random polytopes and the efron stein jackknife inequality let k be a smooth convex body the convex hull of independent random points in k is a random polytope estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained the essential step is the use of the efron stein jackknife inequality for the variance of symmetric statistics consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope a conjecture of barany concerning random and best approximation of convex bodies is confirmed analogous results for random polytopes with vertices on the boundary of the convex body are given

a law of large numbers for nearest neighbour statistics in practical data analysis methods based on proximity near neighbour relationships between sample points are important because these relations can be computed in time n log n as the number of points n associated with such methods are a class of random variables defined to be functions of a given point and its nearest neighbours in the sample if the sample points are independent and identically distributed the associated random variables will also be identically distributed but not independent despite this we show that random variables of this type satisfy a strong law of large numbers in the sense that their sample means converge to their expected values almost surely as the number of sample points n

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

probabilistic index models summary we present a semiparametric statistical model for the probabilistic index which can be defined as p y y a where y and y a are independent random response variables associated with covariate patterns x and x a respectively a link function defines the relationship between the probabilistic index and a linear predictor asymptotic normality of the estimators and consistency of the covariance matrix estimator are established through semiparametric theory the model is illustrated with several examples and the estimation theory is validated in a simulation study

a computationally efficient estimator for mutual information mutual information quantifies the determinism that exists in a relationship between random variables and thus plays an important role in exploratory data analysis we investigate a class of non parametric estimators for mutual information based on the nearest neighbour structure of observations in both the joint and marginal spaces unless both marginal spaces are one dimensional we demonstrate that a well known estimator of this type can be computationally expensive under certain conditions and propose a computationally efficient alternative that has a time complexity of order o n log n a s the number of observations n n

random points and lattice points in convex bodies assume k subset of r d is a convex body and x is a large finite subset of k how many convex polytopes are there whose vertices belong to x is there a typical shape of such polytopes how well does the maximal such polytope which is actually the convex hull of x approximate k we are interested in these questions mainly in two cases the first is when x is a random sample of n uniform independent points from k in this case motivation comes from sylvester s famous four point problem and from the theory of random polytopes the second case is when x k boolean and z d where z d is the lattice of integer points in r d and the questions come from integer programming and geometry of numbers surprisingly or not so surprisingly the answers in the two cases are rather similar

on the physical part of the factorized correlation functions of the xxz chain it was recently shown by jimbo et al 2008 arxiv 0811 0439 that the correlation functions of a generalized xxz chain associated with an inhomogeneous six vertex model with a disorder parameter and with arbitrary inhomogeneities on the horizontal lines factorize and can all be expressed in terms of only two functions and here we approach the description of the same correlation functions and in particular of the function from a different direction we start from a novel multiple integral representation for the density matrix of a finite chain segment of length m in the presence of a disorder field we explicitly factorize the integrals for m 2 based on this we present an alternative description of the function in terms of the solutions of certain linear and nonlinear integral equations we then prove directly that the two definitions of describe the same function the definition in the work of jimbo et al 2008 arxiv 0811 0439 was crucial for the proof of the factorization the definition given here together with the known description of in terms of the solutions of nonlinear integral equations is useful for performing e g the trotter limit in the finite temperature case or for obtaining numerical results for the correlation functions at short distances we also address the issue of the construction of an exponential form of the density matrix for finite

central limit theorems for random polytopes let k be a smooth convex set the convex hull of independent random points in k is a random polytope central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity one essential step is to determine the precise asymptotic order of the occurring variances

number of nearest neighbors in a euclidean code a euclidean code is a finite set of points in n dimensional euclidean space spl rscr sup n the total number of nearest neighbors of a given codepoint in the code is called its touching number we show that the maximum number of codepoints f sub n that can share the same nearest neighbor codepoint is equal to the maximum kissing number spl tau sub n in n dimensions that is the maximum number of unit spheres that can touch a given unit sphere without overlapping we then apply a known upper bound on spl tau sub n to obtain f sub n spl les 2 sup n 0 401 o 1 which improves upon the best known upper known upper bound of f sub n spl les 2 sup n 1 o 1 we also show that the average touching number t of all the points in a euclidean code is upper bounded spl tau sub n

