iterative reweighted algorithms for matrix rank minimization the problem of minimizing the rank of a matrix subject to affine constraints has applications in several areas including machine learning and is known to be np hard a tractable relaxation for this problem is nuclear norm or trace norm minimization which is guaranteed to find the minimum rank matrix under suitable assumptions in this paper we propose a family of iterative reweighted least squares algorithms irls p with 0 p 1 as a computationally efficient way to improve over the performance of nuclear norm minimization the algorithms can be viewed as locally minimizing certain smooth approximations to the rank function when p 1 we give theoretical guarantees similar to those for nuclear norm minimization that is recovery of low rank matrices under certain assumptions on the operator defining the constraints for p 1 irls p shows better empirical performance in terms of recovering low rank matrices than nuclear norm minimization we provide an efficient implementation for irls p and also present a related family of algorithms sirls p these algorithms exhibit competitive run times and improved recovery when compared to existing algorithms for random instances of the matrix completion problem as well as on the movielens movie recommendation data set

random attractors for quasi continuous random dynamical systems and applications to stochastic reaction diffusion equations abstract this paper first introduces the so called quasi continuous random dynamical system rds on a separable banach space the quasi continuity is weaker than all the usual continuities and thus is easier to check in practice we then establish a necessary and sufficient condition for the existence of random attractors for the quasi continuous rds we also give a general method to obtain the random attractors for the rds on the banach space l q d for q 2 as an application it is shown that the rds generated by the stochastic reaction diffusion equation possesses a finite dimensional random attractor in l q d for any q 2 a comparison result of fractal dimensions under the different l q norms is also obtained

a sketched finite element method for elliptic models abstract we consider a sketched implementation of the finite element method for elliptic partial differential equations on high dimensional models motivated by applications in real time simulation and prediction we propose an algorithm that involves projecting the finite element solution onto a low dimensional subspace and sketching the reduced equations using randomised sampling we show that a sampling distribution based on the leverage scores of a tall matrix associated with the discrete laplacian operator can achieve nearly optimal performance and a significant speedup we derive an expression of the complexity of the algorithm in terms of the number of samples that are necessary to meet an error tolerance specification with high probability and an upper bound for the distance between the sketched and the high dimensional solutions our analysis shows that the projection not only reduces the dimension of the problem but also regularises the reduced system against sketching error our numerical simulations suggest speed improvements of two orders of magnitude in exchange for a small loss in the accuracy of the prediction

fractal dimensions of random attractors for stochastic benjamin bona mahony equation on unbounded domains abstract this paper considers random attractor and its fractal dimension for benjamin bona mahony equation driven by additive white noise on unbounded domains firstly we investigate the existence of random attractor for the random dynamical system defined on an unbounded domain secondly we present criterion for estimating an upper bound of the fractal dimension of a random invariant set of a random dynamical system on a separable banach space finally we apply expectations of some random variables and these conditions to prove the finiteness of fractal dimension of the random attractors for stochastic benjamin bona mahony equation driven by additive white noise

spectral regularization algorithms for learning large incomplete matrices we use convex relaxation techniques to provide a sequence of regularized low rank solutions for large scale matrix completion problems using the nuclear norm as a regularizer we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm our algorithm soft impute iteratively replaces the missing elements with those obtained from a soft thresholded svd with warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter the computationally intensive part of our algorithm is in computing a low rank svd of a dense matrix exploiting the problem structure we show that the task can be performed with a complexity of order linear in the matrix dimensions our semidefinite programming algorithm is readily scalable to large matrices for example soft impute takes a few hours to compute low rank approximations of a 106 x 106 incomplete matrix with 107 observed entries and fits a rank 95 approximation to the full netflix training set in 3 3 hours our methods achieve good training and test errors and exhibit superior timings when compared to other competitive state of the art techniques

