higher order properties of gmm and generalized empirical likelihood estimators in an effort to improve the small sample properties of generalized method of moments gmm estimators a number of alternative estimators have been suggested these include empirical likelihood el continuous updating and exponential tilting estimators we show that these estimators share a common structure being members of a class of generalized empirical likelihood gel estimators we use this structure to compare their higher order asymptotic properties we find that gel has no asymptotic bias due to correlation of the moment functions with their jacobian eliminating an important source of bias for gmm in models with endogeneity we also find that el has no asymptotic bias from estimating the optimal weight matrix eliminating a further important source of bias for gmm in panel data models we give bias corrected gmm and gel estimators we also show that bias corrected el inherits the higher order property of maximum likelihood that it is higher order asymptotically efficient relative to the other bias corrected estimators

improving the accuracy of computed eigenvalues and eigenvectors this paper describes a computational method for improving the accuracy of a given eigenvalue and its associated eigenvector the method is analogous to iterative improvement for the solution of linear systems an iterative algorithm using working precision arithmetic is applied to increase the accuracy of the eigenpair the only extended precision computation is the residual calculation the method is related to inverse iteration and to newton s method applied to the eigenvalue problem

rigid body constrained noisy point pattern matching noisy pattern matching problems arise in many areas e g computational vision robotics guidance and control stereophotogrammetry astronomy genetics and high energy physics least squares pattern matching over the euclidean space e sup n for unordered sets of cardinalities p and q is commonly formulated as a combinatorial optimization problem having complexity p p 1 spl middot spl middot spl middot p q 1 q spl les p since p and q may be 10 sup 3 or larger in typical applications less than satisfactory suboptimal methods are usually employed a hybrid approach is described for solving the pattern matching problem under rigid motion constraints which often apply the method reduces the complexity to l sub 21 spl middot n sup 4 l sub 12 spl middot p sup 3 where l sub 12 and l sub 21 are the number of iterations required by steepest ascent and singular value decomposition svd based procedures respectively

understanding the qr algorithm the qr algorithm is currently the most popular method for finding all eigenvalues of a full matrix while qr is now well understood by specialists in eigenvalue computations this understanding is not being conveyed effectively to the mathematical public many accounts present wilkinson s 1965 convergence proof others establish some of the connections between the qr algorithm the power method and inverse iteration usually much emphasis is rightly placed on the refinements such as shifts of origin which are required to make the algorithm competitive but practically all accounts fail to explain the basic meaning of qr iterations as a consequence the qr algorithm is widely thought to be difficult to understand the purpose of this paper is to try to convince the reader that the opposite is true in fact the qr algorithm is neither more nor less than a clever implementation of simultaneous iteration which is itself a natural easily understood extension of the power method this point

the constrained newton method on a lie group and the symmetric eigenvalue problem the newton method for estimating a critical point of a real function is formulated in a coordinate free manner on an arbitrary lie group convergence proofs for the numerical method are given an application of the general approach to computing the eigenvalues of a symmetric matrix is given and the resultant algorithm is compared with the classical shifted qr algorithm properties of the method described suggest that it is of interest for certain computations in online and adaptive environments

convergence of gradient descent for low rank matrix approximation this paper provides a proof of global convergence of gradient search for low rank matrix approximation such approximations have recently been of interest for large scale problems as well as for dictionary learning for sparse signal representations and matrix completion the proof is based on the interpretation of the problem as an optimization on the grassmann manifold and fubiny study distance on this space

dualities between entropy functions and network codes in communications networks the capacity region of multisource network coding is given in terms of the set of entropy functions gamma more broadly determination of gamma would have an impact on converse theorems for multi terminal problems in information theory this paper provides several new dualities between entropy functions and network codes given a function g ges 0 defined on all subsets of n random variables we provide a construction for a network multicast problem which is ldquosolvablerdquo if and only if g is the entropy function of a set of quasi uniform random variables the underlying network topology is fixed and the multicast problem depends on g only through link capacities and source rates a corresponding duality is developed for linear network codes where the constructed multicast problem is linearly solvable if and only if g is linear group characterizable relaxing the requirement that the domain of g be subsets of random variables we obtain a similar duality between polymatroids and the linear programming bound these duality results provide an alternative proof of the insufficiency of linear and abelian network codes and demonstrate the utility of non shannon inequalities to tighten outer bounds on network coding capacity regions

