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

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

scattering from an impedance object at the edge of a perfectly conducting wedge in this study scattering from impedance bodies positioned at the edge of a perfectly electrically conducting wedge is investigated in the treatment of the problem eigenfunction expansion in terms of spherical vector wave functions is employed a complete dyadic green s function for the spherical impedance boss at the edge is developed it is observed that the scattering is highly enhanced by the edge guided waves additionally using t matrix method the solution is extended to the general case of irregularly shaped scatterers the t matrix solution is verified by applying it to the case of a spherical scatterer and results are compared with the dyadic green s function solution

the space complexity of recognizing well parenthesized expressions in the streaming model the index function revisited we show an p n t lower bound for the space required by any unidirectional constanterror randomized t pass streaming algorithm that recognizes whether an expression over two types of parenthesis is well parenthesized this proves a conjecture due to magniez mathieu and nayak 2009 and rigorously establishes the peculiar power of bi directional streams over unidirectional ones observed in the algorithms they present the lower bound is obtained by analysing the information that is necessarily revealed by the players about their respective inputs in a two party communication protocol for a variant of the index function

optimal selection of reduced rank estimators of high dimensional matrices we introduce a new criterion the rank selection criterion rsc for selecting the optimal reduced rank estimator of the coefficient matrix in multivariate response regression models the corresponding rsc estimator minimizes the frobenius norm of the fit plus a regularization term proportional to the number of parameters in the reduced rank model the rank of the rsc estimator provides a consistent estimator of the rank of the coefficient matrix in general the rank of our estimator is a consistent estimate of the effective rank which we define to be the number of singular values of the target matrix that are appropriately large the consistency results are valid not only in the classic asymptotic regime when n the number of responses and p the number of predictors stay bounded and m the number of observations grows but also when either or both n and p grow possibly much faster than m we establish minimax optimal bounds on the mean squared errors of our estimators our finite sample performance bounds for the rsc estimator show that it achieves the optimal balance between the approximation error and the penalty term furthermore our procedure has very low computational complexity linear in the number of candidate models making it particularly appealing for large scale problems we contrast our estimator with the nuclear norm penalized least squares nnp estimator which has an inherently higher computational complexity than rsc for multivariate regression models we show that nnp has estimation properties similar to those of rsc albeit under stronger conditions however it is not as parsimonious as rsc we offer a simple correction of the nnp estimator which leads to consistent rank estimation we verify and illustrate our theoretical findings via an extensive simulation study

tm and te diffraction by a perfectly conducting half plane in presence of a perfectly conducting strip solution by exponentially converging nystrom and galerkin methods we study tm and te diffraction by a perfectly conducting half plane in presence of a perfectly conducting flat strip of finite width and infinite length the edges of the strip are parallel to the edge of the half plane the relevant integral equations with unknowns the induced current densities on the strip are solved by a nystrom and a galerkin method that fully account for the singular nature of the kernels and the singularities of the solution at the edges the proposed algorithms are highly accurate and appear to converge exponentially as shown by detailed numerical examples and case studies the nystrom method is the most efficient of the two methods because of its simple formulation fast computer implementation and much less effort in computing the matrix elements when the strip is coplanar to the half plane an additional formulation method in terms of equivalent surface magnetic currents is possible in the framework of the field equivalence principles this alternative method is applied in order to validate the algorithms and test their accuracy

optimal rates of convergence for sparse covariance matrix estimation this paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and bregman divergence losses a major focus is on the derivation of a rate sharp minimax lower bound the problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems standard techniques fail to yield good results and new tools are thus needed we first develop a lower bound technique that is particularly well suited for treating two directional problems such as estimating sparse covariance matrices the result can be viewed as a generalization of le cam s method in one direction and assouad s lemma in another this lower bound technique is of independent interest and can be used for other matrix estimation problems we then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique a thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm the results are then extended to

