multiscale computation method for parabolic problems of composite materials in this paper we consider the initial boundary value problem of parabolic type equation with rapidly oscillating coefficients in both time and space a multiscale asymptotic expansion of solution for this kind of problem is presented the full discrete finite element method for computing above problem is introduced this method can apply to heat conduction analysis of composite materials the main advantages of this method are that it can greatly save computer memory and cpu time and it has good precision at the same time finally numerical results show that the method presented in this paper is effective and reliable

randomized algorithms for the low rank approximation of matrices we describe two recently proposed randomized algorithms for the construction of low rank approximations to matrices and demonstrate their application inter alia to the evaluation of the singular value decompositions of numerically low rank matrices being probabilistic the schemes described here have a finite probability of failure in most cases this probability is rather negligible 10 17 is a typical value in many situations the new procedures are considerably more efficient and reliable than the classical deterministic ones they also parallelize naturally we present several numerical examples to illustrate the performance of the schemes

the rigorous approximation of long wavelength capillary gravity waves in a previous paper we proved that long wavelength solutions of the water wave problem in the case of zero surface tension split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries kdv equation in this paper we examine the effect of surface tension on this scenario we find that we obtain three different physical regimes depending on the strength of the surface tension for weak surface tension the propagation of the wave packets is very similar to that in the zero surface tension case for strong surface tension the evolution is again governed by a pair of kdv equations but the coefficients in these equations have changed in such a way that the kdv soliton now represents a wave of depression on the fluid surface finally at a special intermediate value of the surface tension where the bond number equals the kdv description breaks down and it is necessary to introduce a new approximating equation the kawahara equation which is a fifth order nonlinear partial differential equation in each of these regimes we give rigorous estimates of the difference between the solution of the appropriate modulation equation and the solution of the true water wave problem

unbiased risk estimates for singular value thresholding and spectral estimators in an increasing number of applications it is of interest to recover an approximately low rank data matrix from noisy observations this paper develops an unbiased risk estimate holding in a gaussian model for any spectral estimator obeying some mild regularity assumptions in particular we give an unbiased risk estimate formula for singular value thresholding svt a popular estimation strategy that applies a soft thresholding rule to the singular values of the noisy observations among other things our formulas offer a principled and automated way of selecting regularization parameters in a variety of problems in particular we demonstrate the utility of the unbiased risk estimation for svt based denoising of real clinical cardiac mri series data we also give new results concerning the differentiability of certain matrix valued functions

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

the hole filling method and multiscale algorithm for the heat transfer performance of periodic porous materials in this paper we consider the transient heat transfer problem with rapidly oscillating coefficients in periodic porous materials the hole filling method through filling all holes with a very compliant material is investigated and as the thermal conductivity parameters of the weak phase go to zero the proof of the limiting process is derived in detail then the multiscale analysis and computations based on the hole filling method for the associated multiphase problem in a domain without holes are proposed finally some numerical results are given which show a good agreement and verify the feasibility of the hole filling method a multiscale analysis and computation based on hole filling method is proposed transient heat transfer problem of periodic porous materials is considered proof of the limiting process as the parameters of the weak phase go to zero is derived some numerical results are given in detail to validate the multiscale method developed

fixed points of generalized approximate message passing with arbitrary matrices the estimation of a random vector with independent components passed through a linear transform followed by a componentwise possibly nonlinear output map arises in a range of applications approximate message passing amp methods based on gaussian approximations of loopy belief propagation have recently attracted considerable attention for such problems for large random transforms these methods exhibit fast convergence and admit precise analytic characterizations with testable conditions for optimality even for certain non convex problem instances however the behavior of amp under general transforms is not fully understood in this paper we consider the generalized amp gamp algorithm and relate the method to more common optimization techniques this analysis enables a precise characterization of the gamp algorithm fixed points that applies to arbitrary transforms in particular we show that the fixed points of the so called max sum gamp algorithm for map estimation are critical points of a constrained maximization of the posterior density the fixed points of the sum product gamp algorithm for estimation of the posterior marginals can be interpreted as critical points of a certain free energy

