machines and regularization networks are extended to the case of multi task learning our analysis shows that the problem of estimating many task functions with regularization can be cast as a single task learning problem if a family of multi task kernel functions we define is used these kernels model relations among the tasks and are derived from a novel form of regularizers specific kernels that can be used for multi task learning are provided and experimentally tested on two real data sets in agreement with past empirical work on multi task learning the experiments show that learning multiple related tasks simultaneously using the proposed approach can significantly outperform standard single task learning particularly when there are many related tasks but few data per task

a monte carlo form finding method for large scale regular and irregular tensegrity structures we propose a monte carlo form finding method that employs a stochastic procedure to determine equilibrium configurations of a tensegrity structure this method does not involve complicated matrix operations or symmetry analysis works for arbitrary initial configurations and can handle large scale regular or irregular tensegrity structures with or without material geometrical constraints

bifurcation analysis of a spruce budworm model with diffusion and physiological structures abstract in this paper the dynamics of a spruce budworm model with diffusion and physiological structures are investigated the stability of steady state and the existence of hopf bifurcation near positive steady state are investigated by analyzing the distribution of eigenvalues the properties of hopf bifurcation are determined by the normal form theory and center manifold reduction for partial functional differential equations and global existence of periodic solutions is established by using the global hopf bifurcation result of wu finally some numerical simulations are carried out to illustrate the analytical results

tensegrity deployment using infinitesimal mechanisms abstract kinematic properties of tensegrity structures reveal that an ideal way of motion is by using their infinitesimal mechanisms for example in motions along infinitesimal mechanisms there is no energy loss due to linearly kinetic tendon damping consequently a deployment strategy which exploits these mechanisms and uses the structure s nonlinear equations of motion is developed desired paths that are tangent to the directions determined by infinitesimal mechanisms are constructed and robust nonlinear feedback control is used for accurate tracking of these paths examples demonstrate the feasibility of this approach and further analysis reveals connections between the power and energy dissipated via damping infinitesimal mechanisms speed of the motion and deployment time

a new approach to collaborative filtering operator estimation with spectral regularization we present a general approach for collaborative filtering cf using spectral regularization to learn linear operators mapping a set of users to a set of possibly desired objects in particular several recent low rank type matrix completion methods for cf are shown to be special cases of our proposed framework unlike existing regularization based cf our approach can be used to incorporate additional information such as attributes of the users objects a feature currently lacking in existing regularization based cf approaches using popular and well known kernel methods we provide novel representer theorems that we use to develop new estimation methods we then provide learning algorithms based on low rank decompositions and test them on a standard cf data set the experiments indicate the advantages of generalizing the existing regularization based cf methods to incorporate related information about users and objects finally we show that certain multi task learning methods can be also seen as special cases of our proposed approach

optimum design of flexible structures under constraints on strain energy and asymptotic stability a new concept and optimization method are presented for designing a flexible structure that allows large deformation constraint is given for the minimum strain energy that can be stored for specified norm of displacement as well as asymptotic stability of the undeformed state within the region defined by the specified displacement norm the optimal solution is obtained by solving a two level optimization problem examples are given for 2 bar and 24 bar trusses asymptotic stability of the optimized truss is verified by carrying out dynamic analysis for given random initial velocity that is equivalent to the specified strain energy

rank regularization and bayesian inference for tensor completion and extrapolation a novel regularizer of the parafac decomposition factors capturing the tensor s rank is proposed in this paper as the key enabler for completion of three way data arrays with missing entries set in a bayesian framework the tensor completion method incorporates prior information to enhance its smoothing and prediction capabilities this probabilistic approach can naturally accommodate general models for the data distribution lending itself to various fitting criteria that yield optimum estimates in the maximum a posteriori sense in particular two algorithms are devised for gaussian and poisson distributed data that minimize the rank regularized least squares error and kullback leibler divergence respectively the proposed technique is able to recover the ground truth tensor rank when tested on synthetic data and to complete brain imaging and yeast gene expression datasets with 50 and 15 of missing entries respectively resulting in recovery errors at 11 db and 15 db

centre manifolds for partial differential equations with delays a centre manifold theory for reaction diffusion equations with temporal delays is developed besides an existence proof we also show that the equation on the centre manifold is a coupled system of scalar ordinary differential equations of higher order as an illustration this reduction procedure is applied to the hutchinson equation with diffusion

