information theoretic determination of minimax rates of convergence we present some general results determining minimax bounds on statistical risk for density estimation based on certain information theoretic considerations these bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence

influential features pca for high dimensional clustering we consider a clustering problem where we observe feature vectors x i in r p i 1 2 ldots n from k possible classes the class labels are unknown and the main interest is to estimate them we are primarily interested in the modern regime of p gg n where classical clustering methods face challenges we propose influential features pca if pca as a new clustering procedure in if pca we select a small fraction of features with the largest kolmogorov smirnov ks scores obtain the first k 1 left singular vectors of the post selection normalized data matrix and then estimate the labels by applying the classical k means procedure to these singular vectors in this procedure the only tuning parameter is the threshold in the feature selection step we set the threshold in a data driven fashion by adapting the recent notion of higher criticism as a result if pca is a tuning free clustering method we apply if pca to 10 gene microarray data sets the method has competitive performance in clustering especially in three of the data sets the error rates of if pca are only 29 or less of the error rates by other methods we have also rediscovered a phenomenon on empirical null by efron j amer statist assoc 99 2004 96 104 on microarray data with delicate analysis especially post selection eigen analysis we derive tight probability bounds on the kolmogorov smirnov statistics and show that if pca yields clustering consistency in a broad context the clustering problem is connected to the problems of sparse pca and low rank matrix recovery but it is different in important ways we reveal an interesting phase transition phenomenon associated with these problems and identify the range of interest for each

a transform theory for a class of group invariant codes a binary cyclic code of length of n can be defined by a set of parity check equations that is invariant under the additive cyclic group generated by a i to i 1 modulo n a class of quasicyclic codes that have parity check equations invariant under the group generated by a subgroup of a and a subgroup of the multiplicative group m i to 2 sup k i modulo n is studied in detail the fourier transform transforms the action of the additive group and a linearized polynomial transform transforms the action of the multiplicative group the form of the transformed code equations permits a bch like bound on the minimum distance of the code the techniques are illustrated by the construction of several codes that are decodable by the berlekamp massey algorithm and that have parameters that approach or surpass those of the best codes known

numerical simulation of 3d viscoelastic flows with free surfaces a numerical model is presented for the simulation of viscoelastic flows with complex free surfaces in three space dimensions the mathematical formulation of the model is similar to that of the volume of fluid vof method but the numerical procedures are different a splitting method is used for the time discretization the prediction step consists in solving three advection problems one for the volume fraction of liquid which allows the new liquid domain to be obtained one for the velocity field one for the extra stress the correction step corresponds to solving an oldroyd b fluid flow problem without advection in the new liquid domain two different grids are used for the space discretization the three advection problems are solved on a fixed structured grid made out of small cubic cells using a forward characteristics method the oldroyd b problem without advection is solved using continuous piecewise linear stabilized finite elements on a fixed unstructured mesh of tetrahedrons efficient post processing algorithms enhance the quality of the numerical solution a hierarchical data structure reduces the memory requirements convergence of the numerical method is checked for the pure extensional flow and the filling of a tube numerical results are presented for the stretching of a filament fingering instabilities are obtained when the aspect ratio is large also results pertaining to jet buckling are reported

constacyclic codes cocycles and a u v u v construction a connection between cohomology cocycles and constacyclic codes is explored it suggests an isomorphism between cyclic codes of length mn and a direct sum of m constacyclic codes of length n the isomorphism is used i to study the discrete fourier transforms and the decomposition of group ring codes ii to give a u v u v construction over gf q when q is odd the u v u v construction gives some of the best ternary cyclic codes and classes of nearly mds cyclic codes of length 2q 2 the symmetry of the construction is used to give a new proof that there are no cyclic self dual codes over gf q when q is odd

computations of multi fluid flows abstract full numerical simulations of three dimensional flows of two or more immiscible fluids of different densities and viscosities separated by a sharp interface with finite surface tension are discussed the method used is based on a finite difference approximation of the full navier stokes equations and explicit tracking of the interface between the fluids preliminary simulations of the rayleigh taylor instability and the motion of bubbles are shown

on consistency and sparsity for principal components analysis in high dimensions principal components analysis pca is a classic method for the reduction of dimensionality of data in the form of n observations or cases of a vector with p variables contemporary datasets often have p comparable with or even much larger than n our main assertions in such settings are a that some initial reduction in dimensionality is desirable before applying any pca type search for principal modes and b the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation we describe a simple asymptotic model in which the estimate of the leading principal component vector via standard pca is consistent if and only if p n n 0 we provide a simple algorithm for selecting a subset of coordinates with largest sample variances and show that if pca is done on the selected subset then consistency is recovered even if p n n