low rank matrix completion by riemannian optimization the matrix completion problem consists of finding or approximating a low rank matrix based on a few samples of this matrix we propose a new algorithm for matrix completion that minimizes the least square distance on the sampling set over the riemannian manifold of fixed rank matrices the algorithm is an adaptation of classical nonlinear conjugate gradients developed within the framework of retraction based optimization on manifolds we describe all the necessary objects from differential geometry necessary to perform optimization over this low rank matrix manifold seen as a submanifold embedded in the space of matrices in particular we describe how metric projection can be used as retraction and how vector transport lets us obtain the conjugate search directions finally we prove convergence of a regularized version of our algorithm under the assumption that the restricted isometry property holds for incoherent matrices throughout the iterations the numerical experiments indicate that our approach

multiscale asymptotic analysis and numerical simulation for the second order helmholtz equations with rapidly oscillating coefficients over general convex domains the multiscale asymptotic analysis and numerical simulation for the second order helmholtz equations with rapidly oscillating coefficients over general convex domains are discussed in this paper a multiscale asymptotic analysis formulation for this problem is presented by constructing properly the boundary layer a multiscale numerical algorithm and a postprocessing technique are given finally numerical results show that the method presented in this paper is effective and reliable

bilinear generalized approximate message passing part ii applications in this paper we extend the generalized approximate message passing g amp approach originally proposed for high dimensional generalized linear regression in the context of compressive sensing to the generalized bilinear case in part i of this two part paper we derived our bilinear g amp big amp algorithm as an approximation of the sum product belief propagation algorithm in the high dimensional limit and proposed an adaptive damping mechanism that aids convergence under finite problem sizes an expectation maximization em based method to automatically tune the parameters of the assumed priors and two rank selection strategies here in part ii we discuss the specializations of big amp to the problems of matrix completion robust pca and dictionary learning and present the results of an extensive empirical study comparing big amp to state of the art algorithms on each problem our numerical results using both synthetic and real world datasets demonstrate that em big amp yields excellent reconstruction accuracy often best in class while maintaining competitive runtimes

blind signal detection in massive mimo exploiting the channel sparsity in practical massive mimo systems a substantial portion of system resources are consumed to acquire channel state information csi leading to a drastically lower system capacity compared with the ideal case where perfect csi is available in this paper we show that the overhead for csi acquisition can be largely compensated by the potential gain due to the sparsity of the massive mimo channel in a certain transformed domain to this end we propose a novel blind detection scheme that simultaneously estimates the channel and data by factorizing the received signal matrix we show that by exploiting the channel sparsity our proposed scheme can achieve a degree of freedom dof very close to the ideal case provided that the channel is sufficiently sparse specifically the achievable dof has a fractional gap of only 1 t from the ideal dof where t is the channel coherence time this is a remarkable advance for understanding the performance limit of the massive mimo system we further show that the performance advantage of our proposed scheme in the asymptotic snr regime carries over to the practical snr regime numerical results demonstrate that our proposed scheme significantly outperforms its counterpart schemes in the practical snr regime under various system configurations

the validity of modulation equations for extended systems with cubic nonlinearities modulation equations play an essential role in the understanding of complicated systems near the threshold of instability here we show that the modulation equation dominates the dynamics of the full problem locally at least over a long time scale for systems with no quadratic interaction term we develop a method which is much simpler than previous ones it involves a careful bookkeeping of errors and an estimate of gronwall type as an example for the dissipative case we find that the ginzburg landau equation is the modulation equation for the swift hohenberg problem moreover the method also enables us to handle hyperbolic problems the nonlinear schrodinger equation is shown to describe the modulation of wave packets in the sine gordon equation

matrix completion from noisy entries given a matrix m of low rank we consider the problem of reconstructing it from noisy observations of a small random subset of its entries the problem arises in a variety of applications from collaborative filtering the netflix problem to structure from motion and positioning we study a low complexity algorithm introduced by keshavan montanari and oh 2010 based on a combination of spectral techniques and manifold optimization that we call here optspace we prove performance guarantees that are order optimal in a number of circumstances