optimal approximation and quantisation abstract a number of interesting functionals f x of a finite point set x in the euclidean space can be represented as integrals of a function that depends on the integration variable y and x restricted onto a certain set c y x that is determined by y and x and satisfies separation and uniform boundedness conditions for instance c y x can be the voronoi cell generated by x that contains y we single out the general properties of c and that ensure that the normalised infimum of f x over all sets x with cardinality n converges to a limit that can be identified using a particular form of the considered functionals include those that appear in quantisation problems for probability measures and in finding the optimal approximation of a function by splines tangent planes and triangulated surfaces for instance it is shown that the minimum l approximation error normalised by n of a sufficiently smooth bivariate convex function f using the convex triangulation converges to the l 1 1 norm of the determinant of the second derivative of f times a certain absolute constant

differentiation and regularity of semi discrete optimal transport with respect to the parameters of the discrete measure this paper aims at determining under which conditions the semi discrete optimal transport is twice dierentiable with respect to the parameters of the discrete measure and exhibits numerical applications the discussion focuses on minimal conditions on the background measure to ensure dierentiability we provide numerical illustrations in stippling and blue noise problems

on normal approximations of distributions in terms of dependency graphs l auteur interprete les bornes de l erreur introduites par stein dans l approximation normale de sommes de variables aleatoires dependantes en termes de graphes de dependance ceci mene a des ameliorations d un theoreme de limite centrale de petrovskaya et leontovich et de recentes applications de baldi et rinatt en particulier on obtient des bornes pour des taux de convergence on etudie ensuite l approximation normale du nombre de maximums locaux d une fonction aleatoire sur un graphe

estimation of nonlinear functionals of densities with confidence this paper introduces a class of k nearest neighbor k nn estimators called bipartite plug in bpi estimators for estimating integrals of nonlinear functions of a probability density such as shannon entropy and renyi entropy the density is assumed to be smooth have bounded support and be uniformly bounded from below on this set unlike previous k nn estimators of nonlinear density functionals the proposed estimator uses data splitting and boundary correction to achieve lower mean square error specifically we assume that t i i d samples xi rd from the density are split into two pieces of cardinality m and n respectively with m samples used for computing a k nn density estimate and the remaining n samples used for empirical estimation of the integral of the density functional by studying the statistical properties of k nn balls explicit rates for the bias and variance of the bpi estimator are derived in terms of the sample size the dimension of the samples and the underlying probability distribution based on these results it is possible to specify optimal choice of tuning parameters m t k for maximizing the rate of decrease of the mean square error the resultant optimized bpi estimator converges faster and achieves lower mean squared error than previous k nn entropy estimators in addition a central limit theorem is established for the bpi estimator that allows us to specify tight asymptotic confidence intervals

normal approximation under local dependence we establish both uniform and nonuniform error bounds of the berry esseen type in normal approximation under local dependence these results are of an order close to the best possible if not best possible they are more general or sharper than many existing ones in the literature the proofs couple stein s method with the concentration inequality approach

pairs of matrices that preserve the value of a generalized matrix function on the set of the upper triangular matrices let h be a subgroup of the symmetric group of degree m and let be an irreducible character of h in this paper we give conditions that characterize the pairs of matrices that leave invariant the value of a generalized matrix function associated with h and on the set of the upper triangular matrices

quantization of probability distributions under norm based distortion measures ii self similar distributions for a probability measure p on rd and n n consider en inf mina v x a dp x where the infimum is taken over all subsets of rd with card n and v is a nondecreasing function under certain conditions on v we derive the precise n asymptotics of en for self similar distributions p and we find the asymptotic performance of optimal quantizers using weighted empirical measures