truncation technique for characterizing linear polymatroids linear polymatroids have a strong connection to network coding the problem of finding the linear network coding capacity region is equivalent to the characterization of all linear polymatroids it is well known that linear polymatroids must satisfy the inequalities of ingleton combin math appln 1971 however it has been an open question for years as to whether these inequalities are sufficient it was until recently that new subspace rank inequalities have been discovered independently by kinser and dougherty in this paper we propose a new approach to investigate properties of linear polymatroids specifically we demonstrate how to construct a new polymatroid that satisfies not only the ingleton and dfz inequalities but also lies outside the minimal closed and convex cone containing all linear polymatroids using this polymatroid we prove that all truncation preserving inequalities including ingleton inequalities and dfz inequalities are insufficient to characterize linear polymatroids

a lie bracket decomposition and its application to flows on symmetric matrices abstract the paper studies the problem of finding a canonical form for differential equations on symmetric matrices that applies in particular to isospectral flows and spectrum increasing flows such a canonical form is obtained using the result for any lie algebra u and elements w z u such that z is ad semisimple there exist unique a b u with w z a b such that a z c for some c u and z b 0 taking u gl n r the ad semisimplicity condition holds for symmetric matrices z and differential equations then have a canonical form z t z t a t b t where a t z c t and z b t 0 a flow on symmetric matrices is isospectral if and only if b t 0 and spectrum increasing if and only if b t is positive definite applying this canonical form twice to the finite nonperiodic toda flow we obtain a double bracket form for this differential equation that differs from one recently found by bloch

differential equations and the qr algorithm in this paper we consider a variety of isospectral flows on the set of n times n matrices these flows arise from lax pairs and can all be interpreted in terms of the qr decomposition for nonsingular matrices the asymptotics of these differential equations are considered in detail and for symmetric matrices these asymptotics provide a new method of solving the eigenvalue problem

local convergence analysis of fastica and related algorithms the fastica algorithm is one of the most prominent methods to solve the problem of linear independent component analysis ica although there have been several attempts to prove local convergence properties of fastica rigorous analysis is still missing in the community the major difficulty of analysis is because of the well known sign flipping phenomenon of fastica which causes the discontinuity of the corresponding fastica map on the unit sphere in this paper by using the concept of principal fiber bundles fastica is proven to be locally quadratically convergent to a correct separation higher order local convergence properties of fastica are also investigated in the framework of a scalar shift strategy moreover as a parallelized version of fastica the so called qr fastica algorithm which employs the qr decomposition gram schmidt orthonormalization process instead of the polar decomposition is shown to share similar local convergence properties with the original fastica

the geometry of weighted low rank approximations the low rank approximation problem is to approximate optimally with respect to some norm a matrix by one of the same dimension but smaller rank it is known that under the frobenius norm the best low rank approximation can be found by using the singular value decomposition svd although this is no longer true under weighted norms in general it is demonstrated here that the weighted low rank approximation problem can be solved by finding the subspace that minimizes a particular cost function a number of advantages of this parameterization over the traditional parameterization are elucidated finding the minimizing subspace is equivalent to minimizing a cost function on the grassmann manifold a general framework for constructing optimization algorithms on manifolds is presented and it is shown that existing algorithms in the literature are special cases of this framework within this framework two novel algorithms a steepest descent algorithm and a newton like algorithm are derived for solving the weighted low rank approximation problem they are compared with other algorithms for low rank approximation as well as with other algorithms for minimizing a cost function on a grassmann manifold