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

multiscale computation method for an advection diffusion equation abstract in this paper we consider the initial boundary value problem of the advection diffusion equation with rapidly oscillating coefficients in both time and space domain a multiscale asymptotic expansion of the solution for this kind of problem is presented a full discrete finite element method for computing above mentioned problem is introduced the proposed method not only can greatly save computer memory and cpu time but also has high precision for approximating the solution the numerical results show that the method presented in this paper is effective and reliable

matrix completion from a few entries let m be an n n matrix of rank r and assume that a uniformly random subset e of its entries is observed we describe an efficient algorithm which we call optspace that reconstructs m from e o rn observed entries with relative root mean square error 1 2 rmse c nr e 1 2 with probability larger than 1 1 n3 further if r o 1 and m is sufficiently unstructured then optspace reconstructs it exactly from e o n log n entries with probability larger than 1 1 n3 this settles in the case of bounded rank a question left open by candes and recht and improves over the guarantees for their reconstruction algorithm the complexity of our algorithm is o e r log n which opens the way to its use for massive data sets in the process of proving these statements we obtain a generalization of a celebrated result by friedman kahn szemeredi and feige ofek on the spectrum of sparse random matrices

recovering low rank and sparse components of matrices from incomplete and noisy observations many problems can be characterized by the task of recovering the low rank and sparse components of a given matrix recently it was discovered that this nondeterministic polynomial time hard np hard task can be well accomplished both theoretically and numerically via heuristically solving a convex relaxation problem where the widely acknowledged nuclear norm and l 1 norm are utilized to induce low rank and sparsity this paper studies the recovery task in the general settings that only a fraction of entries of the matrix can be observed and the observation is corrupted by both impulsive and gaussian noise we show that the resulting model falls into the applicable scope of the classical augmented lagrangian method moreover the separable structure of the new model enables us to solve the involved subproblems more efficiently by splitting the augmented lagrangian function hence some splitting numerical algorithms are developed for solving the new recovery model some preliminary numerical experiments verify that these augmented lagrangian based splitting algorithms are easily implementable and surprisingly efficient for tackling the new recovery model

on l_q optimization and matrix completion rank minimization problems which consist of finding a matrix of minimum rank subject to linear constraints have been proposed in many areas of engineering and science a specific problem is the matrix completion problem in which a low rank data matrix can be recovered from incomplete samples of its entries by solving a rank penalized least squares problem the rank penalty is in fact the l0 norm of the matrix singular values a recent convex relaxation of this penalty is the commonly used l1 norm of the matrix singular values in this paper we bridge the gap between these two penalties and propose the lq 0 q 1 penalized least squares problem for matrix completion an iterative algorithm is developed by solving a non standard optimization problem and a non trivial convergence result is proved we illustrate with simulations comparing the reconstruction quality of the three matrix singular value penalty functions l0 l1 and lq 0 q 1

kawahara dynamics in dispersive media abstract in a previous paper it was shown that in the long wave limit the water wave problem without surface tension can be described approximately by two decoupled kdv equations here we study a model problem and prove that in a degenerate case which occurs for the water wave problem with surface tension and near other codimension two points at which the coefficient in front of the leading order dispersive term in the equation of motion vanishes the long wave limit can be rigorously approximated by two decoupled kawahara equations

near optimal matrix recovery from random linear measurements in matrix recovery from random linear measurements one is interested in recovering an unknown m by n matrix x 0 from n m n measurements y i t r a i x 0 where each a i is an m by n measurement matrix with i i d random entries i 1 n we present a matrix recovery algorithm based on approximate message passing which iteratively applies an optimal singular value shrinker a nonconvex nonlinearity tailored specifically for matrix estimation our algorithm typically converges exponentially fast offering a significant speedup over previously suggested matrix recovery algorithms such as iterative solvers for nuclear norm minimization nnm it is well known that there is a recovery tradeoff between the information content of the object x 0 to be recovered specifically its matrix rank r and the number of linear measurements n from which recovery is to be attempted the precise tradeoff between r and n beyond which recovery by a given algorithm becomes possible traces the so called phase transition curve of that algorithm in the r n plane the phase transition curve of our algorithm is noticeably better than that of nnm interestingly it is close to the information theoretic lower bound for the minimal number of measurements needed for matrix recovery making it not only state of the art in terms of convergence rate but also near optimal in terms of the matrices it successfully recovers

