embedding metrics into ultrametrics and graphs into spanning trees with constant average distortion this paper addresses the basic question of how well a tree can approximate distances of a metric space or a graph given a graph the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design we present scaling distortion embeddings where the distortion scales as a function of epsilon with the guarantee that for each epsilon simultaneously the distortion of a fraction 1 epsilon of all pairs is bounded accordingly quantitatively we prove that any finite metric space embeds into an ultrametric with scaling distortion o sqrt 1 epsilon for the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion o sqrt 1 epsilon these bounds are tight even for embedding into arbitrary trees these results imply that the average distortion of the embedding is constant and that the ell 2 distortion is o sqrt log n for probabilistic embedding into spanning trees we prove a scaling distortion of tilde o log 2 1 epsilon which implies constant ell q distortion for every fixed q infty

net and prune a linear time algorithm for euclidean distance problems we provide a general framework for getting expected linear time constant factor approximations and in many cases fptas s to several well known problems in computational geometry such as k center clustering and farthest nearest neighbor the new approach is robust to variations in the input problem and yet it is simple elegant and practical in particular many of these well studied problems which fit easily into our framework either previously had no linear time approximation algorithm or required rather involved algorithms and analysis a short list of the problems we consider include farthest nearest neighbor k center clustering smallest disk enclosing k points kth largest distance kth smallest m nearest neighbor distance kth heaviest edge in the mst and other spanning forest type problems problems involving upward closed set systems and more finally we show how to extend our framework such that the linear running time bound holds with high probability

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

decomposition of besov and triebel lizorkin spaces on the sphere abstract a discrete system of almost exponentially localized elements needlets on the n dimensional unit sphere s n is constructed it shown that the needlet system can be used for decomposition of besov and triebel lizorkin spaces on the sphere as an application of besov spaces on s n a jackson estimate for nonlinear m term approximation from the needlet system is obtained

sample correlation coefficients based on survey data under regression imputation regression imputation is commonly used to compensate for item nonresponse when auxiliary data are available it is common practice to compute survey estimators by treating imputed values as observed data and using the standard unbiased or nearly unbiased estimation formulas designed for the case of no nonresponse although the commonly used regression imputation method preserves unbiasedness for population marginal totals i e survey estimators computed from imputed data are still nearly unbiased it does not preserve unbiasedness for population correlation coefficients a joint regression imputation method is proposed that preserves unbiasedness for marginal totals second moments and correlation coefficients some simulation results show that the joint regression imputation method produces not only sample correlation coefficients that are nearly unbiased but also estimates that are more stable than those produced by marginal nonrandom regression imputation when correlation coefficients are in a c

direct estimation of the fiber orientation density function from diffusion weighted mri data using spherical deconvolution diffusion weighted magnetic resonance imaging can provide information related to the arrangement of white matter fibers the diffusion tensor is the model most commonly used to derive the orientation of the fibers within a voxel however this model has been shown to fail in regions containing several fiber populations with distinct orientations a number of alternative models have been suggested such as multiple tensor fitting q space and q ball imaging however each of these has inherent limitations in this study we propose a novel method for estimating the fiber orientation distribution directly from high angular resolution diffusion weighted mr data without the need for prior assumptions regarding the number of fiber populations present we assume that all white matter fiber bundles in the brain share identical diffusion characteristics thus implicitly assigning any differences in diffusion anisotropy to partial volume effects the diffusion weighted signal attenuation measured over the surface of a sphere can then be expressed as the convolution over the sphere of a response function the diffusion weighted attenuation profile for a typical fiber bundle with the fiber orientation density function odf the fiber odf the distribution of fiber orientations within the voxel can therefore be obtained using spherical deconvolution the properties of the technique are demonstrated using simulations and on data acquired from a volunteer using a standard 1 5 t clinical scanner the technique can recover the fiber odf in regions of multiple fiber crossing and holds promise for applications such as tractography

multiresolution analysis and adaptive estimation on a sphere using stereographic wavelets abstract we construct an adaptive estimator of a density function on d dimensional unit sphere s d d 2 using a new type of spherical frames the frames or as we call them stereografic wavelets are obtained by transforming a wavelet system namely daubechies using some stereographic operators we prove that our estimator achieves an optimal rate of convergence on some besov type class of functions by adapting to unknown smoothness our new construction of stereografic wavelet system gives us a multiresolution approximation of l 2 s d which can be used in many approximation and estimation problems in this paper we also demonstrate how to implement the density estimator in s 2 and we present a finite sample behavior of that estimator in a numerical experiment