rounds in communication complexity revisited the k round two party communication complexity was studied in the deterministic model by p h papadimitriou and m sipser proc of the 14th stoc 1982 pp 330 337 and p duris z galil and g schnitger proc of the 16th stoc 1984 pp 81 91 and in the probabilistic model by a c yao proc of the 24th focs 1983 pp 420 428 and b halstenberg and r reischuk proc of the 20th stoc 1988 pp 162 172 this paper presents new lower bounds that give 1 randomization is more powerful than determinism in k round protocols and 2 an explicit function which exhibits an exponential gap between its k and k 1 round randomized complexity this paper also studies the three party communication model and exhibits an exponential gap in 3 round protocols that differ in the starting player finally this paper shows new connections of these questions to circuit complexity that motivate further work in this direction

operator norm consistent estimation of large dimensional sparse covariance matrices estimating covariance matrices is a problem of fundamental importance in multivariate statistics in practice it is increasingly frequent to work with data matrices x of dimension n x p where p and n are both large results from random matrix theory show very clearly that in this setting standard estimators like the sample covariance matrix perform in general very poorly in this large n large p setting it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0 and hence this matrix is sparse we develop an estimator to handle this situation the estimator is shown to be consistent in operator norm when for instance we have p n as n oo in other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero this implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues we also propose a notion of sparsity for matrices that is compatible with spectral analysis and is independent of the ordering of the variables

why quantum bit commitment and ideal quantum coin tossing are impossible abstract there had been well known claims of unconditionally secure quantum protocols for bit commitment however we and independently mayers showed that all proposed quantum bit commitment schemes are in principle insecure because the sender alice can almost always cheat successfully by using an einstein podolsky rosen epr type of attack and delaying her measurements one might wonder if secure quantum bit commitment protocols exist at all we answer this question by showing that the same type of attack by alice will in principle break any bit commitment scheme the cheating strategy generally requires a quantum computer we emphasize the generality of this no go theorem unconditionally secure bit commitment schemes based on quantum mechanics fully quantum classical or quantum but with measurements are all ruled out by this result since bit commitment is a useful primitive for building up more sophisticated protocols such as zero knowledge proofs our results cast very serious doubt on the security of quantum cryptography in the so called post cold war applications we also show that ideal quantum coin tossing is impossible because of the epr attack this no go theorem for ideal quantum coin tossing may help to shed some lights on the possibility of non ideal protocols

application and improvement of a recent approximate analytical solution of richards equation a method for developing wetting front profiles for the one dimensional richards equation is given the method is obtained by simplifying and extending a recent approximate solution and brings together features from several different studies difficulties associated with profile development are discussed by applying the method to several examples providing a deeper understanding of the solutions of richards equation for simplicity we illustrate the improvements for soil water diffusivities and conductivities which have a power law dependence on the water content although they should apply in general and the appropriate generalization is given in the conclusion

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

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

exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption several authors have now used the backlund transformation to linearize richard s equation and so obtain analytical solutions for infiltration and redistribution processes for the linearizable class of diffusivities and hydraulic conductivities after transformation a convection diffusion equation is obtained to date all the analytical results making use of the backlund transform have been based on a specified constant flux that is assumed to exist at the surface boundary we generalize the previous work by removing this restriction and obtain a general solution for an arbitrary temporal variation in surface flux the solution contains a function that is specified by a volterra integral equation in practice the volterra equation must be evaluated numerically this task is accomplished using a relatively straightforward algorithm the analytical results for infiltration can be applied to the process of solute transport subject to a nonlinear adsorption isotherm since these two phenomena are equivalent under another mapping our new analytical results therefore also apply to this class of solute transport processes the general form of the adsorption isotherm for which exact results can be obtained is derived this isotherm appears capable of representing a relatively large family of physically relevant cases although this topic remains open for future investigation

approximation dans les espaces metriques et theorie de l estimation we investigate the relations between the speed of estimation and the metric structure of the parameter space especially in the case when its metric dimension is infinite given some distance d on generally hellinger distance in the case of n i i d variables we consider the minimax risk for n observations r n q mathop inf sup limits t n theta in theta text ie theta d q theta t n t n being any estimate of we shall look for functions r such that for positive constants c 1 q and c 2 q c 1 r q n r n q c2 r q n r n is the speed of estimation and we shall show under fairly general conditions including i i d variables and regular cases of markov chains and stationnary gaussian processes that r n is determined up to multiplicative constants by the metric structure of we shall also give a construction for some sort of universal estimates the risk of which is bounded by c 2 r q n in all cases where the preceding theory applies