on normal approximation rates for certain sums of dependent random variables abstract let x1 xn be dependent random variables and set e ni 1xi and 2 var ni 1xi in most of the applications of stein s method for normal approximations the error rate p ni 1xi w w is of the order of 1 2 this rate was improved by stein 1986 and others in some special cases in this paper it is shown that for certain bounded random variables a simple refinement of error term calculations in stein s method leads to improved rates

finite temperature correlation functions from discrete functional equations we present a new approach to the static finite temperature correlation functions of the heisenberg chain based on functional equations an inhomogeneous generalization of the n site density operator is considered the lattice path integral formulation with a finite but arbitrary trotter number allows us to derive a set of discrete functional equations with respect to the spectral parameters we show that these equations yield a unique characterization of the density operator our functional equations are a discrete version of the reduced q knizhnik zamolodchikov equations which played a central role in the study of the zero temperature case as a natural result and independent of the arguments given by jimbo et al 2009 j phys a math theor 42 304018 we prove that the inhomogeneous finite temperature correlation functions have the same remarkable structure as for zero temperature they are a sum of products of nearest neighbour correlators

quantization dimension for condensation systems ii the geometric mean error abstract let f i 1 n be a family of similitudes on r 1 satisfying the strong separation condition and the self similar measure associated with f i 1 n and a probability vector t 1 t n let be the attracting measure of a condensation system associated with f i 1 n and a probability vector p 0 p 1 p n we establish a relationship between the quantization dimension of and its mass distribution on cylinder sets

matrices that preserve the value of the generalized matrix function of the upper triangular matrices abstract let h be a subgroup of the symmetric group s n and an irreducible character of h in this paper we give conditions that characterize matrices that leave invariant the value of a given generalized matrix function on the set of upper triangular matrices in some cases we describe completely these matrices

an efron stein inequality for nonsymmetric statistics on etablit l analogue d un resultat d efron et stein 1981 que l on demontre a l aide d une technique d espace de hilbert introduite par vitale 1984

adaptive reconnection based arbitrary lagrangian eulerian method ew present a new adaptive arbitrary lagrangian eulerian ale method this method is based on the reconnection based ale reale methodology of refs 35 34 6 the main elements in a standard reale method are an explicit lagrangian phase on an arbitrary polygonal in 2d mesh in which the solution and positions of grid nodes are updated a rezoning phase in which a new grid is defined by changing the connectivity using voronoi tessellation but not the number of cells and a remapping phase in which the lagrangian solution is transferred onto the new grid in the standard reale method the rezoned mesh is smoothed by using one or several steps toward centroidal voronoi tessellation but it is not adapted to the solution in any way in the current paper we present a new adaptive reale method a reale that is based on the following design principles first a monitor function or error indicator based on the hessian of some flow parameter s is utilized second an equi distribution principle for the monitor function is used as a criterion for adapting the mesh third a centroidal voronoi tessellation is used to adapt the mesh fourth we scale the monitor function to avoid very small and large cells and then smooth it to permit the use of theoretical results related to weighted centroidal voronoi tessellation in the a reale method both number of cells and their locations are allowed to change at the rezone stage on each time step the number of generators at each time step is chosen to guarantee the required spatial resolution in regions where monitor function reaches its maximum value we present all details required for implementation of new adaptive a reale method and demonstrate its performance in comparison with standard reale method on series of numerical examples

reduced qkz equation and correlation functions of the xxz model correlation functions of the xxz model in the massive and massless regimes are known to satisfy a system of linear equations the main relations among them are the difference equations obtained from the qkz equation by specializing the variables 1 2n as 1 n n 1 1 1 we call it the reduced qkz equation in this article we construct a special family of solutions to this system they can be written as linear combinations of products of two transcendental functions open image in new window with coefficients being rational functions we show that correlation functions of the xxz model in the massive regime are given by these formulas with an appropriate choice of open image in new window we also present a conjectural formula in the massless regime