fitting a cm smooth function to data iii suppose we are given a finite subset e rn and a function f e r how to extend f to a cm function f rn r with cm norm of the smallest possible order of magnitude in this paper and in 20 we tackle this question from the perspective of theoretical computer science we exhibit algorithms for constructing such an extension function f and for computing the order of magnitude of its cm norm the running time of our algorithms is never more than cn log n where n is the cardinality of e and c is a constant depending only on m and n

wavelet deconvolution in a periodic setting in this paper we present an inverse estimation procedure which combines fourier analysis with wavelet expansion in the periodic setting our method can recover a blurred function observed in white noise the blurring process is achieved through a convolution operator which can either be smooth polynomial decay of the fourier transform or irregular such as the convolution with a box car the proposal is non linear and does not require any prior knowledge of the smoothness class it enjoys fast computation and is spatially adaptive this contrasts with more traditional ltering methods which demand a certain amount of regularisation and often fail to recover non homogeneous functions a ne tuning of our method is derived via asymptotic minimax theory which reveals some key dierences with the direct case of donoho et al 1995 a band limited wavelet families have nice theoretical and computing features b the high frequency cut o depends on the spectral characteristics of the convolution kernel c thresholds are level dependent in a geometric fashion we tested our method using simulated lidar data for underwater remote sensing both visual and numerical results show an improvement over existing methods finally the theory behind our estimation paradigm gives a complete characterisation of the maxiset of the method i e the set of functions where the method attains a near optimal rate of convergence for a variety of l p loss functions

optimal euclidean spanners really short thin and lanky the degree the hop diameter and the weight are the most basic and well studied parameters of geometric spanners in a seminal stoc 95 paper titled euclidean spanners short thin and lanky arya et al 1995 devised a construction of euclidean 1 e spanners that achieves constant degree diameter o log n weight o log2n mst and has running time o n log n this construction applies to n point constant dimensional euclidean spaces moreover arya et al conjectured that the weight bound can be improved by a logarithmic factor without increasing the degree and the diameter of the spanner and within the same running time this conjecture of arya et al became one of the most central open problems in the area of euclidean spanners nevertheless the only progress since 1995 towards its resolution was achieved in the lower bounds front any spanner with diameter o log n must incur weight log n mst and this lower bound holds regardless of the stretch or the degree of the spanner dinitz et al 2008 agarwal et al 2005 in this article we resolve the long standing conjecture of arya et al in the affirmative we present a spanner construction with the same stretch degree diameter and running time as in arya et al s result but with optimal weight o log n mst so our spanners are as thin and lanky as those of arya et al but they are really shorte moreover our result is more general in three ways first we demonstrate that the conjecture holds true not only in constant dimensional euclidean spaces but also in doubling metrics second we provide a general trade off between the three involved parameters which is tight in the entire range third we devise a transformation that decreases the lightness of spanners in general metrics while keeping all their other parameters in check our main result is obtained as a corollary of this transformation

concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds let x 1 x n be a random sample from some unknown probability density f defined on a compact homogeneous manifold m of dimension d 1 consider a needlet frame phi j eta describing a localised projection onto the space of eigenfunctions of the laplace operator on m with corresponding eigenvalues less than 22j as constructed in geller and pesenson j geom anal 2011 we prove non asymptotic concentration inequalities for the uniform deviations of the linear needlet density estimator f n j obtained from an empirical estimate of the needlet projection sum eta phi j eta int f phi j eta of f we apply these results to construct risk adaptive estimators and nonasymptotic confidence bands for the unknown density f the confidence bands are adaptive over classes of differentiable and holder continuous functions on m that attain their holder exponents

inference from stratified samples properties of the linearization jackknife and balanced repeated replication methods the asymptotic normality of both linear and nonlinear statistics and the consistency of the variance estimators obtained using the linearization jackknife and balanced repeated replication brr methods in stratified samples are established the results are obtained as l oa within the context of a sequence of finite populations jil with l strata in ji l and are valid for any stratified multistage design in which the primary sampling units psu s are selected with replacement and in which independent subsamples are taken within those psu s selected more than once in addition some exact analytical results on the bias and stability of these alternative variance estimators in the case of ratio estimation are obtained for small l under a general linear regression model

