honey pot constrained searching with local sensory information in this paper we investigate the problem of searching for a hidden target in a bounded region of the plane using an autonomous robot which is only able to use local sensory information the problem is naturally formulated in the continuous domain but the solution proposed is based on an aggregation refinement approach in which the continuous search space is partitioned into a finite collection of regions on which we define a discrete search problem a solution to the original problem is obtained through a refinement procedure that lifts the discrete path into a continuous one we show that the discrete optimization is computationally difficult np hard but there are computationally efficient approximation algorithms for solving it the resulting solution to the continuous problem is in general not optimal but one can construct bounds to gauge the cost penalty incurred due to i the discretization of the problem and ii the attempt to approximately solve the np hard problem in polynomial time numerical simulations show that the algorithms proposed behave significantly better than naive approaches such as a random walk or a greedy algorithm c 2006 elsevier ltd all rights reserved

optimal detection of sparse principal components in high dimension we perform a finite sample analysis of the detection levels for sparse principal components of a high dimensional covariance matrix our minimax optimal test is based on a sparse eigenvalue statistic alas computing this test is known to be np complete in general and we describe a computationally efficient alternative test using convex relaxations our relaxation is also proved to detect sparse principal components at near optimal detection levels and performs very well on simulated datasets ams 2000 subject classifications primary 62h25 secondary 62f04 90c22

gauss curvature flow the fate of the rolling stones we prove firey s 1974 conjecture that convex surfaces moving by their gauss curvature become spherical as they contract to points

object tracking using deformable templates we propose a method for object tracking using prototype based deformable template models to track an object in an image sequence we use a criterion which combines two terms the frame to frame deviations of the object shape and the fidelity of the modeled shape to the input image the deformable template model utilizes the prior shape information which is extracted from the previous frames along with a systematic shape deformation scheme to model the object shape in a new frame the following image information is used in the tracking process 1 edge and gradient information the object boundary consists of pixels with large image gradient 2 region consistency the same object region possesses consistent color and texture throughout the sequence and 3 interframe motion the boundary of a moving object is characterized by large interframe motion the tracking proceeds by optimizing an objective function which combines both the shape deformation and the fidelity of the modeled shape to the current image in terms of gradient texture and interframe motion the inherent structure in the deformable template together with region motion and image gradient cues makes the proposed algorithm relatively insensitive to the adverse effects of weak image features and moderate amounts of occlusion

on the case of equality in the brunn minkowski inequality for capacity suppose that and 1 are convex open subsets of rn denote their convex combination by the brunn minkowski inequality says that vol t 1 t vol 0 1 n t vol for 0 t l moreover if there is equality for some t other than an endpoint then the domains 1 and 0 are translates and dilates of each other borell proved an analogue of the brunn minkowski inequality with capacity defined below in place of volume borel s theorem b says theorem a let t t 1 1 t 0 be a convex combination of two convex subsets of rn n 3 then cap the main purpose of this note is to prove

asymptotic optimality theory for decentralized sequential hypothesis testing in sensor networks the decentralized sequential hypothesis testing problem is studied in sensor networks where a set of sensors receive independent observations and send summary messages to the fusion center which makes a final decision in the scenario where the sensors have full access to their past observations the first asymptotically bayes sequential test is developed having the same asymptotic performance as the optimal centralized test that has access to all sensor observations next in the scenario where the sensors do not have full access to their past observations a simple but asymptotically bayes sequential tests is developed in which sensor message functions are what we call tandem quantizer where each sensor only uses two different sensor quantizers with at most one switch between these two possibilities moreover a new minimax formulation of optimal stationary sensor quantizers is proposed and is studied in detail in the case of additive gaussian sensor noise finally our results show that in the simplest models feedback from the fusion center does not improve asymptotic performance in the scenario with full local memory however even a one shot one bit feedback can significantly improve performance in the case of limited local memory

the spectra of uniform hypertrees abstract in this paper we study the spectra of uniform hypertrees by using the generalized weighted incident matrix we show that is a nonzero eigenvalue of the hypertree h corresponding to an eigenvector with all elements nonzero if and only if is a root of the polynomial h i 0 m 1 i m i x m i r where m i is the number of matchings of order i in h

statistical computational tradeoffs in planted problems and submatrix localization with a growing number of clusters and submatrices we consider two closely related problems planted clustering and submatrix localization in the planted clustering problem a random graph is generated based on an underlying cluster structure of the nodes the task is to recover these clusters given the graph the submatrix localization problem concerns locating hidden submatrices with elevated means inside a large real valued random matrix of particular interest is the setting where the number of clusters submatrices is allowed to grow unbounded with the problem size these formulations cover several classical models such as planted clique planted densest subgraph planted partition planted coloring and the stochastic block model which are widely used for studying community detection graph clustering and bi clustering for both problems we show that the space of the model parameters cluster submatrix size edge probabilities and the mean of the submatrices can be partitioned into four disjoint regions corresponding to decreasing statistical and computational complexities 1 the impossible regime where all algorithms fail 2 the hard regime where the computationally expensive maximum likelihood estimator mle succeeds 3 the easy regime where the polynomial time convexified mle succeeds 4 the simple regime where a local counting thresholding procedure succeeds moreover we show that each of these algorithms provably fails in the harder regimes our results establish the minimax recovery limits which are tight up to universal constants and hold even with a growing number of clusters submatrices and provide order wise stronger performance guarantees for polynomial time algorithms than previously known our study demonstrates the tradeoffs between statistical and computational considerations and suggests that the minimax limits may not be achievable by polynomial time algorithms