bayesian robust principal component analysis a hierarchical bayesian model is considered for decomposing a matrix into low rank and sparse components assuming the observed matrix is a superposition of the two the matrix is assumed noisy with unknown and possibly non stationary noise statistics the bayesian framework infers an approximate representation for the noise statistics while simultaneously inferring the low rank and sparse outlier contributions the model is robust to a broad range of noise levels without having to change model hyperparameter settings in addition the bayesian framework allows exploitation of additional structure in the matrix for example in video applications each row or column corresponds to a video frame and we introduce a markov dependency between consecutive rows in the matrix corresponding to consecutive frames in the video the properties of this markov process are also inferred based on the observed matrix while simultaneously denoising and recovering the low rank and sparse components we compare the bayesian model to a state of the art optimization based implementation of robust pca considering several examples we demonstrate competitive performance of the proposed model

optimal shrinkage of singular values we consider the recovery of low rank matrices from noisy data by shrinkage of singular values in which a single univariate nonlinearity is applied to each of the empirical singular values we adopt an asymptotic framework in which the matrix size is much larger than the rank of the signal matrix to be recovered and the signal to noise ratio of the low rank piece stays constant for a variety of loss functions including mean square error mse square frobenius norm the nuclear norm loss and the operator norm loss we show that in this framework there is a well defined asymptotic loss that we evaluate precisely in each case in fact each of the loss functions we study admits a unique admissible shrinkage nonlinearity dominating all other nonlinearities we provide a general method for evaluating these optimal nonlinearities and demonstrate our framework by working out simple explicit formulas for the optimal nonlinearities in the frobenius nuclear and operator norm cases for example for a square low rank n by n matrix observed in white noise with level sigma the optimal nonlinearity for mse loss simply shrinks each data singular value y to sqrt y 2 4n sigma 2 or to 0 if y this optimal nonlinearity guarantees an asymptotic mse of 2nr sigma 2 which compares favorably with optimally tuned hard thresholding and optimally tuned soft thresholding providing guarantees of 3nr sigma 2 and 6nr sigma 2 respectively our general method also allows one to evaluate optimal shrinkers numerically to arbitrary precision as an example we compute optimal shrinkers for the schatten p norm loss for any p 0

the phase transition of matrix recovery from gaussian measurements matches the minimax mse of matrix denoising let be an unknown m by n matrix in matrix recovery one takes linear measurements of where and each is an m by n matrix a popular approach for matrix recovery is nuclear norm minimization nnm solving the convex optimization problem for all where denotes the nuclear norm namely the sum of singular values empirical work reveals a phase transition curve stated in terms of the undersampling fraction rank fraction and aspect ratio specifically when the measurement matrices ai have independent standard gaussian random entries a curve exists such that if nnm typically succeeds for large m n whereas if it typically fails an apparently quite different problem is matrix denoising in gaussian noise in which an unknown m by n matrix is to be estimated based on direct noisy measurements where the matrix z has independent and identically distributed gaussian entries a popular matrix denoising scheme solves the unconstrained optimization problem when optimally tuned this scheme achieves the asymptotic minimax mean squared error where we report extensive experiments showing that the phase transition in the first problem matrix recovery from gaussian measurements coincides with the minimax risk curve in the second problem matrix denoising in gaussian noise for any rank fraction at each common aspect ratio our experiments considered matrices belonging to two constraint classes real m by n matrices of various ranks and aspect ratios and real symmetric positive semidefinite n by n matrices of various ranks

secular growth estimates for hyperbolic systems abstract we consider a model problem for the secular growth which covers all the cases likely to happen in multi scales bkw expansions for nonlinear hyperbolic systems this model problem consists in studying the growth in time of the solution of a nonhomogeneous hyperbolic system whose source term is a product of various functions which solve homogeneous hyperbolic systems the secular growth is due to resonances that we try to control when this is not possible other tools such as decay properties or strichartz estimates must be used