low rank covariance function estimation for multidimensional functional data multidimensional function data arise from many fields nowadays the covariance function plays an important role in the analysis of such increasingly common data in this article we propose a novel

pairwise support vector machines and their application to large scale problems pairwise classification is the task to predict whether the examples a b of a pair a b belong to the same class or to different classes in particular interclass generalization problems can be treated in this way in pairwise classification the order of the two input examples should not affect the classification result to achieve this particular kernels as well as the use of symmetric training sets in the framework of support vector machines were suggested the paper discusses both approaches in a general way and establishes a strong connection between them in addition an efficient implementation is discussed which allows the training of several millions of pairs the value of these contributions is confirmed by excellent results on the labeled faces in the wild benchmark

stability and bifurcation analysis in hematopoietic stem cell dynamics with multiple delays abstract this paper is devoted to the analysis of a maturity structured system of hematopoietic stem cell hsc populations in the bone marrow the model is a system of differential equations with several time delays we discuss the stability of equilibria and perform the analysis of hopf bifurcation more precisely we first obtain a set of improved sufficient conditions ensuring the global asymptotical stability of the zero solution using the lyapunov method and the embedding technique of asymptotically autonomous semiflows then we prove that there exists at least one positive periodic solution for the n dimensional system as a time delay varies in some region this result is established by combining hopf bifurcation theory the global hopf bifurcation theorem due to wu j wu symmetric functional differential equations and neural networks with memory trans amer math soc 350 1998 4799 4838 and a continuation theorem of coincidence degree theory some numerical simulations are also presented to illustrate the analytic results

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

global dynamics of nicholsonʼs blowflies equation revisited onset and termination of nonlinear oscillations abstract we revisit nicholson s blowflies model with natural death rate incorporated into the delay feedback we consider the delay as a bifurcation parameter and examine the onset and termination of hopf bifurcations of periodic solutions from a positive equilibrium we show that the model has only a finite number of hopf bifurcation values and we describe how branches of hopf bifurcations are paired so the existence of periodic solutions with specific oscillation frequencies occurs only in bounded delay intervals the bifurcation analysis and the matlab package dde biftool developed by engelborghs et al guide some numerical simulations to identify ranges of parameters for coexisting multiple attractive periodic solutions

kernelization of matrix updates when and how abstract we define what it means for a learning algorithm to be kernelizable in the case when the instances are vectors asymmetric matrices and symmetric matrices respectively we can characterize kernelizability in terms of an invariance of the algorithm to certain orthogonal transformations if we assume that the algorithm s action relies on a linear prediction then we can show that in each case the linear parameter vector must be a certain linear combination of the instances we give a number of examples of how to apply our methods in particular we show how to kernelize multiplicative updates for symmetric instance matrices

bifurcation analysis in an approachable haematopoietic stem cells model a haematopoietic stem cells model hsc with one delay is considered at first we investigate the stability and existence of hopf bifurcations by analyzing the distribution of the roots of associated characteristic equation then an explicit formula for determining the stability and the direction of periodic solutions bifurcating from hopf bifurcations is derived using the normal form theory and center manifold argument finally some numerical simulations are carried out for supporting the analytic results

dynamics for a type of general reaction diffusion model abstract in this paper we discuss the following reaction diffusion model which is a general form of many population models u t x t u t x u t x f u t x we study the oscillatory behavior of solutions about the positive equilibrium k of system with neumann boundary conditions sufficient and necessary conditions are obtained for global attractivity of the zero solution and acceptable conditions are established for the global attractivity of k these results improve and complement existing results for system without diffusion moreover when these results are applied to the diffusive nicholson s blowflies model and the diffusive model of hematopoiesis some new results are obtained for the latter

trace norm regularization reformulations algorithms and multi task learning we consider a recently proposed optimization formulation of multi task learning based on trace norm regularized least squares while this problem may be formulated as a semidefinite program sdp its size is beyond general sdp solvers previous solution approaches apply proximal gradient methods to solve the primal problem we derive new primal and dual reformulations of this problem including a reduced dual formulation that involves minimizing a convex quadratic function over an operator norm ball in matrix space this reduced dual problem may be solved by gradient projection methods with each projection involving a singular value decomposition the dual approach is compared with existing approaches and its practical effectiveness is illustrated on simulations and an application to gene expression pattern analysis