an optimal algorithm for approximate nearest neighbor searching fixed dimensions consider a set of s of n data points in real d dimensional space r d where distances are measured using any minkowski metric in nearest neighbor searching we preprocess s into a data structure so that given any query point q r d is the closest point of s to q can be reported quickly given any positive real e data point p is a 1 e approximate nearest neighbor of q if its distance from q is within a factor of 1 e of the distance to the true nearest neighbor we show that it is possible to preprocess a set of n points in r d in o dn log n time and o dn space so that given a query point q r d and e 0 a 1 e approximate nearest neighbor of q can be computed in o c d e log n time where c d e d 1 6d e d is a factor depending only on dimension and e in general we show that given an integer k 1 1 e approximations to the k nearest neighbors of q can be computed in additional o kd log n time

jackknife variance estimation under imputation for estimators using poststratification information abstract poststratified estimators are commonly used in sample surveys to improve the efficiency of estimators and to ensure calibration to known poststrata counts similarly generalized regression estimators are used to handle two or more poststratifiers with known marginal counts in addition weighting adjustment within weighting classes is used to handle unit nonresponse and imputation within imputation classes is used to handle item nonresponse for the full response case asymptotic consistency of the jackknife variance estimator under stratified multistage sampling is established using mild regularity conditions on residuals similar to those of scott and wu for ratio and regression estimation under simple random sampling a jackknife linearization variance estimator obtained by linearizing the jackknife variance estimator is also given for unit nonresponse the general case of poststrata cutting across weighting classes is considered and a jackknife variance estimator and the corresponding

the weighted residual technique for estimating the variance of the general regression estimator of the finite population total summary the paper deals with design based estimation of the variance of the general regression estimator of the finite population total the usual taylor linearization variance estimator is an expression in the design weighted regression residuals in many applications the resulting expression is counterintuitive from a model based standpoint the improved variance estimator in this paper attaches another simple weight called g weight to each individual residual this new variance estimator i gives valid design based confidence intervals ii is nearly unbiased uiider a suitably chosen regression model and iii works well for conditional inference examples are given

localized spherical deconvolution we provide a new algorithm for the treatment of the deconvolution problem on the sphere which combines the traditional svd inversion with an appropriate thresholding technique in a well chosen new basis we establish upper bounds for the behavior of our procedure for any l p loss it is important to emphasize the adaptation properties of our procedures with respect to the regularity sparsity of the object to recover as well as to inhomogeneous smoothness we also perform a numerical study which proves that the procedure shows very promising properties in practice as well

all pairs shortest paths using bridging sets and rectangular matrix multiplication we present two new algorithms for solving the all pairs shortest paths apsp problem for weighted directed graphs both algorithms use fast matrix multiplication algorithms the first algorithm solves the apsp problem for weighted directed graphs in which the edge weights are integers of small absolute value in o n2 time where satisfies the equation 1 1 1 2 and 1 1 is the exponent of the multiplication of an n n matrix by an n n matrix currently the best available bounds on 1 1 obtained by coppersmith imply that 0 is an error parameter and w is the largest edge weight in the graph after the edge weights are scaled so that the smallest non zero edge weight in the graph is 1 it returns estimates of all the distances in the graph with a stretch of at most 1 e corresponding paths can also be found efficiently

1 epsilon beta spanner constructions for general graphs an em alpha beta spanner of a graph g is a subgraph h such that mathit dist h u w le alpha cdot mathit dist t g u w beta for every pair of vertices u w where distg u w denotes the distance between two vertices u and v in g it is known that every graph g has a polynomially constructible 2 kappa 1 0 spanner also known as multiplicative 2 kappa 1 spanner of size o n 1 1 kappa for every integer kappa ge 1 and a polynomially constructible 1 2 spanner also known as additive 2 spanner of size tilde o n 3 2 this paper explores hybrid spanner constructions involving both multiplicative and additive factors for general graphs and shows that the multiplicative factor can be made arbitrarily close to 1 while keeping the spanner size arbitrarily close to o n at the cost of allowing the additive term to be a sufficiently large constant more formally we show that for any constant epsilon lambda 0 there exists a constant beta beta epsilon lambda such that for every n vertex graph g there is an efficiently constructible 1 epsilon beta spanner of size o n 1 lambda

