the adaptive projected subgradient method constrained by families of quasi nonexpansive mappings and its application to online learning many online i e time adaptive inverse problems in signal processing and machine learning fall under the wide umbrella of the asymptotic minimization of a sequence of nonnegative convex and continuous functions to incorporate a priori knowledge into the design the asymptotic minimization task is usually constrained on a fixed closed convex set which is dictated by the available a priori information to increase versatility toward the usage of the available information the present manuscript extends the adaptive projected subgradient method by introducing an algorithmic scheme which incorporates a priori knowledge in the design via a sequence of strongly attracting quasi nonexpansive mappings in a real hilbert space in such a way the benefits offered to online learning tasks by the proposed method unfold in two ways 1 the rich class of quasi nonexpansive mappings provides a plethora of ways to cast a priori knowledge and 2 by introducing a sequence of such mappings the proposed scheme is

lifts of convex sets and cone factorizations in this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone such a representation or lift of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices we show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual this generalizes a theorem of yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix symmetric lifts of convex sets can also be characterized similarly when the cones live in a family our results lead to the definition of the rank of a convex set with respect to this family we present results about this rank in the context of cones of positive semidefinite matrices our methods provide new tools for understanding cone lifts of convex sets

an algebraic translation of cayley dickson linear systems and its applications to online learning the m dimensional cayley dickson number system am is a standard extension of real m 1 complex m 2 quaternion m 22 octonion m 23 and sedenion m 24 etc in this paper we present a systematic algebraic translation of the cayley dickson hypercomplex valued linear systems into a real vector valued linear model this translation is designed by using jointly two new isomorphisms between real vector spaces and enables us to straightforwardly apply the well established schemes in real domain to problems for the hypercomplex linear model we also clarify useful algebraic properties of the proposed translation as an example of many potential algorithms through the proposed algebraic translation we present am adaptive projected subgradient method am apsm for am valued adaptive system identification and show that many hypercomplex adaptive filtering algorithms can be viewed as special cases of this algorithm moreover we also apply the am apsm to nonlinear adaptive filtering by using the kernel trick numerical examples show that the effectiveness of the am apsm in many cayley dickson valued linear system identification and nonlinear channel equalization problems

on polyhedral approximations of the positive semidefinite cone it is well known that state of the art linear programming solvers are more efficient than their semidefinite programming counterparts and can scale to much larger problem sizes this leads us to co

subset algebra lift operators for 0 1 integer programming we extend the sherali adams lovasz schrijver balas ceria cornuejols and lasserre lift and project methods for 0 1 optimization by considering liftings to subset algebras our methods yield polynomial time algorithms for solving a relaxation of a set covering problem at least as strong as that given by the set of all valid inequalities with small coefficients and more generally all valid inequalities where the right hand side is not very large relative to the positive coefficients in the left hand side applied to generalizations of vertex packing problems our methods yield in polynomial time relaxations that have unbounded rank using for example the n operator

efficient enumeration of the vertices of polyhedra associated with network lp s algorithms are given to list the vertices of polyhedra associated with network linear programs and their duals each algorithm has running time which is quadratic in the number of vertices of the polyhedron and does not require that the polyhedron be either bounded or simple the algorithms use characterizations of adjacent vertices in network and dual network lp s to perform an efficient traversal of the edge graph of the polyhedron this contrasts with algorithms for enumerating the vertices of a general polyhedron all of which have worst case complexity which is exponential in the number of vertices of the polyhedron

full mean square performance bounds on quaternion estimators for improper data novel physical insights are provided into the mean square performance bounds of quaternion valued widely linear wl semi widely linear swl and strictly linear estimators for the generality of quaternion valued gaussian data this is achieved by first defining three kinds of complementary mean square errors cmses of these estimators and by further exploiting the corresponding degrees of mathbb h improperness second order noncircularity next the investigation of the bounds of the attainable cmses by these classes of estimators shows that only a joint consideration of the proposed cmse analysis and the standard mse analysis provides enough degrees of freedom for a detailed account of the mse performance the so established framework for the analysis of estimators for mathbb h improper data is shown to be capable of measuring error power distribution for each data channel an important finding which is not possible to obtain through the standard mse analysis only simulations in the system identification setting support the analysis

polytope volume computation a combinatorial form of gram s relation for convex polytopes can be adapted for use in computing polytope volume we present an algorithm for volume computation based on this observation this algorithm is useful in finding the volume of a polytope given as the solution set of a system of linear inequalities p x e r ax b as an illustration we compute a formula for the volume of a projective image of the n cube from this formula we deduce that when a and b have rational entries so that the volume of p is also a rational number the number of binary digits in the denominator of the volume cannot be bounded by a polynomial in the total number of digits in the numerators and denominators of entries of a and b this settles a question posed by dyer and frieze

adaptive learning in complex reproducing kernel hilbert spaces employing wirtinger s subgradients this paper presents a wide framework for non linear online supervised learning tasks in the context of complex valued signal processing the complex input data are mapped into a complex reproducing kernel hilbert space rkhs where the learning phase is taking place both pure complex kernels and real kernels via the complexification trick can be employed moreover any convex continuous and not necessarily differentiable function can be used to measure the loss between the output of the specific system and the desired response the only requirement is the subgradient of the adopted loss function to be available in an analytic form in order to derive analytically the subgradients the principles of the recently developed wirtinger s calculus in complex rkhs are exploited furthermore both linear and widely linear in rkhs estimation filters are considered to cope with the problem of increasing memory requirements which is present in almost all online schemes in rkhs the sparsification scheme based on projection onto closed balls has been adopted we demonstrate the effectiveness of the proposed framework in a non linear channel identification task a non linear channel equalization problem and a quadrature phase shift keying equalization scheme using both circular and non circular synthetic signal sources