low rank matrix recovery via modified schatten p norm minimization with convergence guarantees in recent years low rank matrix recovery problems have attracted much attention in computer vision and machine learning the corresponding rank minimization problems are both combinational and np hard in general which are mainly solved by both nuclear norm and schatten p 0 norm based optimization algorithms however inspired by weighted nuclear norm and schatten p norm as the relaxations of rank function the main merits of this work firstly provide a modified schatten p norm in the affine matrix rank minimization problem denoted as the modified schatten p norm minimization ms p nm secondly its surrogate function is constructed and the equivalence relationship with the ms p nm is further achieved thirdly the iterative singular value thresholding algorithm isvta is devised to optimize it and its accelerated version i e aisvta is also obtained to reduce the number of iterations through the well known nesterov s acceleration strategy most importantly the convergence guarantees and their relationship with objective function stationary point and variable sequence generated by the proposed algorithms are established under some specific assumptions e g kurdyka ojasiewicz k property finally numerical experiments demonstrate the effectiveness of the proposed algorithms in the matrix completion problem for image inpainting and recommender systems it should be noted that the accelerated algorithm has a much faster convergence speed and a very close recovery precision when comparing with the proposed non accelerated one

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

fast monte carlo algorithms for finding low rank approximations we consider the problem of approximating a given m n matrix a by another matrix of specified rank k which is smaller than m and n the singular value decomposition svd can be used to find the best such approximation however it takes time polynomial in m n which is prohibitive for some modern applications in this article we develop an algorithm that is qualitatively faster provided we may sample the entries of the matrix in accordance with a natural probability distribution in many applications such sampling can be done efficiently our main result is a randomized algorithm to find the description of a matrix d of rank at most k so that holds with probability at least 1 where v vf is the frobenius norm the algorithm takes time polynomial in k 1 e log 1 only and is independent of m and n in particular this implies that in constant time it can be determined if a given matrix of arbitrary size has a good low rank approximation

derivatives of spectral functions a spectral function of a hermitian matrix x is a function which depends only on the eigenvalues of x 1x 2x nx and hence may be written f 1x 2x nx for some symmetric function f such functions appear in a wide variety of matrix optimization problems we give a simple proof that this spectral function is differentiable at x if and only if the function f is differentiable at the vector x and we give a concise formula for the derivative we then apply this formula to deduce an analogous expression for the clarke generalized gradient of the spectral function a similar result holds for real symmetric matrices

maximum entropy vector kernels for mimo system identification abstract recent contributions have framed linear system identification as a nonparametric regularized inverse problem relying on l 2 type regularization which accounts for the stability and smoothness of the impulse response to be estimated these approaches have been shown to be competitive w r t classical parametric methods in this paper adopting maximum entropy arguments we derive a new l 2 penalty to do so we exploit the structure of the hankel matrix thus controlling at the same time complexity measured by the mcmillan degree stability and smoothness of the identified models as a special case we recover the nuclear norm penalty on the squared block hankel matrix in contrast with the previous literature on reweighted nuclear norm penalties our kernel is described by a small number of hyper parameters which are iteratively updated through marginal likelihood maximization constraining the structure of the kernel acts as a hyper regularizer which helps controlling the effective degrees of freedom of our estimator to optimize the marginal likelihood we adapt a scaled gradient projection sgp algorithm which is proved to be significantly computationally cheaper than other first and second order off the shelf optimization methods the paper also contains an extensive comparison with many state of the art methods on several monte carlo studies which confirms the effectiveness of our procedure

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

fast computation of low rank matrix approximations given a matrix a it is often desirable to find a good approximation to a that has low rank we introduce a simple technique for accelerating the computation of such approximations when a has strong spectral features that is when the singular values of interest are significantly greater than those of a random matrix with size and entries similar to a our technique amounts to independently sampling and or quantizing the entries of a thus speeding up computation by reducing the number of nonzero entries and or the length of their representation our analysis is based on observing that the acts of sampling and quantization can be viewed as adding a random matrix n to a whose entries are independent random variables with zero mean and bounded variance since with high probability n has very weak spectral features we can prove that the effect of sampling and quantization nearly vanishes when a low rank approximation to a p n is computed we give high probability bounds on the quality of our approximation both in the frobenius and the 2 norm