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

transform domain characterization of abelian codes abelian codes constitute a class of codes that includes cyclic codes as a special case it is shown that the general class of abelian codes can be characterized in the transform domain using the discrete fourier transform dft over finite fields with the appropriate mixed radix number system as the indexing scheme for dft coefficients a simple transform domain description for dual codes of abelian codes is also obtained using this description the idempotent generator of the dual of a given abelian code can be easily obtained finally it is shown that in the case of cyclic codes which can be considered as abelian codes also one can work in smaller extension fields compared to the extension fields if they were considered as cyclic codes only

a class of constacyclic codes a class of q sup m ary constacyclic codes is defined which has the property that the q ary image is equivalent to a class of shortened cyclic codes this description leads to the construction of nonlinear codes as an example one can construct nonlinear codes of length n q sup m 1 over the field gf q sup m m spl les m with a minimum distance greater than n 1 r m m where r is the rate for higher rates these codes are the best known the codes are easily encoded and decoded

classification of pseudo cyclic mds codes linear codes are considered a code is characterized by the length n the dimension k and the minimum distance d an n k d code over the finite field gf q is said to be maximum distance separable mds if d n k 1 a krishna and d v sarwate 1990 investigated the existence of pseudo cyclic mds codes over gf q of length n where n divides q 1 or q 1 it is shown that the pseudo cyclic mds codes constructed by krishna and sarwate are generalised reed solomon codes pseudo cyclic codes are studied over gf q where n and q p sup h are not relatively prime it is proven that pseudo cyclic n k mds codes modulo x sup n a over gf q exist if and only if n p furthermore any pseudo cyclic p k code modulo x sup p a over gf q turns out to be mds and generalized reed solomon it is explicitly proven that some classes of pseudo cyclic n k mds codes over gf q are generalized reed solomon codes furthermore pseudo cyclic q 1 4 mds codes over gf q q even are completely classified

decompositions and extremal type ii codes over z sub 4 in previous work by huffman and by yorgov 1983 a decomposition theory of self dual linear codes c over a finite field f sub q was given when c has a permutation automorphism of prime order r relatively prime to q we extend these results to linear codes over the galois ring z sub 4 and apply the theory to z sub 4 codes of length 24 in particular we obtain 42 inequivalent 24 12 z sub 4 codes of minimum euclidean weight 16 which lead to 42 constructions of the leech lattice

optshrink an algorithm for improved low rank signal matrix denoising by optimal data driven singular value shrinkage the truncated singular value decomposition of the measurement matrix is the optimal solution to the representation problem of how to best approximate a noisy measurement matrix using a low rank matrix here we consider the unobservable denoising problem of how to best approximate a low rank signal matrix buried in noise by optimal re weighting of the singular vectors of the measurement matrix we exploit recent results from random matrix theory to exactly characterize the large matrix limit of the optimal weighting coefficients and show that they can be computed directly from data for a large class of noise models that includes the independent identically distributed gaussian noise case our analysis brings into sharp focus the shrinkage and thresholding form of the optimal weights the nonconvex nature of the associated shrinkage function on the singular values and explains why matrix regularization via singular value thresholding with convex penalty functions such as the nuclear norm will always be suboptimal we validate our theoretical predictions with numerical simulations develop an implementable algorithm optshrink that realizes the predicted performance gains and show how our methods can be used to improve estimation in the setting where the measured matrix has missing entries