on spectral learning in this paper we study the problem of learning a matrix w from a set of linear measurements our formulation consists in solving an optimization problem which involves regularization with a spectral penalty term that is the penalty term is a function of the spectrum of the covariance of w instances of this problem in machine learning include multi task learning collaborative filtering and multi view learning among others our goal is to elucidate the form of the optimal solution of spectral learning the theory of spectral learning relies on the von neumann characterization of orthogonally invariant norms and their association with symmetric gauge functions using this tool we formulate a representer theorem for spectral regularization and specify it to several useful example such as schatten p norms trace norm and spectral norm which should proved useful in applications

topology design of tensegrity structures via mixed integer programming this paper presents a numerical method for finding a tensegrity structure based on the ground structure method we first solve a mixed integer programming mip problem which maximizes the number of struts over the self equilibrium condition of axial forces and the discontinuity condition of struts subsequently we solve the minimization problem of the number of cables in order to remove redundant self equilibrium modes which is also formulated as an mip it is regarded to be advantageous that our method does not require any connectivity information of cables and struts to be known in advance while the obtained tensegrity structure is guaranteed to satisfy the discontinuity condition of struts rigorously

solving inverse problems with piecewise linear estimators from gaussian mixture models to structured sparsity a general framework for solving image inverse problems with piecewise linear estimations is introduced in this paper the approach is based on gaussian mixture models which are estimated via a maximum a posteriori expectation maximization algorithm a dual mathematical interpretation of the proposed framework with a structured sparse estimation is described which shows that the resulting piecewise linear estimate stabilizes the estimation when compared with traditional sparse inverse problem techniques we demonstrate that in a number of image inverse problems including interpolation zooming and deblurring of narrow kernels the same simple and computationally efficient algorithm yields results in the same ballpark as that of the state of the art

learning a robust relevance model for search using kernel methods this paper points out that many search relevance models in information retrieval such as the vector space model bm25 and language models for information retrieval can be viewed as a similarity function between pairs of objects of different types referred to as an s function an s function is specifically defined as the dot product between the images of two objects in a hilbert space mapped from two different input spaces one advantage of taking this view is that one can take a unified and principled approach to address the issues with regard to search relevance the paper then proposes employing a kernel method to learn a robust relevance model as an s function which can effectively deal with the term mismatch problem one of the biggest challenges in search the kernel method exploits a positive semi definite kernel referred to as an s kernel the paper shows that when using an s kernel the model learned by the kernel method is guaranteed to be an s function the paper then gives more general principles for constructing s kernels a specific implementation of the kernel method is proposed using the ranking svm techniques and click through data the proposed approach is employed to learn a relevance model as an extension of bm25 referred to as robust bm25 experimental results on web search and enterprise search data show that robust bm25 significantly outperforms baseline methods and can successfully tackle the term mismatch problem

adaptive force density method for form finding problem of tensegrity structures a numerical method is presented for form finding of tensegrity structures eigenvalue analysis and spectral decomposition are carried out iteratively to find the feasible set of force densities that satisfies the requirement on rank deficiency of the equilibrium matrix with respect to the nodal coordinates the equilibrium matrix is shown to correspond to the geometrical stiffness matrix in the conventional finite element formulation a unique and non degenerate configuration of the structure can then be obtained by specifying an independent set of nodal coordinates a simple explanation is given for the required rank deficiency of the equilibrium matrix that leads to a non degenerate structure several numerical examples are presented to illustrate the robustness as well as the strong ability of searching new configurations of the proposed method

zero stiffness tensegrity structures tension members with a zero rest length allow the construction of tensegrity structures that are in equilibrium along a continuous path of configurations and thus exhibit mechanism like properties equivalently they have zero stiffness the zero stiffness modes are not internal mechanisms as they involve first order changes in member length but are a direct result of the use of the special tension members these modes correspond to an infinitesimal affine transformation of the structure that preserves the length of conventional members they hold over finite displacements and are present if and only if the directional vectors of those members lie on a projective conic this geometric interpretation provides several interesting observations regarding zero stiffness tensegrity structures

