lagrangian hamiltonian unified formalism for field theory the rusk skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems it incorporates all the characteristics of lagrangian and hamiltonian descriptions of these systems including dynamical equations and solutions constraints legendre map evolution operators equivalence etc in this work we extend this unified framework to first order classical field theories and show how this description comprises the main features of the lagrangian and hamiltonian formalisms both for the regular and singular cases this formulation is a first step toward further applications in optimal control theory for partial differential equations

a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

geometric description of vakonomic and nonholonomic dynamics comparison of solutions we treat the vakonomic dynamics with general constraints within a new geometric framework which can be useful in the study of optimal control problems we compare our formulation with the one of vershik and gershkovich in the case of linear constraints we show how nonholonomic mechanics also admits a new geometrical description which allows us to develop an algorithm of comparison between the solutions of both dynamics examples illustrating the theory are treated

phase transitions for high dimensional clustering and related problems consider a two class clustering problem where we observe x i ell i mu z i z i stackrel mathit i i d sim n 0 i p 1 leq i leq n the feature vector mu in r p is unknown but is presumably sparse the class labels ell i in 1 1 are also unknown and the main interest is to estimate them we are interested in the statistical limits in the two dimensional phase space calibrating the rarity and strengths of useful features we find the precise demarcation for the region of impossibility and region of possibility in the former useful features are too rare weak for successful clustering in the latter useful features are strong enough to allow successful clustering the results are extended to the case of colored noise using le cam s idea on comparison of experiments we also extend the study on statistical limits for clustering to that for signal recovery and that for global testing we compare the statistical limits for three problems and expose some interesting insight we propose classical pca and important features pca if pca for clustering for a threshold t 0 if pca clusters by applying classical pca to all columns of x with an l 2 norm larger than t we also propose two aggregation methods for any parameter in the region of possibility some of these methods yield successful clustering we discover a phase transition for if pca for any threshold t 0 let xi t be the first left singular vector of the post selection data matrix the phase space partitions into two different regions in one region there is a t such that cos xi t ell rightarrow 1 and if pca yields successful clustering in the other cos xi t ell leq c 0 0 our results require delicate analysis especially on post selection random matrix theory and on lower bound arguments

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

metropolis monte carlo simulation of lattice animals we propose and study a metropolis monte carlo algorithm for the simulation of weighted lattice animals in in the canonical ensemble with a fixed number of edges we examine the implementation and performance of the algorithm data obtained by sampling uniformly weighted animals and animals weighted as critical percolation clusters are reported and analysed in particular we estimate autocorrelation times and dynamical exponents to study the efficiency of the algorithm and estimate the metric and branch exponents of lattice animals in two and three dimensions

on the upper critical dimension of lattice trees and lattice animals we give a rigorous proof of mean field critical behavior for the susceptibility 1 2 and the correlation length v 1 4 for models of lattice trees and lattice animals in two cases i for the usual model with trees or animals constructed from nearest neighbor bonds in sufficiently high dimensions and ii for a class of spread out or long range models in which trees and animals are constructed from bonds of various lengths above eight dimensions this provides further evidence that for these models the upper critical dimension is equal to eight the proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point and uses an expansion related to the lace expansion for the self avoiding walk

valences of sites in bond percolation presents calculations for the fraction of sites with given valence in finite and infinite clusters in the bethe approximation for a lattice of coordination number q and compares these results with series analysis results for the bond percolation problems on the square and simple cubic lattices

statistics of lattice animals the authors investigate the large n behaviour of the number of lattice animals with n vertices having alpha cycles per vertex they prove concavity and continuity properties for the corresponding growth constant and in particular show that lattice trees are exponentially scarce in the set of lattice animals they also consider the corresponding generating function and prove a number of theorems which bound it and set other limits on its possible behaviour

series estimate of the exponent β for bond percolation on a simple cubic lattice the critical exponent beta for the bond problem on a simple cubic lattice is analysed by a pade approximant technique using high density expansions corresponding to three different definitions of the percolation probability it is found that beta 0 463 or 0 013 12 delta pc where delta pc pc 0 247 the need for more precise estimates of the critical density pc is emphasised