visual data denoising with a unified schatten p norm and lq norm regularized principal component pursuit to address the visual processing problem with corrupted data in this paper we propose a non convex formulation to recover the authentic structure from the corrupted data typically the specific structure is assumed to be low rank which holds for a wide range of data such as images and videos meanwhile the corruption is assumed to be sparse in the literature such a problem is known as robust principal component analysis rpca which usually recovers the low rank structure by approximating the rank function with a nuclear norm and penalizing the error by an 1 norm although rpca is a convex formulation and can be solved effectively the introduced norms are not tight approximations which may cause the solution to deviate from the authentic one therefore we consider here a non convex relaxation consisting of a schatten p norm and an q norm that promote low rank and sparsity respectively we derive a proximal iteratively reweighted algorithm pira to solve the problem our algorithm is based on an alternating direction method of multipliers where in each iteration we linearize the underlying objective function that allows us to have a closed form solution we demonstrate that solutions produced by the linearized approximation always converge and have a tighter approximation than the convex counterpart experiments on benchmarks show encouraging results of our approach highlightswe propose a p q pcp model for low rank and sparse matrix recovery a new proximal iteratively reweighted algorithm is presented to solve the problem we show that our solutions can approximate the original problem experiments on a variety of datasets demonstrate the superiority of our approach

fractal dimension of random attractors for stochastic non autonomous reaction diffusion equations in this paper we first give some conditions for bounding the fractal dimension of a random invariant set for a non autonomous random dynamical system on a separable banach space then we apply these conditions to prove the finiteness of fractal dimension of the random attractors for stochastic reaction diffusion equations with multiplicative white noise and additive white noise

linear beamformer design for interference alignment via rank minimization this paper proposes a new framework for the design of transmit and receive beamformers for interference alignment ia without symbol extensions in multi antenna cellular networks we consider ia in a g cell network with k users cell n antennas at each base station bs and m antennas at each user the proposed framework is developed by recasting the conditions for ia as two sets of rank constraints one on the rank of interference matrices and the other on the transmit beamformers in the uplink the interference matrix consists of all the interfering vectors received at a bs from the out of cell users in the uplink using these conditions and the crucial observation that the rank of interference matrices under alignment can be determined beforehand this paper develops two sets of algorithms for ia the first part of this paper develops rank minimization algorithms for ia by iteratively minimizing a weighted matrix norm of the interference matrix different choices of matrix norms lead to reweighted nuclear norm minimization rnnm or reweighted frobenius norm minimization rfnm algorithms with significantly different per iteration complexities alternately the second part of this paper devises an alternating minimization am algorithm where the rank deficient interference matrices are expressed as a product of two lower dimensional matrices that are then alternately optimized simulation results indicate that rnnm which has a per iteration complexity of a semidefinite program is effective in designing aligned beamformers for proper feasible systems with or without redundant antennas while rfnm and am which have a per iteration complexity of a quadratic program are better suited for systems with redundant antennas

a rank corrected procedure for matrix completion with fixed basis coefficients for the problems of low rank matrix completion the efficiency of the widely used nuclear norm technique may be challenged under many circumstances especially when certain basis coefficients are fixed for example the low rank correlation matrix completion in various fields such as the financial market and the low rank density matrix completion from the quantum state tomography to seek a solution of high recovery quality beyond the reach of the nuclear norm in this paper we propose a rank corrected procedure using a nuclear semi norm to generate a new estimator for this new estimator we establish a non asymptotic recovery error bound more importantly we quantify the reduction of the recovery error bound for this rank corrected procedure compared with the one obtained for the nuclear norm penalized least squares estimator this reduction can be substantial around 50 we also provide necessary and sufficient conditions for rank consistency in the sense of bach j mach learn res 9 1019 1048 2008 very interestingly these conditions are highly related to the concept of constraint nondegeneracy in matrix optimization as a byproduct our results provide a theoretical foundation for the majorized penalty method of gao and sun a majorized penalty approach for calibrating rank constrained correlation matrix problems http www math nus edu sg matsundf majorpen may5 pdf 2010 and gao 2010 for structured low rank matrix optimization problems extensive numerical experiments demonstrate that our proposed rank corrected procedure can simultaneously achieve a high recovery accuracy and capture the low rank structure

random exponential attractor for stochastic reaction diffusion equation with multiplicative noise in r3 abstract in this paper we first improve the existing conditions for the existence of a random exponential attractor for a continuous cocycle on a separable banach space then we consider the existence of a random exponential attractor for stochastic non autonomous reaction diffusion equation with multiplicative noise defined in r 3 which implies the existence of a random attractor with finite fractal dimension the essential difficulty here is the continuity of the spectrum of the linear part of the equation which can be overcome by the tail estimation of solutions of equation and carefully decomposing the solution into a sum of three parts of whose one part is finite dimensional and other two parts are quickly decay in mean sense