local minima and convergence in low rank semidefinite programming the low rank semidefinite programming problem lrsdpr is a restriction of the semidefinite programming problem sdp in which a bound r is imposed on the rank of x and it is well known that lrsdpr is equivalent to sdp if r is not too small in this paper we classify the local minima of lrsdpr and prove the optimal convergence of a slight variant of the successful yet experimental algorithm of burer and monteiro 5 which handles lrsdpr via the nonconvex change of variables x rrt in addition for particular problem classes we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm computational results are presented on a set of combinatorial optimization relaxations including some of the largest quadratic assignment sdps solved to date

sharp oracle inequalities for high dimensional matrix prediction we observe xi yi i 1n where the yi s are real valued outputs and the xi s are m t matrices we observe a new entry x and we want to predict the output y associated with it we focus on the high dimensional setting where mt n this includes the matrix completion problem with noise as well as other problems we consider linear prediction procedures based on different penalizations involving a mixture of several norms the nuclear norm the frobenius norm and the l1 norm for these procedures we prove sharp oracle inequalities using a statistical learning theory point of view a surprising fact in our results is that the rates of convergence do not depend on m and t directly the analysis is conducted without the usually considered incoherency condition on the unknown matrix or restricted isometry condition on the sampling operator moreover our results are the first to give for this problem an analysis of penalization such as nuclear norm penalization as a regularization algorithm our oracle inequalities prove that these procedures have a prediction accuracy close to the deterministic oracle one given that the reguralization parameters are well chosen

stability conditions for tensegrity structures stability conditions for tensegrity structures are derived based on positive definiteness of the tangent stiffness matrix which is the sum of the linear and geometrical stiffness matrices a necessary stability condition is presented by considering the affine motions that lie in the null space of the geometrical stiffness matrix the condition is demonstrated to be equivalent to that derived from the mathematical rigidity theory so as to resolve the discrepancy between the stability theories in the fields of engineering and mathematics furthermore it is shown that the structure is guaranteed to be stable if the structure satisfies the necessary stability condition and the geometrical stiffness matrix is positive semidefinite with the minimum rank deficiency for non degeneracy

geometric stability switch criteria in delay differential equations with two delays and delay dependent parameters abstract most modeling efforts involve multiple physical or biological processes all physical or biological processes take time to complete therefore multiple time delays occur naturally and shall be considered in more advanced modeling efforts carefully formulated models of such natural processes often involve multiple delays and delay dependent parameters however a general and practical theory for the stability analysis of models with more than one discrete delay and delay dependent parameters is nonexistent the main purpose of this paper is to present a practical geometric method to study the stability switching properties of a general transcendental equation which may result from a stability analysis of a model with two discrete time delays and delay dependent parameters that dependent only on one of the time delay in addition to simple and illustrative examples we present a detailed application of our method to the study of a two discrete delay sir model

s shaped and broken s shaped bifurcation diagrams with hysteresis for a multiparameter spruce budworm population problem in one space dimension abstract we study exact multiplicity and bifurcation diagrams of positive solutions for a multiparameter spruce budworm population steady state problem in one space dimension u x r u 1 u q u 2 1 u 2 0 1 x 1 u 1 u 1 0 where u is the population density of the spruce budworm q r are two positive dimensionless parameters and 0 is a bifurcation parameter assume that either r 1 q and q r lies above the curve 1 q r q a 2 a 3 a 2 1 r a 2 a 3 a 2 1 2 1 a 3 or r 2 q for some constants 1 0 0939 and 2 0 0766 then on the u plane we give a classification of three qualitatively different bifurcation diagrams an s shaped curve a broken s shaped curve and a monotone increasing curve our results settle rigorously a long standing open problem in ludwig aronson and weinberger spatial patterning of the spruce budworm j math biol 8 1979 217 258

operator valued kernels for learning from functional response data in this paper we consider the problems of supervised classification and regression in the case where attributes and labels are functions a data is represented by a set of functions and the label is also a function we focus on the use of reproducing kernel hilbert space theory to learn from such functional data basic concepts and properties of kernel based learning are extended to include the estimation of function valued functions in this setting the representer theorem is restated a set of rigorously defined infinite dimensional operator valued kernels that can be valuably applied when the data are functions is described and a learning algorithm for nonlinear functional data analysis is introduced the methodology is illustrated through speech and audio signal processing experiments