bartlett adjustment of empirical discrepancy statistics summary we introduce two classes of empirical discrepancy statistics that extend empirical likelihood and establish simple conditions under which they admit bartlett adjustment

on the largest principal angle between random subspaces formulas are derived for the probability density function and the probability distribution function of the largest canonical angle between two p dirnensional subspaces of r n chosen from the uniform distribution on the grassmann manifold which is the unique distribution invariant by orthogonal transformations of r n the formulas involve the gamma function and the hypergeometric function of a matrix argument c 2005 elsevier inc all rights reserved

networks matroids and non shannon information inequalities we define a class of networks called matroidal networks which includes as special cases all scalar linearly solvable networks and in particular solvable multicast networks we then present a method for constructing matroidal networks from known matroids we specifically construct networks that play an important role in proving results in the literature such as the insufficiency of linear network coding and the unachievability of network coding capacity we also construct a new network from the vamos matroid which we call the vamos network and use it to prove that shannon type information inequalities are in general not sufficient for computing network coding capacities to accomplish this we obtain a capacity upper bound for the vamos network using a non shannon type information inequality discovered in 1998 by zhang and yeung and then show that it is smaller than any such bound derived from shannon type information inequalities this is the first application of a non shannon type inequality to network coding we also compute the exact routing capacity and linear coding capacity of the vamos network finally using a variation of the vamos network we prove that shannon type information inequalities are insufficient even for computing network coding capacities of multiple unicast networks

nonparametric assessment of hedge fund performance abstract we propose a new class of performance measures for hedge fund hf returns based on a family of empirically identifiable stochastic discount factors sdfs the sdf based measures incorporate no arbitrage pricing restrictions and naturally embed information about higher order mixed moments between hf and benchmark factors returns we provide a full asymptotic theory for our sdf estimators to test for the statistical significance of each fund s performance and for the relevance of individual benchmark factors within each proposed measure we apply our methodology to a panel of 4815 individual hedge funds our empirical analysis reveals that fewer funds have a statistically significant positive alpha compared to the jensen s alpha obtained by the traditional linear regression approach moreover the funds rankings vary considerably between the two approaches performance also varies between the members of our family because of a different fund exposure to higher order moments of the benchmark factors highlighting the potential heterogeneity across investors in evaluating performance

on a relation between information inequalities and group theory we establish a one to one correspondence between information inequalities and group inequalities the major implication of our result is that we can prove information inequalities by proving the corresponding group inequalities and vice versa by giving a group theoretic proof for all shannon type inequalities we suggest that new inequalities could be discovered by making use of the rich set of tools in group theory on the other hand via a non shannon type information inequality discovered by zhang and yeung 1997 we obtain a new inequality in group theory whose meaning is yet to be understood

economic implications of nonlinear pricing kernels based on a family of discrepancy functions we derive nonparametric stochastic discount factor bounds that naturally generalize variance entropy and higher moment bounds these bounds are especially useful to identify how parameters affect pricing kernel dispersion in asset pricing models in particular they allow us to distinguish between models where dispersion comes mainly from skewness from models where kurtosis is the primary source of dispersion we analyze the admissibility of disaster disappointment aversion and long run risk models with respect to these bounds this paper was accepted by jerome detemple finance

a framework for linear information inequalities we present a framework for information inequalities namely inequalities involving only shannon s information measures for discrete random variables a region in ir 2 sup n 1 denoted by spl gamma is identified to be the origin of all information inequalities involving n random variables in the sense that all such inequalities are partial characterizations of spl gamma a product from this framework is a simple calculus for verifying all unconstrained and constrained linear information identities and inequalities which can be proved by conventional techniques these include all information identities and inequalities of such types in the literature as a consequence of this work most identities and inequalities involving a definite number of random variables can now be verified by a software called itip which is available on the world wide web our work suggests the possibility of the existence of information inequalities which cannot be proved by conventional techniques we also point out the relation between spl gamma and some important problems in probability theory and information theory