latent variable graphical model selection via convex optimization suppose we observe samples of a subset of a collection of random variables no additional information is provided about the number of latent variables nor of the relationship between the latent and observed variables is it possible to discover the number of latent components and to learn a statistical model over the entire collection of variables we address this question in the setting in which the latent and observed variables are jointly gaussian with the conditional statistics of the observed variables conditioned on the latent variables being specified by a graphical model as a first step we give natural conditions under which such latent variable gaussian graphical models are identifiable given marginal statistics of only the observed variables essentially these conditions require that the conditional graphical model among the observed variables is sparse while the effect of the latent variables is spread out over most of the observed variables next we propose a tractable convex program based on regularized maximum likelihood for model selection in this latent variable setting the regularizer uses both the ell 1 norm and the nuclear norm our modeling framework can be viewed as a combination of dimensionality reduction to identify latent variables and graphical modeling to capture remaining statistical structure not attributable to the latent variables and it consistently estimates both the number of latent components and the conditional graphical model structure among the observed variables these results are applicable in the high dimensional setting in which the number of latent observed variables grows with the number of samples of the observed variables the geometric properties of the algebraic varieties of sparse matrices and of low rank matrices play an important role in our analysis

harmonic mean iteratively reweighted least squares for low rank matrix recovery we propose a new iteratively reweighted least squares irls algorithm for the recovery of a matrix x cd1 d2 of rank r min d1 d2 from incomplete linear observations solving a sequence of low complexity linear problems the easily implementable algorithm which we call harmonic mean iteratively reweighted least squares hm irls optimizes a non convex schatten p quasi norm penalization to promote low rankness and carries three major strengths in particular for the matrix completion setting first we observe a remarkable global convergence behavior of the algorithm s iterates to the low rank matrix for relevant interesting cases for which any other state of the art optimization approach fails the recovery secondly hm irls exhibits an empirical recovery probability close to 1 even for a number of measurements very close to the theoretical lower bound r d1 d2 r i e already for significantly fewer linear observations than any other tractable approach in the literature thirdly hm irls exhibits a locally superlinear rate of convergence of order 2 p if the linear observations fulfill a suitable null space property while for the first two properties we have so far only strong empirical evidence we prove the third property as our main theoretical result

random exponential attractor for cocycle and application to non autonomous stochastic lattice systems with multiplicative white noise abstract we first establish some sufficient conditions for constructing a random exponential attractor for a continuous cocycle on a separable banach space and weighted spaces of infinite sequences then we apply our abstract result to study the existence of random exponential attractors for non autonomous first order dissipative lattice dynamical systems with multiplicative white noise

iterative concave rank approximation for recovering low rank matrices in this paper we propose a new algorithm for recovery of low rank matrices from compressed linear measurements the underlying idea of this algorithm is to closely approximate the rank function with a smooth function of singular values and then minimize the resulting approximation subject to the linear constraints the accuracy of the approximation is controlled via a scaling parameter where a smaller corresponds to a more accurate fitting the consequent optimization problem for any finite is nonconvex therefore to decrease the risk of ending up in local minima a series of optimizations is performed starting with optimizing a rough approximation a large and followed by successively optimizing finer approximations of the rank with smaller s to solve the optimization problem for any 0 it is converted to a new program in which the cost is a function of two auxiliary positive semidefinite variables the paper shows that this new program is concave and applies a majorize minimize technique to solve it which in turn leads to a few convex optimization iterations this optimization scheme is also equivalent to a reweighted nuclear norm minimization nnm for any 0 we derive a necessary and sufficient condition for the exact recovery which are weaker than those corresponding to nnm on the numerical side the proposed algorithm is compared to nnm and a reweighted nnm in solving affine rank minimization and matrix completion problems showing its considerable and consistent superiority in terms of success rate