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

on almost randomizing channels with a short kraus decomposition for large d we study quantum channels on cd obtained by selecting randomly n independent kraus operators according to a probability measure on the unitary group mathcal u d when is the haar measure we show that for n succcurlyeq d varepsilon 2 such a channel is e randomizing with high probability which means that it maps every state within distance e d in operator norm of the maximally mixed state this slightly improves on a result by hayden leung shor and winter by optimizing their discretization argument

constant rate rainfall infiltration a versatile nonlinear model 1 analytic solution analytic solutions are presented for a nonlinear diffusion convection model describing constant rate rainfall infiltration in uniform soils and other porous materials the model is based on the darcy buckingham approach to unsaturated water flow and assumes simple functional forms for the soil water diffusivity d and hydraulic conductivity k which depend on a single free parameter c and readily measured soil hydraulic properties these d and k yield physically reasonable analytic moisture characteristics the relation between this model and other models which give analytic solutions is explored as c the model reduces to the weakly nonlinear burgers equation which has been applied in certain field situations at the other end of the range as c 1 the model approaches a green ampt like model a wide range of realistic soil hydraulic properties is encompassed by varying the c parameter the general features of the analytic solutions are illustrated for selected c values gradual and steep wetting profiles develop during rainfall aspects seen in the laboratory and field in addition the time dependent surface water content and surface water pressure potential are presented explicitly a simple traveling wave approximation is given which agrees closely with the exact solution at comparatively early infiltration times

platelets a multiscale approach for recovering edges and surfaces in photon limited medical imaging the nonparametric multiscale platelet algorithms presented in this paper unlike traditional wavelet based methods are both well suited to photon limited medical imaging applications involving poisson data and capable of better approximating edge contours this paper introduces platelets localized functions at various scales locations and orientations that produce piecewise linear image approximations and a new multiscale image decomposition based on these functions platelets are well suited for approximating images consisting of smooth regions separated by smooth boundaries for smoothness measured in certain holder classes it is shown that the error of m term platelet approximations can decay significantly faster than that of m term approximations in terms of sinusoids wavelets or wedgelets this suggests that platelets may outperform existing techniques for image denoising and reconstruction fast platelet based maximum penalized likelihood methods for photon limited image denoising deblurring and tomographic reconstruction problems are developed because platelet decompositions of poisson distributed images are tractable and computationally efficient existing image reconstruction methods based on expectation maximization type algorithms can be easily enhanced with platelet techniques experimental results suggest that platelet based methods can outperform standard reconstruction methods currently in use in confocal microscopy image restoration and emission tomography

full wave scattering from a grooved cylinder tipped conducting wedge the problem of full wave electromagnetic scattering from a cylinder tipped conducting wedge with either a sectoral or annular groove is formulated using the mode matching technique the mode expansion of the sector involves ratios of bessel functions with large order as the order is inversely proportional to the inner angle of the sector an asymptotic expansion of bessel functions with large orders is introduced in order to overcome the numerical difficulty involved using a regular series expansion numerical results indicate good agreement with data obtained using the finite element method fem