on ingleton violating finite groups given n discrete random variables its entropy vector is the 2 n 1 dimensional vector obtained from the joint entropies of all non empty subsets of the random variables it is well known that there is a close relation between such an entropy vector and a certain group characterizable vector obtained from a finite group and n of its subgroups indeed roughly speaking knowing the region of all such group characterizable vectors is equivalent to knowing the region of all entropy vectors this correspondence may be useful for characterizing the space of entropic vectors and for designing network codes if one restricts attention to abelian groups then not all entropy vectors can be obtained this is an explanation for the fact shown by dougherty et al that linear network codes cannot achieve capacity in general network coding problems since linear network codes come from abelian groups all abelian group characterizable vectors and by fiat all entropy vectors generated by linear network codes satisfy a linear inequality called the ingleton inequality general entropy vectors however do not necessarily have this property it is therefore of interest to identify groups that violate the ingleton inequality in this paper we study the problem of finding nonabelian finite groups that yield characterizable vectors which violate the ingleton inequality using a refined computer search we find the symmetric group s 5 to be the smallest group that violates the ingleton inequality careful study of the structure of this group and its subgroups reveals that it belongs to the ingleton violating family pgl 2 q with a prime power q geq 5 i e the projective group of 2 times 2 nonsingular matrices with entries in mathbb f q we further interpret this family of groups and their subgroups using the theory of group actions and identify the subgroups as certain stabilizers we also extend the construction to more general groups such as pgl n q and gl n q the families of groups identified here are therefore good candidates for constructing network codes more powerful than linear network codes and we discuss some considerations for constructing such group network codes

on the continuous realization of iterative processes many important mathematical problems are solved by iterative methods many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems this paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods

dynamical systems and non hermitian iterative eigensolvers simple preconditioned iterations can provide an efficient alternative to more elaborate eigenvalue algorithms we observe that these simple methods can be viewed as forward euler discretizations of well known autonomous differential equations that enjoy appealing geometric properties this connection facilitates novel results describing convergence of a class of preconditioned eigensolvers to the leftmost eigenvalue provides insight into the role of orthogonality and biorthogonality and suggests the development of new methods and analyses based on more sophisticated discretizations these results also highlight the effect of preconditioning on the convergence and stability of the continuous time system and its discretization

entropy region and convolution the entropy region is constructed from vectors of random variables by collecting shannon entropies of all subvectors its shape is studied here by means of polymatroidal constructions notably by convolution the closure of the region is decomposed into the direct sum of tight and modular parts reducing the study to the tight part the relative interior of the reduction belongs to the entropy region behavior of the decomposition under self adhesivity is clarified results are specialized and extended to the region constructed from four tuples of random variables this and computer experiments help to visualize approximations of a symmetrized part of the entropy region the four atom conjecture on the minimal ingleton score is refuted

a grassmann rayleigh quotient iteration for computing invariant subspaces the classical rayleigh quotient iteration rqi allows one to compute a one dimensional invariant subspace of a symmetric matrix a here we propose a generalization of the rqi which computes a p dimensional invariant subspace of a cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature

the dynamics of rayleigh quotient iteration this paper analyzes the rayleigh quotient iteration algorithm for symmetric matrices dynamical systems techniques are employed to characterize the sets of points for which the algorithm will converge to an eigenvector it is shown that these sets have full measure and that their geometric nature is related to the spacing of the eigenvalues

insufficiency of linear coding in network information flow it is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet it is also known that this result does not generalize to arbitrary networks there are several examples in the literature of solvable networks with no scalar linear solution over any finite field however each example has a linear solution for some vector dimension greater than one it has been conjectured that every solvable network has a linear solution over some finite field alphabet and some vector dimension we provide a counterexample to this conjecture we also show that if a network has no linear solution over any finite field then it has no linear solution over any finite commutative ring with identity our counterexample network has no linear solution even in the more general algebraic context of modules which includes as special cases all finite rings and abelian groups furthermore we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field exactly 10 greater so the network is not even asymptotically linearly solvable it follows that even for more general versions of linearity such as convolutional coding filter bank coding or linear time sharing the network has no linear solution