an ellam scheme for advection diffusion equations in two dimensions we develop an eulerian lagrangian localized adjoint method ellam to solve two dimensional advection diffusion equations with all combinations of inflow and outflow dirichlet neumann and flux boundary conditions the ellam formalism provides a systematic framework for implementation of general boundary conditions leading to mass conservative numerical schemes the computational advantages of the ellam approximation have been demonstrated for a number of one dimensional transport systems practical implementations of ellam schemes in multiple spatial dimensions that require careful algorithm development are discussed in detail in this paper extensive numerical results are presented to compare the ellam scheme with many widely used numerical methods and to demonstrate the strength of the ellam scheme

computing interface motion in compressible gas dynamics a hamilton jacobi level set formulation of the equations of motion for propagating interfaces has been introduced recently by osher and sethian this formulation allows fronts to self intersect develop singularities and change topology the numerical algorithms based on this approach handle topological merging and breaking naturally work in any number of space dimensions and do not require that the moving front be written as a function instead the moving front is embedded as a particular level set in a fixed domain partial differential equation numerical techniques borrowed from hyperbolic conservation laws are then used to accurately capture the complicated surface motion that satisfies the global entropy condition for propagating fronts given by sethian in this paper we analyze the coupling of this level set formulation to a system of conservation laws for compressible gas dynamics we study both conservative and non conservative differencing of the level set function and compare the two approaches to illustrate the capability of the method we study the compressible rayleigh taylor and kelvin helmholtz instabilities for air air and air helium boundaries we perform numerical convergence studies of the method over a range of parameters and analyze the accuracy of this approach applied to these problems

p adic estimates of hamming weights in abelian codes over galois rings a generalization of mceliece s theorem on the p adic valuation of hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by wilson along with a technique known as trace averaging introduced here the original theorem of mceliece concerned cyclic codes over prime fields delsarte and mceliece later extended this to abelian codes over finite fields calderbank li and poonen extended mceliece s original theorem to cover cyclic codes over the rings zopf2 d wilson strengthened their results and extended them to cyclic codes over zopf p d and katz strengthened wilson s results and extended them to abelian codes over zopfp d it is natural to ask whether there is a single analogue of mceliece s theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers in this paper this question is answered affirmatively a single theorem for abelian codes over galois rings is presented this theorem contains all previously mentioned results and more

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

a transform approach to permutation groups of cyclic codes over galois rings berger and charpin see ibid vol 42 p 2194 2209 1996 and des codes cuyptogr vol 18 no 1 3 p 29 53 1999 devised a theoretical method of calculating the permutation group of a primitive cyclic code over a finite field using permutation polynomials and a transform description of such codes we extend this method to cyclic and extended cyclic codes over the galois ring gr p sup a m developing a generalization of the mattson solomon polynomial in particular we classify all affine invariant codes of length 2 sup m over z sub 4 thus generalizing the corresponding result of kasami lin and peterson 1967 and giving an alternative proof to abdukhalikov we give a large class of codes over z sub 4 with large permutation groups which include generalizations of bose chaudhuri hocquenghem bch and reed muller rm codes

large covariance estimation through elliptical factor models we propose a general principal orthogonal complement thresholding poet framework for large scale covariance matrix estimation based on the approximate factor model a set of high level sufficient conditions for the procedure to achieve optimal rates of convergence under different matrix norms is established to better understand how poet works such a framework allows us to recover existing results for sub gaussian data in a more transparent way that only depends on the concentration properties of the sample covariance matrix as a new theoretical contribution for the first time such a framework allows us to exploit conditional sparsity covariance structure for the heavy tailed data in particular for the elliptical distribution we propose a robust estimator based on the marginal and spatial kendall s tau to satisfy these conditions in addition we study conditional graphical model under the same framework the technical tools developed in this paper are of general interest to high dimensional principal component analysis thorough numerical results are also provided to back up the developed theory

robust covariance estimation for approximate factor models in this paper we study robust covariance estimation under the approximate factor model with observed factors we propose a novel framework to first estimate the initial joint covariance matrix of the observed data and the factors and then use it to recover the covariance matrix of the observed data we prove that once the initial matrix estimator is good enough to maintain the element wise optimal rate the whole procedure will generate an estimated covariance with desired properties for data with only bounded fourth moment we propose to use adaptive huber loss minimization to give the initial joint covariance estimation this approach is applicable to a much wider class of distributions beyond sub gaussian and elliptical distributions we also present an asymptotic result for adaptive huber s m estimator with a diverging parameter the conclusions are demonstrated by extensive simulations and real data analysis