bifurcation from rolls to multi pulse planforms via reduction to a parabolic boussinesq model a mechanism is presented for the bifurcation from one dimensional spatially periodic patterns rolls into two dimensional planar states planforms the novelty is twofold the planforms are solutions of a boussinesq partial differential equation pde on a periodic background and secondly explicit formulas for the coefficients in the boussinesq equation are derived based on a form of planar conservation of wave action flux the boussinesq equation is integrable with a vast array of solutions and an example of a new planform bifurcating from rolls which appears to be generic is presented adding in time leads to a new time dependent pde which models the nonlinear behaviour emerging from a generalization of eckhaus instability the class of pdes to which the theory applies is evolution equations whose steady part is a gradient elliptic pde examples are the 2 1 ginzburg landau equation with real coefficients and the 2 1 planar swift hohenberg equation 2014 elsevier b v all rights reserved

hybrid approximate message passing gaussian and quadratic approximations of message passing algorithms on graphs have attracted considerable recent attention due to their computational simplicity analytic tractability and wide applicability in optimization and statistical inference problems this paper presents a systematic framework for incorporating such approximate message passing amp methods in general graphical models the key concept is a partition of dependencies of a general graphical model into strong and weak edges with the weak edges representing small linearizable couplings of variables amp approximations based on the central limit theorem can be readily applied to aggregates of many weak edges and integrated with standard message passing updates on the strong edges the resulting algorithm which we call hybrid generalized approximate message passing hygamp can yield significantly simpler implementations of sum product and max sum loopy belief propagation by varying the partition of strong and weak edges a performance complexity tradeoff can be achieved group sparsity and multinomial logistic regression problems are studied as examples of the proposed methodology

non negative principal component analysis message passing algorithms and sharp asymptotics principal component analysis pca aims at estimating the direction of maximal variability of a high dimensional data set a natural question is does this task become easier and estimation more accurate when we exploit additional knowledge on the principal vector we study the case in which the principal vector is known to lie in the positive orthant similar constraints arise in a number of applications ranging from the analysis of gene expression data to spike sorting in neural signal processing in the unconstrained case the estimation performances of pca have been precisely characterized using the random matrix theory under a statistical model known as the spiked model it is known that the estimation error undergoes a phase transition as the signal to noise ratio crosses a certain threshold unfortunately tools from the random matrix theory have no bearing on the constrained problem despite this challenge we develop an analogous characterization in the constrained case within a one spike model in particular 1 we prove that the estimation error undergoes a similar phase transition albeit at a different thresholds in signal to noise ratio that we determine exactly 2 we prove that unlike in the unconstrained case the estimation error depends on the spike vector and characterize the least favorable vectors and 3 we show that a non negative principal component can be approximately computed under the spiked model in nearly linear time this despite the fact that the problem is non convex and in general np hard to solve exactly

multiscale asymptotic expansion and a post processing algorithm for second order elliptic problems with highly oscillatory coefficients over general convex domains in this paper we consider the boundary value problem of second order elliptic type equation with highly oscillatory coefficients over general lipschitz convex domains and obtain the multiscale asymptotic expansion of solution for this kind of problem at the same time a high accuracy fe computing scheme and a post processing technique are presented finally numerical results support strongly the theoretical analysis reported in this paper

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

on the minimax risk of dictionary learning we consider the problem of learning a dictionary matrix from a number of observed signals which are assumed to be generated via a linear model with a common underlying dictionary in particular we derive lower bounds on the minimum achievable worst case mean squared error mse regardless of computational complexity of the dictionary learning dl schemes by casting dl as a classical or frequentist estimation problem the lower bounds on the worst case mse are derived following an established information theoretic approach to minimax estimation the main contribution of this paper is the adaption of these information theoretic tools to the dl problem in order to derive lower bounds on the worst case mse of any dl algorithm we derive three different lower bounds applying to different generative models for the observed signals the first bound only requires the existence of a covariance matrix of the unknown underlying coefficient vector by specializing this bound to the case of sparse coefficient distributions and assuming the true dictionary satisfies the restricted isometry property we obtain a lower bound on the worst case mse of dl methods in terms of the signal to noise ratio snr the third bound applies to a more restrictive subclass of coefficient distributions by requiring the non zero coefficients to be gaussian although the applicability of this bound is the most limited it is the tightest of the three bounds in the low snr regime a particular use of our lower bounds is the derivation of necessary conditions on the required number of observations sample size such that dl is feasible i e accurate dl schemes might exist by comparing these necessary conditions with sufficient conditions on the sample size such that a particular dl technique is successful we are able to characterize the regimes where those algorithms are optimal in terms of required sample size