efficient svm training using low rank kernel representations svm training is a convex optimization problem which scales with the training set size rather than the feature space dimension while this is usually considered to be a desired quality in large scale problems it may cause training to be impractical the common techniques to handle this difficulty basically build a solution by solving a sequence of small scale subproblems our current effort is concentrated on the rank of the kernel matrix as a source for further enhancement of the training procedure we first show that for a low rank kernel matrix it is possible to design a better interior point method ipm in terms of storage requirements as well as computational complexity we then suggest an efficient use of a known factorization technique to approximate a given kernel matrix by a low rank matrix which in turn will be used to feed the optimizer finally we derive an upper bound on the change in the objective function value based on the approximation error and the number of active constraints support vectors this bound is general in the sense that it holds regardless of the approximation method

matrix completion with covariate information this article investigates the problem of matrix completion from the corrupted data when the additional covariates are available despite being seldomly considered in the matrix completion literatu

when is there a representer theorem vector versus matrix regularizers we consider a general class of regularization methods which learn a vector of parameters on the basis of linear measurements it is well known that if the regularizer is a nondecreasing function of the l2 norm then the learned vector is a linear combination of the input data this result known as the representer theorem lies at the basis of kernel based methods in machine learning in this paper we prove the necessity of the above condition in the case of differentiable regularizers we further extend our analysis to regularization methods which learn a matrix a problem which is motivated by the application to multi task learning in this context we study a more general representer theorem which holds for a larger class of regularizers we provide a necessary and sufficient condition characterizing this class of matrix regularizers and we highlight some concrete examples of practical importance our analysis uses basic principles from matrix theory especially the useful notion of matrix nondecreasing functions

nonlinear programming approach to form finding and folding analysis of tensegrity structures using fictitious material properties abstract an optimization approach is presented for form finding of tensegrity structures it is shown that various equilibrium shapes can be easily found by solving a forced deformation analysis problem formulated as a minimization problem considering the nodal coordinates as design variables the objective function is defined in terms of the member lengths and it can be regarded as the total strain energy corresponding to fictitious elastic material properties the self equilibrium forces can be found from the optimality conditions of the nonlinear programming problem stability of the self equilibrium shape is investigated based on the local convexity of the objective function similarity between form finding problem of a structure with zero unstressed length cables and the problem of minimum square length network is also discussed furthermore folding of a structure with small unstressed length cables is approximately simulated using affine transformation of equilibrium shape

global hopf bifurcation of a general predator prey system with diffusion and stage structures abstract in this paper the dynamics of a general predator prey system with diffusion and stage structures are investigated under some assumptions the stability of positive steady state and the existence of local hopf bifurcation are obtained by analyzing the distribution of eigenvalues moreover the global existence of positive periodic solution is established by using the global hopf bifurcation result of wu finally an example and some numerical simulations are carried out for illustrating the theoretical results

normal forms and hopf bifurcation for partial differential equations with delays the paper addresses the computation of normal forms for some partial functional differential equations pfdes near equilibria the analysis is based on the theory previously developed for autonomous retarded functional differential equations and on the existence of center or other invariant manifolds as an illustration of this procedure two examples of pfdes where a hopf singularity occurs on the center manifold are considered

canard solutions and travelling waves in the spruce budworm population model abstract the spruce budworm and forest population models are investigated both in the form of ode s and in the form of pde s in the former the canard solutions were detected in the latter the canard travelling waves were discovered we investigate mainly the curvilinear waves and we suggest an approach which gives us the possibility to overcome the difficulties arising due to different diffusion coefficients

learning the kernel matrix with semidefinite programming kernel based learning algorithms work by embedding the data into a euclidean space and then searching for linear relations among the embedded data points the embedding is performed implicitly by specifying the inner products between each pair of points in the embedding space this information is contained in the so called kernel matrix a symmetric and positive semidefinite matrix that encodes the relative positions of all points specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space classical model selection problems in machine learning in this paper we show how the kernel matrix can be learned from data via semidefinite programming sdp techniques when applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm using the labeled part of the data one can learn an embedding also for the unlabeled part the similarity between test points is inferred from training points and their labels importantly these learning problems are convex so we obtain a method for learning both the model class and the function without local minima furthermore this approach leads directly to a convex method for learning the 2 norm soft margin parameter in support vector machines solving an important open problem

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states