efficient regression in metric spaces via approximate lipschitz extension we present a framework for performing efficient regression in general metric spaces roughly speaking our regressor predicts the value at a new point by computing an approximate lipschitz extension the smoothest function consistent with the observed data after performing structural risk minimization to avoid overfitting we obtain finite sample risk bounds with minimal structural and noise assumptions and a natural runtime precision tradeoff the offline learning and online prediction stages can be solved by convex programming but this naive approach has runtime complexity o n 3 which is prohibitive for large data sets we design instead a regression algorithm whose speed and generalization performance depend on the intrinsic dimension of the data to which the algorithm adapts while our main innovation is algorithmic the statistical results may also be of independent interest

gaussian approximation of nonlinear statistics on the sphere abstract we show how it is possible to assess the rate of convergence in the gaussian approximation of triangular arrays of u statistics built from wavelets coefficients evaluated on a spherical poisson field of arbitrary dimension for this purpose we exploit the stein malliavin approach introduced in the seminal paper by peccati sole taqqu and utzet 2011 we focus in particular on statistical applications covering evaluation of variance in non parametric density estimation and sobolev tests for uniformity

random rotations characters and random walks on so n we analyze a random walk on the orthogonal group so n given by repeatedly rotating by a fixed angle through randomly chosen planes of r we derive estimates of the rate at which this random walk will converge to haar measure on so n using character theory and the upper bound lemma of diaconis and shashahani in some cases we are able to establish the existence of a cut off phenomenon for the random walk this is the first interesting such result on a non finite group

new imputation methods for missing data using quantiles the problem of missing values commonly arises in data sets and imputation is usually employed to compensate for non response we propose a novel imputation method based on quantiles which can be implemented with or without the presence of auxiliary information the proposed method is extended to unequal sampling designs and non uniform response mechanisms iterative algorithms to compute the proposed imputation methods are presented monte carlo simulations are conducted to assess the performance of the proposed imputation methods with respect to alternative imputation methods simulation results indicate that the proposed methods perform competitively in terms of relative bias and relative root mean square error

space efficient path reporting approximate distance oracles we consider approximate path reporting distance oracles distance labeling and labeled routing with extremely low space requirements for general undirected graphs for distance oracles we show how to break the nlog n space bound of thorup and zwick if approximate paths rather than distances need to be reported for approximate distance labeling and labeled routing we break the previously best known space bound of o log n words per vertex the cost for such space efficiency is an increased stretch

fast construction of nets in low dimensional metrics and their applications we present a near linear time algorithm for constructing hierarchical nets in finite metric spaces with constant doubling dimension this data structure is then applied to obtain improved algorithms for the following problems approximate nearest neighbor search well separated pair decomposition spanner construction compact representation scheme doubling measure and computation of the approximate lipschitz constant of a function in all cases the running preprocessing time is near linear and the space being used is linear

compact oracles for reachability and approximate distances in planar digraphs it is shown that a planar digraph can be preprocessed in near linear time producing a near linear space oracle that can answer reachability queries in constant time the oracle can be distributed as an o log n space label for each vertex and then we can determine if one vertex can reach another considering their two labels only the approach generalizes to give a near linear space approximate distances oracle for a weighted planar digraph with weights drawn from 0 n it approximates distances within a factor 1 e in o log log nn 1 e time our scheme can be extended to find and route along correspondingly short dipaths

fast algorithms for constructing t spanners and paths with stretch t the distance between two vertices in a weighted graph is the weight of a minimum weight path between them where the weight of a path is the sum of the weights of the edges in the path a path has stretch t if its weight is at most t times the distance between its end points we present algorithms that compute paths of stretch 2 leq t leq log n on undirected graphs g v e with nonnegative weights the stretch t is of the form t beta 2 epsilon where beta is integral and epsilon 0 is at least as large as some fixed epsilon 0 we present an tilde o m k n 2 epsilon t time randomized algorithm that finds paths between k specified pairs of vertices and an tilde o m ns n 2 1 log n m epsilon t deterministic algorithm that finds paths from s specified sources to all other vertices for any fixed epsilon 0 where n v and m e this improves significantly over the slower tilde o min k n m exact shortest paths algorithms and a previous tilde o mn 64 t kn 32 t time algorithm by awerbuch et al proc 34th ieee annual symposium on foundations of computer science ieee piscataway nj 1993 pp 638 647 a t spanner of a graph g is a set of weighted edges on the vertices of g such that distances in the spanner are not smaller and within a factor of t from the corresponding distances in g previous work was concerned with bounding the size and efficiently constructing t spanners we construct t spanners of size tilde o n 1 2 epsilon t in tilde o mn 2 epsilon t expected time for any fixed epsilon 0 which constitutes a faster construction by a factor of n3 2 t m of sparser spanners than was previously attainable we also provide efficient parallel constructions our algorithms are based on pairwise covers and a novel approach to construct them efficiently