quantum spin chains and riemann zeta function with odd arguments the riemann zeta function is an important object of number theory we argue that it is related to the heisenberg spin 1 2 anti ferromagnet in the xxx spin chain we study the probability of formation of a ferromagnetic string in the anti ferromagnetic ground state in the thermodynamics limit we prove that for short strings the probability can be expressed in terms of the riemann zeta function with odd arguments

practical bounds on image denoising from estimation to information recently in a previous work we proposed a way to bound how well any given image can be denoised the bound was computed directly from the noise free image that was assumed to be available in this work we extend the formulation to the more practical case where no ground truth is available we show that the parameters of the bounds namely the cluster covariances and level of redundancy for patches in the image can be estimated directly from the noise corrupted image further we analyze the bounds formulation to show that these two parameters are interdependent and they along with the bounds formulation as a whole have a nice information theoretic interpretation as well the results are verified through a variety of well motivated experiments

holonomy of the ising model form factors we study the ising model two point diagonal correlation function c n n by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of wu mccoy tracy and barouch we extend this expansion weighting by powers of a variable the j particle contributions f j n n in the form factor expansion the corresponding extension of the two point diagonal correlation function c n n is shown for arbitrary to be a solution of the sigma form of the painleve vi equation introduced by jimbo and miwa in their isomonodromic approach to the ising model fuchsian linear differential equations for the form factors f j n n are obtained for j 9 and shown to have both a russian doll nesting and a decomposition of the corresponding linear differential operators as a direct sum of operators equivalent to symmetric powers of the second order linear differential operator associated with the elliptic integral e from this we show that each f j n n is unexpectedly simple being expressed polynomially in terms of the elliptic integrals e and k in contrast we exhibit some mathematical objects built from these form factors f j n n which break the direct sum of symmetric powers decomposition with its associated polynomial expressions first we show that the scaling limit of these differential operators and form factors breaks the direct sum structure but not the russian doll structure secondly we show that the previous extension of two point diagonal correlation functions c n n is for singled out values cos m n m n integers also solutions of fuchsian linear differential equations these solutions of painleve vi are not polynomial in e and k but are actually algebraic functions being associated with modular curves

bounds on the geometric complexity of optimal centroidal voronoi tesselations in 3d gersho s conjecture in 3d asserts the asymptotic periodicity and structure of the optimal centroidal voronoi tessellation this relatively simple crystallization problem remains to date open we prove bounds on the geometric complexity of optimal centroidal voronoi tessellations as the number of generators tends to infinity combined with an approach of gruber in 2d these bounds reduce the resolution of the 3d gersho s conjecture to a finite albeit very large computation of an explicit convex problem in finitely many variables

the hexagon theorem we wish to int s z f z 2 s the unit square over all choices of f z which take just n values for large n it is shown that f z approaches that function which on each face of a hexagonal mesh has the value of the centroid of that face this problem in the design of quantizers for a pulse code modulation pcm system arose in which pairs of adjacent samples are quantized in an effort to capitalize on the correlation between samples and on the geometry of the plane

some remarks on absolute continuity and quantization of probability measures we introduce a notion of monotonicity of dimensions of measures we show that the upper and lower quantization dimensions are not monotone we give sufficient conditions in terms of so called vanishing rates such that implies d r d r as an application we determine the quantization dimension of a class of measures which are absolutely continuous w r t some self similar measure with the corresponding radon nikodym derivative bounded or unbounded we study the set of quantization dimensions of measures which are absolutely continuous w r t a given probability measure we prove that the infimum on this set coincides with the lower packing dimension of furthermore this infimum can be attained provided that the upper and lower packing dimensions of are equal

non parametric estimation of residual moments and covariance the aim of non parametric regression is to model the behaviour of a response vector y in terms of an explanatory vector x based only on a finite set of empirical observations this is usually perf