an efficient robust adaptive filtering algorithm based on parallel subgradient projection techniques this paper presents a novel robust adaptive filtering scheme based on the interactive use of statistical noise information and the ideas developed originally for efficient algorithmic solutions to the convex feasibility problems the statistical noise information is quantitatively formulated as stochastic property closed convex sets by the simple design formulae developed in this paper a simple set theoretic inspection also leads to an important statistical reason for the sensitivity to noise of the affine projection algorithm apa the proposed adaptive algorithm is computationally efficient and robust to noise because it requires only an iterative parallel projection onto a series of closed half spaces that are highly expected to contain the unknown system to be identified and is free from the computational load of solving a system of linear equations the numerical examples show that the proposed adaptive filtering scheme realizes dramatically fast and stable convergence for highly colored excited speech like input signals in severe noise situations

computational complexity of inner and outer j radii of polytopes in finite dimensional normed spaces this paper studies the complexity of computing or approximating or bounding the various inner and outer radii of ann dimensional convex polytope in the space n equipped with an lp norm or a polytopal norm the polytopep is assumed to be presented as the convex hull of finitely many points with rational coordinates v presented or as the intersection of finitely many closed halfspaces defined by linear inequalities with rational coefficients presented the innerj radius ofp is the radius of a largestj ball contained inp it isp s inradius whenj n and half ofp s diameter whenj 1 the outerj radius measures how wellp can be approximated in a minimax sense by an n j flat it isp s circumradius whenj n and half ofp s width whenj 1 the binary turing machine model of computation is employed the primary concern is not with finding optimal algorithms but with establishing polynomial time computability or np hardness special attention is paid to the case in whichp is centrally symmetric when the dimensionn is permitted to vary the situation is roughly as follows a for general presented polytopes in lp spaces with 1

a representation of an efficiency equivalent polyhedron for the objective set of a multiple objective linear program abstract associated with the objective set y of a linear k objective minimization problem is the efficiency equivalent polyhedron y y r k since y has the same efficient structure as y and since every extreme point of y is efficient this polyhedron provides a promising avenue for the analysis of the given multiple objective linear program molp however in order to fully explore this avenue a representation of y as a system of linear inequalities is needed in this paper an algorithm is given to construct a matrix h and a vector g such that y has the representation hy g and it is shown that no inequality in this representation is redundant the input data for the algorithm are a finite set of points of y containing the efficient extreme points and a finite set of recession directions for y containing the directions associated with unbounded efficient edges these data which can be obtained using standard molp software packages are used to form a polar polyhedron whose extreme points are precisely what is needed to define h and g

on the geometric interpretation of the nonnegative rank the nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank one factors needed to reconstruct it exactly the problem of determining this rank and computing the corresponding nonnegative factors is difficult however it has many potential applications e g in data mining graph theory and computational geometry in particular it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program in this paper we introduce and study a related quantity called the restricted nonnegative rank we show that computing this quantity is equivalent to a problem in polyhedral combinatorics and fully characterize its computational complexity this in turn sheds new light on the nonnegative rank problem and in particular allows us to provide new improved lower bounds based on its geometric interpretation we apply these results to slack matrices and linear euclidean distance matrices and obtain counter examples to two conjectures of beasly and laffey namely we show that the nonnegative rank of linear euclidean distance matrices is not necessarily equal to their dimension and that the rank of a matrix is not always greater than the nonnegative rank of its square

quantum learning of classical stochastic processes the completely positive realization problem among several tasks in machine learning a specially important one is the problem of inferring the latent variables of a system and their causal relations with the observed behavior a paradigmatic instance of this is the task of inferring the hidden markov model underlying a given stochastic process this is known as the positive realization problem prp l benvenuti and l farina ieee trans autom control 49 5 651 664 2004 and constitutes a central problem in machine learning the prp and its solutions have far reaching consequences in many areas of systems and control theory and is nowadays an important piece in the broad field of positive systems theory we consider the scenario where the latent variables are quantum i e quantum states of a finite dimensional system and the system dynamics is constrained only by physical transformations on the quantum system the observable dynamics is then described by a quantum instrument and the task is to determine which quantum instrument if any yields the process at hand by iterative application we take as a starting point the theory of quasi realizations whence a description of the dynamics of the process is given in terms of linear maps on state vectors and probabilities are given by linear functionals on the state vectors this description despite its remarkable resemblance with the hidden markov model or the iterated quantum instrument is however devoid of any stochastic or quantum mechanical interpretation as said maps fail to satisfy any positivity conditions the completely positive realization problem then consists in determining whether an equivalent quantum mechanical description of the same process exists we generalize some key results of stochastic realization theory and show that the problem has deep connections with operator systems theory giving possible insight to the lifting problem in quotient operator systems our results have potential applications in quantum machine learning device independent characterization and reverse engineering of stochastic processes and quantum processors and more generally of dynamical processes with quantum memory m gu phys rev a 83 6 062324 2011 m gu and n yamamoto e print arxiv 1303 3771 2013

widely linear estimation with complex data mean square estimation of complex and normal data is not linear as in the real case but widely linear the purpose of the paper is to calculate the optimum widely linear mean square estimate and to present its main properties the advantage with respect to the linear procedure is especially analyzed

