shrinkage priors for bayesian prediction we investigate shrinkage priors for constructing bayesian predictive distributions it is shown that there exist shrinkage predictive distributions asymptotically dominating bayesian predictive distributions based on the jeffreys prior or other vague priors if the model manifold satisfies some differential geometric conditions kullback leibler divergence from the true distribution to a predictive distribution is adopted as a loss function conformal transformations of model manifolds corresponding to vague priors are introduced we show several examples where shrinkage predictive distributions dominate bayesian predictive distributions based on vague priors

motion of a set by the curvature of its boundary abstract the study of a crystal shrinking or growing in a melt gives rise to equations relating the normal velocity of the motion to the curvature of the crystal boundary often these equations are anisotropic indicating the preferred directions of the crystal structure in the isotropic case this equation is called the curve shortening or the mean curvature flow equation and has been studied by differential geometric tools in general there are no classical solutions to these equations in this paper we develop a weak theory for the generalized mean curvature equation using the newly developed theory of viscosity solutions our approach is closely related to that of osher and sethian chen giga and goto and evans and spruck who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation although we use their results in an essential way we give an intrinsic definition our main results are the existence of a solution large time asymptotics of this solution and its connection to the level set solution of osher and sethian chen giga and goto and evans and spruck in general there is no uniqueness even for classical solutions but we prove a uniqueness result under restrictive assumptions we also construct a class of explicit solutions which are dilations of wulff crystals

singular value shrinkage priors for bayesian prediction we develop singular value shrinkage priors for the mean matrix parameters in the matrix variate normal model with known covariance matrices introduced priors are superharmonic and put more weight on matrices with smaller singular values they are a natural generalization of the stein prior bayes estimators and bayesian predictive densities based on introduced priors are minimax and dominate those based on the uniform prior in finite samples the risk reduction is large when the true value of the parameter has small singular values in particular introduced priors perform stably well in the low rank case we apply this result to multivariate linear regression problems by considering priors depending on the future samples introduced priors are compatible with reduced rank regression

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

time variant channel estimation using discrete prolate spheroidal sequences we propose and analyze a low complexity channel estimator for a multiuser multicarrier code division multiple access mc cdma downlink in a time variant frequency selective channel mc cdma is based on orthogonal frequency division multiplexing ofdm the time variant channel is estimated individually for every flat fading subcarrier assuming small intercarrier interference the temporal variation of every subcarrier over the duration of a data block is upper bounded by the doppler bandwidth determined by the maximum velocity of the users slepian showed that time limited snapshots of bandlimited sequences span a low dimensional subspace this subspace is also spanned by discrete prolate spheroidal dps sequences we expand the time variant subcarrier coefficients in terms of orthogonal dps sequences we call slepian basis expansion this enables a time variant channel description that avoids the frequency leakage effect of the fourier basis expansion the square bias of the slepian basis expansion per subcarrier is three magnitudes smaller than the square bias of the fourier basis expansion we show simulation results for a fully loaded mc cdma downlink with classic linear minimum mean square error multiuser detection the users are moving with 19 4 m s using the slepian basis expansion channel estimator and a pilot ratio of only 2 we achieve a bit error rate performance as with perfect channel knowledge

front propagation problems with nonlocal terms ii abstract we study the propagation of hyper surfaces t of r n satisfying the equation v h x t where v is the normal velocity of t at x t is the interior of t and h is a given evolution law we prove the existence of generalized solutions and an inclusion principle for these generalized solutions copyright 2001 academic press dans cet article nous etudions certains problemes de propagation d hypersurfaces t de r n suivant des lois de la forme v h x t ou v est la vitesse normale a t en x t est l interieur de t et h est une loi d evolution on definit une notion de solutions generalisees et on montre l existence de telles solutions de plus on demontre un principe d inclusion pour ces solutions

inadmissibility of the best equivariant predictive density in the unknown variance case where is an estimator of and the expectation is taken with respect to random quantities usedin the construction of this phenomenon the stein e ect was rst brought to lightin the context of parameter estimation for such a problem many classes of estimatorsdominating the average have been proposed in parallel a similar phenomenon has been observed for predictive density estimation for instance let us observe nindependent d dimensional vectors x

simultaneous prediction of independent poisson observables simultaneous predictive distributions for independent poisson observables are investigated a class of improper prior distributions for poisson means is introduced the bayesian predictive distributions based on priors from the introduced class are shown to be admissible under the kullback leibler loss a bayesian predictive distribution based on a prior in this class dominates the bayesian predictive distribution based on the jeffreys prior