unimodality convexity and applications properties of univariate unimodal distributions concepts of multivariate unimodality some more notions of unimodality unimodality for discrete distributions unimodality of infinitely divisible distributions unimodality and notions of dependence ordering of distributions by peakedness applications of unimodality in statistical inference convexity in reliability theory appendix

false discovery control for random fields this article extends false discovery rates to random fields for which there are uncountably many hypothesis tests we develop a method for finding regions in the field s domain where there is a significant signal while controlling either the proportion of area or the proportion of clusters in which false rejections occur the method produces confidence envelopes for the proportion of false discoveries as a function of the rejection threshold from the confidence envelopes we derive threshold procedures to control either the mean or the specified tail probabilities of the false discovery proportion an essential ingredient of this construnction is a new algorithm to compute a confidence superset for the set of all true null locations we demonstrate our method with applications to scan statistics and functional neuroimaging

some applications of mass transport to gaussian type inequalities as discovered by brenier mapping through a convex gradient gives the optimal transport in n in the present article this map is used in the setting of gaussian like measures to derive an inequality linking entropy with mass displacement by a straightforward argument as a consequence logarithmic sobolev and transport inequalities are recovered finally a result of caffarelli on the brenier map is used to obtain gaussian correlation inequalities

innovated higher criticism for detecting sparse signals in correlated noise higher criticism is a method for detecting signals that are both sparse and weak although first proposed in cases where the noise variables are independent higher criticism also has reasonable performance in settings where those variables are correlated in this paper we show that by exploiting the nature of the correlation performance can be improved by using a modified approach which exploits the potential advantages that correlation has to offer indeed it turns out that the case of independent noise is the most difficult of all from a statistical viewpoint and that more accurate signal detection for a given level of signal sparsity and strength can be obtained when correlation is present we characterize the advantages of correlation by showing how to incorporate them into the definition of an optimal detection boundary the boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is toeplitz

unpredictable paths and percolation we construct a nearest neighbor process s n on z that is less predictable than simple random walk in the sense that given the process until time n the conditional probability that s n k x is uniformly bounded by ck for some 1 2 from this process we obtain a probability measure on oriented paths in z 3 such that the number of intersections of two paths chosen independently according to has an exponential tail for d 4 the uniform measure on oriented paths from the origin in z d has this property we show that on any graph where such a measure on paths exists oriented percolation clusters are transient if the retention parameter p is close enough to 1 this yields an extension of a theorem of grimmett kesten and zhang who proved that supercritical percolation clusters in z d are transient for all d 3

searching for a trail of evidence in a maze consider a graph with a set of vertices and oriented edges connecting pairs of vertices each vertex is associated with a random variable and these are assumed to be independent in this setting suppose we wish to solve the following hypothesis testing problem under the null the random variables have common distribution n 0 1 while under the alternative there is an unknown path along which random variables have distribution n 1 0 and distribution n 0 1 away from it for which values of the mean shiftcan one reliably detect and for which values is this impossible this paper develops detection thresholds for two types of common graphs which exhibit a different behavior the first is the usual regular lattice with vertices of the form i j 0 i i j i and j has the parity of i and oriented edges i j i 1 j s where s 1 we show that for paths of length m start ing at the origin the hypotheses become distinguishable in a minimax sense ifm log m while they are not ifm log m we derive equivalent results in a bayesian setting where one assumes that all paths are equally likely there the asymptotic threshold ism m 1 4 we obtain corresponding results for trees where the threshold is of order 1 and independent of the size of the tree for distributions other than the gaussian and for other graphs the concept of predictability profile first introduced by benjamini pemantle and peres plays a crucial role in our analysis

