on the energy equality for solutions to newtonian and non newtonian fluids abstract we are concerned with the energy equality for weak solutions to newtonian and non newtonian incompressible fluids furthermore a new result allows an interpretation of the classical shinbrot condition as a measure of strength which looks coherent to the more recent results in the literature

rigorous remarks about scaling laws in turbulent fluids a definition of scaling law for suitable families of measures is given and investigated first a number of necessary conditions are proved they imply the absence of scaling laws for 2d stochastic navier stokes equations and for the stochastic stokes linear problem in any dimension while they imply a lower bound on the mean vortex stretching in 3d second for the 3d stochastic navier stokes equations necessary and sufficient conditions for scaling laws to hold are given translating the problem into bounds for energy and enstrophy of high and low modes respectively unlike in the 2d case the validity or invalidity of such conditions in 3d remains open

primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

analytic solutions for a stefan problem with gibbs thomson correction we provide existence of a unique smooth solution for a class of oneand two phase stefan problems with gibbs thomson correction in arbitrary space dimensions in addition it is shown that the moving interface depends analytically on the temporal and spatial variables of crucial importance for the analysis is the property of maximal lpregularity for the linearized problem which is fully developed in this paper as well

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

kronecker basis representation based tensor sparsity and its applications to tensor recovery as a promising way for analyzing data sparse modeling has achieved great success throughout science and engineering it is well known that the sparsity low rank of a vector matrix can be rationally measured by nonzero entries number l 0 norm nonzero singular values number rank respectively however data from real applications are often generated by the interaction of multiple factors which obviously cannot be sufficiently represented by a vector matrix while a high order tensor is expected to provide more faithful representation to deliver the intrinsic structure underlying such data ensembles unlike the vector matrix case constructing a rational high order sparsity measure for tensor is a relatively harder task to this aim in this paper we propose a measure for tensor sparsity called kronecker basis representation based tensor sparsity measure kbr briefly which encodes both sparsity insights delivered by tucker and candecomp parafac cp low rank decompositions for a general tensor then we study the kbr regularization minimization kbrm problem and design an effective admm algorithm for solving it where each involved parameter can be updated with closed form equations such an efficient solver makes it possible to extend kbr to various tasks like tensor completion and tensor robust principal component analysis a series of experiments including multispectral image msi denoising msi completion and background subtraction substantiate the superiority of the proposed methods beyond state of the arts

bilinear factor matrix norm minimization for robust pca algorithms and applications the heavy tailed distributions of corrupted outliers and singular values of all channels in low level vision have proven effective priors for many applications such as background modeling photometric stereo and image alignment and they can be well modeled by a hyper laplacian however the use of such distributions generally leads to challenging non convex non smooth and non lipschitz problems and makes existing algorithms very slow for large scale applications together with the analytic solutions to ell p norm minimization with two specific values of p i e p 1 2 and p 2 3 we propose two novel bilinear factor matrix norm minimization models for robust principal component analysis we first define the double nuclear norm and frobenius nuclear hybrid norm penalties and then prove that they are in essence the schatten 1 2 and 2 3 quasi norms respectively which lead to much more tractable and scalable lipschitz optimization problems our experimental analysis shows that both our methods yield more accurate solutions than original schatten quasi norm minimization even when the number of observations is very limited finally we apply our penalties to various low level vision problems e g text removal moving object detection image alignment and inpainting and show that our methods usually outperform the state of the art methods

convex low rank approximation low rank approximation is an important tool in many applications given an observed matrix with elements corrupted by gaussian noise it is possible to find the best approximating matrix of a given rank through singular value decomposition however due to the non convexity of the formulation it is not possible to incorporate any additional knowledge of the sought matrix without resorting to heuristic optimization techniques in this paper we propose a convex formulation that is more flexible in that it can be combined with any other convex constraints and penalty functions the formulation uses the so called convex envelope which is the provably best possible convex relaxation we show that for a general class of problems the envelope can be efficiently computed and may in some cases even have a closed form expression we test the algorithm on a number of real and synthetic data sets and show state of the art results

a simpler approach to matrix completion this paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix these results improve on prior work by candes and recht 2009 candes and tao 2009 and keshavan et al 2009 the reconstruction is accomplished by minimizing the nuclear norm or sum of the singular values of the hidden matrix subject to agreement with the provided entries if the underlying matrix satisfies a certain incoherence condition then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition the proof of this assertion is short self contained and uses very elementary analysis the novel techniques herein are based on recent work in quantum information theory