a new approach to front propagation problems theory and applications in this paper we present a new definition for the global in time propagation motion of fronts hypersurfaces boundaries with a prescribed normal velocity past the first time they develop singularities we show that if this propagation satisfies a geometric maximum principle inclusion avoidance type property then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures this new approach which is more geometric and as it turns out equivalent to the level set method is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction diffusion equations we finally present a number of new asymptotic results among them are the asymptotics of i reaction diffusion equations with rapidly oscillating coefficients ii fully nonlinear nonlocal integral differential equations and iii stochastic ising models with long range anisotropic interactions and general spin flip dynamics

pilot assisted channel estimation based on second order statistics a survey about linear channel estimation exploiting slowly varying channel properties is given for receivers with multiple antenna elements the slowly varying channel properties are described by the channel s second order statistics first a detailed comparison of classical linear pilot based channel estimators a novel matched filter and reduced rank rr approaches is made in a common framework with respect to the model they assume for the channel their performance mse and complexity the matched filter channel estimator is introduced which exploits second order statistics with quadratic order of complexity for flat correlated rayleigh fading channels an analytical performance comparison of all estimators in terms of the uncoded bit error probability is provided it is a generalization of previous results and shows that the matched filter is a low complexity alternative with good performance at an interesting signal to noise ratio snr range the effects of these approaches on linear equalization are briefly discussed in the context of direct sequence code division multiple access ds cdma where the focus is on a generalization of the rake receiver in space and time which reduces the channel rank based on its second order statistics resulting in a complexity reduction of the equalizer for this discussion a new notation for the generalized rake is presented which allows for its interpretation in the context of channel estimation and reveals alternatives for implementation we conclude that exploiting second order channel statistics results in significant performance gains and rr channel estimation should only be used together with equalization in the reduced signal subspace

asymptotically minimax bayes predictive densities given a random sample from a distribution with density function that depends on an unknown parameter 0 we are interested in accurately estimating the true parametric density function at a future observation from the same distribution the asymptotic risk of bayes predictive density estimates with kullback leibler loss function d f f f f log f f is used to examine various ways of choosing prior distributions the principal type of choice studied is minimax we seek asymptotically least favorable predictive densities for which the corresponding asymptotic risk is minimax a result resembling stein s paradox for estimating normal means by maximum likelihood holds for the uniform prior in the multivariate location family case when the dimensionality of the model is at least three the jeffreys prior is minimax though inadmissible the jeffreys prior is both admissible and minimax for one and two dimensional location problems

robust precoding with bayesian error modeling for limited feedback mu miso systems we consider the robust precoder design for multiuser multiple input single output mu miso systems where the channel state information csi is fed back from the single antenna receivers to the centralized transmitter equipped with multiple antennas we propose to compress the feedback data by projecting the channel estimates onto a vector basis known at the receivers and the transmitter and quantizing the resulting coefficients the channel estimator and the basis for the rank reduction are jointly optimized by minimizing the mean square error mse between the true and the rank reduced csi expressions for the conditional mean and the conditional covariance of the channel are derived which are necessary for the robust precoder design these expressions take into account the following sources of error channel estimation truncation for rank reduction quantization and feedback channel delay as an example for the robust problem formulation vector precoding vp is designed based on the expectation of the mse conditioned on the fed back csi our results show that robust precoding based on fed back csi clearly outperforms conventional precoding designs which do not take into account the errors in the csi

admissible predictive density estimation let xj np v xi and y j np v yi be independent p dimensional multivariate normal vectors with common unknown mean based on observing x x we consider the problem of estimating the true predictive density p yj of y under expected kullback leibler loss our focus here is the characterization of admissible procedures for this problem we show that the class of all generalized bayes rules is a complete class and that the easily interpretable conditions of brown and hwang 1982 are sucient for a formal bayes rule to be admissible 1 introduction let xj np v xi and yj np v yi be independent p dimensional multivariate normal vectors with a common unknown mean 2 r p we assume that vx 0 and vy 0 are known we let p xj and p yj denote the conditional densities of x and y suppressing the dependence onvx and vy throughout based on observing only x x we consider the problem of estimating the density p yj of y the natural action spacea0 consists of all proper densities on r p i e

harmonic bayesian prediction under alpha divergence we investigate bayesian shrinkage methods for constructing predictive distributions we consider the multivariate normal model with a known covariance matrix and show that the bayesian predictive density with respect to stein s harmonic prior dominates the best invariant bayesian predictive density when the dimension is greater than or equal to 3 alpha divergence from the true distribution to a predictive distribution is adopted as a loss function