energy landscape for large average submatrix detection problems in gaussian random matrices the problem of finding large average submatrices of a real valued matrix arises in the exploratory analysis of data from a variety of disciplines ranging from genomics to social sciences in this paper we provide a detailed asymptotic analysis of large average submatrices of an n times n gaussian random matrix the first part of the paper addresses global maxima for fixed k we identify the average and the joint distribution of the k times k submatrix having largest average value as a dual result we establish that the size of the largest square sub matrix with average bigger than a fixed positive constant is with high probability equal to one of two consecutive integers that depend on the threshold and the matrix dimension n the second part of the paper addresses local maxima specifically we consider submatrices with dominant row and column sums that arise as the local optima of iterative search procedures for large average submatrices for fixed k we identify the limiting average value and joint distribution of a k times k submatrix conditioned to be a local maxima in order to understand the density of such local optima and explain the quick convergence of such iterative procedures we analyze the number l n k of local maxima beginning with exact asymptotic expressions for the mean and fluctuation behavior of l n k for fixed k the mean of l n k is theta n k log n k 1 2 while the standard deviation is theta n 2k 2 k 1 log n k 2 k 1 our principal result is a gaussian central limit theorem for l n k that is based on a new variant of stein s method

detection of correlations we consider the hypothesis testing problem of deciding whether an observed high dimensional vector has independent normal components or alternatively if it has a small subset of correlated components the correlated components may have a certain combinatorial structure known to the statistician we establish upper and lower bounds for the worst case minimax risk in terms of the size of the correlated subset the level of correlation and the structure of the class of possibly correlated sets we show that some simple tests have near optimal performance in many cases while the generalized likelihood ratio test is suboptimal in some important cases

on the _ minkowski problem a volume normalized formulation of the lp minkowski problem is presented this formulation has the advantage that a solution is possible for all p 1 including the degenerate case where the index p is equal to the dimension of the ambient space a new approach to the lp minkowski problem is presented which solves the volume normalized formulation for even data and all p 1

computational and statistical boundaries for submatrix localization in a large noisy matrix we study in this paper computational and statistical boundaries for submatrix localization given one observation of one or multiple nonoverlapping signal submatrix of magnitude lambda and size k m times k n embedded in a large noise matrix of size m times n the goal is to optimal identify the support of the signal submatrix computationally and statistically two transition thresholds for the signal to noise ratio lambda sigma are established in terms of m n k m and k n the first threshold sf snr c corresponds to the computational boundary we introduce a new linear time spectral algorithm that identifies the submatrix with high probability when the signal strength is above the threshold sf snr c below this threshold it is shown that no polynomial time algorithm can succeed in identifying the submatrix under the hidden clique hypothesis the second threshold sf snr s captures the statistical boundary below which no method can succeed in localization with probability going to one in the minimax sense the exhaustive search method successfully finds the submatrix above this threshold in marked contrast to submatrix detection and sparse pca the results show an interesting phenomenon that sf snr c is always significantly larger than sf snr s under the sub gaussian error model which implies an essential gap between statistical optimality and computational efficiency for submatrix localization

distinguishing infections on different graph topologies the history of infections and epidemics holds famous examples where understanding containing and ultimately treating an outbreak began with understanding its mode of spread influenza hiv and most computer viruses spread person to person device to device and through contact networks cholera cancer and seasonal allergies on the other hand do not in this paper we study two fundamental questions of detection first given a snapshot view of a perhaps vanishingly small fraction of those infected under what conditions is an epidemic spreading via contact e g influenza distinguishable from a random illness operating independently of any contact network e g seasonal allergies second if we do have an epidemic under what conditions is it possible to determine which network of interactions is the main cause of the spread the causative network without any knowledge of the epidemic other than the identity of a minuscule subsample of infected nodes the core therefore of this paper is to obtain an understanding of the diagnostic power of network information we derive sufficient conditions that networks must satisfy for these problems to be identifiable and produce efficient highly scalable algorithms that solve these problems we show that the identifiability condition we give is fairly mild and in particular is satisfied by two common graph topologies the d dimensional grid and the erdos renyi graphs

shadows of convex bodies it is proved that if c is a convex body in rn then c has an affine image c of nonzero volume so that if p is any 1 codimensional orthogonal projection pca 1 cl n l n it is also shown that there is a pathological body k all of whose orthogonal projections have volume about vx times as large as iki n 1 n 0 introduction the problems discussed in this paper concern the areas of shadows orthogonal projections of convex bodies and to a lesser extent the surface areas of such bodies if c is a convex body in irn and 0 a unit vector p c will denote the orthogonal projection of c onto the 1 codimensional space perpendicular to 0 volumes and areas of convex bodies and their surfaces will be denoted with i 1 the relationship between shadows and surface areas of convex bodies is expressed in cauchy s well known formula for each n e n let vn be the volume of the n dimensional euclidean unit ball and let a an be the rotationally invariant probability on the unit sphere sn cauchy s formula states that if c is a convex body in rn then its surface area is ac i s p6ci dcr o vni jn i the classical isoperimetric inequality in rn states that any body has surface area at least as large as a euclidean ball of the same volume the first section of this paper is devoted to the proof of a local isoperimetric inequality showing that all bodies have large shadows rather than merely large surface area or average shadow the principal motivation for this result is its relationship to a conjecture of vaaler and the important problems surrounding it this theorem and its connection with vaaler s conjecture are described at the beginning of 1 an important role is played in the theory of convex bodies by the so called projection body of a convex body it is easily seen by considering polytopes received by the editors september 12 1989 1980 mathematics subject classification 1985 revision primary 52a20 52a40 the author was supported in part by nsf dms 8807243 1991 american mathematical society 0002 9947 91 1 00 25 per page