estimating extremal eigenvalues and condition numbers of matrices the paper describes probabilistic algorithms which may be used to make rough estimates of the largest and smallest eigenvalues of a positive definite matrix and the condition number of a nonsingular matrix in the 2 norm given varepsilon 0 and a prescribed relative error the algorithms compute estimates which with probability at least 1 varepsilon have relative errors less than that prescribed in particular the method gives a reliable way to estimate the condition number of a matrix of large degree

a nonconvex splitting method for symmetric nonnegative matrix factorization convergence analysis and optimality symmetric nonnegative matrix factorization symnmf has important applications in data analytics problems such as document clustering community detection and image segmentation in this paper we propose a novel nonconvex variable splitting method for solving symnmf the proposed algorithm is guaranteed to converge to the set of karush kuhn tucker kkt points of the nonconvex symnmf problem furthermore it achieves a global sublinear convergence rate we also show that the algorithm can be efficiently implemented in parallel further sufficient conditions are provided that guarantee the global and local optimality of the obtained solutions extensive numerical results performed on both synthetic and real datasets suggest that the proposed algorithm converges quickly to a local minimum solution

minimax risk of matrix denoising by singular value thresholding an unknown m by n matrix x 0 is to be estimated from noisy measurements y x 0 z where the noise matrix z has i i d gaussian entries a popular matrix denoising scheme solves the nuclear norm penalization problem operatorname min x y x f 2 2 lambda x where x denotes the nuclear norm sum of singular values this is the analog for matrices of ell 1 penalization in the vector case it has been empirically observed that if x 0 has low rank it may be recovered quite accurately from the noisy measurement y in a proportional growth framework where the rank r n number of rows m n and number of columns n all tend to infty proportionally to each other r n m n rightarrow rho m n n rightarrow beta we evaluate the asymptotic minimax mse mathcal m rho beta lim m n n rightarrow infty inf lambda sup operatorname rank x leq r n operatorname mse x 0 hat x lambda our formulas involve incomplete moments of the quarter and semi circle laws beta 1 square case and the marcenko pastur law beta 1 nonsquare case for finite m and n we show that mse increases as the nonzero singular values of x 0 grow larger as a result the finite n worst case mse a quantity which can be evaluated numerically is achieved when the signal x 0 is infinitely strong the nuclear norm penalization problem is solved by applying soft thresholding to the singular values of y we also derive the minimax threshold namely the value lambda rho which is the optimal place to threshold the singular values all these results are obtained for general nonsquare nonsymmetric real matrices comparable results are obtained for square symmetric nonnegative definite matrices

finite sample analysis of approximate message passing algorithms approximate message passing amp refers to a class of efficient algorithms for statistical estimation in high dimensional problems such as compressed sensing and low rank matrix estimation this paper analyzes the performance of amp in the regime where the problem dimension is large but finite for concreteness we consider the setting of high dimensional regression where the goal is to estimate a high dimensional vector beta 0 from a noisy measurement y a beta 0 w amp is a low complexity scalable algorithm for this problem under suitable assumptions on the measurement matrix a amp has the attractive feature that its performance can be accurately characterized in the large system limit by a simple scalar iteration called state evolution previous proofs of the validity of state evolution have all been asymptotic convergence results in this paper we derive a concentration inequality for amp with gaussian matrices with independent and identically distributed i i d entries and finite dimension n times n the result shows that the probability of deviation from the state evolution prediction falls exponentially in n this provides theoretical support for empirical findings that have demonstrated excellent agreement of amp performance with state evolution predictions for moderately large dimensions the concentration inequality also indicates that the number of amp iterations t can grow no faster than order log n log log n for the performance to be close to the state evolution predictions with high probability the analysis can be extended to obtain similar non asymptotic results for amp in other settings such as low rank matrix estimation