multidimensional linearized nonsteady infiltration with prescribed boundary conditions at the soil surface the present work explores a class of analytical solutions of moisture movement in unsaturated porous media characterized by an exponential dependence of the hydraulic conductivity and the moisture content on water pressure the green s function method is used to derive a general analytical model pertaining to multidimensional nonsteady infiltration in a semi infinite flow domain with arbitrary initial conditions boundary conditions and root uptake forcing functions and for various simple source geometry the general solution is expressed in integral form from which particular analytical solutions pertaining to cases of surface and subsurface irrigation evaporation root uptake and moisture redistribution can be easily deduced from the general analytical model the model offers the analyst significant flexibility in deriving results and analyzing infiltration phenomena of practical interest new explicit solutions have been obtained for one dimensional infiltration under various prescribed time dependent flux boundary conditions and for two and three dimensional moisture redistribution for constant initial or boundary conditions the multidimensional solution is essentially the product of two or three time dependent terms with each term being a function of only one space variable

lower bounds for the minimax risk using f divergences and applications lower bounds involving f divergences between the underlying probability measures are proved for the minimax risk in estimation problems our proofs just use simple convexity facts special cases and straightforward corollaries of our bounds include well known inequalities for establishing minimax lower bounds such as fano s inequality pinsker s inequality and inequalities based on global entropy conditions two applications are provided a new minimax lower bound for the reconstruction of convex bodies from noisy support function measurements and a different proof of a recent minimax lower bound for the estimation of a covariance matrix

clean two and three dimensional analytical solutions of richards equation for testing numerical solvers 1 this technical note derives clean analytical solutions of richards equation for three dimensional unsaturated groundwater flow clean means that the boundary conditions and steady state solutions are closed form expressions and the transient solutions have relatively simple additional fourier series terms two dimensional versions of these solutions are also given the primary purpose for the solutions is to test linear and nonlinear solvers in finite difference volume element computer programs for accuracy and scalability using architectures ranging from pcs to parallel high performance computers this derivation starts from the quasi linear assumption of relative hydraulic conductivity varying exponentially with pressure head and the separate approximation that relative hydraulic conductivity varies linearly with moisture content this allows a transformation to be used to create a linear partial differential equation separation of variables and fourier series are then used to obtain the final solution physically reasonable material properties are also used a total of four solutions are given in this technical note steady state and transient solutions for two different boundary conditions of the sample problem

optimal sparse singular value decomposition for high dimensional high order data in this article we consider the sparse tensor singular value decomposition which aims for dimension reduction on high dimensional high order data with certain sparsity structure a method named s

hypothesis tests for large density matrices of quantum systems based on pauli measurements abstract for a quantum system its density matrix usually has a size growing exponentially with the number of particles in the system and quantum state tomography techniques often encounter an exponential complexity problem for recovering the density matrix based on experimental data recent statistical methods for estimating a large density matrix have been developed for the cases that i the entries of the density matrix with respect to the pauli basis are sparse or ii the density matrix has a low rank and its eigenvectors are sparse their performances depend on the assumed structures and it is important to test for the structures and choose appropriate estimation methods accordingly this paper investigates hypothesis tests for sparsity specifically we propose hypothesis test procedures and establish asymptotic theories for the proposed tests numerical studies are conducted to check the finite sample performances of the proposed hypothesis tests

statistical analysis of compressive low rank tomography with random measurements we consider the statistical problem of compressive estimation of low rank states r d with random basis measurements where r d are the rank and dimension of the state respectively we investigate whether for a fixed sample size n the estimation error associated with a compressive measurement setup is close to that of the setting where a large number of bases are measured we generalise and extend previous results and show that the mean square error mse associated with the frobenius norm attains the optimal rate rd n with only o rlogd random basis measurements for all states an important tool in the analysis is the concentration of the fisher information matrix fim we demonstrate that although a concentration of the mse follows from a concentration of the fim for most states the fim fails to concentrate for states with eigenvalues close to zero we analyse this phenomenon in the case of a single qubit and demonstrate a concentration of the mse about its optimal despite a lack of concentration of the fim for states close to the boundary of the bloch sphere we also consider the estimation error in terms of a different metric the quantum infidelity we show that a concentration in the mean infidelity minf does not exist uniformly over all states highlighting the importance of loss function choice specifically we show that for states that are nearly pure the minf scales as 1 n but the constant converges to zero as the number of settings is increased this demonstrates a lack of compressive recovery for nearly pure states in this metric