self avoiding walk in 5 or more dimensions using an expansion based on the renormalization group philosophy we prove that for a t step weakly self avoiding random walk in five or more dimensions the variance of the endpoint is of order t and the scaling limit is gaussian as t

generalized hamiltonian dynamics i formulation on t q tq the dirac bergmann generalized hamiltonian dynamics for a degenerate lagrangian system is formulated on the whitney sum t q tq of the phase space t q and the velocity space tq over the configuration space q the formulation is related to those on t q and tq some ambiguities concerning generalized dynamics that have appeared in the literature are clarified

unified formalism for nonautonomous mechanical systems we present a unified geometric framework for describing both the lagrangian and hamiltonian formalisms of regular and non regular time dependent mechanical systems which is based on the approach of skinner and rusk 18 the dynamical equations of motion and their compatibility and consistency are carefully studied making clear that all the characteristics of the lagrangian and the hamiltonian formalisms are recovered in this formulation as an example it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism

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

series study of random percolation in three dimensions new high density series data for the mean number percolation probability and susceptibility of finite clusters are presented for bond and site percolation on four standard three dimensional lattices a pade approximant analysis of both high and low density series makes particular use of the rather precise unbiased estimates of the percolation threshold pc obtained recently by heermann and stauffer 1981 for the simple cubic lattice for this lattice the authors obtain the biased estimate gamma 1 73 or 0 03 for the site problem and a similar estimate but with larger uncertainties for the bond problem such a value is significantly larger than earlier series estimates assuming gamma to be universal they obtain precise although biased estimates of pc for both bond and site percolation on all four lattices using the bond estimates of pc they find an overall biased estimate of beta 0 454 or 0 008 for bond percolation on all three dimensional lattices the corresponding site problem requires further study scaling estimates of other critical exponents are alpha 0 64 or 0 05 delta 4 81 or 0 14 delta 2 18 or 0 04 nu 0 88 or 0 02 and nu 0 03 or 0 03

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

a new simulation of branched polymers the scanning method for the simulation of linear chain is extended to general models of branched polymers without loops a branched chain grows in time namely a number of steps from the origin therefore i in contrast to other simulation techniques which are of a relaxation type the chains are statistically independent and the statistical error can reliably be estimated ii the probability of a chain is known and hence the entropy and iii the scanning construction enables one to study geometrical properties which depend on time for self avoiding trees on a square lattice the author obtains the relatively accurate estimates for the static critical exponents nu 0 640 or 0 004 and theta 1 003 or 0 02 and for the connective constant mu 5 1419 or 0 003 the author also obtains critical exponents gamma t approximately 1 26 and nu t approximately 0 83 which characterise the growth in time of the number of bonds and the gyration radius respectively application of the scanning method to more complex branched polymers is discussed

approximate momentum conservation for spatial semidiscretizations of semilinear wave equations we prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions analytic nonlinearity and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic the method of proof is that of backward error analysis whereby we construct a modified equation which is itself lagrangian and translation invariant and therefore also conserves momentum this modified equation interpolates the semidiscrete system for all time and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system these properties directly imply approximate momentum conservation for the semidiscrete system we also consider discretizations that are not variational as well as discretizations on non uniform grids through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

universality in branched polymers on d dimensional hypercubic lattices the authors have derived series for weakly and strongly embeddable trees in d dimensional simple hypercubic lattices for arbitrary integral d for d 2 3 9 they present series evidence that such trees are in the same universality class as lattice animals in addition they have derived expansions in inverse powers of sigma 2d 1 for the growth parameters for bond and site trees and compare these with the corresponding results for animals

gunther s formalism k symplectic formalism in classical field theory skinner rusk approach and the evolution operator the first aim of this paper is to extend the skinner rusk formalism on classical mechanics for first order field theories the second is to generalize the definition and properties of the evolution k operator on classical mechanics for first order field theories using in both cases gunther s formalism k symplectic formalism