a new ellipsoid associated with convex bodies it is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the legendre ellipsoid of classical mechanics sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid these inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid among other things this leads to a new approach to establishing ball s maximal shadows conjecture for symmetric convex bodies corresponding to each origin symmetric convex or more general subset of euclidean n space r there is a unique ellipsoid with the following property the moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1 dimensional subspace of r this ellipsoid is called the legendre ellipsoid of the convex set the legendre ellipsoid and its polar the binet ellipsoid are well known concepts from classical mechanics see milman and pajor mpa1 mpa2 lindenstrauss and milman lim and leichtweis le for some historical references it has slowly come to be recognized that along side the brunn minkowski theory there is a dual theory the nature of the duality between this dual brunn minkowski theory and the brunn minkowski theory is subtle and not yet understood it is easily seen that the legendre and binet ellipsoid is an object of this dual brunnminkowski theory this observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical legendre ellipsoid in the brunn minkowski theory it is the aim of this paper to demonstrate the existence of precisely this dual object in retrospect one may well wonder why research supported in part by nsf grant dms 9803261 typeset by ams tex 1 2 a new ellipsoid associated with convex bodies the new ellipsoid presented in this note was not discovered long ago the simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of l2 curvature in hand however the brunn minkowski theory was only recently extended to incorporate the new notion of lp curvature see l2 l3 a positive definite n n real symmetric matrix a generates an ellipsoid a in r defined by a x r x ax 1 where x ax denotes the standard inner product of x and ax in r associated with a star shaped about the origin set k r is its legedre ellipsoid 2k which is generated by the matrix mij k where mij k n 2 v k k ei x ej x dx with e1 en denoting the standard basis for r and v k denoting the ndimensional volume of k we will associate a new ellipsoid 2k with each convex body k r one approach to defining 2k without introducing new notation is to first define it for polytopes and then use approximation with respect to the hausdorff metric to extend the definition to all convex bodies suppose p r is a polytope that contains the origin in its interior let u1 un denote the outer unit normals to the faces of p let a1 an denote the areas i e n 1 dimensional volumes of the corresponding faces and let h1 hn denote the distances from the origin to the corresponding faces the ellipsoid 2p is generated by the matrix mij p where mij p 1 v p n l 1 al hl ei ul ej ul an alternate definition of the operator 2 will be given after additional notation is introduced the easily established affine nature of the operator 2 is formally stated in lemma 1 if k r is star shaped about the origin then for each gl n

dimension reduction for hypothesis testing in worst case scenarios this paper considers a one among many detection problem where one has to discriminate between pure noise and one among l alternatives that are known up to an amplitude factor two issues linked to high dimensionality arise in this framework first the computational complexity associated to the generalized likelihood ratio glr with the constraint of sparsity one inflates linearly with l which can be an obstacle when multiple data sets have to be tested second standard procedures based on dictionary learning aimed at reducing the dimensionality may suffer from severe power losses for some alternatives thus suggesting a worst case scenario strategy in the case where the learned dictionary has k 1 column we show that the exact solution of the resulting detection problem which can be formulated as a minimax problem can be obtained by quadratic programming because it allows a better sampling of the diversity of the alternatives the case k 1 is expected to improve the detection performances over the case k 1 the worst case analysis of this case which is more involved leads us to propose two minimax learning algorithms numerical results show that these algorithms indeed allow to increase performances over the k 1 case and are in fact comparable to the glr using the full set of alternatives while being computationally simpler

maximal meaningful events and applications to image analysis we discuss the mathematical properties of a recently introduced method for computing geometric structures in a digital image without any a priori information this method is based on a basic principle of perception which we call the helmholtz principle according to this principle an observed geometric structure is perceptually meaningful if the expectation of its number of occurrences in other words its number of false alarms nf is very small in a random image it is maximal meaningful if its nf is minimal among the meaningful structures of the same kind which it contains or is contained in this definition meets the gestalt theory requirement that parts of a whole are not perceived we explain by large deviation estimates why this definition leads to an a priori knowledge free method compatible with phenomenology we state a principle according to which maximal structures do not meet we prove this principle in the large deviations framework in the case of alignments in a digital image we show why these results make maximal meaningful structures computable and display several applications

greedy lattice animals negative values and unconstrained maxima let x z d be i i d random variables and s x be the weight of a lattice animal let n n max s n and contains the origin and g n max s n n d we show that regardless of the negative tail of the distribution of x if e x d log x d a 0 then first lim n n 1 n n n exists is finite and constant a e and second there is a transition in the asymptotic behavior of g n depending on the sign of n if n 0 then g n n d and if n 0 the exact behavior of g n in this last case depends on the positive tail of the distribution of x we show that if it is nontrivial and has exponential moments then g n log n with a transition from g n n d occurring in general not as predicted by large deviations estimates finally if x d 1 f x as x oo then no transition takes place