recovering the long range links in augmented graphs the augmented graph model as introduced in kleinberg stoc 2000 23 is an appealing model for analyzing navigability in social networks informally this model is defined by a pair h f where h is a graph in which inter node distances are supposed to be easy to compute or at least easy to estimate this graph is augmented by links called long range links that are selected according to the probability distribution f the augmented graph model enables the analysis of greedy routing in augmented graphs email protected h f in greedy routing each intermediate node handling a message for a target t selects among all its neighbors in g the one that is the closest to t in h and forwards the message to it this paper addresses the problem of checking whether a given graph g is an augmented graph it answers part of the questions raised by kleinberg in his problem 9 int congress of math 2006 more precisely given email protected h f we aim at extracting the base graph h and the long range links r out of g we prove that if h has a high clustering coefficient and h has bounded doubling dimension then a simple local maximum likelihood algorithm enables us to partition the edges of g into two sets h and r such that e h h and the edges in h e h are of small stretch i e the map h is not perturbed too greatly by undetected long range links remaining in h the perturbation is actually so small that we can prove that the expected performances of greedy routing in g using the distances in h are close to the expected performances of greedy routing using the distances in h although this latter result may appear intuitively straightforward since h e h it is not as we also show that routing with a map more precise than h may actually damage greedy routing significantly finally we show that in the absence of a hypothesis regarding the high clustering coefficient any local maximum likelihood algorithm extracting the long range links can miss the detection of w n 5 e logn long range links of stretch w n 1 5 e for any 0 e 1 5 and thus the map h cannot be recovered with good accuracy

space time tradeoffs for approximate nearest neighbor searching nearest neighbor searching is the problem of preprocessing a set of n point points in d dimensional space so that given any query point q it is possible to report the closest point to q rapidly in approximate nearest neighbor searching a parameter e 0 is given and a multiplicative error of 1 e is allowed we assume that the dimension d is a constant and treat n and e as asymptotic quantities numerous solutions have been proposed ranging from low space solutions having space o n and query time o log n 1 ed 1 to high space solutions having space roughly o n log n ed and query time o log n e we show that there is a single approach to this fundamental problem which both improves upon existing results and spans the spectrum of space time tradeoffs given a tradeoff parameter where 2 1 e we show that there exists a data structure of space o n d 1 log 1 e that can answer queries in time o log n 1 e d 1 2 when e 2 this yields a data structure of space o n log 1 e that can answer queries in time o log n 1 e d 1 2 when e 1 e it provides a data structure of space o n ed 1 log 1 e that can answer queries in time o log n e our results are based on a data structure called a t e avd which is a hierarchical quadtree based subdivision of space into cells each cell stores up to t representative points of the set such that for any query point q in the cell at least one of these points is an approximate nearest neighbor of q we provide new algorithms for constructing avds and tools for analyzing their total space requirements we also establish lower bounds on the space complexity of avds and show that up to a factor of o log 1 e our space bounds are asymptotically tight in the two extremes e 2 and e 1 e

near optimal sample compression for nearest neighbors we present the first sample compression algorithm for nearest neighbors with non trivial performance guarantees we complement these guarantees by demonstrating almost matching hardness lower bounds which show that our performance bound is nearly optimal our result yields new insight into margin based nearest neighbor classification in metric spaces and allows us to significantly sharpen and simplify existing bounds some encouraging empirical results are also presented

clustering to minimize the maximum intercluster distance the problem of clustering a set of points so as to minimize the maximum intercluster distance is studied an o kn approximation algorithm where n is the number of points and k is the number of clusters that guarantees solutions with an objective function value within two times the optimal solution value is presented this approximation algorithm succeeds as long as the set of points satisfies the triangular inequality we also show that our approximation algorithm is best possible with respect to the approximation bound if pz np