asymptotic uniformity of the quantization error of self similar measures let be a self similar measure on mathbb r d associated with a family of contractive similitudes s1 sn and a probability vector p1 pn let alpha n n 1 infty be a sequence of n optimal sets for of order r for each n we denote by p a alpha n a in alpha n a voronoi partition of mathbb r d with respect to n under the strong separation condition for s1 sn we show that the nth quantization error of of order r in 1 infty satisfies the following asymptotic uniformity property int limits p a alpha n rm d x a rd mu x asymp frac 1 n v n r mu quad rm for all a in alpha n

equality of decomposable symmetrized tensors and matrix groups let g be a subgroup of the full symmetric group sn and a character of g a matrix can be defined as an n n matrix b which satisfies dgx bx dgx x for every n n matrix x they form a multiplicative group denoted s g x which plays a fundamental role in the study of equality of two decomposable symmetrized tensors the main result of this paper theorems 2 2 2 3 and 2 4 is a complete description of the matrices in s g x this description has many consequences that we present there are also results on related questions

quantum kz equation with q 1 and correlation functions of the xxz model in the gapless regime an integral solution to the quantum knizhnik zamolodchikov qkz equation with q 1 is presented upon specialization it leads to a conjectural formula for correlation functions of the xxz model in the gapless regime the validity of this conjecture is verified in special cases including the nearest neighbour correlator with an arbitrary coupling constant and general correlators in the xxx and xy limits

convergence of the lloyd algorithm for computing centroidal voronoi tessellations centroidal voronoi tessellations cvts are voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids mass centers of the corresponding voronoi regions with respect to a given density function centroidal voronoi tessellations may also be defined in more abstract and more general settings due to the natural optimization properties enjoyed by cvts they have many applications in diverse fields the lloyd algorithm is one of the most popular iterative schemes for computing the cvts but its theoretical analysis is far from complete in this paper some new analytical results on the local and global convergence of the lloyd algorithm are presented these results are derived through careful utilization of the optimization properties shared by cvts numerical experiments are also provided to substantiate the theoretical analysis

an analytic treatment of finite size corrections in the spin 1 antiferromagnetic xxz chain it is known that the standard method for calculating finite size corrections in bethe ansatz solvable systems is not applicable to the takhtajan babujian model and its anisotropic xxz generalisation the authors develop a new analytic method explicitly avoiding root densities and associated problems nonlinear integral equations are derived whose solutions yield the correct central charges c 1 and c 3 2 for the spin 1 2 and spin 1 xxz chains respectively in the spin 1 case the authors obtain as a by product the finite size deviation of the bethe ansatz roots from the 2 string formation

density matrices for finite segments of heisenberg chains of arbitrary length we derive a multiple integral representing the ground state density matrix of a segment of length m of the xxz spin chain on l lattice sites which depends on l only parametrically this allows us to treat chains of arbitrary finite length specializing to the isotropic limit of the xxx chain we show for small m that the multiple integrals factorize we conjecture that this property holds for arbitrary m and suggest an exponential formula for the density matrix which involves only a double cauchy type integral in the exponent we demonstrate the efficiency of our formula by computing the next to nearest neighbour zz correlation function for chain lengths ranging from two to macroscopic numbers

sums of functions of nearest neighbor distances moment bounds limit theorems and a goodness of fit test abstract the limiting behavior of sums of functions of nearest neighbor distances is studied for an m dimensional sample a central limit theorem and moment bounds for such sums and an invariance principle for the empirical process of nearest neighbor distances are both established as a consequence the asymptotic behavior of a practicable goodness of fit test is obtained based on nearest neighbor distances

optimum quantization and its applications abstract minimum sums of moments or equivalently distortion of optimum quantizers play an important role in several branches of mathematics fejes toth s inequality for sums of moments in the plane and zador s asymptotic formula for minimum distortion in euclidean d space are the first precise pertinent results in dimension d 2 in this article these results are generalized in the form of asymptotic formulae for minimum sums of moments resp distortion of optimum quantizers on riemannian d manifolds and normed d spaces in addition we provide geometric and analytic information on the structure of optimum configurations our results are then used to obtain information on i the minimum distortion of high resolution vector quantization and optimum quantizers ii the error of best approximation of probability measures by discrete measures and support sets of best approximating discrete measures iii the minimum error of numerical integration formulae for classes of holder continuous functions and optimum sets of nodes iv best volume approximation of convex bodies by circumscribed convex polytopes and the form of best approximating polytopes and v the minimum isoperimetric quotient of convex polytopes in minkowski spaces and the form of the minimizing polytopes