electromagnetic scattering from a slotted conducting wedge electromagnetic wave scattering from a slotted conducting wedge is studied the kontorovich lebedev transform and the mode matching are utilized to obtain a convergent series solution computations are performed to illustrate scattering from slotted conducting wedges grooved conducting wedges are also treated as a special case the mode matching formulation provides a stable and convergent series solution

lower bounds on communication complexity abstract we prove the following four results on communication complexity 1 for every k 2 the language of encodings of directed graphs of out degree one that contain a path of length k 1 from the first vertex to the last vertex can be recognized by exchanging o k log n 1 bits using a simple k round protocol and requires the exchange of n 1 2 k 4 log 3 n bits by any k 1 round protocol 2 for every k 1 and for infinitely many n 1 there exists a collection of sets lkn 0 1 2n that can be recognized by exchanging o k log n bits using a k round protocol and any k 1 round protocol recognizing lkn requires the exchange of n k bits 3 given a set l 0 1 2n there is a set l 0 1 8n such that any k round protocol recognizing l can be transformed to a k round fixed partition protocol recognizing l with the same communication complexity and vice versa 4 for every integer function f 1 f n n there are languages recognizable by a one round deterministic protocol exchanging f n bits but not by any nondeterministic protocol exchanging f n 1 bits the first two results show in an incomparable way an exponential gap between k 1 round and k round protocols settling a conjecture by papadimitriou and sipser the third result shows that as long as we are interested in existence proofs a fixed partition of the input is not a restriction the fourth result extends a result by papadimitriou and sipser who showed that for every integer function f 1 f n n there is a language accepted by a deterministic protocol exchanging f n bits but not by any deterministic protocol exchanging f n 1 bits

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

minimax risk over l p balls for l p error consider estimating the mean vector from datan n 2 i withl q norm loss q 1 when is known to lie in ann dimensionall p ball p 0 for largen the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp q obvious exceptions aside the limiting ratio equals 1 only ifp q 2 our arguments are mostly indirect involving a reduction to a univariate bayes minimax problem whenp q simple non linear co ordinatewise threshold rules are asymptotically minimax at small signal to noise ratios and within a bounded factor of asymptotic minimaxity in general we also give asymptotic evaluations of the minimax linear risk our results are basic to a theory of estimation in besov spaces using wavelet bases to appear elsewhere

the quantum communication complexity of sampling sampling is an important primitive in probabilistic and quantum algorithms in the spirit of communication complexity given a function f x times y rightarrow 0 1 and a probability distribution cal d over x times y we define the sampling complexity of f cal d as the minimum number of bits that alice and bob must communicate for alice to pick x in x and bob to pick y in y as well as a value z such that the resulting distribution of x y z is close to the distribution cal d f cal d in this paper we initiate the study of sampling complexity in both the classical and quantum models we give several variants of a definition we completely characterize some of these variants and give upper and lower bounds on others in particular this allows us to establish an exponential gap between quantum and classical sampling complexity for the set disjointness function

dense quantum coding and quantum finite automata we consider the possibility of encoding m classical bits into many fewer n quantum bits qubits so that an arbitrary bit from the original m bits can be recovered with good probability we show that nontrivial quantum codes exist that have no classical counterparts on the other hand we show that quantum encoding cannot save more than a logarithmic additive factor over the best classical encoding the proof is based on an entropy coalescence principle that is obtained by viewing holevo s theorem from a new perspective in the existing implementations of quantum computing qubits are a very expensive resource moreover it is difficult to reinitialize existing bits during the computation in particular reinitialization is impossible in nmr quantum computing which is perhaps the most advanced implementation of quantum computing at the moment this motivates the study of quantum computation with restricted memory and no reinitialization that is of quantum finite automata it was known that there are languages that are recognized by quantum finite automata with sizes exponentially smaller than those of corresponding classical automata here we apply our technique to show the surprising result that there are languages for which quantum finite automata take exponentially more states than those of corresponding classical automata