tensor svd statistical and computational limits in this paper we propose a general framework for tensor singular value decomposition tensor singular value decomposition svd which focuses on the methodology and theory for extracting the hidden low rank structure from high dimensional tensor data comprehensive results are developed on both the statistical and computational limits for tensor svd this problem exhibits three different phases according to the signal to noise ratio snr in particular with strong snr we show that the classical higher order orthogonal iteration achieves the minimax optimal rate of convergence in estimation with weak snr the information theoretical lower bound implies that it is impossible to have consistent estimation in general with moderate snr we show that the non convex maximum likelihood estimation provides optimal solution but with np hard computational cost moreover under the hardness hypothesis of hypergraphic planted clique detection there are no polynomial time algorithms performing consistently in general

consta abelian codes over galois rings we study n length consta abelian codes a generalization of the well known abelian codes and constacyclic codes over galois rings of characteristic p sup a where n and p are coprime a twisted discrete fourier transform dft is used to generalize transform domain results of abelian and constacyclic codes to consta abelian codes further we characterize consta abelian codes invariant under two kinds of monomials whose underlying permutations are effected by i multiplying the coordinates with a unit in the appropriate mixed radix representation of the coordinate positions and ii shifting the coordinates by t positions all the codes studied here belong to the class of quasi twisted codes which are known to contain some good codes we show that the dual of a consta abelian code invariant under the two monomials is also a consta abelian code closed under both monomials

sparse pca optimal rates and adaptive estimation principal component analysis pca is one of the most commonly used statistical procedures with a wide range of applications this paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting under mild technical conditions we first establish the optimal rates of convergence for estimating the principal subspace which are sharp with respect to all the parameters thus providing a complete characterization of the difficulty of the estimation problem in term of the convergence rate the lower bound is obtained by calculating the local metric entropy and an application of fano s lemma the rate optimal estimator is constructed using aggregation which however might not be computationally feasible we then introduce an adaptive procedure for estimating the principal subspace which is fully data driven and can be computed efficiently it is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces a key idea in our construction is a reduction scheme which reduces the sparse pca problem to a high dimensional multivariate regression problem this method is potentially also useful for other related problems

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

discrete equations for physical and numerical compressible multiphase mixtures we have recently proposed in 21 a compressible two phase unconditionally hyperbolic model able to deal with a wide range of applications interfaces between compressible materials shock waves in condensed multiphase mixtures homogeneous two phase flows bubbly and droplet flows and cavitation in liquids one of the difficulties of the model as always in this type of physical problems was the occurrence of non conservative products in 21 we have proposed a discretisation technique that was without any ambiguity only in the case of material interfaces not in the case of shock waves this model was extended to several space dimensions in 24 in this paper thanks to a deeper analysis of the model we propose a class of schemes that are able to converge to the correct solution even when shock waves interact with volume fraction discontinuities this analysis provides a more accurate estimate of closure terms but also an accurate resolution method for the conservative fluxes as well as non conservative terms even for situations involving discontinuous solutions the accuracy of the model and method is clearly demonstrated on a sequence of difficult test problems

reconstructing volume tracking a new algorithm for the volume tracking of interfaces in two dimensions is presented the algorithm is based upon a well defined second order geometric solution of a volume evolution equation the method utilizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology a linearity preserving piecewise linear interface geometry approximation ensures that solutions generated retain second order spatial accuracy second order temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme we detail our geometrically based solution method in which material volume fluxes are computed systematically with a set of simple geometric tasks we then interrogate the method by testing its ability to track interfaces through large controlled topology changes whereby an initially simple interface configuration is subjected to vortical flows numerical results for these strenuous test problems provide evidence for the algorithm s improved solution quality and accuracy