a proof of onsager s conjecture for any 1 3 we construct weak solutions to the 3d incompressible euler equations in the class c tc x that have nonempty compact support in time on r t 3 and therefore fail to conserve the total kinetic energy this result together with the proof of energy conservation for 1 3 due to eyink and constantin e titi solves onsager s conjecture that the exponent 1 3 marks the threshold for conservation of energy for weak solutions in the class l t c x the previous best results were solutions in the classc tc x for 1 5 due to isett and in the class l t 1c x for 1 3 due to buckmaster de lellis szekelyhidi both based on the method of convex integration developed for the incompressible euler equations by de lellis szekelyhidi the present proof combines the method of convex integration and a new gluing approximation technique the convex integration part of the proof relies on the mikado flows introduced by daneri szekelyhidi and the framework of estimates developed in the author s previous work

nonconvex nonsmooth low rank minimization via iteratively reweighted nuclear norm the nuclear norm is widely used as a convex surrogate of the rank function in compressive sensing for low rank matrix recovery with its applications in image recovery and signal processing however solving the nuclear norm based relaxed convex problem usually leads to a suboptimal solution of the original rank minimization problem in this paper we propose to use a family of nonconvex surrogates of l 0 norm on the singular values of a matrix to approximate the rank function this leads to a nonconvex nonsmooth minimization problem then we propose to solve the problem by an iteratively reweighted nuclear norm irnn algorithm irnn iteratively solves a weighted singular value thresholding problem which has a closed form solution due to the special properties of the nonconvex surrogate functions we also extend irnn to solve the nonconvex problem with two or more blocks of variables in theory we prove that the irnn decreases the objective function value monotonically and any limit point is a stationary point extensive experiments on both synthesized data and real images demonstrate that irnn enhances the low rank matrix recovery compared with the state of the art convex algorithms

sensitivity to basis mismatch in compressed sensing the theory of compressed sensing suggests that successful inversion of an image of the physical world broadly defined to include speech signals radar sonar returns vibration records sensor array snapshot vectors 2 d images and so on for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis provided that the image is sparse in an apriori known basis for imaging problems in spectrum analysis and passive and active radar sonar this basis is usually taken to be a dft basis however in reality no physical field is sparse in the dft basis or in any apriori known basis no matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity in this paper we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases we start by analyzing the effect of basis mismatch on the best k term approximation error which is central to providing exact sparse recovery guarantees we establish achievable bounds for the l1 error of the best k term approximation and show that these bounds grow linearly with the image or grid dimension and the mismatch level between the assumed and actual bases for sparsity we then derive bounds with similar growth behavior for the basis pursuit l1 recovery error indicating that the sparse recovery may suffer large errors in the presence of basis mismatch although we present our results in the context of basis pursuit our analysis applies to any sparse recovery principle that relies on the accuracy of best k term approximations for its performance guarantees we particularly highlight the problematic nature of basis mismatch in fourier imaging where spillage from off grid dft components turns a sparse representation into an incompressible one we substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch for problem sizes common to radar and sonar

onsager s conjecture on the energy conservation for solutions of euler s equation we give a simple proof of a result conjectured by onsager 1 on energy conservation for weak solutions of euler s equation

robust matrix factorization by majorization minimization l 1 norm based low rank matrix factorization in the presence of missing data and outliers remains a hot topic in computer vision due to non convexity and non smoothness all the existing methods either lack scalability or robustness or have no theoretical guarantee on convergence in this paper we apply the majorization minimization technique to solve this problem at each iteration we upper bound the original function with a strongly convex surrogate by minimizing the surrogate and updating the iterates accordingly the objective function has sufficient decrease which is stronger than just being non increasing that other methods could offer as a consequence without extra assumptions we prove that any limit point of the iterates is a stationary point of the objective function in comparison other methods either do not have such a convergence guarantee or require extra critical assumptions extensive experiments on both synthetic and real data sets testify to the effectiveness of our algorithm the speed of our method is also highly competitive