estimation of variance of the ratio estimator summary a general class of estimators of the variance of the ratio estimator is considered which includes two standard estimators vo and v2 and approximates another estimator vh suggested by royall eberhardt 1975 asymptotic expansions for the variances and biases of the proposed estimators are obtained based on this we obtain optimal variance estimators in the class and compare the relative merits of three estimators vo v1 and v2 without any model assumption under a simple regression model a more definite comparison of vo v1 and v2 is made in terms of variance and bias some key word8 asymptotic expansion bias optimal variance estimator ratio estimator superpopulation variance

the prediction approach to robust variance estimation in two stage cluster sampling abstract robust statistics are derived for estimating the error variance when a finite population total or a ratio of totals is estimated from a two stage cluster sample in which the first stage sampling fraction is small the analysis is based on prediction superpopulation models that allow for very general correlation structure within clusters results are obtained for when all cluster sizes are known and for when sizes are known only for sample clusters ratio regression and mean of ratios statistics are in the class of population total estimators considered

data driven density estimation in the presence of additive noise with unknown distribution we study the following model of deconvolution y x varepsilon with i i d observations y 1 dots y n and varepsilon 1 dots varepsilon m the x j 1 leq j leq n are i i d with density f independent of the varepsilon j the aim of the paper is to estimate f without knowing the density f varepsilon of the varepsilon j we first define a projection estimator for which we provide bounds for the pointwise and the integrated l 2 risk we consider ordinary smooth and supersmooth noise varepsilon with regard to ordinary smooth and supersmooth densities f then we present an adaptive estimator of the density of f this estimator is obtained by penalization of the projection contrast which provides model selection lastly we present simulation experiments to illustrate the good performances of our estimator and study from the empirical point of view the importance of theoretical constraints

a decomposition of multidimensional point sets with applications to k nearest neighbors and n body potential fields we define the notion of a well separated pair decomposition of points in d dimensional space we then develop efficient sequential and parallel algorithms for computing such a decomposition we apply the resulting decomposition to the efficient computation of k nearest neighbors and n body potential fields

balanced repeated replication for stratified multistage survey data under imputation abstract balanced repeated replication brr is a popular method for variance estimation in surveys the standard brr method works by first creating a set of balanced pseudoreplicated datasets from the original dataset for a survey estimator the brr variance estimator is the average of squared deviations r where r is the same as but based on the data in the rth pseudoreplicated dataset only but when there are a large number of imputed missing values nonrespondents treating the imputed values as observed data and applying the standard brr variance estimation formula does not produce valid variance estimators intuitively the variation due to imputation can be captured by the brr method if every pseudoreplicated dataset is imputed in exactly the same way as the original dataset is imputed assuming that the dataset contains flags for nonrespondents but when a random imputation method such as random hot deck imputation random ratio imputation or random regression imputation is u

approximate nearest neighbor search for l p spaces 2 p via embeddings abstract while the problem of approximate nearest neighbor search has been well studied for euclidean space and l 1 few non trivial algorithms are known for l p when 2 p in this paper we revisit this fundamental problem and present approximate nearest neighbor search algorithms which give the best known approximation factor guarantees in this setting

poststratification and conditional variance estimation poststratification estimation is a technique used in sample surveys to improve efficiency of estimators survey weights are adjusted to force the estimated numbers of units in each of a set of estimation cells to be equal to known population totals the resulting weights are then used in forming estimates of means or totals of variables collected in the survey for example in a household survey the estimation cells may be based on age race sex categories of individuals and the known totals may come from the most recent population census although the variance of a poststratified estimator can be computed over all possible sample configurations inferences made conditionally on the achieved sample configuration are desirable theory and a simulation study using data from the u s current population survey are presented to study both the conditional bias and variance of the poststratified estimator of a total the linearization balanced repeated replication and jackknife variance estimators are also exa

fast estimation of diameter and shortest paths without matrix multiplication in the recent past there has been considerable progress in devising algorithms for the all pairs shortest paths apsp problem running in time significantly smaller than the obvious time bound of o n3 unfortunately all the new algorithms are based on fast matrix multiplication algorithms that are notoriously impractical our work is motivated by the goal of devising purely combinatorial algorithms that match these improved running times our results come close to achieving this goal in that we present algorithms with a small additive error in the length of the paths obtained our algorithms are easy to implement have the desired property of being combinatorial in nature and the hidden constants in the running time bound are fairly small our main result is an algorithm which solves the apsp problem in unweighted undirected graphs with an additive error of 2 in time o n 2 5 sqrt log n this algorithm returns actual paths and not just the distances in addition we give more efficient algorithms with running time footnotesize o n 1 5 sqrt k log n n 2 log 2 n for the case where we are only required to determine shortest paths between k specified pairs of vertices rather than all pairs of vertices the starting point for all our results is an o m sqrt n log n algorithm for distinguishing between graphs of diameter 2 and 4 and this is later extended to obtaining a ratio 2 3 approximation to the diameter in time o m sqrt n log n n 2 log n unlike in the case of apsp our results for approximate diameter computation can be extended to the case of directed graphs with arbitrary positive real weights on the edges