an information theoretic alternative to generalized method of moments estimation while optimally weighted gmm estimation has desirable large sample properties its small sample performance is poor in some applications we propose a computationally simple alternative for weakly dependent data generating mechanisms based on minimization of the kullback leibler information criterion conditions are derived under which the large sample properties of this estimator are similar to gmm i e the estimator will be consistent and asymptotically normal with the same asymptotic covariance matrix as gmm in addition we propose overidentifying and parametric restrictions tests as alternatives to analogous gmm procedures

a framework for generalising the newton method and other iterative methods from euclidean space to manifolds the newton iteration is a popular method for minimising a cost function on euclidean space various generalisations to cost functions defined on manifolds appear in the literature in each case the convergence rate of the generalised newton iteration needed establishing from first principles the present paper presents a framework for generalising iterative methods from euclidean space to manifolds that ensures local convergence rates are preserved it applies to any memoryless iterative method computing a coordinate independent property of a function such as a zero or a local minimum all possible newton methods on manifolds are believed to come under this framework changes of coordinates and not any riemannian structure are shown to play a natural role in lifting the newton method to a manifold the framework also gives new insight into the design of newton methods in general

assessing misspecified asset pricing models with empirical likelihood estimators abstract hansen and jagannathan 1997 compare misspecified asset pricing models based on least square projections on a family of admissible stochastic discount factors we extend their fundamental contribution by considering minimum discrepancy projections where misspecification is measured by a family of convex functions that take into account higher moments of asset returns the minimum discrepancy problems are solved on dual spaces producing a family of estimators that captures the least square problem as a particular case we derive the asymptotic distributions of the estimators for the cressie read family of discrepancies and illustrate their use with an assessment of the consumption asset pricing model

least squares matching problems abstract a large class of matching problems can be thought of in the following way one is given a group g which acts on a riemannian manifold and one is given two points x 1 and x 2 in the manifold the problem is to find g g such that d g x 1 x 2 is as small as possible if the group has finite cardinality this is a combinatorial optimization problem whereas if g is a lie group one can use the calculus to obtain necessary conditions our main results establish circumstances under which one can relax the combinatorial problem to an appropriate lie group problem and then solve it by the method of steepest descent an interesting class of applications involves embedding the permutation group on n letters in the orthogonal group o n with the latter acting on r n n 1 2 via a symmetricized tensor product the convergence properties of the descent equation which arises in this way are analyzed in some detail

block newton method and block rayleigh quotient iteration for computing invariant subspaces of general complex matrices abstract we consider the block newton method and a modification of it the block rayleigh quotient iteration for approximating a simple p dimensional invariant subspace x im x and the corresponding eigenvalues collected in the projection l x h x 1 x h a x of an arbitrary complex matrix a c n n both methods generate a sequence of bases u k such that the subspaces im u k approximate the target subspace x the block newton method is newton s method applied to the extended system a x x l 0 w h x i with respect to x l where w is a normalization matrix we give a direct convergence analysis which in contrast to the approach by beyn kless thummler 2001 explicitly exploits the product form of the second derivative and therefore leads to new and essentially sharper error recursions for each of the sequences u k and k where u k and k are the k th x and l iterates resp then following losche schwetlick timmermann 1998 where the case of real symmetric a is considered we modify the block newton method as follows in step k we set w u k and use only the x update from the newton method to improve u k and then with the improved basis perform a rayleigh schur process and take the orthonormal basis and the generalized rayleigh quotient matrix obtained from it as new iterates so the method can be considered as a block rayleigh quotient iteration we show that for a sufficiently good initial approximation u 0 the method converges in that the sequence sin k with k im u k x converges q quadratically to zero provided that the invariant subspace x is simple i e that the eigenvalues corresponding to x which are the eigenvalues of the projection l x h x 1 x h a x of a onto x are separated from those corresponding to x which are given by a l the convergence results are directly proven without exploiting standard newton convergence results by using the angles between the corresponding subspaces instead of working with the norm differences of the bases as done by beyn kless thummler 2001 as a byproduct the method is shown to converge cubically if a a h as in the case p 1