scattering from a grooved conducting wedge electromagnetic scattering from a grooved perfectly conducting wedge is investigated using the kontorovich lebedev transform residue calculus is utilized to obtain a fast convergent series solution that yields accurate results the behavior of a scattered field is presented in terms of wedge geometries and incidence angles

a property of quantum relative entropy with an application to privacy in quantum communication we prove the following information theoretic property about quantum states substate theorem let and be quantum states in the same hilbert space with relative entropy s c tr log log c then for all 0 there is a state such that the trace distance tr tr 2 and 2o c 2 it states that if the relative entropy of and is small then there is a state close to i e with small trace distance tr that when scaled down by a factor 2o c sits inside or becomes a substate of this result has several applications in quantum communication complexity and cryptography using the substate theorem we derive a privacy trade off for the set membership problem in the two party quantum communication model here alice is given a subset a n bob an input i i n and they need to determine if i i a privacy trade off for set membership in any two party quantum communication protocol for the set membership problem if bob reveals only k bits of information about his input then alice must reveal at least n 2o k bits of information about her input we also discuss relationships between various information theoretic quantities that arise naturally in the context of the substate theorem

rapidly convergent eigenfunction expansions of green functions for a perfectly conducting wedge existing green functions for a conducting wedge in the form of infinite series of orthogonal eigenfunctions are known to suffer from slow and conditional convergence 1 near the source point due to the singularity of the green functions at source points and 2 when the source point is near the surface of the wedge due to the presence of a nearby image source as a result such expansions are unsuited for the exact solution of singular integral equations wherein values of the green functions at the source point do appear inside the integral in this paper alternative expansions are proposed which converge rapidly for any combination of the source and observation points the main idea is to extract in closed form both the singular term and the nearby image term together with certain simple asymptotic terms this helps isolate the inherent singularities and thus transform the remaining part of the green function into a rapidly converging series of elementary terms numerical examples and case studies illustrate the stability and high accuracy of the new expansions application to the solution of an integral equation associated with wave diffraction by perfectly conducting curved strips in the vicinity of a wedge is exemplified

volume ratio sparsity and minimaxity under unitarily invariant norms this paper studies non asymptotic minimax estimation of high dimensional matrices and provides tight minimax rates for a large collection of loss functions in a variety of problems via information theoretic methods based on the convex geometry of finite dimensional banach spaces we first develop a volume ratio approach for determining minimax estimation rates of unconstrained mean matrices under all unitarily invariant norm losses which turn out to only depend on the norm of identity matrix in addition we establish the minimax rates for estimating normal mean matrices with submatrix sparsity where the sparsity constraint introduces an additional term in the rate which in contrast to the unconstrained case is determined by the smoothness lipschitz constant of the norm this method is also applicable to the low rank matrix completion problem and extends well beyond the additive noise model in particular it yields tight rates in covariance matrix estimation and poisson rate matrix estimation problems for all unitarily invariant norms

a comparison of picard and newton iteration in the numerical solution of multidimensional variably saturated flow problems picard iteration is a widely used procedure for solving the nonlinear equation governing flow in variably saturated porous media the method is simple to code and computationally cheap but has been known to fail or converge slowly the newton method is more complex and expensive on a per iteration basis than picard and as such has not received very much attention its robustness and higher rate of convergence however make it an attractive alternative to the picard method particularly for strongly nonlinear problems in this paper the picard and newton schemes are implemented and compared in one two and three dimensional finite element simulations involving both steady state and transient flow the eight test cases presented highlight different aspects of the performance of the two iterative methods and the different factors that can affect their convergence and efficiency including problem size spatial and temporal discretization initial solution estimates convergence error norm mass lumping time weighting conductivity and moisture content characteristics boundary conditions seepage faces and the extent of fully saturated zones in the soil previous strategies for enhancing the performance of the picard and newton schemes are revisited and new ones are suggested the strategies include chord slope approximations for the derivatives of the characteristic equations relaxing convergence requirements along seepage faces dynamic time step control nonlinear relaxation and a mixed picard newton approach the tests show that the picard or relaxed picard schemes are often adequate for solving richards equation but that in cases where these fail to converge or converge slowly the newton method should be used the mixed picard newton approach can effectively overcome the newton scheme s sensitivity to initial solution estimates while comparatively poor performance is reported for the various chord slope approximations finally given the reliability and efficiency of current conjugate gradient like methods for solving linear nonsymmetric systems the only real drawback of using newton rather than picard iteration is the algebraic complexity and computational cost of assembling the derivative terms of the jacobian matrix and it is suggested that both methods can be effectively implemented and used in numerical models of richards equation