near linear time construction of sparse neighborhood covers this paper introduces a near linear time sequential algorithm for constructing a sparse neighborhood cover this implies analogous improvements from quadratic to near linear time for any problem whose solution relies on network decompositions including small edge cuts in planar graphs approximate shortest paths and weight and distance preserving graph spanners in particular an o log n approximation to the k shortest paths problem on an n vertex e edge graph is obtained that runs in soh n e k time

generalized variance functions in stratified two stage sampling abstract generalized variance functions gvf s are used in a number of sample surveys as a convenient method of publishing sampling errors the method consists of estimating the relative variance relvariance of an estimator of a total t by using a model with the form a b t using the prediction approach to finite population sampling some asymptotic theory is presented for estimators of totals that are linear combinations of sample cluster means from stratified two stage cluster samples one choice of gvf estimator is shown to be consistent under a particular class of prediction models the theory is illustrated by an empirical study in which two stage stratified samples are selected from a population of households the prediction model is one in which units within a stratum have a common mean and variance units in the same cluster are correlated but units in different clusters are not and in which the common variance is a quadratic function of the common mean in a stratum bernoulli and poisson r

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

normal approximations for wavelet coefficients on spherical poisson fields abstract we compute explicit upper bounds on the distance between the law of a multivariate gaussian distribution and the joint law of wavelet needlet coefficients based on a homogeneous spherical poisson field in particular we develop some results from peccati and zheng 2010 42 based on malliavin calculus and stein s methods to assess the rate of convergence to gaussianity for a triangular array of needlet coefficients with growing dimensions our results are motivated by astrophysical and cosmological applications in particular related to the search for point sources in cosmic rays data

moving beyond the conventional stratified analysis to estimate an overall treatment efficacy with the data from a comparative randomized clinical study for a two group comparative study a stratified inference procedure is routinely used to estimate an overall group contrast to increase the precision of the simple two sample estimator unfortunately most commonly used methods including the cochran mantel haenszel statistic for a binary outcome and the stratified cox procedure for the event time endpoint do not serve this purpose well in fact these procedures may be worse than their two sample counterparts even when the observed treatment allocations are imbalanced across strata various procedures beyond the conventional stratified methods have been proposed to increase the precision of estimation when the naive estimator is consistent in this paper we are interested in the case when the treatment allocation proportions vary markedly across strata we study the stochastic properties of the two sample naive estimator conditional on the ancillary statistics the observed treatment allocation proportions and or the stratum sizes and present a biased adjusted estimator this adjusted estimator is asymptotically equivalent to the augmentation estimators proposed under the unconditional setting moreover this consistent estimation procedure is also equivalent to a rather simple procedure which estimates the mean response of each treatment group first via a stratum size weighted average and then constructs the group contrast estimate this simple procedure is flexible and readily applicable to any target patient population by choosing appropriate stratum weights all the proposals are illustrated with the data from a cardiovascular clinical trial whose treatment allocations are imbalanced

variance estimation for survey data with composite imputation and nonnegligible sampling fractions abstract this article considers variance estimation for horvitz thompson type estimated totals based on survey data with imputed non respondents and with nonnegligible sampling fractions a method based on a variance decomposition is proposed our method can be applied to complicated situations where a composite of some deterministic and or random imputation methods is used including using imputed data in subsequent imputations although here linearization or taylor expansion type techniques are adopted replication methods such as the jackknife balanced repeated replication and random groups can also be used in applying our method to derive variance estimators using our method variance estimators can be derived under either the customary design based approach or the model assisted approach and are asymptotically unbiased and consistent the transportation annual survey conducted at the u s census bureau in which nonrespondents are imputed using a composite of cold deck and ratio type imputation m

adaptive density estimation for directional data using needlets this paper is concerned with density estimation of directional data on the sphere we introduce a procedure based on thresholding on a new type of spherical wavelets called needlets we establish a minimax result and prove its optimality we are motivated by astrophysical applications in particular in connection with the analysis of ultra high energy cosmic rays