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

trace norm regularization reformulations algorithms and multi task learning we consider a recently proposed optimization formulation of multi task learning based on trace norm regularized least squares while this problem may be formulated as a semidefinite program sdp its size is beyond general sdp solvers previous solution approaches apply proximal gradient methods to solve the primal problem we derive new primal and dual reformulations of this problem including a reduced dual formulation that involves minimizing a convex quadratic function over an operator norm ball in matrix space this reduced dual problem may be solved by gradient projection methods with each projection involving a singular value decomposition the dual approach is compared with existing approaches and its practical effectiveness is illustrated on simulations and an application to gene expression pattern analysis

reduction of average path length in binary decision diagrams by spectral methods this paper deals with analytic methods for the calculation and reduction of the average path lengths apls in binary decision diagrams bdds usually information theoretic measures and information theoretic techniques are used to construct bdds of minimal apls specifically the mutual information between a boolean function and its variables and the shannon fano prefix coding are utilized this paper deals with the problem of the apl reduction by using spectral techniques particularly methods based on the properties of the walsh spectrum of a boolean function and its autocorrelation function are discussed it is shown that the apl is a linear function of the autocorrelation values that is the apl depends on the boolean function s properties in the time domain autocorrelation rather on the boolean function s properties in the frequency domain walsh spectrum in addition it is shown that information theoretic criteria like the mutual information or the conditional entropy are equivalent to frequency domain criteria consequently existing information theoretic approaches for apl reduction that are based on mutual information criterion or the walsh transform coefficients may derive a suboptimal apl the representation of the apl as a function of the autocorrelation values opens a way to determine the optimal ordering of the input variables analytically two procedures for apl reduction by ordering and by using linear combinations of the input variables are presented 1 minimization by using the autocorrelation values and 2 minimization by using the mutual information between the boolean function and a linear function of the input variables the time domain approach may derive a linearized bdd of a lower apl whereas the information theoretic approach has a lower computational complexity with comparable performance experimental results show the efficiency of the suggested techniques

on the space time regularity of c 0 t ln very weak solutions to the navier stokes equations abstract in this paper we consider the initial boundary value problem for the navier stokes equations in a domain r n we prove the space time regularity of very weak solutions belonging to c 0 t l n in particular we prove regularity for a class of solutions that are possibly not weak solutions in the sense of leray hopf

discrete transforms semidefinite programming and sum of squares representations of nonnegative polynomials we present a new semidefinite programming formulation of sum of squares representations of nonnegative polynomials cosine polynomials and trigonometric polynomials of one variable the parametrization is based on discrete transforms specifically the discrete fourier cosine and polynomial transforms and has a simple structure that can be exploited by straightforward modifications of standard interior point algorithms

low rank approximation via generalized reweighted iterative nuclear and frobenius norms the low rank approximation problem has recently attracted wide concern due to its excellent performance in real world applications such as image restoration traffic monitoring and face recognition compared with the classic nuclear norm the schatten p norm is stated to be a closer approximation to restrain the singular values for practical applications in the real world however schatten p norm minimization is a challenging non convex non smooth and non lipschitz problem in this paper inspired by the reweighted ell 1 and ell 2 norm for compressive sensing the generalized iterative reweighted nuclear norm girnn and the generalized iterative reweighted frobenius norm girfn algorithms are proposed to approximate schatten p norm minimization by involving the proposed algorithms the problem becomes more tractable and the closed solutions are derived from the iteratively reweighted subproblems in addition we prove that both proposed algorithms converge at a linear rate to a bounded optimum numerical experiments for the practical matrix completion mc robust principal component analysis rpca and image decomposition problems are illustrated to validate the superior performance of both algorithms over some common state of the art methods

matrix completion by least square low rank and sparse self representations abstract conventional matrix completion methods are generally based on rank minimization these methods assume that the given matrix is of low rank and the data points are drawn from a single subspace of low dimensionality therefore they are not effective in completing matrices where the data are drawn from multiple subspaces in this paper we establish a novel matrix completion framework that is based on self representation specifically least square low rank and sparse self representations based matrix completion algorithms are provided the underlying idea is that one data point can be efficiently reconstructed by other data points belonging to a common subspace where the missing entries are recovered so as to fit the common subspace the proposed algorithms actually maximize the weighted correlations among the columns of a given matrix we prove that the proposed algorithms are approximations for rank minimization of the incomplete matrix in addition they are able to complete high rank or even full rank matrices when the data are drawn from multiple subspaces comparative studies are conducted on synthetic datasets natural image inpainting tasks temperature prediction task and collaborative filtering tasks the results show that the proposed algorithms often outperform other state of the art algorithms in various tasks