outlier robust matrix completion via ell _p minimization matrix completion refers to recovering a low rank matrix from only a subset of its possibly noisy entries and has a variety of important applications because many real world signals can be modeled by a n 1 times n 2 matrix with rank r ll min n 1 n 2 most existing techniques for matrix completion assume gaussian noise and thus they are not robust to outliers in this paper we devise two algorithms for robust matrix completion based on low rank matrix factorization and ell p norm minimization of the fitting error with 0 the first method tackles the low rank matrix factorization with missing data by iteratively solving n 1 n 2 linear ell p regression problems whereas the second applies the alternating direction method of multipliers admm in the ell p space at each iteration of the admm it requires performing a least squares ls matrix factorization and calculating the proximity operator of the p th power of the ell p norm the ls factorization is efficiently solved using linear ls regression while the proximity operator has closed form solution for p 1 or can be obtained by root finding of a scalar nonlinear equation for other values of p the two proposed algorithms have comparable recovery capability and computational complexity of mathcal o k omega r 2 where omega is the number of observed entries and k is a fixed constant of several hundreds to thousands and dimension independent it is demonstrated that they are superior to the singular value thresholding singular value projection and alternating projection schemes in terms of computational simplicity statistical accuracy and outlier robustness

limiting dynamics for stochastic wave equations abstract in this paper relations between the asymptotic behavior for a stochastic wave equation and a heat equation are considered by introducing almost surely d contracting property for random dynamical systems we obtain a global random attractor of the stochastic wave equation u t t u t u f u w endowed with dirichlet boundary condition for any 0 1 the upper semicontinuity of this global random attractor and the global attractor of the heat equation z t z f z 0 with dirichlet boundary condition as goes to zero is investigated furthermore we show the stationary solutions of the stochastic wave equation converge in probability to some stationary solution of the heat equation

truncated low rank and total p variation constrained color image completion and its moreau approximation algorithm recently low rank lr and total variation tv constrained tensor completion algorithms have been broadly studied for image restoration these algorithms however ignore the difference of the intrinsic properties along spatial structure spectral correlation and unfolded mode in this paper we go further by providing a detailed comparison of the lr and tv properties in matrix and tensor cases and figure out the lrtv constraints for pixel matrices are more evident and accordant than for others this inspires us to develop a simple yet effective multichannel lrtv model that is capable of genuinely discovering the intrinsic properties with reduced computational cost moreover due to the suboptimality of nuclear norm and l 1 norm in approximating the essential low rank and low gradient properties we employ two enhanced constraints i e truncated nuclear norm tnn and total p variation text t p text v for a better performance this results in a challenging problem since that both tnn and text t p text v are nonsmooth and nonconvex observing that the moreau approximation of text t p text v constraint is a continuous difference of convex function we then develop a first order method by repeatedly computing two simple proximal operators under mild assumption we further prove that the sequence generated by our method clusters at a stationary point extensive experimental results on color image completion show the efficacy and efficiency of our method over state of the art competitors

finite fractal dimension of random attractor for stochastic non autonomous strongly damped wave equation abstract in this paper we first prove the existence of a random attractor for stochastic non autonomous strongly damped wave equations with additive white noise then we apply a criteria to obtain an upper bound of fractal dimension of the random attractor of considered system

smoothed low rank and sparse matrix recovery by iteratively reweighted least squares minimization this paper presents a general framework for solving the low rank and or sparse matrix minimization problems which may involve multiple nonsmooth terms the iteratively reweighted least squares irlss method is a fast solver which smooths the objective function and minimizes it by alternately updating the variables and their weights however the traditional irls can only solve a sparse only or low rank only minimization problem with squared loss or an affine constraint this paper generalizes irls to solve joint mixed low rank and sparse minimization problems which are essential formulations for many tasks as a concrete example we solve the schatten p norm and ell 2 q norm regularized low rank representation problem by irls and theoretically prove that the derived solution is a stationary point globally optimal if p q geq 1 our convergence proof of irls is more general than previous one that depends on the special properties of the schatten p norm and ell 2 q norm extensive experiments on both synthetic and real data sets demonstrate that our irls is much more efficient

generalized unitarily invariant gauge regularization for fast low rank matrix recovery spectral regularization is a widely used approach for low rank matrix recovery lrmr by regularizing matrix singular values most of the existing lrmr solvers iteratively compute the singular values via applying singular value decomposition svd on a dense matrix which is computationally expensive and severely limits their applications to large scale problems to address this issue we present a generalized unitarily invariant gauge guig function for lrmr the proposed guig function does not act on the singular values however we show that it generalizes the well known spectral functions including the rank function the schatten p quasi norm and logsum of singular values the proposed guig regularization model can be formulated as a bilinear variational problem which can be efficiently solved without computing svd such a property makes it well suited for large scale lrmr problems we apply the proposed guig model to matrix completion and robust principal component analysis and prove the convergence of the algorithms experimental results demonstrate that the proposed guig method is not only more accurate but also much faster than the state of the art algorithms especially on large scale problems