jackknife variance estimation for nearest neighbor imputation nearest neighbor imputation is a popular hot deck imputation method used to compensate for nonresponse in sample surveys although this method has a long history of application the problem of variance estimation after nearest neighbor imputation has not been fully investigated because nearest neighbor imputation is a nonparametric method a nonparametric variance estimation technique such as the jackknife is desired we show that the naive jackknife that treats imputed values as observed data produces serious underestimation we also show that rao and shao s adjusted jackknife or the jackknife with each pseudoreplicate reimputed which produces asymptotically unbiased and consistent jackknife variance estimators for other imputation methods such as mean imputation random hot deck imputation and ratio or regression imputation produces serious overestimation in the case of nearest neighbor imputation two partially reimputed and a partially adjusted jackknife variance estimators are proposed and s

ultra low dimensional embeddings for doubling metrics we consider the problem of embedding a metric into low dimensional euclidean space the classical theorems of bourgain and of johnson and lindenstrauss say that any metric on n points embeds into an o log n dimensional euclidean space with o log n distortion moreover a simple volume argument shows that this bound is nearly tight a uniform metric on n points requires nearly logarithmic number of dimensions to embed with logarithmic distortion it is natural to ask whether such a volume restriction is the only hurdle to low dimensional embeddings in other words do doubling metrics that do not have large uniform submetrics and thus no volume hurdles to low dimensional embeddings embed in low dimensional euclidean spaces with small distortionq in this article we give a positive answer to this question we show how to embed any doubling metrics into o log log n dimensions with o log n distortion this is the first embedding for doubling metrics into fewer than logarithmic number of dimensions even allowing for logarithmic distortion this result is one extreme point of our general trade off between distortion and dimension given an n point metric v d with doubling dimension dimd and any target dimension t in the range dimd log log n t o log n we show that the metric embeds into euclidean space rt with o log n s dimd t distortion

approximate distance oracles let g v e be an undirected weighted graph with vvv n and vev m let k 1 be an integer we show that g v e can be preprocessed in o kmn1 k expected time constructing a data structure of size o kn1p1 k such that any subsequent distance query can be answered approximately in o k time the approximate distance returned is of stretch at most 2k 1 that is the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k 1 a 1963 girth conjecture of erdos implies that n1p1 k space is needed in the worst case for any real stretch strictly smaller than 2kp1 the space requirement of our algorithm is therefore essentially optimal the most impressive feature of our data structure is its constant query time hence the name oracle previously data structures that used only o n1p1 k space had a query time of n1 k our algorithms are extremely simple and easy to implement efficiently they also provide faster constructions of sparse spanners of weighted graphs and improved tree covers and distance labelings of weighted or unweighted graphs

an empirical study of the ratio estimator and estimators of its variance abstract this paper reports results from an empirical study of the ratio estimator for a finite population total from each of six real populations 1 000 simple random samples 1 000 restricted random samples and three nonrandom samples of size 32 are drawn performance of the ratio estimator and of five estimators of its variance is compared with theoretical results generated using a prediction superpopulation models and b probability sampling distributions the results presented graphically show that theory based on prediction models can reveal relationships that are essential in making inferences but that are concealed in probability sampling analyses

undirected single source shortest paths with positive integer weights in linear time the single source shortest paths problem sssp is one of the classic problems in algorithmic graph theory given a positively weighted graph g with a source vertex s find the shortest path from s to all other vertices in the graph since 1959 all theoretical developments in sssp for general directed and undirected graphs have been based on dijkstra s algorithm visiting the vertices in order of increasing distance from s thus any implementation of dijkstra s algorithm sorts the vertices according to their distances from s however we do not know how to sort in linear time here a deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with positive integer weights the algorithm avoids the sorting bottleneck by building a hierarchical bucketing structure identifying vertex pairs that may be visited in any order

a new series of dense graphs of high girth let k 1 be an odd integer t b k 2 4 c and q be a prime power we construct a bipartite q regular edge transitive graph cd k q of order v 2qk t 1 and girth g k 5 if e is the the number of edges of cd k q then e v 1 k t 1 these graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g g 5 g 6 11 12 for g 24 this represents a slight improvement on bounds established by margulis and lubotzky phillips sarnak for 5 g 23 g 6 11 12 it improves on or ties existing bounds