a sharp interface method for incompressible two phase flows we present a sharp interface method for computing incompressible immiscible two phase flows it couples the level set and volume of fluid techniques and retains their advantages while overcoming their weaknesses it is stable and robust even for large density and viscosity ratios on the order of 1000 to 1 the numerical method is an extension of the second order method presented by sussman m sussman a second order coupled levelset and volume of fluid method for computing growth and collapse of vapor bubbles journal of computational physics 187 2003 110 136 in which the previous method treated the gas pressure as spatially constant and the present method treats the gas as a second incompressible fluid the new method yields solutions in the zero gas density limit which are comparable in accuracy to the method in which the gas pressure was treated as spatially constant this improvement in accuracy allows one to compute accurate solutions on relatively coarse grids thereby providing a speed up over continuum or ghost fluid methods

a generalized dft for abelian codes over z sub m a generalized discrete fourier transform defined over an appropriate extension ring is given that is suitable to characterize abelian codes over residue class integer rings z sub m the characterization is in terms of generalized discrete fourier transform components taking values from certain ideals of the extension ring it is shown that the results known for cyclic codes over z sub m like the simple characterization of dual and self dual codes and the nonexistence of self dual codes for certain values of code parameters extend to abelian codes over z sub m as well

abelian codes over galois rings closed under certain permutations we study n length abelian codes over galois rings with characteristic p sup a where n and p are relatively prime having the additional structure of being closed under the following two permutations i permutation effected by multiplying the coordinates with a unit in the appropriate mixed radix representation of the coordinate positions and ii shifting the coordinates by t positions a code is t quasi cyclic t qc if t is an integer such that cyclic shift of a codeword by t positions gives another codeword we call the abelian codes closed under the first permutation as unit invariant abelian codes and those closed under the second as quasi cyclic abelian qca codes using a generalized discrete fourier transform gdft defined over an appropriate extension of the galois ring we show that unit invariant abelian and qca codes can be easily characterized in the transform domain for t 1 qca codes coincide with those that are cyclic as well as abelian the number of such codes for a specified size and length is obtained and we also show that the dual of an unit invariant t qca code is also an unit invariant t qca code unit invariant abelian hence unit invariant cyclic and t qca codes over galois field f sub p sup l and over the integer residue rings are obtainable as special cases

hamming metric decoding of alternant codes over galois rings the standard decoding procedure for alternant codes over fields centers on solving a key equation which relates an error locator polynomial and an error evaluator polynomial by a syndrome sequence we extend this technique to decode alternant codes over galois rings we consider the module m a b as spl equiv b mod x sup r of all solutions to the key equation where s is the syndrome polynomial and r is the number of rows in a parity check matrix for the code in decoding we seek a particular solution spl sigma spl omega spl isin m which we prove can be found in a grobner basis for m we present an iterative algorithm which generates a grobner basis modulo x sup k 1 from a given basis modulo x sup k at the rth step a grobner basis for m is found and the required solution recovered

new quasi twisted quaternary linear codes let n k d sub q codes be linear codes of length n dimension k and minimum hamming distance d over gf q in this article 31 new quaternary q 4 codes are constructed which improve the respective lower bounds on minimum distance in brouwer s tables

estimation of functionals of sparse covariance matrices high dimensional statistical tests often ignore correlations to gain simplicity and stability leading to null distributions that depend on functionals of correlation matrices such as their frobenius norm and other l r norms motivated by the computation of critical values of such tests we investigate the difficulty of estimation the functionals of sparse correlation matrices specifically we show that simple plug in procedures based on thresholded estimators of correlation matrices are sparsity adaptive and minimax optimal over a large class of correlation matrices akin to previous results on functional estimation the minimax rates exhibit an elbow phenomenon our results are further illustrated in simulated data as well as an empirical study of data arising in financial econometrics

volume of fluid interface tracking with smoothed surface stress methods for three dimensional flows motivated by the need for three dimensional methods for interface calculations that can deal with topology changes we describe a numerical scheme built from a volume of fluid interface tracking technique that uses a piecewise linear interface calculation in each cell momentum balance is computed using explicit finite volume finite differences on a regular cubic grid surface tension is implemented by the continuous surface stress or continuous surface force method examples and verifications of the method are given by comparing simulations to analytical results and experiments for sedimenting droplet arrays and capillary waves at finite reynolds number in the case of a pinching pendant drop both three dimensional and axisymmetric simulations are compared to experiments agreement is found both before and after the reconnections