on the energy equality for the 3d navier stokes equations abstract in this paper we study the problem of energy conservation for the solutions of the initial boundary value problem associated to the 3d navier stokes equations with dirichlet boundary conditions first we consider leray hopf weak solutions and we prove some new criteria involving the gradient of the velocity next we compare them with the existing literature in scaling invariant spaces and with the onsager conjecture then we consider the problem of energy conservation for very weak solutions proving energy equality for distributional solutions belonging to the so called shinbrot class a possible explanation of the role of this classical class of solutions which is not scaling invariant is also given

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

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

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

principal component analysis with missing data and its application to polyhedral object modeling observation based object modeling often requires integration of shape descriptions from different views to overcome the problems of errors and their accumulation we have developed a weighted least squares wls approach which simultaneously recovers object shape and transformation among different views without recovering interframe motion we show that object modeling from a range image sequence is a problem of principal component analysis with missing data pcamd which can be generalized as a wls minimization problem an efficient algorithm is devised after we have segmented planar surface regions in each view and tracked them over the image sequence we construct a normal measurement matrix of surface normals and a distance measurement matrix of normal distances to the origin for all visible regions over the whole sequence of views respectively these two matrices which have many missing elements due to noise occlusion and mismatching enable us to formulate multiple view merging as a combination of two wls problems a two step algorithm is presented after surface equations are extracted spatial connectivity among the surfaces is established to enable the polyhedral object model to be constructed experiments using synthetic data and real range images show that our approach is robust against noise and mismatching and generates accurate polyhedral object models

improving the variable ordering of obdds is np complete ordered binary decision diagrams are a useful representation of boolean functions if a good variable ordering is known variable orderings are computed by heuristic algorithms and then improved with local search and simulated annealing algorithms this approach is based on the conjecture that the following problem is np complete given an obdd g representing f and a size bound s does there exist an obdd g respecting an arbitrary variable ordering representing f with at most s nodes this conjecture is proved

alternating direction method with gaussian back substitution for separable convex programming we consider the linearly constrained separable convex minimization problem whose objective function is separable into m individual convex functions with nonoverlapping variables a douglas rachford alternating direction method of multipliers adm has been well studied in the literature for the special case of m 2 but the convergence of extending adm to the general case of m ge 3 is still open in this paper we show that the straightforward extension of adm is valid for the general case of m ge 3 if it is combined with a gaussian back substitution procedure the resulting adm with gaussian back substitution is a novel approach towards the extension of adm from m 2 to m ge 3 and its algorithmic framework is new in the literature for the adm with gaussian back substitution we prove its convergence via the analytic framework of contractive type methods and we show its numerical efficiency by some application problems

local minima and convergence in low rank semidefinite programming the low rank semidefinite programming problem lrsdpr is a restriction of the semidefinite programming problem sdp in which a bound r is imposed on the rank of x and it is well known that lrsdpr is equivalent to sdp if r is not too small in this paper we classify the local minima of lrsdpr and prove the optimal convergence of a slight variant of the successful yet experimental algorithm of burer and monteiro 5 which handles lrsdpr via the nonconvex change of variables x rrt in addition for particular problem classes we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm computational results are presented on a set of combinatorial optimization relaxations including some of the largest quadratic assignment sdps solved to date

efficient recovery of low rank matrix via double nonconvex nonsmooth rank minimization recently there is a rapidly increasing attraction for the efficient recovery of low rank matrix in computer vision and machine learning the popular convex solution of rank minimization is nuclear norm based minimization nnm which usually leads to a biased solution since nnm tends to overshrink the rank components and treats each rank component equally to address this issue some nonconvex nonsmooth rank nnr relaxations have been exploited widely different from these convex and nonconvex rank substitutes this paper first introduces a general and flexible rank relaxation function named weighted nnr relaxation function which is actually derived from the initial double nnr dnnr relaxations i e dnnr relaxation function acts on the nonconvex singular values function svf an iteratively reweighted svf optimization algorithm with continuation technology through computing the supergradient values to define the weighting vector is devised to solve the dnnr minimization problem and the closed form solution of the subproblem can be efficiently obtained by a general proximal operator in which each element of the desired weighting vector usually satisfies the nondecreasing order we next prove that the objective function values decrease monotonically and any limit point of the generated subsequence is a critical point combining the kurdyka ojasiewicz property with some milder assumptions we further give its global convergence guarantee as an application in the matrix completion problem experimental results on both synthetic data and real world data can show that our methods are competitive with several state of the art convex and nonconvex matrix completion methods