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

a method of transferring polyhedron between the intersection form and the sum form abstract a bounded polyhedron or a polyhedral cone can be given by either an intersection of linear inequalities which is thus called intersection form or a convex combination or nonnegative linear combination of some given points which is thus called sum form this paper first proposes a simple and effective method to transfer a bounded polyhedron or a polyhedral cone from the intersection form to the sum form a new method is proposed to transfer a bounded polyhedron or a polyhedral cone from the sum form to the intersection form both procedures can be completed in finite number of iterations numerical examples are given for illustration

extension of wirtinger s calculus to reproducing kernel hilbert spaces and the complex kernel lms over the last decade kernel methods for nonlinear processing have successfully been used in the machine learning community the primary mathematical tool employed in these methods is the notion of the reproducing kernel hilbert space rkhs however so far the emphasis has been on batch techniques it is only recently that online techniques have been considered in the context of adaptive signal processing tasks moreover these efforts have only been focussed on real valued data sequences to the best of our knowledge no adaptive kernel based strategy has been developed so far for complex valued signals furthermore although the real reproducing kernels are used in an increasing number of machine learning problems complex kernels have not yet been used in spite of their potential interest in applications that deal with complex signals with communications being a typical example in this paper we present a general framework to attack the problem of adaptive filtering of complex signals using either real reproducing kernels taking advantage of a technique called complexification of real rkhss or complex reproducing kernels highlighting the use of the complex gaussian kernel in order to derive gradients of operators that need to be defined on the associated complex rkhss we employ the powerful tool of wirtinger s calculus which has recently attracted attention in the signal processing community wirtinger s calculus simplifies computations and offers an elegant tool for treating complex signals to this end in this paper the notion of wirtinger s calculus is extended for the first time to include complex rkhss and use it to derive several realizations of the complex kernel least mean square cklms algorithm experiments verify that the cklms offers significant performance improvements over several linear and nonlinear algorithms when dealing with nonlinearities

a convex analysis framework for blind separation of non negative sources this paper presents a new framework for blind source separation bss of non negative source signals the proposed framework referred herein to as convex analysis of mixtures of non negative sources camns is deterministic requiring no source independence assumption the entrenched premise in many existing usually statistical bss frameworks the development is based on a special assumption called local dominance it is a good assumption for source signals exhibiting sparsity or high contrast and thus is considered realistic to many real world problems such as multichannel biomedical imaging under local dominance and several standard assumptions we apply convex analysis to establish a new bss criterion which states that the source signals can be perfectly identified in a blind fashion by finding the extreme points of an observation constructed polyhedral set methods for fulfilling the camns criterion are also derived using either linear programming or simplex geometry simulation results on several data sets are presented to demonstrate the efficacy of the proposed method over several other reported bss methods

complex valued signal processing the proper way to deal with impropriety complex valued signals occur in many areas of science and engineering and are thus of fundamental interest in the past it has often been assumed usually implicitly that complex random signals are proper or circular a proper complex random variable is uncorrelated with its complex conjugate and a circular complex random variable has a probability distribution that is invariant under rotation in the complex plane while these assumptions are convenient because they simplify computations there are many cases where proper and circular random signals are very poor models of the underlying physics when taking impropriety and noncircularity into account the right type of processing can provide significant performance gains there are two key ingredients in the statistical signal processing of complex valued data 1 utilizing the complete statistical characterization of complex valued random signals and 2 the optimization of real valued cost functions with respect to complex parameters in this overview article we review the necessary tools among which are widely linear transformations augmented statistical descriptions and wirtinger calculus we also present some selected recent developments in the field of complex valued signal processing addressing the topics of model selection filtering and source separation

exponential lower bounds for polytopes in combinatorial optimization we solve a 20 year old problem posed by yannakakis and prove that no polynomial size linear program lp exists whose associated polytope projects to the traveling salesman polytope even if the lp is not required to be symmetric moreover we prove that this holds also for the cut polytope and the stable set polytope these results were discovered through a new connection that we make between one way quantum communication protocols and semidefinite programming reformulations of lps

on the complexity of nonnegative matrix factorization nonnegative matrix factorization nmf has become a prominent technique for the analysis of image databases text databases and other information retrieval and clustering applications the problem is most naturally posed as continuous optimization in this report we define an exact version of nmf then we establish several results about exact nmf i that it is equivalent to a problem in polyhedral combinatorics ii that it is np hard and iii that a polynomial time local search heuristic exists

a quaternion widely linear adaptive filter a quaternion widely linear qwl model for quaternion valued mean square error mse estimation is proposed the augmented statistics are first introduced into the field of quaternions and it is demonstrated that this allows for capturing the complete second order statistics available the qwl model is next incorporated into the quaternion least mean square qlms algorithm to yield the widely linear qlms wl qlms this allows for a unified approach to adaptive filtering of both q proper and q improper signals leading to improved accuracies compared to the qlms class of algorithms simulations on both benchmark and real world data support the analysis

balance laws and centro velocity in dissipative systems starting with a density that is conserved for a dynamical system when dissipation is ignored a local conservation law is derived for which the total flux integrated over the spatial domain is unique when dissipation is incorporated the conservation law becomes a balance law the contribution due to dissipation in this balance law is split in a unique way in a part that is proportional to the density and in a divergence expression that adds to the original conservative flux density the total additional flux is uniquely defined it is shown that these total fluxes appear in the expression for the centro velocity i e in the velocity of the center of gravity of the density which shows that this velocity can be defined in a unique way in contrast to a local velocity applications to the korteweg de vries burgers equations and to the incompressible navier stokes equations are given