metric marginal problems for set valued or non measurable variables in a separable metric space if two borel probability measures laws are nearby in a suitable metric then there exist random variables with those laws which are nearby in probability specifically by a well known theorem of strassen the prohorov distance between two laws is the infimum of ky fan distances of random variables with those laws the present paper considers possible extensions of strassen s theorem to two random elements one of which may be compact set valued and or non measurable there are positive results in finite dimensional spaces but with factors depending on the dimension examples show that such factors cannot entirely be avoided so that the extension of strassen s theorem to the present situation fails in infinite dimensions

volume of mixed bodies by using inequalities obtained for the volume of mixed bodies and the petty projection inequality sharp isoperimetric inequalities are derived for the projection measures quermassintegrale of a convex body these projection measure inequalities which involve mixed projection bodies zonoids are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body the inequality obtained for the volume of mixed bodies is also used to derive a form of the brunn minkowski inequality involving mixed bodies as an application inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies o introduction in 24 it was shown that the petty projection inequality 26 will yield extensions of the known isoperimetric inequalities between the volume of a convex body and its projection measures quermassintegrale in this article we show that the petty projection inequality can be used to obtain sharp isoperimetric inequalities corollary 5 20 which are strengthened forms of the classical inequalities between the projection measures of a convex body this is accomplished by using the so called mixed convex bodies introduced by firey 17 geometric inequalities are derived for the volume of mixed bodies theorems 4 1 and 4 3 which when combined with the petty projection inequality yield inequalities involving projection bodies zonoids which are related to the inequalities between the projection measures of a convex body the inequalities for mixed bodies are also used to obtain a version of the brunn minkowski inequality theorem 4 2 involving mixed bodies also derived are inequalities between the volume of convex bodies and their associated mixed projection integral theorems 6 4 and 6 4 that are extensions and improvements of some previously established inequalities since some of our results concern projection bodies a topic of interest to workers outside the area of geometric convexity we have restated a few basic definitions and standard results from the field of geometric inequalities in 1 we collect for quick reference some well known results concerning mixed volumes and mixed area measures which will be required later references for material presented in this section are bonnesen and fenchel 7 for mixed volumes busemann 9 and leichtweiss 20 the original works of aleksandrov 1 4 and fenchel and jessen 15 supplemented by the survey of schneider 28 are recommended references for mixed area measures 2 contains introductory material regarding blaschke addition received by the editors january 8 1985 1980 mcithematics subject classihecition primary 52a40 secondary 53a15 46b20

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

on a reverse form of the brascamp lieb inequality we prove a reverse form of the multidimensional brascamp lieb inequality our method also gives a new way to derive the brascamp lieb inequality and is rather convenient for the study of equality cases

snakes shapes and gradient vector flow snakes or active contours are used extensively in computer vision and image processing applications particularly to locate object boundaries problems associated with initialization and poor convergence to boundary concavities however have limited their utility this paper presents a new external force for active contours largely solving both problems this external force which we call gradient vector flow gvf is computed as a diffusion of the gradient vectors of a gray level or binary edge map derived from the image it differs fundamentally from traditional snake external forces in that it cannot be written as the negative gradient of a potential function and the corresponding snake is formulated directly from a force balance condition rather than a variational formulation using several two dimensional 2 d examples and one three dimensional 3 d example we show that gvf has a large capture range and is able to move snakes into boundary concavities

fischer type determinantal inequalities for accretive dissipative matrices abstract let a a 11 a 12 a 21 a 22 be an n n accretive dissipative matrix k and l be the orders of a 11 and a 22 respectively and let m min k l then det a a det a 11 det a 22 where a 2 3 m 2 if m n 3 2 n 2 if n 3 m n 2 this improves a result of ikramov

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

the complexity of the a b c problem we present a deterministic polynomial time algorithm for the a b c problem which is the membership problem for 2 generated commutative linear semigroups over an algebraic number field we also obtain a polynomial time algorithm for the easier membership problem for 2 generated abelian linear groups furthermore we provide a polynomial sized encoding for the set of all solutions

properties of higher criticism under strong dependence the problem of signal detection using sparse faint information is closely related to a variety of contemporary statistical problems including the control of false discovery rate and classification using very high dimensional data each problem can be solved by conducting a large number of simultaneous hypothesis tests the properties of which are readily accessed under the assumption of independence in this paper we address the case of dependent data in the context of higher criticism methods for signal detection short range dependence has no first order impact on performance but the situation changes dramatically under strong dependence there although higher criticism can continue to perform well it can be bettered using methods based on differences of signal values or on the maximum of the data the relatively inferior performance of higher criticism in such cases can be explained in terms of the fact that under strong dependence the higher criticism statistic behaves as though the data were partitioned into very large blocks with all but a single representative of each block being eliminated from the dataset