quaternion based bilinear factor matrix norm minimization for color image inpainting as a new color image representation tool quaternion has achieved excellent results in the color image processing because it treats the color image as a whole rather than as a separate color space component thus it can make full use of the high correlation among rgb channels recently low rank quaternion matrix completion lrqmc methods have proven very useful for color image inpainting in this article we propose three novel lrqmc methods based on three quaternion based bilinear factor qbf matrix norm minimization models specifically we define quaternion double frobenius norm q dfn quaternion double nuclear norm q dnn and quaternion frobenius nuclear norm q fnn and then show their relationship with quaternion based matrix schatten p q schatten p norm for certain p values the proposed methods can avoid computing quaternion singular value decompositions qsvd for large quaternion matrices and thus can effectively reduce the calculation time compared with existing lrqmc methods furthermore we also analyze the convergence of the algorithms and introduce a rank estimation method for the quaternion matrix the experimental results demonstrate the superior performance of the proposed methods over some state of the art low rank quaternion matrix completion methods

on the o 1 n convergence rate of the douglas rachford alternating direction method alternating direction methods adms have been well studied in the literature and they have found many efficient applications in various fields in this note we focus on the douglas rachford adm scheme proposed by glowinski and marrocco and we aim at providing a simple approach to estimating its convergence rate in terms of the iteration number the linearized version of this adm scheme which is known as the split inexact uzawa method in the image processing literature is also discussed

rank minimization with applications to image noise removal abstract rank minimization problem has a wide range of applications in different areas however since this problem is np hard and non convex the frequently used method is to replace the matrix rank minimization with nuclear norm minimization nuclear norm is the convex envelope of the matrix rank and it is more computationally tractable matrix completion is a special case of rank minimization problem in this paper we consider directly using matrix rank as the regularization term instead of nuclear norm in the cost function for matrix completion problem the solution is analyzed and obtained by a hard thresholding operation on the singular values of the observed matrix then by exploiting patch based nonlocal self similarity scheme we apply the proposed rank minimization algorithm to remove white gaussian additive noise in images gamma multiplicative noise is also removed in logarithm domain the experimental results illustrate that the proposed algorithm can remove noises in images more efficiently than nuclear norm can do and the results are also competitive with those obtained by using the existing state of the art noise removal methods in the literature

fast and accurate matrix completion via truncated nuclear norm regularization recovering a large matrix from a small subset of its entries is a challenging problem arising in many real applications such as image inpainting and recommender systems many existing approaches formulate this problem as a general low rank matrix approximation problem since the rank operator is nonconvex and discontinuous most of the recent theoretical studies use the nuclear norm as a convex relaxation one major limitation of the existing approaches based on nuclear norm minimization is that all the singular values are simultaneously minimized and thus the rank may not be well approximated in practice in this paper we propose to achieve a better approximation to the rank of matrix by truncated nuclear norm which is given by the nuclear norm subtracted by the sum of the largest few singular values in addition we develop a novel matrix completion algorithm by minimizing the truncated nuclear norm we further develop three efficient iterative procedures tnnr admm tnnr apgl and tnnr admmap to solve the optimization problem tnnr admm utilizes the alternating direction method of multipliers admm while tnnr agpl applies the accelerated proximal gradient line search method apgl for the final optimization for tnnr admmap we make use of an adaptive penalty according to a novel update rule for admm to achieve a faster convergence rate our empirical study shows encouraging results of the proposed algorithms in comparison to the state of the art matrix completion algorithms on both synthetic and real visual datasets