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

a vector perturbation technique for near capacity multiantenna multiuser communication part ii perturbation recent theoretical results describing the sum capacity when using multiple antennas to communicate with multiple users in a known rich scattering environment have not yet been followed with practical transmission schemes that achieve this capacity we introduce a simple encoding algorithm that achieves near capacity at sum rates of tens of bits channel use the algorithm is a variation on channel inversion that regularizes the inverse and uses a sphere encoder to perturb the data to reduce the energy of the transmitted signal the paper is comprised of two parts in this second part we show that after the regularization of the channel inverse introduced in the first part a certain perturbation of the data using a sphere encoder can be chosen to further reduce the energy of the transmitted signal the performance difference with and without this perturbation is shown to be dramatic with the perturbation we achieve excellent performance at all signal to noise ratios the results of both uncoded and turbo coded simulations are presented

optimal regularity for the convex envelope and semiconvex functions related to supersolutions of fully nonlinear elliptic equations in this paper we prove optimal regularity for the convex envelope of supersolutions to general fully nonlinear elliptic equations with unbounded coefficients more precisely we deal with coefficients and right hand sides rhs in lq with q geq n this extends the result of caffarelli on the c loc 1 1 regularity of the convex envelope of supersolutions of fully nonlinear elliptic equations with bounded rhs moreover we also provide a regularity result with estimates for omega semiconvex functions that are supersolutions to the same type of equations with unbounded rhs i e rhs in l q q geq n by a completely different method our results here extend the recent regularity results obtained by braga et al adv math 334 184 242 2018 for q n as far as fully nonlinear pdes are concerned these results include in particular the apriori estimate obtained by caffarelli et al commun pure appl math 38 2 209 252 1985 on the modulus of continuity of the gradient of omega semiconvex supersolutions for linear equations and bounded rhs that have a holder modulus of semiconvexity

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

on asymptotic properties of predictive distributions summary second order asymptotic theory of predictive distributions is investigated it is shown that estimative distributions with asymptotically efficient estimators can be improved to predictive distributions that do not belong to the model the model is assumed to be a multidimensional curved exponential family predictive distributions can be constructed by shifting estimative distributions in a direction orthogonal to the model the average kullback leibler divergence from the true distribution to a predictive distribution is represented as the sum of two components one of them depends on the choice of the estimative distribution and the other depends on the shift orthogonal to the model the optimal shift orthogonal to the model is obtained a quantity that can be regarded as the mixture mean curvature of the model in the space of all probability distributions is introduced the difference in average kullback leibler divergence between an estimative distribution and the optimal predictive distribution obtained by shifting it can be regarded as the mixture mean curvature of the model an asymptotic expression for bayesian predictive distributions is also obtained

nonlocal curvature flows this paper aims at building a unified framework to deal with a wide class of local and nonlocal translation invariant geometric flows we introduce a class of nonlocal generalized mean curvatures and prove the existence and uniqueness for the level set formulation of the corresponding geometric flows we then introduce a class of generalized perimeters whose first variation is an admissible generalized curvature within this class we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set pde we also describe several examples and applications besides recovering and presenting in a unified way existence uniqueness and approximation results for several geometric motions already studied and scattered in the literature the theory developed in this paper also allows us to establish new results

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

nonlocal geometric expansion of convex planar curves abstract we consider a class of nonlocal geometric equations for expanding curves in the plane arising in the study of evolutions governed by monge kantorovich mass transfer we construct convex solutions given convex initial data in order to obtain such solutions we develop a new version of perron s method we give applications to the problem of characterizing fast slow diffusion limits

exact minimax estimation of the predictive density in sparse gaussian models we consider estimating the predictive density under kullback leibler loss in an l0 sparse gaussian sequence model explicit expressions of the first order minimax risk along with its exact constant asymptotically least favorable priors and optimal predictive density estimates are derived compared to the sparse recovery results involving point estimation of the normal mean new decision theoretic phenomena are seen suboptimal performance of the class of plug in density estimates reflects the predictive nature of the problem and optimal strategies need diversification of the future risk we find that minimax optimal strategies lie outside the gaussian family but can be constructed with threshold predictive density estimates novel minimax techniques involving simultaneous calibration of the sparsity adjustment and the risk diversification mechanisms are used to design optimal predictive density estimates