some nonasymptotic results on resampling in high dimension i confidence regions we study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure the random vector is supposed to be either gaussian or to have a symmetric and bounded distribution the dimensionality of the vector can possibly be much larger than the number of observations and we focus on a non asymptotic control of the confidence level following ideas inspired by recent results in learning theory we consider two approaches the first based on a concentration principle valid for a large class of resampling weights and the second on a direct resampled quantile specifically using rademacher weights several intermediate results established in the approach based on concentration principles are of self interest we also discuss the question of accuracy when using monte carlo approximations of the resampled quantities we present an application of these results to the one sided and two sided multiple testing problem in which we derive several resampling based step down procedures providing a non asymptotic fwer control we compare our different procedures in a simulation study and we show that they can outperform bonferroni s or holm s procedures as soon as the observed vector has sufficiently correlated coordinates

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

on the limitation of spectral methods from the gaussian hidden clique problem to rank one perturbations of gaussian tensors we consider the following detection problem given a realization of a symmetric matrix x of dimension n distinguish between the hypothesis that all upper triangular variables are independent and identically distributed i i d gaussians variables with mean 0 and variance 1 and the hypothesis where x is the sum of such matrix and an independent rank one perturbation this setup applies to the situation where under the alternative there is a planted principal submatrix b of size l for which all upper triangular variables are i i d gaussians with mean 1 and variance 1 whereas all other upper triangular elements of x not in b are i i d gaussians variables with mean 0 and variance 1 we refer to this as the gaussian hidden clique problem when l 1 epsilon sqrt n epsilon 0 it is possible to solve this detection problem with probability 1 o n 1 by computing the spectrum of x and considering the largest eigenvalue of x we prove that this condition is tight in the following sense when l no algorithm that examines only the eigenvalues of x can detect the existence of a hidden gaussian clique with error probability vanishing as n to infty we prove this result as an immediate consequence of a more general result on rank one perturbations of k dimensional gaussian tensors in this context we establish a lower bound on the critical signal to noise ratio below which a rank one signal cannot be detected

feeling the bern adaptive estimators for bernoulli probabilities of pairwise comparisons we study methods for aggregating pairwise comparison data among a collection of n items with the goal of estimating the outcome probabilities for future comparisons working within a flexible model that only imposes a form of strong stochastic transitivity we introduce an adaptivity index which compares the risk of our estimator to that of an oracle over appropriate sub models where the oracle knows the specific sub model in the ground truth in addition to measuring the usual worst case risk of an estimator this adaptivity index also captures the extent to which the estimator adapts to instance specific difficulty relative to an oracle estimator first we propose a three step estimator termed count randomize least squares and show that it has adaptivity index upper bounded by sqrt n up to logarithmic factors we then show that conditional on the planted clique hypothesis no computationally efficient estimator can achieve an adaptivity index smaller than sqrt n second we show that a regularized least squares estimator can achieve a poly logarithmic adaptivity index thereby demonstrating a sqrt n gap between optimal and computationally achievable adaptivity finally we prove that the standard least squares estimator which is known to be optimally adaptive in several closely related problems fails to adapt in the context of estimating pairwise probabilities

a tree based scaling exponent for random cluster models for models of random trees in the hypercubic lattice or more general random clusters to which tree structure may be added the authors introduce a novel scaling exponent for the particular model of uniform random spanning trees in two and three dimensions its value is estimated by monte carlo simulation

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

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

hamiltonian dynamics on the symplectic extended phase space for autonomous and non autonomous systems we will present a consistent description of hamiltonian dynamics on the symplectic extended phase space that is analogous to that of a time independent hamiltonian system on the conventional symplectic phase space the extended hamiltonian h1 and the pertaining extended symplectic structure that establish the proper canonical extension of a conventional hamiltonian h will be derived from a generalized formulation of hamilton s variational principle the extended canonical transformation theory then naturally permits transformations that also map the time scales of the original and destination system while preserving the extended hamiltonian h1 and hence the form of the canonical equations derived from h1 the lorentz transformation as well as time scaling transformations in celestial mechanics will be shown to represent particular canonical transformations in the symplectic extended phase space furthermore the generalized canonical transformation approach allows us to directly map explicitly time dependent hamiltonians into time independent ones an extended generating function that defines transformations of this kind will be presented for the time dependent damped harmonic oscillator and for a general class of explicitly time dependent potentials in the appendix we will re establish the proper form of the extended hamiltonian h1 by means of a legendre transformation of the extended lagrangian l1