mutual information metric entropy and cumulative relative entropy risk assume p theta theta epsilon theta is a set of probability distributions with a common dominating measure on a complete separable metric space y a state theta epsilon theta is chosen by nature a statistician obtains n independent observations y 1 dots y n from y distributed according to p theta for each time t between 1 and n based on the observations y 1 dots y t 1 the statistician produces an estimated distribution hat p t for p theta and suffers a loss l p theta hat p t the cumulative risk for the statistician is the average total loss up to time n of special interest in information theory data compression mathematical finance computational learning theory and statistical mechanics is the special case when the loss l p theta hat p t is the relative entropy between the true distribution p theta and the estimated distribution hat p t here the cumulative bayes risk from time 1 to n is the mutual information between the random parameter theta and the observations y 1 dots y n new bounds on this mutual information are given in terms of the laplace transform of the hellinger distance between pairs of distributions indexed by parameters in theta from these bounds on the cumulative minimax risk are given in terms of the metric entropy of theta with respect to the hellinger distance the assumptions required for these bounds are very general and do not depend on the choice of the dominating measure they apply to both finite and infinite dimensional theta they apply in some cases where y is infinite dimensional in some cases where y is not compact in some cases where the distributions are not smooth and in some parametric cases where asymptotic normality of the posterior distribution fails

wavelets ridgelets and curvelets for poisson noise removal in order to denoise poisson count data we introduce a variance stabilizing transform vst applied on a filtered discrete poisson process yielding a near gaussian process with asymptotic constant variance this new transform which can be deemed as an extension of the anscombe transform to filtered data is simple fast and efficient in very low count situations we combine this vst with the filter banks of wavelets ridgelets and curvelets leading to multiscale vsts ms vsts and nonlinear decomposition schemes by doing so the noise contaminated coefficients of these ms vst modified transforms are asymptotically normally distributed with known variances a classical hypothesis testing framework is adopted to detect the significant coefficients and a sparsity driven iterative scheme reconstructs properly the final estimate a range of examples show the power of this ms vst approach for recovering important structures of various morphologies in very low count images these results also demonstrate that the ms vst approach is competitive relative to many existing denoising methods

cryptographic distinguishability measures for quantum mechanical states this paper mostly expository in nature surveys four measures of distinguishability for quantum mechanical states this is done from the point of view of the cryptographer with a particular eye on applications in quantum cryptography each of the measures considered is rooted in an analogous classical measure of distinguishability for probability distributions namely the probability of an identification error the kolmogorov distance the bhattacharyya coefficient and the shannon 1948 distinguishability as defined through mutual information these measures have a long history of use in statistical pattern recognition and classical cryptography we obtain several inequalities that relate the quantum distinguishability measures to each other one of which may be crucial for proving the security of quantum cryptographic key distribution in another vein these measures and their connecting inequalities are used to define a single notion of cryptographic exponential indistinguishability for two families of quantum states this is a tool that may prove useful in the analysis of various quantum cryptographic protocols