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

low complexity and high resolution line spectral estimation using cyclic minimization the line spectral estimation problem has applications in radar wireless communications spectroscopy and power electronics among others the signal is modeled as a sparse linear combination of complex sinusoids and the problem target is to estimate the number of sinusoids in the mixture and respective parameters of each individual sinusoid such as magnitude phase and frequency in this work we first introduce a novel formulation of the rank function which involves the solution of a multi convex optimization problem using the multi convex reformulation of the rank function an over parameterization of the line estimation problem is proposed together with a cyclic minimization procedure to obtain a solution the cyclic method iterates between the optimization of the signal subspace and null space and stops when the two are orthogonal every limit point of the sequence of iterates is shown to be a stationary point of the original problem numerical experiments show that only a small number of iterations is required for convergence the signal subspace optimization is a semidefinite program sdp and the null space optimization has a closed form solution the cost per iteration of a general purpose interior point method ipm to solve the sdp is a quartic function of the problem dimension it is shown that by exploiting the problem structure the cost per iteration of the ipm may be reduced to a cubic function of the problem dimension comparable to the cost of the alternating direction method of multipliers admm however contrarily to the latter the former achieves the precision necessary for fine frequency localization a large set of numerical experiments show the effectiveness of the proposed approach

majorization minimization algorithms for wavelet based image restoration standard formulations of image signal deconvolution under wavelet based priors regularizers lead to very high dimensional optimization problems involving the following difficulties the non gaussian heavy tailed wavelet priors lead to objective functions which are nonquadratic usually nondifferentiable and sometimes even nonconvex the presence of the convolution operator destroys the separability which underlies the simplicity of wavelet based denoising this paper presents a unified view of several recently proposed algorithms for handling this class of optimization problems placing them in a common majorization minimization mm framework one of the classes of algorithms considered when using quadratic bounds on nondifferentiable log priors shares the infamous singularity issue si of iteratively re weighted least squares irls algorithms the possibility of having to handle infinite weights which may cause both numerical and convergence issues in this paper we prove several new results which strongly support the claim that the si does not compromise the usefulness of this class of algorithms exploiting the unified mm perspective we introduce a new algorithm resulting from using bounds for nonconvex regularizers the experiments confirm the superior performance of this method when compared to the one based on quadratic majorization finally an experimental comparison of the several algorithms reveals their relative merits for different standard types of scenarios

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

matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise a study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and or undamped sinusoids in noise is presented comparison of this method to a polynomial method svd prony method shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes it is found through perturbation analysis and simulation that for signals with unknown damping factors the pencil method is less sensitive to noise than the polynomial method an expression of the cramer rao bound for the exponential signals is presented

navier stokes equations and area of interfaces we present new a priori estimates for the vorticity of solutions of the three dimensional navier stokes equations these estimates imply that thel 1 norm of the vorticity is a priori bounded in time and that the time average of the 4 3 e power of thel 4 3 e spatial norm of the gradient of the vorticity is a priori bounded using these bounds we construct global leray weak solutions of the navier stokes equations which satisfy these inequalities in particular it follows that vortex sheet vortex line and even more general vortex structures with arbitrarily large vortex strengths are initial data which give rise to global weak solutions of this type of the navier stokes equations next we apply these inequalities in conjunction with geometric measure theoretical arguments to study the two dimensional hausdorff measure of level sets of the vorticity magnitude we obtain a priori bounds on an average such measure when expressed in terms of the reynolds number and the kolmogorov dissipation length these bounds are left langle mu right rangle leqq frac l 3 eta left 1 operatorname re 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 the right hand side of this inequality has a simple geometrical interpretation it represents the area of a union of non overlapping spheres of radii which fill a fraction of the spatial domain as the reynolds number increases this fraction decreases we study also the area of level sets of scalars and in particular isotherms in rayleigh benard convection we define a quantity r t x 0 describing an average value of the area of a portion of a level set contained in a small ball of radiusr about the pointx 0 we obtain the inequality left langle mu right rangle r t left x 0 right leqq c kappa 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 r 5 mathord left vphantom 5 2 right kern nulldelimiterspace 2 left langle v left x 0 right right rangle 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 wherek is the diffusivity constant and is the velocity the inequality is valid forr small but larger than a small scale k 1 in the case of turbulent velocities this scale is smaller than the smallest significant physical scale suggesting that 2 5 is a lower bound for the fractal dimension of an ensemble average interface in turbulent flow a fact which agrees with the experimental lower bound of 2 35 we define a similar quantity t representing an average value of the area of a portion of a level set contained in the region of spaced x d dist x d whered is the domain of aspect ratio of order one and diameterl where the convection takes place we obtain the inequality left langle mu right rangle delta t leqq c left l 3 delta 1 l 5 mathord left vphantom 5 2 right kern nulldelimiterspace 2 delta 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 ra 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right where ra is the rayleigh number