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

detection of correlations with adaptive sensing the problem of detecting correlations from samples of a high dimensional gaussian vector has recently received a lot of attention in most existing work detection procedures are provided with a full sample however following common wisdom in experimental design the experimenter may have the capacity to make targeted measurements in an on line and adaptive manner in this paper we investigate such adaptive sensing procedures for detecting positive correlations it is shown that using the same number of measurements adaptive procedures are able to detect significantly weaker correlations than their nonadaptive counterparts we also establish minimax lower bounds that show the limitations of any procedure keywords sequential testing adaptive sensing high dimensional detection highdimensional detection sequential testing sparse covariance matrices sparse principal component analysis

extended legendre transformation approach to the time dependent hamiltonian formalism the hamiltonian formalism for time dependent systems is developed using vector fields defined in the extended cotangent bundle t q r the extended hamiltonian h is obtained using the extended legendre transformation associated to an extended lagrangian l previously defined in t q r and then the techniques of symplectic mechanics are employed in t q r to study the properties of the extended hamiltonian system finally this approach is related with the multisymplectic formalism and other extended approaches

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

some nonasymptotic results on resampling in high dimension ii multiple tests in the context of correlated multiple tests we aim to nonasymptotically control the family wise error rate fwer using resampling type procedures we observe repeated realizations of a gaussian random vector in possibly high dimension and with an unknown covariance matrix and consider the one and two sided multiple testing problem for the mean values of its coordinates we address this problem by using the confidence regions developed in the companion paper ann statist 2009 to appear which lead directly to single step procedures these can then be improved using step down algorithms following an established general methodology laid down by romano and wolf j amer statist assoc 100 2005 94 108 this gives rise to several different procedures whose performances are compared using simulated data

computational barriers in minimax submatrix detection this paper studies the minimax detection of a small submatrix of elevated mean in a large matrix contaminated by additive gaussian noise to investigate the tradeoff between statistical performance and computational cost from a complexity theoretic perspective we consider a sequence of discretized models which are asymptotically equivalent to the gaussian model under the hypothesis that the planted clique detection problem cannot be solved in randomized polynomial time when the clique size is of smaller order than the square root of the graph size the following phase transition phenomenon is established when the size of the large matrix p to infty if the submatrix size k theta p alpha for any alpha in 0 2 3 computational complexity constraints can incur a severe penalty on the statistical performance in the sense that any randomized polynomial time test is minimax suboptimal by a polynomial factor in p if k theta p alpha for any alpha in 2 3 1 minimax optimal detection can be attained within constant factors in linear time using schatten norm loss as a representative example we show that the hardness of attaining the minimax estimation rate can crucially depend on the loss function implications on the hardness of support recovery are also obtained

on the necessity of irrelevant variables this work explores the effects of relevant and irrelevant boolean variables on the accuracy of classifiers the analysis uses the assumption that the variables are conditionally independent given the class and focuses on a natural family of learning algorithms for such sources when the relevant variables have a small advantage over random guessing the main result is that algorithms relying predominately on irrelevant variables have error probabilities that quickly go to 0 in situations where algorithms that limit the use of irrelevant variables have errors bounded below by a positive constant we also show that accurate learning is possible even when there are so few examples that one cannot determine with high confidence whether or not any individual variable is relevant

skinner rusk unified formalism for optimal control systems and applications a geometric approach to time dependent optimal control problems is proposed this formulation is based on the skinner and rusk formalism for lagrangian and hamiltonian systems the corresponding unified formalism developed for optimal control systems allows us to formulate geometrically the necessary conditions given by a weak form of pontryagin s maximum principle provided that the differentiability with respect to controls is assumed and the space of controls is open furthermore our method is also valid for implicit optimal control systems and in particular for the so called descriptor systems optimal control problems including both differential and algebraic equations