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

smallest compact formulation for the permutahedron in this note we consider the permutahedron the convex hull of all permutations of 1 2 ldots n 1 2 n we show how to obtain an extended formulation for this polytope from any sorting network by using the optimal ajtai komlos szemeredi sorting network this extended formulation has vartheta n log n nlogn variables and inequalities furthermore from basic polyhedral arguments we show that this is best possible up to a multiplicative constant since any extended formulation has at least varomega n log n nlogn inequalities the results easily extend to the generalized permutahedron

approximation limits of linear programs beyond hierarchies we develop a framework for proving approximation limits of polynomial size linear programs lps from lower bounds on the nonnegative ranks of suitably defined matrices this framework yields unconditional impossibility results that are applicable to any lp as opposed to only programs generated by hierarchies using our framework we prove that o n1 2 approximations for clique require lps of size 2n this lower bound applies to lps using a certain encoding of clique as a linear optimization problem moreover we establish a similar result for approximations of semidefinite programs by lps our main technical ingredient is a quantitative improvement of razborov s 38 rectangle corruption lemma for the high error regime which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix

variational principles and conservation laws in the derivation of radiation boundary conditions for wave equations radiation boundary conditions are derived for partial differential equations which describe wave phenomena assuming the evolution of the system to be governed by a lagrangian variational principle boundary conditions are obtained with noether s theorem from the requirement that they transmit some appropriate density such as the energy density as well as possible the theory is applied to a nonlinear version of the klein gordon equation for this application numerical test results are presented in an accompanying paper the theory is applied to a two dimensional wave equation

on the copositive representation of binary and continuous nonconvex quadratic programs in this paper we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices this result can be viewed as an extension of earlier separate results which have established the copositive representation of a small collection of np hard problems a simplification which reduces the dimension of the linear conic program and an extension to complementarity constraints are established and computational issues are discussed

on polyhedral approximations of the second order cone we demonstrate that a conic quadratic problem is polynomially reducible to linear programming we demonstrate this by constructing for every an lp program explicitly given in terms of e and the data of cqp with the following properties i the number dimx dimu of variables and the number dimp of constraints in lp do not exceed ii every feasible solutionx to cqp can be extended to a feasible solution x u to lp iii if x u is feasible for lp thenx satisfies the e relaxed constraints of cqp namely

solution of the multisource weber and conditional weberproblems by d c programming d c programming is a recent technique of global optimization that allows the solution of problems whose objective function and constraints can be expressed as differences of convex i e d c functions many such problems arise in continuous location theory the problem first considered is to locate a known number of source facilities to minimize the sum of weighted euclidean distances between a user s fixed location and the source facility closest to the location of each user we also apply d c programming to the solution of the conditional weber problem an extension of the multisource weber problem in which some facilities are assumed to be already established in addition we consider a generalization of weber s problem the facility location problem with limited distances where the effective service distance becomes a constant when the actual distance attains a given value computational results are reported for problems with up to 10 000 users and two new facilities 50 users and three new facilities 1 000 users 20 existing facilities and one new facility or 200 users 10 existing and two new facilities

on the connection of facially exposed and nice cones abstract a closed convex cone k in a finite dimensional euclidean space is called nice if the set k f is closed for all f faces of k where k is the dual cone of k and f is the orthogonal complement of the linear span of f the niceness property plays a role in the facial reduction algorithm of borwein and wolkowicz and the question of whether the linear image of the dual of a nice cone is closed also has a simple answer we prove several characterizations of nice cones and show a strong connection with facial exposedness we prove that a nice cone must be facially exposed conversely facial exposedness with an added condition implies niceness we conjecture that nice and facially exposed cones are actually the same and give supporting evidence

hypercomplex widely linear estimation through the lens of underpinning geometry we provide a rigorous account of the equivalence between the complex valued widely linear estimation method and the quaternion involution widely linear estimation method with their vector valued real linear estimation counterparts this is achieved by an account of degrees of freedom and by providing matrix mappings between a complex variable and an isomorphic bivariate real vector and a quaternion variable versus a quadri variate real vector furthermore we show that the parameters in the complex valued linear estimation method the complex valued widely linear estimation method the quaternion linear estimation method the quaternion semi widely linear estimation method and the quaternion involution widely linear estimation method include distinct geometric structures imposed on complex numbers and quaternions respectively whereas the real valued linear estimation methods do not exhibit any structure this key difference explains both in theoretical and practical terms the advantage of estimation in division algebras complex quaternion over their multivariate real vector counterparts in addition we discuss the computational complexities of the estimators of the hypercomplex widely linear estimation methods

optimizing the double description method for normal surface enumeration many key algorithms in 3 manifold topology involve the enumeration of normal surfaces which is based upon the double description method for finding the vertices of a convex polytope typically we are only interested in a small subset of these vertices many key algorithms in 3 manifold topology involve the enumeration of normal surfaces which is based upon the double description method for finding the vertices of a convex polytope typically we are only interested in a small subset of these vertices thus opening the way for substantial optimization here we give an account of the vertex enumeration problem as it applies to normal surfaces and present new optimizations that yield strong improvements in both running time and memory consumption the resulting algorithms are tested using the freely available software package regina

complexity of approximating the vertex centroid of a polyhedron let p be an h polytope in r d with vertex set v the vertex centroid is defined as the average of the vertices in v we first prove that computing the vertex centroid of anh polytope or even just checking whether it lies in a given halfspace is p hard we also consider the problem of approximating the vertex centroid by finding a point within an e distance from it and prove this problem to be p easy in the sense that it can be solved efficiently using an oracle for some p complete problem in particular we show that given an oracle for counting the number of vertices of an h polytope one can approximate the vertex centroid in polynomial time counting the number of vertices of a polytope defined as the intersection of halfspaces is known to be p complete we also show that any algorithm approximating the vertex centroid to any sufficiently non trivial for example constant distance can be used to construct a fully polynomial time approximation scheme for approximating the centroid and also an output sensitive polynomial algorithm for the vertex enumeration problem finally we show that for unbounded polyhedra the vertex centroid cannot be approximated to a distance of d 1 2 d for any fixed constant d 0 unless p np