multi scale method for the quasi periodic structures of composite materials in this paper we shall present a new second order and two scale approximate solution of boundary value problems for the linear elasticity systems with quasi periodic structures by multi scale analysis the computation of strain and stress is presented here this idea can apply to the damage computation of composite materials because the computation of the local stress by this method having more precision it can provide some inspiration for construct optimal design finally numerical results show that the method presented in this paper is effective and reliable

bilinear generalized approximate message passing part i derivation in this paper we extend the generalized approximate message passing g amp approach originally proposed for high dimensional generalized linear regression in the context of compressive sensing to the generalized bilinear case which enables its application to matrix completion robust pca dictionary learning and related matrix factorization problems here in part i of a two part paper we derive our bilinear g amp big amp algorithm as an approximation of the sum product belief propagation algorithm in the high dimensional limit where central limit theorem arguments and taylor series approximations apply and under the assumption of statistically independent matrix entries with known priors in addition we propose an adaptive damping mechanism that aids convergence under finite problem sizes an expectation maximization em based method to automatically tune the parameters of the assumed priors and two rank selection strategies in part ii of the paper we will discuss the specializations of em big amp to the problems of matrix completion robust pca and dictionary learning and we will present the results of an extensive empirical study comparing em big amp to state of the art algorithms on each problem

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

the long wave limit for the water wave problem i the case of zero surface tension the korteweg de vries equation boussinesq equation and many other equations can be formally derived as approximate equations for the two dimensional water wave problem in the limit of long waves here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension we prove that the solutions of the water wave problem in the long wave limit split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries equation our result allows us to describe the nonlinear interaction of solitary waves 2000 john wiley sons inc

fast low rank bayesian matrix completion with hierarchical gaussian prior models the problem of low rank matrix completion is considered in this paper to exploit the underlying low rank structure of the data matrix we propose a hierarchical gaussian prior model where columns of the low rank matrix are assumed to follow a gaussian distribution with zero mean and a common precision matrix and a wishart distribution is specified as a hyperprior over the precision matrix we show that such a hierarchical gaussian prior has the potential to encourage a low rank solution based on the proposed hierarchical prior model we develop a variational bayesian matrix completion method which embeds the generalized approximate massage passing technique to circumvent cumbersome matrix inverse operations simulation results show that our proposed method demonstrates superiority over some state of the art matrix completion methods

recovery of low rank matrices under affine constraints via a smoothed rank function in this paper the problem of matrix rank minimization under affine constraints is addressed the state of the art algorithms can recover matrices with a rank much less than what is sufficient for the uniqueness of the solution of this optimization problem we propose an algorithm based on a smooth approximation of the rank function which practically improves recovery limits on the rank of the solution this approximation leads to a non convex program thus to avoid getting trapped in local solutions we use the following scheme initially a rough approximation of the rank function subject to the affine constraints is optimized as the algorithm proceeds finer approximations of the rank are optimized and the solver is initialized with the solution of the previous approximation until reaching the desired accuracy on the theoretical side benefiting from the spherical section property we will show that the sequence of the solutions of the approximating programs converges to the minimum rank solution on the experimental side it will be shown that the proposed algorithm termed srf standing for smoothed rank function can recover matrices which are unique solutions of the rank minimization problem and yet not recoverable by nuclear norm minimization furthermore it will be demonstrated that in completing partially observed matrices the accuracy of srf is considerably and consistently better than some famous algorithms when the number of revealed entries is close to the minimum number of parameters that uniquely represent a low rank matrix

the long wave limit for a boussinesq equation we consider the long wave limit for a boussinesq equation it is shown that for spatially localized initial conditions the dynamics are under a korteweg de vries kdv regime on a large time interval i e in this limit the solutions split up into two counterpropagating wave packets where each of the wave packets evolve independently and approximately as a solution of a kdv equation for the proof exact estimates between the long wave solutions of the boussinesq equation and the approximations obtained via the decoupled set of kdv equations are demonstrated we expect that this result holds for the water wave problem too