compressed sensing off the grid this paper investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples unlike previous work in compressed sensing the frequencies are not assumed to lie on a grid but can assume any values in the normalized frequency domain 0 1 an atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies which is then reformulated as an exact semidefinite program even with this continuous dictionary it is shown that o slog s log n random samples are sufficient to guarantee exact frequency localization with high probability provided the frequencies are well separated extensive numerical experiments are performed to illustrate the effectiveness of the proposed method

reduction of sizes of decision diagrams by autocorrelation functions this paper discusses optimization of decisions diagrams dds by total autocorrelation functions we present an efficient algorithm for construction of linearly transformed binary decision diagrams lt bdds and linearly transformed multiterminal bdds lt mtbdds for systems of boolean functions based on linearization of these functions by the corresponding autocorrelation functions then we present a method for reduction of sizes of dds by a level by level reduction of the width of dds using the total autocorrelation functions the approach provides for a simple procedure for minimization of lt bdds and lt mtbdds and upper bounds on their sizes experimental results for benchmarks illustrate that the proposed method on average is very efficient

detection of the number of signals a predicted eigen threshold approach a novel method for detecting the number of signals incident upon an array of sensors is described the method is based on finding upper thresholds for the observed eigenvalues of the correlation matrix of the array output the asymptotic normality of the multifold eigenvalue estimate is used to derive a prediction formula for the thresholds due to the fact that the detection error of the method can be controlled by a parameter t the performance of the method is superior to the mdl minimum description length under low snr signal to noise ratio and superior to an aic akaike information criterion under high snr the distribution functions of both missing and false alarm errors are evaluated so that the performance of the method can be analyzed and so that t can be chosen simulation results are presented to confirm the analysis

on the shinbrot s criteria for energy equality to newtonian fluids a simplified proof and an extension of the range of application abstract we show that the classical shinbrot s criteria to guarantee that a leray hopf solution satisfies the energy equality follows trivially from the l 4 0 t lions prodi particular case moreover we extend shinbrot s result to space coefficients r 3 4 in this last case our condition coincides with shinbrot condition for r 4 but for r 4 it is more restrictive than the classical one 2 p 2 r 1 it looks significant that in correspondence to the extreme values r 3 and r and just for these two values the conditions become respectively u l l 3 and u l 2 l which imply regularity by appealing to classical ladyzhenskaya prodi serrin l p s type conditions however for values r 3 the l p s condition does not apply even for the more demanding case 3 r 4 the proofs are quite trivial by appealing to interpolation with l l 2 in the first case and with l 2 l 6 in the second case the central position of this old classical problem in fluid mechanics together with the simplicity of the proofs in particular the novelty of the second result looks at least curious this may be considered a merit of this very short note

some inequalities for the eigenvalues of the product of positive semidefinite hermitian matrices let 1 a n a denote the eigenvalues of a hermitian n by n matrix a and let 1 i1 ik n our main results are t 1k t gh t 1k it g n it 1 h and t 1k it gh t 1k it g n t 1 h here g and h are n by n positive semidefinite hermitian matrices these results extend marshall and olkin s inequality t 1k t gh t 1k t g n t 1 h we also present analogous results for singular values

computationally efficient truncated nuclear norm minimization for high dynamic range imaging matrix completion is a rank minimization problem to recover a low rank data matrix from a small subset of its entries since the matrix rank is nonconvex and discrete many existing approaches approximate the matrix rank as the nuclear norm however the truncated nuclear norm is known to be a better approximation to the matrix rank than the nuclear norm exploiting a priori target rank information about the problem in rank minimization in this paper we propose a computationally efficient truncated nuclear norm minimization algorithm for matrix completion which we call tnnm alm we reformulate the original optimization problem by introducing slack variables and considering noise in the observation the central contribution of this paper is to solve it efficiently via the augmented lagrange multiplier alm method where the optimization variables are updated by closed form solutions we apply the proposed tnnm alm algorithm to ghost free high dynamic range imaging by exploiting the low rank structure of irradiance maps from low dynamic range images experimental results on both synthetic and real visual data show that the proposed algorithm achieves significantly lower reconstruction errors and superior robustness against noise than the conventional approaches while providing substantial improvement in speed thereby applicable to a wide range of imaging applications