the kernel of the modular representation and the galois action in rcft it is shown that for the modular representations associated to rational conformal field theories the kernel is a congruence subgroup whose level equals the order of the dehn twist an explicit algebraic characterization of the kernel is given it is also shown that the order of the dehn twist is bounded by a function of the number of primary fields allowing for a systematic enumeration of the modular representations coming from rcfts restrictions on the spectrum of the dehn twist and arithmetic properties of modular matrix elements are presented

on rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions we consider different definitions of the correlation dimension and find some relationships between them and other characteristics of dimension type such as hausdorff dimension box dimension etc we also introduce different ways to define and study the generalized spectrum for dimensions a one parameter family of characteristics of dimension type

sampling of planar curves theory and fast algorithms we introduce a continuous domain framework for the recovery of a planar curve from a few samples we model the curve as the zero level set of a trigonometric polynomial we show that the exponential feature maps of the points on the curve lie on a low dimensional subspace we show that the null space vector of the feature matrix can be used to uniquely identify the curve given a sufficient number of samples the worst case theoretical guarantees show that the number of samples required for unique recovery depends on the bandwidth of the underlying trigonometric polynomial which is a measure of the complexity of the curve we introduce an iterative algorithm that relies on the low rank property of the feature maps to recover the curves when the samples are noisy or when the true bandwidth of the curve is unknown we also demonstrate the preliminary utility of the proposed curve representation in the context of image segmentation

sufficient and necessary criteria for existence of pullback attractors for non compact random dynamical systems abstract we study pullback attractors of non autonomous non compact dynamical systems generated by differential equations with non autonomous deterministic as well as stochastic forcing terms we first introduce the concepts of pullback attractors and asymptotic compactness for such systems we then prove a sufficient and necessary condition for existence of pullback attractors we also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors for random systems containing periodic deterministic forcing terms we show the pullback attractors are also periodic under certain conditions as an application of the abstract theory we prove the existence of a unique pullback attractor for reaction diffusion equations on r n with both deterministic and random external terms since sobolev embeddings are not compact on unbounded domains the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions

random attractors of reaction diffusion equations with multiplicative noise in lp in this paper we study the random dynamical system rds generated by the reaction diffusion equation with multiplicative noise and prove the existence of a random attractor for such rds in l p d for any p 2

fc sets and twisters the basics of orbifold deconstruction we present a detailed account of the properties of text twister s and their generalizations text fc set s which are essential ingredients of the orbifold deconstruction procedure aimed at recognizing whether a given conformal model may be obtained as an orbifold of another one and if so to identify the twist group and the original model the close analogy with the character theory of finite groups is discussed and its origin explained

weighted nuclear norm minimization and its applications to low level vision as a convex relaxation of the rank minimization model the nuclear norm minimization nnm problem has been attracting significant research interest in recent years the standard nnm regularizes each singular value equally composing an easily calculated convex norm however this restricts its capability and flexibility in dealing with many practical problems where the singular values have clear physical meanings and should be treated differently in this paper we study the weighted nuclear norm minimization wnnm problem which adaptively assigns weights on different singular values as the key step of solving general wnnm models the theoretical properties of the weighted nuclear norm proximal wnnp operator are investigated albeit nonconvex we prove that wnnp is equivalent to a standard quadratic programming problem with linear constrains which facilitates solving the original problem with off the shelf convex optimization solvers in particular when the weights are sorted in a non descending order its optimal solution can be easily obtained in closed form with wnnp the solving strategies for multiple extensions of wnnm including robust pca and matrix completion can be readily constructed under the alternating direction method of multipliers paradigm furthermore inspired by the reweighted sparse coding scheme we present an automatic weight setting method which greatly facilitates the practical implementation of wnnm the proposed wnnm methods achieve state of the art performance in typical low level vision tasks including image denoising background subtraction and image inpainting