curvature driven flows a variational approach this paper introduces a new mathematical approach to the study of time evolutions of solids k t in n space whose boundaries move with velocity equal to the weighted mean curvature derived from the boundary surface energy phi partial k t these flat phi curvature flows are limits of sequences of solutions to variational problems in which a sum of surface and bulk energy is minimized the construction works equally well for smooth elliptic phi s for nondifferentiable crystalline phi s and for anything in between the flows agree with classical smooth flows when the data is smooth and elliptic in any dimension and coincide with motion by crystalline curvature for polyhedral curves in the plane

improved minimax predictive densities under kullback leibler loss let x n p u x i and y n p v y i be independentp dimensional multivariate normal vectors with common unknown mean based on only observing x x we consider the problem of obtaining a predictive density p y x for y that is close to p y as measured by expected kullback leibler loss a natural procedure for this problem is the formal bayes predictive density p u y x under the uniform prior u 1 which is best invariant and minimax we show that any bayes predictive density will be minimax if it is obtained by a prior yielding a marginal that is superharmonic or whose square root is superharmonic this yields wide classes of minimax procedures that dominate p u y x including bayes predictive densities under superharmonic priors fundamental similarities and differences with the parallel theory of estimating a multivariate normal mean under quadratic loss are described

a variational inequality approach to hele shaw flow with a moving boundary this paper is concerned with the study of a mathematical model of the injection of fluid into a finite hele shaw cell the mathematical problem is one of solving laplace s equation in an unknown region whose boundary changes with time by a transformation of the dependent variable an elliptic variational inequality formulation of the moving boundary problem is obtained the variational inequality is shown to have a unique solution up to the time at which the cell is filled regularity results for the solution of the inequality are obtained by studying a penalty approximation of the inequality

a shrinkage predictive distribution for multivariate normal observables summary we investigate shrinkage methods for constructing predictive distributions we consider the multivariate normal model with a known covariance matrix and show that there exists a shrinkage predictive distribution dominating the bayesian predictive distribution based on the vague prior when the dimension is not less than three kullback leibler divergence from the true distribution to a predictive distribution is adopted as a loss function somiie key w ords invariance james stein estimator kullback leibler divergence stein s prior vague prior

inhomogeneous hopf oleĭnik lemma and regularity of semiconvex supersolutions via new barriers for the pucci extremal operators abstract in this paper we construct new barriers for the pucci extremal operators with unbounded rhs the geometry of these barriers is given by a harnack inequality up to the boundary type estimate under the possession of these barriers we prove a new quantitative version of the hopf ole nik lemma for quasilinear elliptic equations with g laplace type growth finally we prove sharp regularity for semiconvex supersolutions for some nonlinear pdes these results are new even for second order linear elliptic equations in nondivergence form moreover these estimates extend and improve a classical a priori estimate proven by l caffarelli j j kohn j spruck and l nirenberg in 13 in 1985 as well as a more recent result on the c 1 1 regularity for convex supersolutions obtained by c imbert in 33 in 2006

some flows in shape optimization geometric flows related to shape optimization problems of the bernoulli type are investigated the evolution law is the sum of a curvature term and a nonlocal term of hele shaw type we introduce generalized set solutions the definition of which is widely inspired by viscosity solutions the main result is an inclusion preservation principle for generalized solutions as a consequence we obtain existence uniqueness and stability of solutions asymptotic behavior for the flow is discussed we prove that the solutions converge to a generalized bernoulli exterior free boundary problem

linear transmit processing in mimo communications systems we examine and compare the different types of linear transmit processing for multiple input multiple output systems where we assume that the receive filter is independent of the transmit filter contrary to the joint optimization of transmit and receive filters we can identify three filter types similar to receive processing the transmit matched filter the transmit zero forcing filter and the transmit wiener filter we show that the transmit filters are based on similar optimizations as the respective receive filters with an additional constraint for the transmit power moreover the transmit wiener filter has similar convergence properties as the receive wiener filter i e it converges to the matched filter and the zero forcing filter for low and high signal to noise ratio respectively we give closed form solutions for all transmit filters and present the fundamental result that their mean square errors are equal to the errors of the respective receive filters if the information symbols and the additive noise are uncorrelated however our simulations reveal that the bit error ratio results of the transmit filters differ from the results for the respective receive filters