the kernel least mean square algorithm the combination of the famed kernel trick and the least mean square lms algorithm provides an interesting sample by sample update for an adaptive filter in reproducing kernel hilbert spaces rkhs which is named in this paper the klms unlike the accepted view in kernel methods this paper shows that in the finite training data case the klms algorithm is well posed in rkhs without the addition of an extra regularization term to penalize solution norms as was suggested by kivinen kivinen smola and williamson ldquoonline learning with kernels rdquo ieee transactions on signal processing vol 52 no 8 pp 2165 2176 aug 2004 and smale smale and yao ldquoonline learning algorithms rdquo foundations in computational mathematics vol 6 no 2 pp 145 176 2006 this result is the main contribution of the paper and enhances the present understanding of the lms algorithm with a machine learning perspective the effect of the klms step size is also studied from the viewpoint of regularization two experiments are presented to support our conclusion that with finite data the klms algorithm can be readily used in high dimensional spaces and particularly in rkhs to derive nonlinear stable algorithms with comparable performance to batch regularized solutions

conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone we investigate the completely positive semidefinite cone mathcal cs n a new matrix cone consisting of all n times n matrices that admit a gram representation by positive semidefinite matrices of any size in particular we study relationships between this cone and the completely positive and the doubly nonnegative cone and between its dual cone and trace positive noncommutative polynomials we use this new cone to model quantum analogues of the classical independence and chromatic graph parameters alpha g and chi g which are roughly obtained by allowing variables to be positive semidefinite matrices instead of 0 1 scalars in the programs defining the classical parameters we can formulate these quantum parameters as conic linear programs over the cone mathcal cs n using this conic approach we can recover the bounds in terms of the theta number and define further approximations by exploiting the link to trace positive polynomials

rank functions of tropical matrices abstract we consider rank functions important in tropical linear algebra the tropical rank equal to the topological dimension of the tropical linear span of the columns of a given matrix the factor rank equal to the smallest number of vectors containing these columns in their span the kapranov rank related to problems of tropical algebraic geometry and the determinantal and gondran minoux ranks which generalize the classical linear algebraic notion of rank we are interested in studying the arithmetic properties of the rank functions their mutual behavior and the related computational problems we discuss different open problems related to rank functions of tropical matrices and illustrate them by the number of examples

the complexity of vertex enumeration methods the computational complexity of problems relating to the enumeration of all the vertices of a convex polyhedron defined by linear inequalities is examined several published approaches are evaluated in this light a new algorithm is described and shown to have a better complexity estimate than existing methods empirical evidence supporting the theoretical superiority is presented finally vertex enumeration is discussed when the space containing the polyhedra is of fixed dimension and only the size of the inequality system is permitted to vary

theta bodies for polynomial ideals inspired by a question of lovasz we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal called theta bodies of the ideal these relaxations generalize lovasz s construction of the theta body of a graph we establish a relationship between theta bodies and lasserre s relaxations for real varieties which allows in many cases for theta bodies to be expressed as feasible regions of semidefinite programs examples from combinatorial optimization are given lovasz asked to characterize ideals for which the first theta body equals the closure of the convex hull of its real variety we answer this question for vanishing ideals of finite point sets via several equivalent characterizations we also give a geometric description of the first theta body for all ideals

nonnegative ranks decompositions and factorizations of nonnegative matrices the nonnegative rank of a nonnegative matrix is the smallest number of nonnegative rank one matrices into which the matrix can be decomposed additively such decompositions are useful in diverse scientific disciplines we obtain characterizations and bounds and show that the nonnegative rank can be computed exactly over the reals by a finite algorithm

time asymptotics and the self organization hypothesis for 2d navier stokes equations in this paper we study the long time behaviour of solutions with a two dimensional structure of the navier stokes equations on a periodic grid from a rigorous investigation of the decrease of the energy e and the enstrophy w it follows that the rayleigh quotient image is monotonically decreasing on solutions this is shown to imply that the spatial structure of any solution tends to a critical point of q which is the structure of some planar vortex and that all these structures are unstable except for the fundamental one which has the longest wavelength the time dependence of the approach towards this self organized state is investigated in some detail in the spectral plane the mean squared wave numbers of both the spectral energy and enstrophy density are shown to decrease as a function of time for the enstrophy this implies that the normal cascade does not match the discriminating effect of the viscous dissipation

segments in enumerating faces we introduce the concept of a segment of a degenerate convex polytope specified by a system of linear constraints and explain its importance in developing algorithms for enumerating the faces using segments we describe an algorithm that enumerates all the faces in time polynomial in their number the role of segments in the unsolved problem of enumerating the extreme points of a convex polytope specified by a degenerate system of linear constraints in time polynomial in the number of extreme points is discussed

methods for global concave minimization a bibliographic survey the global concave minimization problem is to find the constrained global minimum of a concave function since such a function may have many local minima finding the global minimum is a computationally difficult problem in this bibliographic survey which includes most of the recent papers on constrained global concave minimization we have attempted to briefly summarize the main ideas in each paper these recent papers include those concerned with large scale global concave minimization and bilinear programming

completely positive semidefinite rank an n times n matrix x is called completely positive semidefinite cpsd if there exist d times d hermitian positive semidefinite matrices p i i 1 n for some d ge 1 such that x ij mathrm tr p ip j for all i j in 1 ldots n the cpsd rank of a cpsd matrix is the smallest d ge 1 for which such a representation is possible in this work we initiate the study of the cpsd rank which we motivate in two ways first the cpsd rank is a natural non commutative analogue of the completely positive rank of a completely positive matrix second we show that the cpsd rank is physically motivated as it can be used to upper and lower bound the size of a quantum system needed to generate a quantum behavior in this work we present several properties of the cpsd rank unlike the completely positive rank which is at most quadratic in the size of the matrix no general upper bound is known on the cpsd rank of a cpsd matrix in fact we show that the cpsd rank can be sub exponential in terms of the size specifically for any n ge 1 we construct a cpsd matrix of size 2n whose cpsd rank is 2 varomega sqrt n our construction is based on gram matrices of lorentz cone vectors which we show are cpsd the proof relies crucially on the connection between the cpsd rank and quantum behaviors in particular we use a known lower bound on the size of matrix representations of extremal quantum correlations which we apply to high rank extreme points of the n dimensional elliptope lastly we study cpsd graphs i e graphs g with the property that every doubly nonnegative matrix whose support is given by g is cpsd we show that a graph is cpsd if and only if it has no odd cycle of length at least 5 as a subgraph this coincides with the characterization of cp graphs