a mixed markers and volume of fluid method for the reconstruction and advection of interfaces in two phase and free boundary flows in this work we present a new mixed markers and volume of fluid vof algorithm for the reconstruction and advection of interfaces in the two dimensional space the interface is described by using both the volume fraction function c as in vof methods and surface markers which locate the interface within the computational cells the c field and the markers are advected by following the streamlines new markers are determined by computing the intersections of the advected interface with the grid lines then other markers are added inside each cut cell to conserve the volume fraction c a smooth motion of the interface is obtained typical of the marker approach with a good volume conservation as in standard vof methods in this article we consider a few typical two dimensional tests and compare the results of the mixed algorithm with those obtained with vof methods translations rotations and vortex tests are performed showing that many problems of the vof technique can be solved and a good accuracy in the geometrical motion and mass conservation can be achieved

numerical study of flows of two immiscible liquids at low reynolds number we review our recent results on fingering bamboo waves and drop breakup in low reynolds number flows composed of two viscous liquids under shear

adaptive thresholding for sparse covariance matrix estimation in this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries the estimators are fully data driven and demonstrate excellent performance both theoretically and numerically it is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm in contrast the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces support recovery is discussed as well the adaptive thresholding estimators are easy to implement the numerical performance of the estimators is studied using both simulated and real data simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators the method is also illustrated in an analysis on a dataset from a small round blue cell tumor microarray experiment a supplement to this ar

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

minimax bounds for sparse pca with noisy high dimensional data we study the problem of estimating the leading eigenvectors of a high dimensional population covariance matrix based on independent gaussian observations we establish a lower bound on the minimax risk of estimators under the l2 loss in the joint limit as dimension and sample size increase to infinity under various models of sparsity for the population eigenvectors the lower bound on the risk points to the existence of different regimes of sparsity of the eigenvectors we also propose a new method for estimating the eigenvectors by a two stage coordinate selection scheme

statistical and computational trade offs in estimation of sparse principal components in recent years sparse principal component analysis has emerged as an extremely popular dimension reduction technique for high dimensional data the theoretical challenge in the simplest case is to estimate the leading eigenvector of a population covariance matrix under the assumption that this eigenvector is sparse an impressive range of estimators have been proposed some of these are fast to compute while others are known to achieve the minimax optimal rate over certain gaussian or sub gaussian classes in this paper we show that under a widely believed assumption from computational complexity theory there is a fundamental trade off between statistical and computational performance in this problem more precisely working with new larger classes satisfying a restricted covariance concentration condition we show that there is an effective sample size regime in which no randomised polynomial time algorithm can achieve the minimax optimal rate we also study the theoretical performance of a polynomial time variant of the well known semidefinite relaxation estimator revealing a subtle interplay between statistical and computational efficiency

the spectral norm of random inner product kernel matrices we study an inner product kernel random matrix model whose empirical spectral distribution was shown by xiuyuan cheng and amit singer to converge to a deterministic measure in the large n and p limit we provide an interpretation of this limit measure as the additive free convolution of a semicircle law and a marchenko pastur law by comparing the tracial moments of this matrix to an additive deformation of a wigner matrix we establish that for odd kernel functions the spectral norm of this matrix converges almost surely to the edge of the limiting spectrum our study is motivated by the analysis of a covariance thresholding procedure for the statistical detection and estimation of sparse principal components and our results characterize the limit of the largest eigenvalue of the thresholded sample covariance matrix in the null setting

optimal rates of convergence for covariance matrix estimation covariance matrix plays a central role in multivariate statistical analysis significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices however a minimax theory has yet been developed in this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and frobenius norm it is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation the minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties a key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound the technical analysis requires new ideas that are quite different from those used in the more conventional function sequence estimation problems

a numerical method for two phase flow consisting of separate compressible and incompressible regions abstract we propose a numerical method for modeling two phase flow consisting of separate compressible and incompressible regions this is of interest for example when the combustion of fuel droplets or the shock induced mixing of liquids is numerically modeled we use the level set method to track the interface between the compressible and incompressible regions as well as the ghost fluid method gfm to create accurate discretizations across the interface the gfm is particularly effective here since the equations differ in both number and type across the interface the numerical method is presented in two spatial dimensions with numerical examples in both one and two spatial dimensions while three dimensional extensions are straightforward