the projected gradient methods for least squares matrix approximations with spectral constraints the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure this paper describes a general procedure in using the projected gradient method it is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly this gives rise to the construction of a descent flow that can be followed numerically the explicit form also facilitates the computation of the second order optimality conditions examples of applications are discussed with slight modifications the procedure can be extended to solve least squares problems for general matrices subject to singular value constraints

duality relationships for entropy like minimization problems this paper considers the minimization of a convex integral functional over the positive cone of an l p space subject to a finite number of linear equality constraints such problems arise in spectral estimation where the bjective function is often entropy like and in constrained approximation the lagrangian dual problem is finite dimensional and unconstrained under a quasi interior constraint qualification the primal and dual values are equal with dual attainment examples show the primal value may not be attained conditions are given that ensure that the primal optimal solution can be calculated directly from a dual optimum these conditions are satisfied in many examples

self similar flows abstract large classes of self similar isospectral flows can be viewed as continuous analogues of certain matrix eigenvalue algorithms in particular there exist families of flows associated with the qr lr and cholesky eigenvalue algorithms this paper uses lie theory to develop a general theory of self similar flows which includes the qr lr and cholesky flows as special cases also included are new families of flows associated with the sr and hr eigenvalue algorithms the basic theory produces analogues of unshifted single step eigenvalue algorithms but it is also shown how the theory can be extended to include flows which are continuous analogues of shifted and multiple step eigenvalue algorithms

two sided grassmann rayleigh quotient iteration the two sided rayleigh quotient iteration proposed by ostrowski computes a pair of corresponding left right eigenvectors of a matrix c we propose a grassmannian version of this iteration i e its iterates are pairs of p dimensional subspaces instead of one dimensional subspaces in the classical case the new iteration generically converges locally cubically to the pairs of left right p dimensional invariant subspaces of c moreover grassmannian versions of the rayleigh quotient iteration are given for the generalized hermitian eigenproblem the hamiltonian eigenproblem and the skew hamiltonian eigenproblem

the continuous time rayleigh quotient flow on the sphere a continuous time differential equation inspired by the rayleigh quotient iteration for a symmetric matrix a is studied the set of all continuous solutions termed the rayleigh quotient flow is shown to be a time scaled version of the newton flow for rayleigh quotient cost functional the scaling factor ensures that the rate of variation of the rayleigh quotient is constant and positive along solutions this interpretation leads to a precise phase portrait for the rayleigh quotient flow it is shown that a complete non degenerate solution of the rayleigh quotient flow visits each of the eigenvectors of a in ascending order

ordinary differential equations and the symmetric eigenvalue problem in this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations new algorithms are suggested and old algorithms including qr are interpreted

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

generalized spectral decomposition method for solving stochastic finite element equations invariant subspace problem and dedicated algorithms stochastic galerkin methods have become a significant tool for the resolution of stochastic partial differential equations spde however they suffer from prohibitive computational times and memory requirements when dealing with large scale applications and high stochastic dimensionality some alternative techniques based on the construction of suitable reduced deterministic or stochastic bases have been proposed in order to reduce these computational costs recently a new approach based on the concept of generalized spectral decomposition gsd has been introduced for the definition and the automatic construction of reduced bases in this paper the concept of gsd initially introduced for a class of linear elliptic spde is extended to a wider class of stochastic problems the proposed definition of the gsd leads to the resolution of an invariant subspace problem which is interpreted as an eigen like problem this interpretation allows the construction of efficient numerical algorithms for building optimal reduced bases which are associated with dominant generalized eigenspaces the proposed algorithms by separating the resolution of reduced stochastic and deterministic problems lead to drastic computational savings their efficiency is illustrated on several examples where they are compared to classical resolution techniques