quaternion valued nonlinear adaptive filtering a class of nonlinear quaternion valued adaptive filtering algorithms is proposed based on locally analytic nonlinear activation functions to circumvent the stringent standard analyticity conditions which are prohibitive to the development of nonlinear adaptive quaternion valued estimation models we use the fact that stochastic gradient learning algorithms require only local analyticity at the operating point in the estimation space it is shown that the quaternion valued exponential function is locally analytic and since local analyticity extends to polynomials products and ratios we show that a class of transcendental nonlinear functions can serve as activation functions in nonlinear and neural adaptive models this provides a unifying framework for the derivation of gradient based learning algorithms in the quaternion domain and the derived algorithms are shown to have the same generic form as their real and complex valued counterparts to make such models second order optimal for the generality of quaternion signals both circular and noncircular we use recent developments in augmented quaternion statistics to introduce widely linear versions of the proposed nonlinear adaptive quaternion valued filters this allows full exploitation of second order information in the data contained both in the covariance and pseudocovariances to cater rigorously for second order noncircularity improperness and the corresponding power mismatch in the signal components simulations over a range of circular and noncircular synthetic processes and a real world 3 d noncircular wind signal support the approach

complexity and algorithms for computing voronoi cells of lattices in this paper we are concerned with finding the vertices of the voronoi cell of a euclidean lattice given a basis of a lattice we prove that computing the number of vertices is a p hard problem on the other hand we describe an algorithm for this problem which is especially suited for low dimensional say dimensions at most 12 and for highly symmetric lattices we use our implementation which drastically outperforms those of current computer algebra systems to find the vertices of voronoi cells and quantizer constants of some prominent lattices

equivariant semidefinite lifts of regular polygons given a polytope p n we say that p has a positive semidefinite lift psd lift of size d if one can express p as the projection of an affine slice of the d d positive semidefinite cone such a representation allows us to solve linear optimization problems over p using a semidefinite program of size d and can be useful in practice when d is much smaller than the number of facets of p if a polytope p has symmetry we can consider equivariant psd lifts i e those psd lifts that respect the symmetries of p one of the simplest families of polytopes with interesting symmetries is regular polygons in the plane in this paper we give tight lower and upper bounds on the size of equivariant psd lifts for regular polygons we give an explicit construction of an equivariant psd lift of the regular 2n gon of size 2n 1 and we prove that our construction is essentially optimal by proving a lower bound on the size of any equivariant psd lift of the regular n gon that is logarithmic in n our construction is

which nonnegative matrices are slack matrices abstract in this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices this leads to an algorithm for deciding whether a given matrix is a slack matrix the underlying decision problem is equivalent to the polyhedral verification problem whose complexity is unknown

disjoint products and efficient computation of reliability in this paper we analyze the problem of computing the probability of the union of a set of events where each event is given as the product of a set of boolean variables each boolean variable represents the operation or failure of a particular component the problem has direct applications to the reliability analysis of complex systems as well as more general applications after showing that the problem is np hard in general we define simple computationally efficient procedures for generating upper and lower bounds on the required probability we show that these procedures produce the exact answer i e the upper bound equals the lower bounds for two classes of systems matroids and threshold systems these results draw on the relationship between this problem and the notion of shellability studied in the context of simplicial polytopes shellability is shown to have a very interesting and useful interpretation in this problem setting

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

enumeration of all extreme equilibria of bimatrix games the set of equilibrium points of a bimatrix game is the union of polytopes that are not necessarily disjoint knowledge of the vertices of these polytopes extreme equilibria is sufficient to identify all equilibria we present an algorithm that enumerates all extreme equilibria by exploiting complementary slackness optimality conditions of two pairs of parameterized linear programming problems the algorithm is applied to randomly generated problems of size up to 29 29 when both dimensions are equal and up to 700 5 when the second dimension is fixed the number of extreme equilibria grows exponentially with problem size but remains moderate for the instances considered therefore the results could be useful for further study of refinements of nash equilibria

the vector linear program solver bensolve notes on theoretical background bensolve is an open source implementation of benson s algorithm and its dual variant both algorithms compute primal and dual solutions of vector linear programs vlp which include the subclass of multiple objective linear programs molp the recent version of bensolve can treat arbitrary vector linear programs whose upper image does not contain lines this article surveys the theoretical background of the implementation in particular the role of vlp duality for the implementation is pointed out some numerical examples are provided in contrast to the existing literature we consider a less restrictive class of vector linear programs

affine reductions for lps and sdps we define a reduction mechanism for lp and sdp formulations that degrades approximation factors in a controlled fashion our reduction mechanism is a minor restriction of classical hardness reductions requiring an additional independence assumption and it allows for reusing many hardness reductions that have been used to show inapproximability in the context of pcp theorems as a consequence we establish strong linear programming inapproximability for lps with a polynomial number of constraints for many problems in particular we obtain a frac 3 2 varepsilon inapproximability for answering an open question in chan et al proceedings of focs pp 350 359 2013 https doi org 10 1109 focs 2013 45 and prove an inapproximability factor of frac 1 2 varepsilon for bounded degree in the case of sdps we obtain inapproximability results for these problems relative to the sdp inapproximability of textsf maxcut moreover using our reduction framework we are able to reproduce various results for csps from chan et al proceedings of focs pp 350 359 2013 https doi org 10 1109 focs 2013 45 via simple reductions from max 2 xor