optimal estimation and rank detection for sparse spiked covariance matrices this paper considers a sparse spiked covariance matrix model in the high dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection the optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariance matrices we also establish the minimax rate under the spectral norm for estimating the principal subspace the primary object of interest in principal component analysis in addition the optimal rate for the rank detection boundary is obtained this result also resolves the gap in a recent paper by berthet and rigollet ann stat 41 4 1780 1815 2013 where the special case of rank one is considered

regular article numerical simulation of free surface flows a numerical model is presented for the simulation of complex fluid flows with free surfaces the unknowns are the velocity and pressure fields in the liquid region together with a function defining the volume fraction of liquid although the mathematical formulation of the model is similar to the volume of fluid vof method the numerical schemes used to solve the problem are different a splitting method is used for the time discretization at each time step two advection problems and a generalized stokes problem are to be solved two different grids are used for the space discretization the two advection problems are solved on a fixed structured grid made out of small rectangular cells using a forward characteristic method the generalized stokes problem is solved using a finite element method on a fixed unstructured mesh numerical results are presented for several test cases the filling of an s shaped channel the filling of a disk with core the broken dam in a confined domain

rate optimal posterior contraction for sparse pca principal component analysis pca is possibly one of the most widely used statistical tools to recover a low rank structure of the data in the high dimensional settings the leading eigenvector of the sample covariance can be nearly orthogonal to the true eigenvector a sparse structure is then commonly assumed along with a low rank structure recently minimax estimation rates of sparse pca were established under various interesting settings on the other side bayesian methods are becoming more and more popular in high dimensional estimation but there is little work to connect frequentist properties and bayesian methodologies for high dimensional data analysis in this paper we propose a prior for the sparse pca problem and analyze its theoretical properties the prior adapts to both sparsity and rank the posterior distribution is shown to contract to the truth at optimal minimax rates in addition a computationally efficient strategy for the rank one case is discussed

numerical simulation of free surface incompressible liquid flows surrounded by compressible gas a numerical model for the three dimensional simulation of liquid gas flows with free surfaces is presented the incompressible navier stokes equations are assumed to hold in the liquid domain in the gas domain the velocity is disregarded the pressure is supposed to be constant in each connected component of the gas domain and follows the ideal gas law the gas pressure is imposed as a normal force on the liquid gas interface an implicit splitting scheme is used to decouple the physical phenomena given the gas pressure on the interface the method described in j comput phys 155 1999 439 int j numer meth fluids 42 7 2003 697 is used to track the liquid domain and to compute the velocity and pressure fields in the liquid then the connected components of the gas domain are found using an original numbering algorithm finally the gas pressure is updated from the ideal gas law in each connected component of gas the implementation is validated in the frame of mould filling numerical results in two and three space dimensions show that the effect of pressure in the bubbles of gas trapped by the liquid cannot be neglected

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

sparse generalized eigenvalue problem optimal statistical rates via truncated rayleigh flow the sparse generalized eigenvalue problem gep plays a pivotal role in a large family of high dimensional statistical models including sparse fisher s discriminant analysis canonical correlation analysis and sufficient dimension reduction the sparse gep involves solving a non convex optimization problem most existing methods and theory in the context of specific statistical models that are special cases of the sparse gep require restrictive structural assumptions on the input matrices we propose a two stage computational framework to solve the sparse gep at the first stage we solve a convex relaxation of the sparse gep taking the solution as an initial value we then exploit a non convex optimization perspective and propose the truncated rayleigh flow method which we call rifle to estimate the leading generalized eigenvector we show that rifle converges linearly to a solution with the optimal statistical rate of convergence theoretically our method significantly improves on the existing literature by eliminating structural assumptions on the input matrices to achieve this our analysis involves two key ingredients a new analysis of the gradient based method on non convex objective functions and a fine grained characterization of the evolution of sparsity patterns along the solution path thorough numerical studies are provided to validate the theoretical results