generalizations of the brunn minkowski inequality abstract yuan and leng 2007 gave a generalization of the matrix form of the brunn minkowski inequality in this note we first give a simple proof of this inequality and then show a generalization of this to a larger class of matrices namely matrices whose numerical ranges are contained in a sector

detection of an anomalous cluster in a network we consider the problem of detecting whether or not in a given sensor network there is a cluster of sensors which exhibit an unusual behavior formally suppose we are given a set of nodes and attach a random variable to each node we observe a realization of this process and want to decide between the following two hypotheses under the null the variables are i i d standard normal under the alternative there is a cluster of variables that are i i d normal with positive mean and unit variance while the rest are i i d standard normal we also address surveillance settings where each sensor in the network collects information over time the resulting model is similar now with a time series attached to each node we again observe the process over time and want to decide between the null where all the variables are i i d standard normal and the alternative where there is an emerging cluster of i i d normal variables with positive mean and unit variance the growth models used to represent the emerging cluster are quite general and in particular include cellular automata used in modelling epidemics in both settings we consider classes of clusters that are quite general for which we obtain a lower bound on their respective minimax detection rate and show that some form of scan statistic by far the most popular method in practice achieves that same rate within a logarithmic factor our results are not limited to the normal location model but generalize to any one parameter exponential family when the anoma lous clusters are large enough

inequalities for mixed projection bodies mixed projection bodies are related to ordinary projection bodies zonoids in the same way that mixed volumes are related to ordinary volume analogs of the classical inequalities from the brunn minkowski theory such as the minkowski brunn minkowski and aleksandrov fenchel inequalities are developed for projection and mixed projection bodies

membership testing in commutative tranformation semigroups abstract given a finite set x of states a finite set of commuting transformations of x generators and another transformation f of x we analyze the complexity of deciding whether f can be obtained by composition of the generators looking first at the action of a commutative semigroup of transformations of a finite set we obtain an algorithm for membership testing valid for arbitrary commutative semigroups we then show that the complexity of the problem varies with the threshold of the semigroup polynomial time nc 3 in parallel with threshold zero or one and np complete otherwise

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

the wulff theorem revisited the parametrized indicator measures and the brunn minkowski inequality are used to prove that the wulff set w is a minimizer for the surface energy where the density is the support function of w the support of the indicator measures associated to minimizing sequences is characterized it is shown that if w is polyhedral then minimizing sequences cannot oscillate

equilibrium shapes for planar crystals in an external field the equilibrium shape of a two dimensional crystal in a convex background potential g x is analyzed for g 0 the shape of minimum energy may be deduced from surface tension via the wulff construction but if g is not constant little is known beyond the case of a crystal sitting in a uniform field only an unpublished result of okikiolu shows each connected component of the equilibrium crystal to be convex here it will be shown that any such component minimizes energy uniquely among convex sets of its area if the wulff shape and g x are symmetric under x x it follows that the equilibrium crystal is unique convex and connected this last result leads to a new proof that convex crystals away from equilibrium remain convex as they evolve by curvature driven flow subsequent work with felix otto shows without assuming symmetry that no equilibrium crystal has more than two convex components

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

multiple hypothesis mapping of functional mri data in orthogonal and complex wavelet domains we are interested in methods for multiple hypothesis testing that optimize power to refute the null hypothesis while controlling the false discovery rate fdr the wavelet transform of a spatial map of brain activation statistics can be tested in two stages to achieve this objective first a set of possible wavelet coefficients to test is reduced and second each hypothesis in the remaining subset is formally tested we show that a bayesian bivariate shrinkage operator baybishrink for the first step provides a powerful and expedient alternative to a subband adaptive chi squared test or an enhanced fdr algorithm based on the generalized degrees of freedom we also investigate the dual tree complex wavelet transform cwt as an alternative basis to the orthogonal discrete wavelet transform dwt we design and validate a test for activation based on the magnitude of the complex wavelet coefficients and show that this confers improved specificity for mapping spatial signals the methods are applied to simulated and experimental data including a pharmacological magnetic resonance imaging mri study we conclude that using baybishrink to define a reduced set of complex wavelet coefficients and testing the magnitude of each complex pair to control the fdr represents a competitive solution for multiple hypothesis mapping in fmri

on some generalizations of the brunn minkowski inequality abstract recently liu 2016 9 obtained some determinantal inequalities for sector matrices in this paper we make considerable improvement on liu s results

gaussian kernels have only gaussian maximizers a gaussian integral kernel g x y on rn x rn is the exponential of a quadratic form in x and y the fourier transform kernel is an example the problem addressed here is to find the sharp bound of g as an operator from l p rp to l q rn and to prove that the l p rn functions that saturate the bound are necessarily gaussians this is accomplished generally for 1 q in some special cases besides greatly extending previous results in this area the proof technique is also essentially different from earlier ones a corollary of these results is a fully multidimensional multilinear generalization of young s inequality