on representing the positive semidefinite cone using the second order cone the second order cone plays an important role in convex optimization and has strong expressive abilities despite its apparent simplicity second order cone formulations can also be solved more efficiently than semidefinite programming problems in general we consider the following question posed by lewis and glineur parrilo saunderson is it possible to express the general positive semidefinite cone using second order cones we provide a negative answer to this question and show that the 3 times 3 positive semidefinite cone does not admit any second order cone representation in fact we show that the slice consisting of 3 times 3 positive semidefinite hankel matrices does not admit a second order cone representation our proof relies on exhibiting a sequence of submatrices of the slack matrix of the 3 times 3 positive semidefinite cone whose second order cone rank grows to infinity

on the existence of 0 1 polytopes with high semidefinite extension complexity in rothvo math program 142 1 2 255 268 2013 it was shown that there exists a 0 1 polytope a polytope whose vertices are in 0 1 n 0 1 n such that any higher dimensional polytope projecting to it must have 2 varomega n 2 n facets i e its linear extension complexity is exponential the question whether there exists a 0 1 polytope with high positive semidefinite extension complexity was left open we answer this question in the affirmative by showing that there is a 0 1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension 2 varomega n 2 n and an affine space our proof relies on a new technique to rescale semidefinite factorizations

performance analysis of deficient length quaternion least mean square adaptive filters quaternion adaptive filters have been widely used for processing 3d and 4d phenomena deficient length quaternion adaptive filters are explicitly or implicitly used in many practical applications where the length of system impulse response is large or unknown however their statistical behaviors are yet to be fully understood as theoretical results on the class of full length quaternion least mean square qlms algorithms do not necessarily apply to their deficient length versions this article fills this void and analyses the mean and mean square convergence of the deficient length qlms algorithms both for the strictly linear and widely linear cases transient and steady state performance is characterised by exploiting the augmented statistics of noncircular quaternion random vectors a novel decorrelation technique in the quaternion domain is shown to allow for the development of intuitive closed form solutions for correlated quaternion gaussian inputs thus unveiling the relationship between the algorithm behaviour and the noncircularity of quaternion input data the analysis also provides a general framework whereby the strictly linear and semi widely linear qlms algorithms can be seen as deficient length versions of the widely linear qlms numerical simulations validate the accuracy of the theoretical results and support the behaviour of the considered algorithms

on the linear extension complexity of regular n gons in this paper we propose new lower and upper bounds on the linear extension complexity of regular n gons our bounds are based on the equivalence between the computation of i an extended formulation of size r of a polytope p and ii a rank r nonnegative factorization of a slack matrix of the polytope p the lower bound is based on an improved bound for the rectangle covering number also known as the boolean rank of the slack matrix of the n gons the upper bound is a slight improvement of the result of fiorini rothvoss and tiwary 2012 9 the difference with their result is twofold i our proof uses a purely algebraic argument while fiorini et al used a geometric argument and ii we improve the base case allowing us to reduce their upper bound 2 log2 n by one when 2k 1 n 2k 1 2k 2 for some integer k we conjecture that this new upper bound is tight which is suggested by numerical experiments for small n moreover this improved upper bound allows us to close the gap with the best known lower bound for certain regular n gons namely 9 n 13 and 21 n 24 hence allowing for the first time to determine their extension complexity

generalized probabilistic theories and conic extensions of polytopes generalized probabilistic theories gpt provide a general framework that includes classical and quantum theories it is described by a cone c and its dual c we show that whether some one way communication complexity problems can be solved within a gpt is equivalent to the recently introduced cone factorization of the corresponding communication matrix m we also prove an analogue of holevo s theorem when the cone c is contained in the classical capacity of the channel realized by sending gpt states and measuring them is bounded by polytopes and optimising functions over polytopes arise in many areas of discrete mathematics a conic extension of a polytope is the intersection of a cone c with an affine subspace whose projection onto the original space yields the desired polytope extensions of polytopes can sometimes be much simpler geometric objects than the polytope itself the existence of a conic extension of a polytope is equivalent to that of a cone factorization of the slack matrix of the polytope on the same cone we show that all polytopes whose vertices can be recognized by a polynomial size circuit which includes as a special case the travelling salesman polytope and many other polytopes from combinatorial optimization have small conic extension complexity when the cone is the completely positive cone using recent exponential lower bounds on the linear extension complexity of polytopes this provides an exponential gap between the communication complexity of gpt based on the completely positive cone and classical communication complexity and a conjectured exponential gap with quantum communication complexity our work thus relates the communication complexity of generalizations of quantum theory to questions of mainstream interest in the area of combinatorial optimization

h pid controller design for lur e systems and its application to a ball and wheel apparatus in this paper we consider the problem of synthesizing proportional integral derivative pid controllers for a given lur e system in the presence of exogenous energy bounded disturbance based on the circle criterion sufficient conditions for guaranteeing closed loop absolute stability and achieving disturbance attenuation are given in terms of the multi objective h specifications the results from the earlier work are then used to solve the resulting multi objective h pid design problem it is shown that for a fixed proportional gain and by sweeping over a variable the set of admissible integral and derivative gain values can be determined constructively using linear programming techniques the proposed synthesis method is used to design a pid controller for the ball and wheel system and the experimental results are presented to show effectiveness of the proposed method

new lower bounds for convex hull problems in odd dimensions we show that in the worst case omega n ceil d 2 1 n log n sidedness queries are required to determine whether the convex hull of n points in real d is simplicial or to determine the number of convex hull facets this lower bound matches known upper bounds in any odd dimension our result follows from a straightforward adversary argument a key step in the proof is the construction of a quasi simplicial n vertex polytope with omega n ceil d 2 1 degenerate facets while it has been known for several years that d dimensional convex hulls can have omega n floor d 2 facets the previously best lower bound for these problems is only omega n log n using similar techniques we also obtain simple and correct proofs of erickson and seidel s lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane as a related result we show that detecting simplicial convex hulls in real d is ceil d 2 sum hard in the sense of gajentaan and overmars