minimax sparse principal subspace estimation in high dimensions we study sparse principal components analysis in high dimensions where p the number of variables can be much larger than n the number of observations and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix we prove optimal non asymptotic lower and upper bounds on the minimax subspace estimation error under two different but related notions of lq subspace sparsity for 0 q 1 our upper bounds apply to general classes of covariance matrices and they show that lq constrained estimates can achieve optimal minimax rates without restrictive spiked covariance conditions 1 introduction principal components analysis pca was introduced in the early 20th century pearson 1901 hotelling 1933 and is arguably the most well known and widely used technique for dimension reduction it is part of the mainstream statistical repertoire and is routinely used in numerous and diverse areas of application however contemporary applications often involve much higher dimensional data than envisioned by the early developers of pca in such high dimensional situations where the number of variables p is of the same order or much larger than the number of observations n serious difficulties emerge standard pca can produce inconsistent estimates of the principal directions of variation and lead to unreliable conclusions johnstone and lu 2009 paul 2007 nadler 2008 the principal directions of variation correspond to the eigenvectors of the covariance matrix and in high dimensions consistent estimation of the eigenvectors is generally not possible without additional assumptions about the covariance matrix or its eigenstructure much of the recent development in pca has focused on methodology that applies the concept of sparsity to eigenvector estimation some examples include jolliffe trendafilov and uddin

generalized thresholding of large covariance matrices we propose a new class of generalized thresholding operators that combine thresholding with shrinkage and study generalized thresholding of the sample covariance matrix in high dimensions generalized thresholding of the covariance matrix has good theoretical properties and carries almost no computational burden we obtain an explicit convergence rate in the operator norm that shows the tradeoff between the sparsity of the true model dimension and the sample size and shows that generalized thresholding is consistent over a large class of models as long as the dimension p and the sample size n satisfy log p n 0 in addition we show that generalized thresholding has the sparsistency property meaning it estimates true zeros as zeros with probability tending to 1 and under an additional mild condition is sign consistent for nonzero elements we show that generalized thresholding covers as special cases hard and soft thresholding smoothly clipped absolute deviation and adaptive lasso and comp

a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow

pseudocyclic maximum distance separable codes the n k pseudocyclic maximum distance separable mds codes modulo x sup n a over gf q are considered suppose that n is a divisor of q 1 if n is odd pseudocyclic mds codes exist for all k however if n is even nontrivial pseudocyclic mds codes exist for odd k but not for even k if a is a quadratic residue in gf q and they exist for even k but not for odd k if a is not a quadratic residue in gf q also considered is the case when n is a divisor of q 1 and it is shown that pseudocyclic mds codes exist if and only if the multiplicative order of a divides q 1 n and that when this condition is satisfied such codes exist for all k if the condition is not satisfied every pseudocyclic code of length n is the result of interleaving a shorter pseudocyclic code

minimax estimation in sparse canonical correlation analysis canonical correlation analysis is a widely used multivariate statistical technique for exploring the relation between two sets of variables this paper considers the problem of estimating the leading canonical correlation directions in high dimensional settings recently under the assumption that the leading canonical correlation directions are sparse various procedures have been proposed for many high dimensional applications involving massive data sets however there has been few theoretical justification available in the literature in this paper we establish rate optimal nonasymptotic minimax estimation with respect to an appropriate loss function for a wide range of model spaces two interesting phenomena are observed first the minimax rates are not affected by the presence of nuisance parameters namely the covariance matrices of the two sets of random variables though they need to be estimated in the canonical correlation analysis problem second we allow the presence of the residual canonical correlation directions however they do not influence the minimax rates under a mild condition on eigengap a generalized sin theta theorem and an empirical process bound for gaussian quadratic forms under rank constraint are used to establish the minimax upper bounds which may be of independent interest

a link between quasi cyclic codes and convolutional codes given a quasi cyclic code with minimum hamming distance d a set of convolutional codes is derived with free distance equal to d it is shown that an increase in the rate of these codes results in a decrease in the memory length the connection between these codes is illustrated with several well known quasi cyclic codes the free distance of some partial unit memory convolutional codes can be determined using the results in this correspondence