random growth models with polygonal shapes we consider discrete time random perturbations of monotone cellular automata ca in two dimensions under general conditions we prove the existence of half space velocities and then establish the validity of the wulff construction for asymptotic shapes arising from finite initial seeds such a shape converges to the polygonal invariant shape of the corresponding deterministic model as the perturbation decreases in many cases exact stability is observed that is for small perturbations the shapes of the deterministic and random processes agree exactly we give a complete characterization of such cases and show that they are prevalent among threshold growth ca with box neighborhood we also design a nontrivial family of ca in which the shape is exactly computable for all values of its probability parameter

on combinatorial testing problems we study a class of hypothesis testing problems in which upon observing the realization of an n dimensional gaussian vector one has to decide whether the vector was drawn from a standard normal distribution or alternatively whether there is a subset of the components belonging to a certain given class of sets whose elements have been contaminated that is have a mean different from zero we establish some general conditions under which testing is possible and others under which testing is hopeless with a small risk the combinatorial and geometric structure of the class of sets is shown to play a crucial role the bounds are illustrated on various examples

on the complexity of the orbit problem we consider higher dimensional versions of kannan and lipton s orbit problem determining whether a target vector space may be reached from a starting point x under repeated applications of a linear transformation a answering two questions posed by kannan and lipton in the 1980s we show that when has dimension one this problem is solvable in polynomial time and when has dimension two or three the problem is in nprp

multiscale change point inference summary we introduce a new estimator the simultaneous multiscale change point estimator smuce for the change point problem in exponential family regression an unknown step function is estimated by minimizing the number of change points over the acceptance region of a multiscale test at a level the probability of overestimating the true number of change points k is controlled by the asymptotic null distribution of the multiscale test statistic further we derive exponential bounds for the probability of underestimating k by balancing these quantities will be chosen such that the probability of correctly estimating k is maximized all results are even non asymptotic for the normal case on the basis of these bounds we construct asymptotically honest confidence sets for the unknown step function and its change points at the same time we obtain exponential bounds for estimating the change point locations which for example yield the minimax rate up to a log term finally the simultaneous multiscale change point estimator achieves the optimal detection rate of vanishing signals as n even for an unbounded number of change points we illustrate how dynamic programming techniques can be employed for efficient computation of estimators and confidence regions the performance of the multiscale approach proposed is illustrated by simulations and in two cutting edge applications from genetic engineering and photoemission spectroscopy

detecting anomalous activity on networks with the graph fourier scan statistic we consider the problem of deciding based on a single noisy measurement at each vertex of a given graph whether the underlying unknown signal is constant over the graph or there exists a cluster of vertices with anomalous activation this problem is relevant to several applications such as surveillance disease outbreak detection biomedical imaging environmental monitoring etc since the activations in these problems often tend to be localized to small groups of vertices in the graphs we model such activity by a class of signals that are elevated over a possibly disconnected cluster with low cut size relative to its size we analyze the corresponding generalized likelihood ratio glr statistics and relate it to the problem of finding a sparsest cut in the graph we develop a convex relaxation of the glr statistic based on spectral graph theory which we call the graph fourier scan statistic gfss in our main theoretical result we show that the performance of the gfss depends explicitly on the spectral properties of the graph to assess the optimality of the gfss we prove an information theoretic lower bound for the detection of anomalous activity on graphs because the gfss requires the specification of a tuning parameter we develop an adaptive version of the gfss using these results we are able to characterize in a very explicit form the performance of the gfss on a few notable graph topologies we demonstrate that the gfss can efficiently detect a simulated arsenic contamination in groundwater

an analytic solution to the busemann petty problem on sections of convex bodies we derive a formula connecting the derivatives of parallel section functions of an origin symmetric star body in r n with the fourier transform of powers of the radial function of the body a parallel section function or ni 1 dimensional x ray gives the ni 1 dimensional volumes of all hyperplane sections of the body orthogonal to a given direction this formula provides a new characterization of intersection bodies in r n and leads to a unifled analytic solution to the busemann petty problem suppose that k and l are two origin symmetric convex bodies inr n such that the ni 1 dimensional volume of each central hyperplane section of k is smaller than the volume of the corresponding section of l is the n dimensional volume of k smaller than the volume of l in conjunction with earlier established connections between the busemann petty problem intersection bodies and positive deflnite distributions our formula shows that the answer to the problem depends on the behavior of the ni 2 nd derivative of the parallel section functions the a rmative answer to the busemann petty problem for n 4 and the negative answer for n 5 now follow from the fact that convexity controls the second derivatives but does not control the derivatives of higher orders