rank constrained optimization and its applications abstract this paper investigates an iterative approach to solve the general rank constrained optimization problems rcops defined to optimize a convex objective function subject to a set of convex constraints and rank constraints on unknown rectangular matrices in addition rank minimization problems rmps are introduced and equivalently transformed into rcops by introducing a quadratic matrix equality constraint the rank function is discontinuous and nonconvex thus the general rcops are classified as np hard in most of the cases an iterative rank minimization irm method with convex formulation at each iteration is proposed to gradually approach the constrained rank the proposed irm method aims at solving rcops with rank inequalities constrained by upper or lower bounds as well as rank equality constraints proof of the convergence to a local minimizer with at least a sublinear convergence rate is provided four representative applications of rcops and rmps including system identification output feedback stabilization and structured h 2 controller design problems are presented with comparative simulation results to verify the feasibility and improved performance of the proposed irm method

asymptotic behavior of stochastic wave equations with critical exponents on mathbb r 3 the existence of a random attractor in h 1 3 x l 2 3 is proved for the damped semilinear stochastic wave equation defined on the entire space 3 the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent the uniform pullback estimates on the tails of solutions for large space variables are established the pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method

random attractor for stochastic reaction diffusion equation with multiplicative noise on unbounded domains abstract in this paper we study the asymptotic dynamics for stochastic reaction diffusion equation with multiplicative noise defined on unbounded domains we investigate the existence of a random attractor for the random dynamical system associated with the equation the asymptotic compactness of the random dynamical system is established by using uniform a priori estimates for far field values of solutions and a cut off technique

the coordinatization of arguesian lattices we show that the auxiliary planar ternary ring of an n frame in an arguesian lattice n 3 is indeed an associative ring with unit the addition of two weak necessary conditions allows us to coordinatize a hyperplane of this n frame this generalizes the classical work of von neumann baer inaba j6nsson and j6nsson monk

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

recovery of damped exponentials using structured low rank matrix completion we introduce a structured low rank matrix completion algorithm to recover a series of images from their under sampled measurements where the signal along the parameter dimension at every pixel is described by a linear combination of exponentials we exploit the exponential behavior of the signal at every pixel along with the spatial smoothness of the exponential parameters to derive an annihilation relation in the fourier domain this relation translates to a low rank property on a structured matrix constructed from the fourier samples we enforce the low rank property of the structured matrix as a regularization prior to recover the images since the direct use of current low rank matrix recovery schemes to this problem is associated with high computational complexity and memory demand we adopt an iterative re weighted least squares algorithm which facilitates the exploitation of the convolutional structure of the matrix novel approximations involving 2 d fast fourier transforms are introduced to drastically reduce the memory demand and computational complexity which facilitates the extension of structured low rank methods to large scale 3 d problems we demonstrate our algorithm in the mr parameter mapping setting and show improvement over the state of the art methods

l2 lp random attractors for stochastic reaction diffusion equation on unbounded domains abstract in this paper the existence of l 2 l p random attractor is established for a stochastic reaction diffusion equation on the whole space r n this random attractor is a compact and invariant tempered set which attracts every tempered random subset of l 2 in the topology of l p the nonlinearity f is supposed to satisfy some growth of arbitrary order p 1 where p 2 the l 2 l p asymptotic compactness of the random dynamical system is proved by an asymptotic a priori estimate of the unbounded part of solutions

correspondences of ribbon categories much of algebra and representation theory can be formulated in the general framework of tensor categories the aim of this paper is to further develop this theory for braided tensor categories several results are established that do not have a substantial counterpart for symmetric tensor categories in particular we exhibit various equivalences involving categories of modules over algebras in ribbon categories finally we establish a correspondence of ribbon categories that can be applied to and is in fact motivated by the coset construction in conformal quantum field theory

weighted schatten p norm minimization for image denoising and background subtraction low rank matrix approximation lrma which aims to recover the underlying low rank matrix from its degraded observation has a wide range of applications in computer vision the latest lrma methods resort to using the nuclear norm minimization nnm as a convex relaxation of the nonconvex rank minimization however nnm tends to over shrink the rank components and treats the different rank components equally limiting its flexibility in practical applications we propose a more flexible model namely the weighted schatten p norm minimization wsnm to generalize the nnm to the schatten p norm minimization with weights assigned to different singular values the proposed wsnm not only gives better approximation to the original low rank assumption but also considers the importance of different rank components we analyze the solution of wsnm and prove that under certain weights permutation wsnm can be equivalently transformed into independent non convex l p norm subproblems whose global optimum can be efficiently solved by generalized iterated shrinkage algorithm we apply wsnm to typical low level vision problems e g image denoising and background subtraction extensive experimental results show both qualitatively and quantitatively that the proposed wsnm can more effectively remove noise and model the complex and dynamic scenes compared with state of the art methods