tropical lower bounds for extended formulations the tropical arithmetic operations on mathbb r r defined as oplus a b rightarrow min a b a b min a b and otimes a b rightarrow a b a b a b arise from studying the geometry over non archimedean fields we present an application of tropical methods to the study of extended formulations for convex polytopes we propose a non archimedean generalization of the well known boolean rank bound for the extension complexity we show how to construct a real polytope with the same extension complexity and combinatorial type as a given non archimedean polytope our results allow us to develop a method of constructing real polytopes with large extension complexity

lower bounds on nonnegative rank via nonnegative nuclear norms the nonnegative rank of an entrywise nonnegative matrix a in mathbb r m times n a r m n is the smallest integer r r such that a a can be written as a uv a uv where u in mathbb r m times r u r m r and v in mathbb r r times n v r r n are both nonnegative the nonnegative rank arises in different areas such as combinatorial optimization and communication complexity computing this quantity is np hard in general and it is thus important to find efficient bounding techniques especially in the context of the aforementioned applications in this paper we propose a new lower bound on the nonnegative rank which unlike most existing lower bounds does not solely rely on the matrix sparsity pattern and applies to nonnegative matrices with arbitrary support the idea involves computing a certain nuclear norm with nonnegativity constraints which allows to lower bound the nonnegative rank in the same way the standard nuclear norm gives lower bounds on the standard rank our lower bound is expressed as the solution of a copositive programming problem and can be relaxed to obtain polynomial time computable lower bounds using semidefinite programming we compare our lower bound with existing ones and we show examples of matrices where our lower bound performs better than currently known ones

a laplace transform algorithm for the volume of a convex polytope we provide two algorithms for computing the volume of the convex polytope x n ax b for a m n b n the computational complexity of both algorithms is essentially described by nm which makes them especially attractive for large n and relatively small m when the other methods with o mn complexity fail the methodology which differs from previous existing methods uses a laplace transform technique that is well suited to the half space representation of

cones of matrices and successive convex relaxations of nonconvex sets let f be a compact subset of the n dimensional euclidean space rn represented by finitely or infinitely many quadratic inequalities we propose two methods one based on successive semidefinite programming sdp relaxations and the other on successive linear programming lp relaxations each of our methods generates a sequence of compact convex subsets ck k 1 2 of rn such that a the convex hull of f subseteq c k 1 subseteq c k monotonicity b cap k 1 infty c k text the convex hull of f asymptotic convergence our methods are extensions of the corresponding lovasz schrijver lift and project procedures with the use of sdp or lp relaxation applied to general quadratic optimization problems qops with infinitely many quadratic inequality constraints utilizing descriptions of sets based on cones of matrices and their duals we establish the exact equivalence of the sdp relaxation and the semi infinite convex qop relaxation proposed originally by fujie and kojima using this equivalence we investigate some fundamental features of the two methods including a and b above

dual domain adaptive beamformer under linearly and quadratically constrained minimum variance in this paper a novel adaptive beamforming algorithm is proposed under a linearly and quadratically constrained minimum variance lqcmv beamforming framework based on a dual domain projection approach that can efficiently implement a quadratic inequality constraint with a possibly rank deficient positive semi definite matrix and the properties of the proposed algorithm are analyzed as an application relaxed zero forcing rzf beamforming is presented which adopts a specific quadratic constraint that bounds the power of residual interference in the beamformer output with the aid of interference channel side information available typically in wireless multiple access systems the dual domain projection in this case plays a role in guiding the adaptive algorithm towards a better direction to minimize the interference and noise leading to considerably faster convergence the robustness issue against channel mismatch and ill posedness is also addressed numerical examples show that the efficient use of interference side information brings considerable gains

regularizing the abstract convex program abstract characterizations of optimality for the abstract convex program inf p x g x s x p where s is an arbitrary convex cone in a finite dimensional space is a convex set and p and g are respectively convex and s convex on were given in 10 these characterizations hold without any constraint qualification they use the minimal cone s f of p and the cone of directions of constancy d g s f in the faithfully convex case these cones can be used to regularize p i e transform p into an equivalent program p r for which slater s condition holds we present an algorithm that finds both s f and d g s f the main step of the algorithm consists in solving a particular complementarity problem we also present a characterization of optimality for p in terms of the cone of directions of constancy of a convex functional d g rather than d g s f

wave diffusion dualism of the neutral fractional processes in this paper a neutral fractional equation is introduced and analyzed in contrast to the general time and space fractional diffusion equation the neutral fractional equation contains fractional derivatives of the same order 1 2 both in space and in time as it has been shown earlier solutions of the neutral fractional equation can be interpreted as damped waves with the constant propagation velocities that means that this equation inherits some characteristics of the wave equation otherwise the first fundamental solution of the one dimensional neutral fractional equation is known to be a spatial probability density function evolving in time and is thus related to the diffusion processes in this paper we investigate the entropy and the entropy production rate of the neutral fractional equation and show that both of them are strongly connected to those of the diffusion processes thus a wave diffusion dualism of the processes described by the neutral fractional equation is established

multi criteria analysis with partial information about the weighting coefficients abstract in this paper we address the problem of ranking a set of alternatives with partial information about the weighting coefficients we introduce a family of quasiorders that are easily interpretable and manageable which includes among others the natural quasiorder in r n and other well known preference structures in the literature the enrichment of the preference structure with respect to the natural quasiorder is measured by means of an absolute measure we introduce