capacitary inequalities of the brunn minkowski type the main purpose of this paper is to show how so called a concave capacities may be built up from newton capacity c in lr n 3 in much the same way as concave measures from lebesgue measure m n in lr n 1 throughout o f2 o stand for non empty open and convex subsets of ir and j o denotes the class of all convex bodies contained in f2 recall that a positive radon measure in f2 is said to be concave if a and

higher criticism for detecting sparse heterogeneous mixtures higher criticism or second level significance testing is a multiplecomparisons concept mentioned in passing by tukey it concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis tukey suggested comparing the fraction of observed significances at a given level to the expected fraction under the joint null in fact he suggested standardizing the difference of the two quantities and forming a z score the resulting z score tests the significance of the body of significance tests we consider a generalization where we maximize this z score over a range of significance levels 0 0 we are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem testing whether n normal means are all zero versus the alternative that a small fraction is nonzero the subtlety of this sparse normal means testing problem can be seen from work of ingster and jin who studied such problems in great detail in their studies they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p values either for the bulk of the tests or for the few most highly significant tests in this range when the amplitude of nonzero means is calibrated with the fraction of nonzero means the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses we show that the higher criticism is successful throughout the same region of amplitude sparsity where the likelihood ratio test

nonparametric detection of geometric structures over networks nonparametric detection of the possible existence of an anomalous structure over a network is investigated nodes corresponding to the anomalous structure if one exists receive samples generated by a distribution q which is different from a distribution p generating samples for other nodes if an anomalous structure does not exist all nodes receive samples generated by p it is assumed that the distributions p and q are arbitrary and unknown the goal is to design statistically consistent tests with probability of errors converging to zero as the network size becomes asymptotically large kernel based tests are proposed based on maximum mean discrepancy which measures the distance between mean embeddings of distributions into a reproducing kernel hilbert space detection of an anomalous interval over a line network is first studied sufficient conditions on minimum and maximum sizes of candidate anomalous intervals are characterized in order to guarantee that the proposed test is consistent it is also shown that certain necessary conditions must hold in order to guarantee that any test is universally consistent comparison of sufficient and necessary conditions yields that the proposed test is order level optimal and nearly optimal respectively in terms of minimum and maximum sizes of candidate anomalous intervals generalization of the results to other networks is further developed numerical results are provided to demonstrate the performance of the proposed tests

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

optimal and fast detection of spatial clusters with scan statistics we consider the detection of multivariate spatial clusters in the bernoulli model with n locations where the design distribution has weakly dependent marginals the locations are scanned with a rectangular window with sides parallel to the axes and with varying sizes and aspect ratios multivariate scan statistics pose a statistical problem due to the multiple testing over many scan windows as well as a computational problem because statistics have to be evaluated on many windows this paper introduces methodology that leads to both statistically optimal inference and computationally efficient algorithms the main difference to the traditional calibration of scan statistics is the concept of grouping scan windows according to their sizes and then applying different critical values to different groups it is shown that this calibration of the scan statistic results in optimal inference for spatial clusters on both small scales and on large scales as well as in the case where the cluster lives on one of the marginals methodology is introduced that allows for an efficient approximation of the set of all rectangles while still guaranteeing the statistical optimality results described above it is shown that the resulting scan statistic has a computational complexity that is almost linear in n

an elliptic spatial scan statistic the spatial scan statistic is commonly used for geographical disease cluster detection cluster evaluation and disease surveillance the most commonly used shape of the scanning window is circular in this paper we explore an elliptic version of the spatial scan statistic using a scanning window of variable location shape eccentricity angle and size and with and without an eccentricity penalty the method is applied to breast cancer mortality data from northeastern united states and female oral cancer mortality in the united states power comparisons are made with the circular scan statistic

on the decidability of membership in matrix exponential semigroups we consider the decidability of the membership problem for matrix exponential semigroups given k n and square matrices a1 ak c all of the same dimension and with real algebraic entries decide whether c is contained in the semigroup generated by the matrix exponentials exp ai t where i 1 k and t 0 this problem can be seen as a continuous analog of babai et al s and cai et al s problem of solving multiplicative matrix equations and has applications to reachability analysis of linear hybrid automata and switching systems our main results are that the semigroup membership problem is undecidable in general but decidable if we assume that a1 ak commute the decidability proof is by reduction to a version of integer programming that has transcendental constants we give a decision procedure for the latter using baker s theorem on linear forms in logarithms of algebraic numbers among other tools the undecidability result is shown by reduction from hilbert s tenth problem

the complexity of grigorchuk groups with application to cryptography abstract the turing complexity of the word problems of a class of groups introduced by grigorchuk 1985 is examined in particular it is shown that such problems of permutation groups of the infinite complete binary tree yield natural complete sets that separate time and space complexity classes if they are distinct a refinement of savitch s translation theorem as well as a similar result restricted for time complexity follow new families of nonfinitely presented groups are shown to have word problems uniformly solvable in simultaneous logspace and quadratic time a new family of public key cryptosystems based on these word problems is constructed