low rank decomposition based restoration of compressed images via adaptive noise estimation images coded at low bit rates in real world applications usually suffer from significant compression noise which significantly degrades the visual quality traditional denoising methods are not suitable for the content dependent compression noise which usually assume that noise is independent and with identical distribution in this paper we propose a unified framework of content adaptive estimation and reduction for compression noise via low rank decomposition of similar image patches we first formulate the framework of compression noise reduction based upon low rank decomposition compression noises are removed by soft thresholding the singular values in singular value decomposition of every group of similar image patches for each group of similar patches the thresholds are adaptively determined according to compression noise levels and singular values we analyze the relationship of image statistical characteristics in spatial and transform domains and estimate compression noise level for every group of similar patches from statistics in both domains jointly with quantization steps finally quantization constraint is applied to estimated images to avoid over smoothing extensive experimental results show that the proposed method not only improves the quality of compressed images obviously for post processing but are also helpful for computer vision tasks as a pre processing method

fast monte carlo algorithms for matrices ii computing a low rank approximation to a matrix in many applications the data consist of or may be naturally formulated as an m times n matrix a it is often of interest to find a low rank approximation to a i e an approximation d to the matrix a of rank not greater than a specified rank k where k is much smaller than m and n methods such as the singular value decomposition svd may be used to find an approximation to a which is the best in a well defined sense these methods require memory and time which are superlinear in m and n for many applications in which the data sets are very large this is prohibitive two simple and intuitive algorithms are presented which when given an m times n matrix a compute a description of a low rank approximation d to a and which are qualitatively faster than the svd both algorithms have provable bounds for the error matrix a d for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively in the first algorithm c columns of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling then it is shown that from c tc approximations to the top singular values and corresponding singular vectors may be computed from the computed singular vectors a description d of the matrix a may be computed such that mathrm rank d le k and such that left a d right xi 2 le min d mathrm rank d le k left a d right xi 2 poly k 1 c left a right 2 f holds with high probability for both xi 2 f this algorithm may be implemented without storing the matrix a in random access memory ram provided it can make two passes over the matrix stored in external memory and use o cm c 2 additional ram the second algorithm is similar except that it further approximates the matrix c by randomly sampling r rows of c to form a r times c matrix w thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a 2 f both algorithms take time which is polynomial in k 1 epsilon and log 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n our bounds improve previously published results with respect to the rank parameter k for both the frobenius and spectral norms in addition the proofs for the error bounds use a novel method that makes important use of matrix perturbation theory the probability distribution over columns of a and the rescaling are crucial features of the algorithms which must be chosen judiciously

attractors for the stochastic reaction diffusion equation driven by linear multiplicative noise with a variable coefficient abstract by the classic law of iterated logarithm we get a new way to study the existence of random attractors for the stochastic reaction diffusion equation driven by linear multiplicative noise with a variable coefficient and estimate the hausdorff dimensions of the random attractors

dynamical behavior of non autonomous fractional stochastic reaction diffusion equations abstract in this paper we investigate the asymptotic behavior of solutions for the non autonomous fractional stochastic reaction diffusion equations driven by multiplicative noise with s 0 1 we prove the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain u by the compactness of sobolev embedding h s u l 2 u which is different from the previous work 46

regularity and backward compactness of attractors for non autonomous lattice systems with random coefficients abstract we study longtime behavior for the non autonomous lattice model with multiplicative white noise and a random coefficient in the discrete laplace operator we first show existence of a bi spatial attractor when the initial space is the weighted square summation space and the terminal space is the weighted p times summation space for p 2 we then show backward compactness of the attractor in both initial and terminal spaces if the time indexed forces are backward tempered and backward null finally by proving identity of the attractors on the different universes of tempered or backward tempered sets we show measurability of the attractor in the initial space and in the terminal space respectively