a truncated nuclear norm regularization method based on weighted residual error for matrix completion low rank matrix completion aims to recover a matrix from a small subset of its entries and has received much attention in the field of computer vision most existing methods formulate the task as a low rank matrix approximation problem a truncated nuclear norm has recently been proposed as a better approximation to the rank of matrix than a nuclear norm the corresponding optimization method truncated nuclear norm regularization tnnr converges better than the nuclear norm minimization based methods however it is not robust to the number of subtracted singular values and requires a large number of iterations to converge in this paper a tnnr method based on weighted residual error tnnr wre for matrix completion and its extension model etnnr wre are proposed tnnr wre assigns different weights to the rows of the residual error matrix in an augmented lagrange function to accelerate the convergence of the tnnr method the etnnr wre is much more robust to the number of subtracted singular values than the tnnr wre tnnr alternating direction method of multipliers and tnnr accelerated proximal gradient with line search methods experimental results using both synthetic and real visual data sets show that the proposed tnnr wre and etnnr wre methods perform better than tnnr and iteratively reweighted nuclear norm irnn methods

convergence of proximal iteratively reweighted nuclear norm algorithm for image processing the nonsmooth and nonconvex regularization has many applications in imaging science and machine learning research due to its excellent recovery performance a proximal iteratively reweighted nuclear norm algorithm has been proposed for the nonsmooth and nonconvex matrix minimizations in this paper we aim to investigate the convergence of the algorithm with the kurdyka ojasiewicz property we prove the algorithm globally converges to a critical point of the objective function the numerical results presented in this paper coincide with our theoretical findings

majorized proximal alternating imputation for regularized rank constrained matrix completion abstract low rank approximation in the presence of missing data is ubiquitous in many areas of science and engineering as an np hard problem spectral regularization e g nuclear norm schatten p norm is widely used however solving this problem directly can be computationally expensive due to the unknown rank of variables and full rank singular value decompositions svd bilinear factorization is efficient and effectively benefits from the fast numerical optimization however most of existing methods require explicit knowledge of the rank and often tend to be flatlining or over fitting to balance and take advantage of both we formulate the matrix completion problem under the regularized rank constrained conditions we relax this problem to the surrogates with majorization minimization mm by employing a proximal alternating minimization method to optimize the corresponding surrogates we propose an efficient and accurate algorithm named majorized proximal alternating imputation mpa impute algorithm the proposed method iteratively replaces the missing data at each step with the most recent estimate and only involves simple matrix computation and small scale svd theoretically the proposed method is guaranteed to converge to the stationary point under mild conditions moreover the proposed method can be generalized to a family of non convex regularization problems extensive experiments over synthetic data and real world dataset demonstrate the superior performance of our method

turbulent flows as generalized kelvin voigt materials modeling and analysis we model a 3d turbulent fluid evolving toward a statistical equilibrium by adding to the equations for the mean field v p a term like x dv t which is of the kelvin voigt form where the prandtl mixing length is not constant and vanishes at the solid walls we get estimates for velocity v in l t h 1 x w 1 2 t h 1 2 x that allow us to prove the existence and uniqueness of a regular weak solutions v p to the resulting system for a given fixed eddy viscosity we then prove a structural compactness result that highlights the robustness of the model this allows us to pass to the limit in the quadratic source term in the equation for the turbulent kinetic energy k which yields the existence of a weak solution to the corresponding rans system satisfied by v p k

manifold ranking based matrix factorization for saliency detection saliency detection is used to identify the most important and informative area in a scene and it is widely used in various vision tasks including image quality assessment image matching and object recognition manifold ranking mr has been used to great effect for the saliency detection since it not only incorporates the local spatial information but also utilizes the labeling information from background queries however mr completely ignores the feature information extracted from each superpixel in this paper we propose an mr based matrix factorization mrmf method to overcome this limitation mrmf models the ranking problem in the matrix factorization framework and embeds query sample labels in the coefficients by incorporating spatial information and embedding labels mrmf enforces similar saliency values on neighboring superpixels and ranks superpixels according to the learned coefficients we prove that the mrmf has good generalizability and develops an efficient optimization algorithm based on the nesterov method experiments using popular benchmark data sets illustrate the promise of mrmf compared with the other state of the art saliency detection methods