confidence region of singular subspaces for low rank matrix regression low rank matrix regression refers to the instances of recovering a low rank matrix based on specially designed measurements and the corresponding noisy outcomes numerous statistical methods have been developed over the recent decade for efficiently reconstructing the unknown low rank matrices it is often interesting in certain applications to estimate the unknown singular subspaces in this paper we revisit the low rank matrix regression model and introduce a two step procedure to construct confidence regions of the singular subspaces we investigate distributions of the joint projection distance between the empirical singular subspaces and the unknown true singular subspaces we prove asymptotical normality of the joint projection distance with data dependent centering and normalization when r 3 2 m 1 m 2 3 2 o n log n where m 1 m 2 denote the matrix row and column sizes r is the rank and n is the number of independent random measurements consequently data dependent confidence regions of the true singular subspaces are established which attain pre determined confidence levels asymptotically additionally non asymptotic convergence rates are also established numerical results are presented to show the merits of our methods

von neumann entropy penalization and low rank matrix estimation we study a problem of estimation of a hermitian nonnegatively definite matrix of unit trace for instance a density matrix of a quantum system based on n i i d measurements x1 y1 xn yn where

fast state tomography with optimal error bounds projected least squares pls is an intuitive and numerically cheap technique for quantum state tomography the method first computes the least squares estimator or a linear inversion estimator and then projects the initial estimate onto the space of states the main result of this paper equips this point estimator with a rigorous non asymptotic confidence region expressed in terms of the trace distance the analysis holds for a variety of measurements including 2 designs and pauli measurements the sample complexity of the estimator is comparable to the strongest convergence guarantees available in the literature and in the case of measuring the uniform povm saturates fundamental lower bounds the results are derived by reinterpreting the least squares estimator as a sum of random matrices and applying a matrix valued concentration inequality the theory is supported by numerical simulations for mutually unbiased bases pauli observables and pauli basis measurements

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

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

interaction in quantum communication in some scenarios there are ways of conveying information with many fewer even exponentially fewer qubits than possible classically moreover some of these methods have a very simple structure they involve only few message exchanges between the communicating parties it is therefore natural to ask whether every classical protocol may be transformed to a simpler quantum protocol one that has similar efficiency but uses fewer message exchanges we show that for any constant k there is a problem such that its k 1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity this in particular proves a round hierarchy theorem for quantum communication complexity and implies via a simple reduction an omega n1k lower bound for k message quantum protocols for set disjointness for constant k enroute we prove information theoretic lemmas and define a related measure of correlation the informational distance that we believe may be of significance in other contexts as well

optimal estimation of low rank density matrices the density matrices are positively semi definite hermitian matrices of unit trace that describe the state of a quantum system the goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography in particular in the case of pauli measurements with explicit dependence of the bounds on the rank and other complexity parameters such bounds are established for several statistically relevant distances including quantum versions of kullback leibler divergence relative entropy distance and of hellinger distance so called bures distance and schatten p norm distances sharp upper bounds and oracle inequalities for least squares estimator with von neumann entropy penalization are obtained showing that minimax lower bounds are attained up to logarithmic factors for these distances

a perturbative view on the subsurface water pressure response at hillslope scale 1 this work deals with a rearrangement of richards equation to obtain a better understanding of the components of the subsurface water and pressure flows and their interactions in a hillslope the basic idea proposed in this paper is the normalization of the spatial and temporal coordinates according to the physical knowledge of the processes and the geometry of a hillslope the pressure head and all the other quantities contained in richards equation are rescaled to obtain a dimensionless equation the pressure head is also split into an equilibrium component and a transient component and expanded in a perturbation series subsequently richards equation itself is conveniently split into two equations connected by a source sink term results of these manipulations show that for the first order approximation the water and pressure flow is slope normal while slope parallel effects only occur at successive orders integrating the perturbed equation obtained for the long term pressure head component results in the well known boussinesq equation for the water table level further integrations then lead to the hillslope storage boussinesq model and finally to a generalization of the o loughlin formula for the slope parallel long term subsurface flow the introduced source sink term is estimated at the first order of approximation to be proportional to the temporal variation of water pressure head at the bottom of the integration domain the treatment of the equations clarifies some of the mechanisms acting in water pressure redistribution at hillslope scale even without solving the equation