integral representations for correlation functions of the xxz chain at finite temperature we derive a novel multiple integral representation for a generating function of the z z correlation functions of the spin xxz chain at finite temperature and finite longitudinal magnetic field our work combines algebraic bethe ansatz techniques for the calculation of matrix elements with the quantum transfer matrix approach to thermodynamics

nondegeneracy and weak global convergence of the lloyd algorithm in mathbb r d the lloyd algorithm originated in the context of optimal quantization and represents a fixed point iteration for computing an optimal quantizer reducing average distortion at every step it constructs a voronoi partition of the domain and replaces each generator with the centroid of the corresponding voronoi cell optimal quantization is obtained in the case of a centroidal voronoi tessellation cvt which is a special voronoi tessellation of a domain omega in mathbb r d having the property that the generators of the voronoi diagram are also the centers of mass with respect to a given density function rho geq 0 of the corresponding voronoi cells the lloyd iteration is currently the most popular and elegant algorithm for computing cvts and optimal quantizers but many questions remain about its convergence especially in d dimensional spaces d 1 in this paper we prove that any limit point of the lloyd iteration in any dimensional spaces is nondegenerate provided that omega is a convex and bounded set and rho belongs to l 1 omega and is positive almost everywhere this ensures that the fixed point map remains closed and hence the standard theory of descent methods guarantees weak global convergence of the lloyd iteration to the set of nondegenerate fixed point quantizers while previously only conjectured the convergence properties of the lloyd iteration are rigorously justified under such minimal regularity assumptions on the density functional the results presented in this paper go beyond existing convergence theories for cvts and optimal quantization related algorithms and should be of interest to both the mathematical and engineering communities

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

a new method of normal approximation we introduce a new version of stein s method that reduces a large class of normal approximation problems to variance bounding exercises thus making a connection between central limit theorems and concentration of measure unlike skorokhod embeddings the object whose variance must be bounded has an explicit formula that makes it possible to carry out the program more easily as an application we derive a general clt for functions that are obtained as combinations of many local contributions where the definition of local itself depends on the data several examples are given including the solution to a nearest neighbor clt problem posed by p bickel

asymptotic moments of near neighbour distance distributions let c be a compact convex body in r m and consider a set of points selected at random from c according to some well behaved sampling distribution we obtain an asymptotic expression for the positive moments of the k th near neighbour distance distribution as the number of points increases to infinity

linearn a new approach to nearest neighbour density estimator abstract despite their wide spread use nearest neighbour density estimators have two fundamental limitations o n 2 time complexity and o n space complexity both limitations constrain nearest neighbour density estimators to small data sets only recent progress using indexing schemes has improved to near linear time complexity only we propose a new approach called linearn for linear time nearest neighbour algorithm that yields the first nearest neighbour density estimator having o n time complexity and constant space complexity as far as we know this is achieved without using any indexing scheme because linearn uses a subsampling approach for which the subsample values are significantly less than the data size like existing density estimators our asymptotic analysis reveals that the new density estimator has a parameter to trade off between bias and variance we show that algorithms based on the new nearest neighbour density estimator can easily scale up to data sets with millions of instances in anomaly detection and clustering tasks

stein s method for concentration inequalities we introduce a version of stein s method for proving concentration and moment inequalities in problems with dependence simple illustrative examples from combinatorics physics and mathematical statistics are provided

integral representation of the density matrix of the xxz chain at finite temperatures we present an integral formula for the density matrix of a finite segment of the infinitely long spin xxz chain this formula is valid for any temperature and any longitudinal magnetic field

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments