rip based performance guarantee for low tubal rank tensor recovery abstract the essential task of tensor data analysis focuses on the tensor decomposition and the corresponding notion of rank in this paper by introducing the notion of tensor singular value decomposition t svd we establish a regularized tensor nuclear norm minimization rtnnm model for low tubal rank tensor recovery as we know that many variants of the restricted isometry property rip have proven to be crucial analysis tools for sparse recovery in the t svd framework we initiatively define a novel tensor restricted isometry property t rip furthermore we show that any third order tensor x can stably be recovered from few linear noise measurements under some certain t rip conditions via the rtnnm model we note that as far as the authors are aware such kind of result has not previously been reported in the literature

reconstruction of 3 d seismic signals irregularly sampled along two spatial coordinates seismic signals are often irregularly sampled along spatial coordinates leading to suboptimal processing and imaging results least squares estimation of fourier components is used for the reconstruction of band limited seismic signals that are irregularly sampled along two spatial coordinates a simple and efficient diagonal weighting scheme based on the areas surrounding the spatial samples takes the properties of the noise signal outside the bandwidth into account in an approximate sense diagonal stabilization based on the energies of the signal and the noise ensures robust estimation reconstruction by temporal frequency component allows the specification of varying bandwidth in two dimensions depending on the minimum apparent velocity this parameterization improves the reconstruction capability for lower temporal frequencies the shape of the spatial aperture affects the method of sampling the fourier domain taking into account this property a larger bandwidth can be recovered the properti

framelet representation of tensor nuclear norm for third order tensor completion the main aim of this paper is to develop a framelet representation of the tensor nuclear norm for third order tensor recovery in the literature the tensor nuclear norm can be computed by using tensor singular value decomposition based on the discrete fourier transform matrix and tensor completion can be performed by the minimization of the tensor nuclear norm which is the relaxation of the sum of matrix ranks from all fourier transformed matrix frontal slices these fourier transformed matrix frontal slices are obtained by applying the discrete fourier transform on the tubes of the original tensor in this paper we propose to employ the framelet representation of each tube so that a framelet transformed tensor can be constructed because of framelet basis redundancy the representation of each tube is sparsely represented when the matrix slices of the original tensor are highly correlated we expect the corresponding sum of matrix ranks from all framelet transformed matrix frontal slices would be small and the resulting tensor completion can be performed much better the proposed minimization model is convex and global minimizers can be obtained numerical results on several types of multi dimensional data videos multispectral images and magnetic resonance imaging data have tested and shown that the proposed method outperformed the other testing methods

tensor n tubal rank and its convex relaxation for low rank tensor recovery abstract the recent popular tensor tubal rank defined based on tensor singular value decomposition t svd yields promising results however its framework is applicable only to three way tensors and lacks the flexibility necessary tohandle different correlations along different modes to tackle these two issues we define a new tensor unfolding operator named mode k 1 k 2 tensor unfolding as the process of lexicographically stacking all mode k 1 k 2 slices of an n way tensor into a three way tensor which is a three way extension of the well known mode k tensor matricization on this basis we define a novel tensor rank named the tensor n tubal rank as a vector consisting of the tubal ranks of all mode k 1 k 2 unfolding tensors to depict the correlations along different modes to efficiently minimize the proposed n tubal rank we establish its convex relaxation the weighted sum of the tensor nuclear norm wstnn then we apply the wstnn to low rank tensor completion lrtc and tensor robust principal component analysis trpca the corresponding wstnn based lrtc and trpca models are proposed and two efficient alternating direction method of multipliers admm based algorithms are developed to solve the proposed models numerical experiments demonstrate that the proposed models significantly outperform the compared ones

multistep autoregressive reconstruction of seismic records linear prediction filters in the f x domain are widely used to interpolate regularly sampled data we study the problem of reconstructing irregularly missing data on a regular grid using linear prediction filters we propose a two stage algorithm first we reconstruct the unaliased part of the data spectrum using a fourier method minimum weighted norm interpolation then prediction filters for all the frequencies are extracted from the reconstructed low frequencies the latter is implemented via a multistep autoregressive msar algorithm finally these prediction filters are used to reconstruct the complete data in the f x domain the applicability of the proposed method is examined using synthetic and field data examples

generalized inverses and their application to applied probability problems the main aim of this paper is to examine the applicability of generalized inverses to a wide variety of problems in applied probability where a markov chain is present either directly or indirectly through some form of imbedding by characterizing all generalized inverses of i p where p is the transition matrix of a finite irreducible discrete time markov chain we are able to obtain general procedures for finding stationary distributions moments of the first passage time distributions and asymp totic forms for the moments of the occupation time random variables it is shown that all known explicit methods for examining these problems can be expressed in this generalized inverse framework more generally in the context of a markov renewal process setting the aforementioned problems are also examined using generalized inverses of i p as a special case markov chains in continuous time are considered and we show that the generalized inverse technique can be applied directly to the infinitesimal generator of the process instead of to i p where p is the transition matrix of the discrete time jump markov chain

polar n complex and n bicomplex singular value decomposition and principal component pursuit informed by recent work on tensor singular value decomposition and circulant algebra matrices this paper presents a new theoretical bridge that unifies the hypercomplex and tensor based approaches to singular value decomposition and robust principal component analysis we begin our work by extending the principal component pursuit to olariu s polar n complex numbers as well as their bicomplex counterparts in doing so we have derived the polar n complex and n bicomplex proximity operators for both the ell 1 and trace norm regularizers which can be used by proximal optimization methods such as the alternating direction method of multipliers experimental results on two sets of audio data show that our algebraically informed formulation outperforms tensor robust principal component analysis we conclude with the message that an informed definition of the trace norm can bridge the gap between the hypercomplex and tensor based approaches our approach can be seen as a general methodology for generating other principal component pursuit algorithms with proper algebraic structures

difference equations for generalized meixner polynomials abstract in this paper is introduced a system of polynomials orthogonal with respect to the classical discrete weight function for meixner polynomials with an extra point mass added at x 0 a difference operator of infinite order is constructed for which these new polynomials are eigenfunctions and a second order difference equation is given with polynomial coefficients n dependent and of at most degree 2 which these polynomials satisfy

tensor tensor products with invertible linear transforms abstract research in tensor representation and analysis has been rising in popularity in direct response to a the increased ability of data collection systems to store huge volumes of multidimensional data and b the recognition of potential modeling accuracy that can be provided by leaving the data and or the operator in its natural multidimensional form in recent work 1 the authors introduced the notion of the t product a generalization of matrix multiplication for tensors of order three which can be extended to multiply tensors of arbitrary order 2 the multiplication is based on a convolution like operation which can be implemented efficiently using the fast fourier transform fft the corresponding linear algebraic framework from the original work was further developed in 3 and it allows one to elegantly generalize all classical algorithms from numerical linear algebra in this paper we extend this development so that tensor tensor products can be defined in a so called transform domain for any invertible linear transform in order to properly motivate this transform based approach we begin by defining a new tensor tensor product alternative to the t product we then show that it can be implemented efficiently using dcts and that subsequent definitions and factorizations can be formulated by appealing to the transform domain using this new product as our guide we then generalize the transform based approach to any invertible linear transform we introduce the algebraic structures induced by each new multiplication in the family which is that of c algebras and modules finally in the spirit of 4 we give a matrix algebra based interpretation of the new family of tensor tensor products and from an applied perspective we briefly discuss how to choose a transform we demonstrate the convenience of our new framework within the context of an image deblurring problem and we show the potential for using one of these new tensor tensor products and resulting tensor svd for hyperspectral image compression

factorization strategies for third order tensors abstract operations with tensors or multiway arrays have become increasingly prevalent in recent years traditionally tensors are represented or decomposed as a sum of rank 1 outer products using either the candecomp parafac cp or the tucker models or some variation thereof such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array the specifics of the approximate representation such as how many terms to use in the sum orthogonality constraints etc depend on the application in this paper we explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem reminiscent of matrix factorizations we present a new factorization of a tensor as a product of tensors to derive the new factorization we define a closed multiplication operation between tensors a major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications specifically this new multiplication allows us to introduce concepts such as tensor transpose inverse and identity which lead to the notion of an orthogonal tensor the multiplication also gives rise to a linear operator and the null space of the resulting operator is identified we extend the concept of outer products of vectors to outer products of matrices all derivations are presented for third order tensors however they can be easily extended to the order p p 3 case we conclude with an application in image deblurring

tensor based formulation and nuclear norm regularization for multienergy computed tomography the development of energy selective photon counting x ray detectors allows for a wide range of new possibilities in the area of computed tomographic image formation under the assumption of perfect energy resolution here we propose a tensor based iterative algorithm that simultaneously reconstructs the x ray attenuation distribution for each energy we use a multilinear image model rather than a more standard stacked vector representation in order to develop novel tensor based regularizers in particular we model the multispectral unknown as a three way tensor where the first two dimensions are space and the third dimension is energy this approach allows for the design of tensor nuclear norm regularizers which like its 2d counterpart is a convex function of the multispectral unknown the solution to the resulting convex optimization problem is obtained using an alternating direction method of multipliers approach simulation results show that the generalized tensor nuclear norm can be used as a standalone regularization technique for the energy selective spectral computed tomography problem and when combined with total variation regularization it enhances the regularization capabilities especially at low energy images where the effects of noise are most prominent

exact tensor completion using t svd in this paper we focus on the problem of completion of multidimensional arrays also referred to as tensors in particular three dimensional 3 d arrays from limited sampling our approach is based on a recently proposed tensor algebraic framework where 3 d tensors are treated as linear operators over the set of 2 d tensors in this framework one can obtain a factorization for 3 d data referred to as the tensor singular value decomposition t svd which is similar to the svd for matrices t svd results in a notion of rank referred to as the tubal rank using this approach we consider the problem of sampling and recovery of 3 d arrays with low tubal rank we show that by solving a convex optimization problem which minimizes a convex surrogate to the tubal rank one can guarantee exact recovery with high probability as long as number of samples is of the order o rnk log nk given a tensor of size n times n times k with tubal rank r the conditions under which this result holds are similar to the incoherence conditions for low rank matrix completion under random sampling the difference is that we define incoherence under the algebraic setup of t svd which is different from the standard matrix incoherence conditions we also compare the numerical performance of the proposed algorithm with some state of the art approaches on real world datasets

improved iteratively reweighted least squares for unconstrained smoothed ell_q minimization in this paper we first study ell q minimization and its associated iterative reweighted algorithm for recovering sparse vectors unlike most existing work we focus on unconstrained ell q minimization for which we show a few advantages on noisy measurements and or approximately sparse vectors inspired by the results in daubechies et al comm pure appl math 63 2010 pp 1 38 for constrained ell q minimization we start with a preliminary yet novel analysis for unconstrained ell q minimization which includes convergence error bound and local convergence behavior then the algorithm and analysis are extended to the recovery of low rank matrices the algorithms for both vector and matrix recovery have been compared to some state of the art algorithms and show superior performance on recovering sparse vectors and low rank matrices

general tensor discriminant analysis and gabor features for gait recognition traditional image representations are not suited to conventional classification methods such as the linear discriminant analysis lda because of the undersample problem usp the dimensionality of the feature space is much higher than the number of training samples motivated by the successes of the two dimensional lda 2dlda for face recognition we develop a general tensor discriminant analysis gtda as a preprocessing step for lda the benefits of gtda compared with existing preprocessing methods such as the principal components analysis pca and 2dlda include the following 1 the usp is reduced in subsequent classification by for example lda 2 the discriminative information in the training tensors is preserved and 3 gtda provides stable recognition rates because the alternating projection optimization algorithm to obtain a solution of gtda converges whereas that of 2dlda does not we use human gait recognition to validate the proposed gtda the averaged gait images are utilized for gait representation given the popularity of gabor function based image decompositions for image understanding and object recognition we develop three different gabor function based image representations 1 gabord is the sum of gabor filter responses over directions 2 gabors is the sum of gabor filter responses over scales and 3 gaborsd is the sum of gabor filter responses over scales and directions the gabord gabors and gaborsd representations are applied to the problem of recognizing people from their averaged gait images a large number of experiments were carried out to evaluate the effectiveness recognition rate of gait recognition based on first obtaining a gabor gabord gabors or gaborsd image representation then using gdta to extract features and finally using lda for classification the proposed methods achieved good performance for gait recognition based on image sequences from the university of south florida usf humanid database experimental comparisons are made with nine state of the art classification methods in gait recognition

wave equation trace interpolation spatial aliasing in multichannel seismic data can be overcome by solving an inversion in which the model is the section that would be recorded in a well sampled zero offset experiment and the data are seismic data after normal moveout nmo the formulation of the linear relation between the data and the model is based on the wave equation and on fourier analysis of aliasing a processing sequence in which one treats missing data as zero data and performs partial migration before stacking is equivalent to application of the transpose of the operator that actually needs to be inverted the inverse of that operator cannot be uniquely determined but it can be estimated using spatial spectral balancing in a conjugate gradient iterative scheme the first iteration is conventional processing including prestack partial migration as shown in a field data example in which severe spatial aliasing was simulated a few more iterations are necessary to achieve significantly better results

bayesian robust tensor factorization for incomplete multiway data we propose a generative model for robust tensor factorization in the presence of both missing data and outliers the objective is to explicitly infer the underlying low candecomp parafac cp rank tensor capturing the global information and a sparse tensor capturing the local information also considered as outliers thus providing the robust predictive distribution over missing entries the low cp rank tensor is modeled by multilinear interactions between multiple latent factors on which the column sparsity is enforced by a hierarchical prior while the sparse tensor is modeled by a hierarchical view of student t distribution that associates an individual hyperparameter with each element independently for model learning we develop an efficient variational inference under a fully bayesian treatment which can effectively prevent the overfitting problem and scales linearly with data size in contrast to existing related works our method can perform model selection automatically and implicitly without the need of tuning parameters more specifically it can discover the groundtruth of cp rank and automatically adapt the sparsity inducing priors to various types of outliers in addition the tradeoff between the low rank approximation and the sparse representation can be optimized in the sense of maximum model evidence the extensive experiments and comparisons with many state of the art algorithms on both synthetic and real world data sets demonstrate the superiorities of our method from several perspectives

an approach to n mode components analysis as an extension of lastovicka s four mode components analysis ann mode components analysis is developed using a convenient notation both a canonical and a least squares solution are derived the relation between both solutions and their computational aspects are discussed

tensor decompositions for learning latent variable models this work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models including gaussian mixture models hidden markov models and latent dirichlet allocation which exploits a certain tensor structure in their low order observable moments typically of second and third order specifically parameter estimation is reduced to the problem of extracting a certain orthogonal decomposition of a symmetric tensor derived from the moments this decomposition can be viewed as a natural generalization of the singular value decomposition for matrices although tensor decompositions are generally intractable to compute the decomposition of these specially structured tensors can be efficiently obtained by a variety of approaches including power iterations and maximization approaches similar to the case of matrices a detailed analysis of a robust tensor power method is provided establishing an analogue of wedin s perturbation theorem for the singular vectors of matrices this implies a robust and computationally tractable estimation approach for several popular latent variable models

fourier reconstruction of nonuniformly sampled aliased seismic data there are numerous methods for interpolating uniformly sampled aliased seismic data but few can handle the combination of nonuniform sampling and aliasing we combine the principles of fourier reconstruction of nonaliased nonuniformly sampled data with the ideas of frequency wavenumber f k interpolation of aliased uniformly sampled data in a new two step algorithm in the first step we estimate the fourier coefficients at the lower nonaliased temporal frequencies from the nonuniformly sampled data the coefficients are then used in the second step as an a priori model to distinguish between aliased and nonaliased energy at the higher aliased temporal frequencies by using a nonquadratic model penalty in the inversion both the artifacts in the fourier domain from nonuniform sampling and the aliased energy are suppressed the underlying assumption is that events are planar therefore the algorithm is applied to seismic data in overlapping spatiotemporal windows

image classification using correlation tensor analysis images as high dimensional data usually embody large variabilities to classify images for versatile applications an effective algorithm is necessarily designed by systematically considering the data structure similarity metric discriminant subspace and classifier in this paper we provide evidence that besides the fisher criterion graph embedding and tensorization used in many existing methods the correlation based similarity metric embodied in supervised multilinear discriminant subspace learning can additionally improve the classification performance in particular a novel discriminant subspace learning algorithm called correlation tensor analysis cta is designed to incorporate both graph embedded correlational mapping and discriminant analysis in a fisher type of learning manner the correlation metric can estimate intrinsic angles and distances for the locally isometric embedding which can deal with the case when euclidean metric is incapable of capturing the intrinsic similarities between data points cta learns multiple interrelated subspaces to obtain a low dimensional data representation reflecting both class label information and intrinsic geometric structure of the data distribution extensive comparisons with most popular subspace learning methods on face recognition evaluation demonstrate the effectiveness and superiority of cta parameter analysis also reveals its robustness

a tensor higher order singular value decomposition for prestack seismic data noise reduction and interpolation a patch of prestack data depends on four spatial dimensions x y midpoints and x y offsets and frequency the spatial data at one temporal frequency can be represented by a fourth order tensor in ideal conditions of high signal to noise ratio and complete sampling one can assume that the seismic data can be approximated via a low rank fourth order tensor missing samples were recovered by reinserting data obtained by approximating the original noisy and incomplete data volume with new observations obtained via the rank reduction process the higher order singular value decompostion was used to reduce the rank of the prestack seismic tensor synthetic data demonstrated the ability of the proposed seismic data completion algorithm to reconstruct events with curvature the synthetic example allowed to quantify the quality of the reconstruction for different levels of noise and survey sparsity we also provided a real data example from the western canadian sedimentary basin

dual contouring for domains with topology ambiguity this paper describes an automatic and robust approach to generate quality triangular meshes for complicated domains with topology ambiguity in previous works we developed an octree based dual contouring dc method to construct triangular meshes for complicated domains however topology ambiguity exists and causes non conformal meshes in this study we discuss all possible topology configurations and develop an extension of dc which guarantees the correct topology we first generate one base mesh with the previous dc method then we analyze all the octree leaf cells and categorize them into 31 topology groups in order to discriminate these cells we compute the values of their face and body saddle points based on a tri linear representation inside the cells knowing the correct categorization we are able to modify the base mesh and introduce more minimizer points within the same cell with these minimizer points we update the mesh connectivities to preserve the correct topology finally we use a laplacian smoothing technique to improve the mesh quality our main contribution is the topology categorization and mesh modification we have applied our algorithm to three complicated domains and obtained good results

j orthogonal matrices properties and generation a real square matrix q is j orthogonal if qtjq j where the signature matrix j diag pm 1 j orthogonal matrices arise in the analysis and numerical solution of various matrix problems involving indefinite inner products including in particular the downdating of cholesky factorizations we present techniques and tools useful in the analysis application and construction of these matrices giving a self contained treatment that provides new insights first we define and explore the properties of the exchange operator which maps j orthogonal matrices to orthogonal matrices and vice versa then we show how the exchange operator can be used to obtain a hyperbolic cs decomposition of a j orthogonal matrix directly from the usual cs decomposition of an orthogonal matrix we employ the decomposition to derive an algorithm for constructing random j orthogonal matrices with specified norm and condition number we also give a short proof of the fact that j orthogonal matrices are optimally scaled und

the drazin inverse of an even order tensor and its application to singular tensor equations abstract the notion of the moore penrose inverses of matrices was recently extended from matrix space to even order tensor space with einstein product in the literature in this paper we further study the properties of even order tensors with einstein product we define the index and characterize the invertibility of an even order square tensor we also extend the notion of the drazin inverse of a square matrix to an even order square tensor an expression for the drazin inverse through the core nilpotent decomposition for a tensor of even order is obtained as an application the drazin inverse solution of the singular linear tensor equation a x b will also be included

on the krall type discrete polynomials in this paper we present a unified theory for studying the so called krall type discrete orthogonal polynomials in particular the three term recurrence relation lowering and raising operators as well as the second order linear difference equation that the sequences of monic orthogonal polynomials satisfy are established some relevant examples of q krall polynomials are considered in detail

the twist tensor nuclear norm for video completion in this paper we propose a new low rank tensor model based on the circulant algebra namely twist tensor nuclear norm t tnn the twist tensor denotes a three way tensor representation to laterally store 2 d data slices in order on one hand t tnn convexly relaxes the tensor multirank of the twist tensor in the fourier domain which allows an efficient computation using fast fourier transform on the other t tnn is equal to the nuclear norm of block circulant matricization of the twist tensor in the original domain which extends the traditional matrix nuclear norm in a block circulant way we test the t tnn model on a video completion application that aims to fill missing values and the experiment results validate its effectiveness especially when dealing with video recorded by a nonstationary panning camera the block circulant matricization of the twist tensor can be transformed into a circulant block representation with nuclear norm invariance this representation after transformation exploits the horizontal translation relationship between the frames in a video and endows the t tnn model with a more powerful ability to reconstruct panning videos than the existing state of the art low rank models

robust low rank tensor minimization via a new tensor spectral k support norm recently based on a new tensor algebraic framework for third order tensors the tensor singular value decomposition t svd and its associated tubal rank definition have shed new light on low rank tensor modeling its applications to robust image video recovery and background modeling show promising performance due to its superior capability in modeling cross channel frame information under the t svd framework we propose a new tensor norm called tensor spectral k support norm tsp k by an alternative convex relaxation as an interpolation between the existing tensor nuclear norm tnn and tensor frobenius norm tfn it is able to simultaneously drive minor singular values to zero to induce low rankness and to capture more global information for better preserving intrinsic structure we provide the proximal operator and the polar operator for the tsp k norm as key optimization blocks along with two showcase optimization algorithms for medium and large size tensors experiments on synthetic image and video datasets in medium and large sizes all verify the superiority of the tsp k norm and the effectiveness of both optimization methods in comparison with the existing counterparts

fastderain a novel video rain streak removal method using directional gradient priors rain streaks removal is an important issue in outdoor vision systems and has recently been investigated extensively in this paper we propose a novel video rain streak removal approach fastderain which fully considers the discriminative characteristics of rain streaks and the clean video in the gradient domain specifically on the one hand rain streaks are sparse and smooth along the direction of the raindrops whereas on the other hand clean videos exhibit piecewise smoothness along the rain perpendicular direction and continuity along the temporal direction theses smoothness and continuity result in the sparse distribution in the different directional gradient domain thus we minimize 1 the ell 1 norm to enhance the sparsity of the underlying rain streaks 2 two ell 1 norm of unidirectional total variation regularizers to guarantee the anisotropic spatial smoothness and 3 an ell 1 norm of the time directional difference operator to characterize the temporal continuity a split augmented lagrangian shrinkage algorithm based algorithm is designed to solve the proposed minimization model experiments conducted on synthetic and real data demonstrate the effectiveness and efficiency of the proposed method according to the comprehensive quantitative performance measures our approach outperforms other state of the art methods especially on account of the running time the code of fastderain can be downloaded at https github com taixiangjiang fastderain

seismic trace interpolation in the f x domain interpolation of seismic traces is an effective means of improving migration when the data set exhibits spatial aliasing a major difficulty of standard interpolation methods is that they depend on the degree of reliability with which the various geological events can be separated in this respect a multichannel interpolation method is described which requires neither a priori knowledge of the directions of lateral coherence of the events nor estimation of these directions the method is based on the fact that linear events present in a section made of equally spaced traces may be interpolated exactly regardless of the original spatial interval without any attempt to determine their true dips the predictability of linear events in the f x domain allows the missing traces to be expressed as the output of a linear system the input of which consists of the recorded traces the interpolation operator is obtained by solving a set of linear equations whose coefficients depend only on the spectrum of the spatial prediction filter defined by the recorded traces synthetic examples show that this method is insensitive to random noise and that it correctly handles curvatures and lateral amplitude variations assessment of the method with a real data set shows that the interpolation yields an improved migrated section

quasi orthogonality and differential equations abstract the following theorem is proved quasi orthogonal polynomials corresponding to the semi classical or laguerre hahn families and the orthogonal polynomials themselves verify differential equations of same order reducible to two or four by the hahn theorem

robust low rank tensor recovery with regularized redescending m estimator this paper addresses the robust low rank tensor recovery problems tensor recovery aims at reconstructing a low rank tensor from some linear measurements which finds applications in image processing pattern recognition multitask learning and so on in real world applications data might be contaminated by sparse gross errors however the existing approaches may not be very robust to outliers to resolve this problem this paper proposes approaches based on the regularized redescending m estimators which have been introduced in robust statistics the robustness of the proposed approaches is achieved by the regularized redescending m estimators however the nonconvexity also leads to a computational difficulty to handle this problem we develop algorithms based on proximal and linearized block coordinate descent methods by explicitly deriving the lipschitz constant of the gradient of the data fitting risk the descent property of the algorithms is present moreover we verify that the objective functions of the proposed approaches satisfy the kurdyka ojasiewicz property which establishes the global convergence of the algorithms the numerical experiments on synthetic data as well as real data verify that our approaches are robust in the presence of outliers and still effective in the absence of outliers

symmetric convolution and the discrete sine and cosine transforms this paper discusses the use of symmetric convolution and the discrete sine and cosine transforms dsts and dcts for general digital signal processing the operation of symmetric convolution is a formalized approach to convolving symmetrically extended sequences the result is the same as that obtained by taking an inverse discrete trigonometric transform dtt of the product of the forward dtts of those two sequences there are 16 members in the family of dtts each provides a representation for a corresponding distinct type of symmetric periodic sequence the author defines symmetric convolution relates the dsts and dcts to symmetric periodic sequences and then use these principles to develop simple but powerful convolution multiplication properties for the entire family of dsts and dcts symmetric convolution can be used for discrete linear filtering when the filter is symmetric or antisymmetric the filtering will be efficient because fast algorithms exist for all versions of the dtts conventional linear convolution is possible if one first zero pad the input data symmetric convolution and its fast implementation using dtts are now an alternative to circular convolution and the dft

correlation metric for generalized feature extraction beyond linear and kernel based feature extraction we propose in this paper the generalized feature extraction formulation based on the so called graph embedding framework two novel correlation metric based algorithms are presented based on this formulation correlation embedding analysis cea which incorporates both correlational mapping and discriminating analysis boosts the discriminating power by mapping data from a high dimensional hypersphere onto another low dimensional hypersphere and preserving the intrinsic neighbor relations with local graph modeling correlational principal component analysis cpca generalizes the conventional principal component analysis pca algorithm to the case with data distributed on a high dimensional hypersphere their advantages stem from two facts 1 tailored to normalized data which are often the outputs from the data preprocessing step and 2 directly designed with correlation metric which shows to be generally better than euclidean distance for classification purpose extensive comparisons with existing algorithms on visual classification experiments demonstrate the effectiveness of the proposed algorithms

numerical cp decomposition of some difficult tensors in this paper a numerical method is proposed for canonical polyadic cp decomposition of small size tensors the focus is primarily on decomposition of tensors that correspond to small matrix multiplications here rank of the tensors is equal to the smallest number of scalar multiplications that are necessary to accomplish the matrix multiplication the proposed method is based on a constrained levenbergmarquardt optimization numerical results indicate the rank and border ranks of tensors that correspond to multiplication of matrices of the size 23 and 32 33 and 32 33 and 33 and 34 and 43 the ranks are 11 15 23 and 29 respectively in particular a novel algorithm for computing product of matrices of the sizes 34 and 43 using 29 multiplications is presented

multi dimensional imaging data recovery via minimizing the partial sum of tubal nuclear norm abstract in this paper we investigate tensor recovery problems within the tensor singular value decomposition t svd framework we propose the partial sum of the tubal nuclear norm pstnn of a tensor the pstnn is a surrogate of the tensor tubal multi rank we build two pstnn based minimization models for two typical tensor recovery problems i e the tensor completion and the tensor principal component analysis we give two algorithms based on the alternating direction method of multipliers admm to solve proposed pstnn based tensor recovery models experimental results on the synthetic data and real world data reveal the superior of the proposed pstnn

5d seismic data completion and denoising using a novel class of tensor decompositions abstractwe have developed a novel strategy for simultaneous interpolation and denoising of prestack seismic data most seismic surveys fail to cover all possible source receiver combinations leading to missing data especially in the midpoint offset domain this undersampling can complicate certain data processing steps such as amplitude variation with offset analysis and migration data interpolation can mitigate the impact of missing traces we considered the prestack data as a 5d multidimensional array or otherwise referred to as a 5d tensor using synthetic data sets we first found that prestack data can be well approximated by a low rank tensor under a recently proposed framework for tensor singular value decomposition tsvd under this low rank assumption we proposed a complexity penalized algorithm for the recovery of missing traces and data denoising in this algorithm the complexity regularization was controlled by tuning a single regularization parameter using a statistical test we tested t

a proof of conjecture on restricted isometry property constants delta _ tk left 0 t frac 4 3 right in this paper we give a complete answer to the conjecture on restricted isometry property rip constants delta tk 0 which was proposed by t cai and a zhang we have shown that when 0 the condition delta tk is sufficient to guarantee the exact recovery for all k sparse signals in the noiseless case via the constrained ell 1 norm minimization these bounds are sharp in the sense that for any epsilon 0 delta tk cannot guarantee the exact recovery of some k sparse signals furthermore it will be shown that similar characterizations also hold for low rank matrix recovery thus combined with t cai and a zhang s work a complete characterization for sharp rip constants delta tk for all t 0 is obtained to guarantee the exact recovery of all k sparse signals and matrices with rank at most k by ell 1 norm minimization and nuclear norm minimization respectively noisy cases and approximately sparse cases are also considered to solve the conjecture we construct a few identities so that rip of order tk which is the target of our main results can be perfectly applied to them

a fused cp factorization method for incomplete tensors low rank tensor completion methods have been advanced recently for modeling sparsely observed data with a multimode structure however low rank priors may fail to interpret the model factors of general tensor objects the most common method to address this drawback is to use regularizations together with the low rank priors however due to the complex nature and diverse characteristics of real world multiway data the use of a single or a few regularizations remains far from efficient and there are limited systematic experimental reports on the advantages of these regularizations for tensor completion to fill these gaps we propose a modified cp tensor factorization framework that fuses the l 2 norm constraint sparseness l 1 norm manifold and smooth information simultaneously the factorization problem is addressed through a combination of nesterov s optimal gradient descent method and block coordinate descent here we construct a smooth approximation to the l 1 norm and tv norm regularizations and then the tensor factor is updated using the projected gradient method where the step size is determined by the lipschitz constant extensive experiments on simulation data visual data completion intelligent transportation systems and gps data of user involvement are conducted and the efficiency of our method is confirmed by the results moreover the obtained results reveal the characteristics of these commonly used regularizations for tensor completion in a certain sense and give experimental guidance concerning how to use them

seislet transform and seislet frame we introduce a digital waveletlike transform which is tailored specifically for representing seismic data the transform provides a multiscale orthogonal basis with basis functions aligned along seismic events in the input data it is defined with the help of the wavelet lifting scheme combined with local plane wave destruction in the 1d case the seislet transform is designed to follow locally sinusoidal components in the 2d case it is designed to follow local plane wave components with smoothly variable slopes if more than one component is present the transform turns into an overcomplete representation or a tight frame in these terms the classic digital wavelet transform is simply a seislet transform for a zero frequency in one dimension or zero slope in two dimensions the main objective of the new transform is an effective seismic data compression for designing efficient data analysis algorithms traditional signal processing tasks such as noise attenuation and trace interpolation become simply defined in the seislet domain when applied in the offset direction on common midpoint or common image point gathers the seislet transform finds an additional application in optimal stacking of seismic records

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

generalized higher order orthogonal iteration for tensor learning and decomposition low rank tensor completion lrtc has successfully been applied to a wide range of real world problems despite the broad successful applications existing lrtc methods may become very slow or even not applicable for large scale problems to address this issue a novel core tensor trace norm minimization ctnm method is proposed for simultaneous tensor learning and decomposition and has a much lower computational complexity in our solution first the equivalence relation of trace norm of a low rank tensor and its core tensor is induced second the trace norm of the core tensor is used to replace that of the whole tensor which leads to two much smaller scale matrix tnm problems finally an efficient alternating direction augmented lagrangian method is developed to solve our problems our ctnm formulation needs only o r n nri log sqrt i n observations to reliably recover an n th order i times i times cdots times i tensor of n rank r r ldots r compared with o ri n 1 observations required by those tensor tnm methods i gg r geq r extensive experimental results show that ctnm is usually more accurate than them and is orders of magnitude faster

resolving topology ambiguity for multiple material domains abstract this paper describes a novel approach to resolve topology ambiguity in generating quality triangular and tetrahedral meshes for multiple material domains in previous works we developed an octree based dual contouring method to construct surface and volumetric meshes for complicated domains with topology ambiguity resolved however this method is based on enumeration and cannot handle domains with multiple materials as a follow up in this study we develop an automatic and efficient approach to resolve topology ambiguity for multiple material domains we first use an indicator variable to identify the topology inside each octree leaf cell whenever one cell is ambiguous we split the cell into 12 tetrahedra in the following we analyze each material change edge to create triangles or boundary tetrahedra with surrounding minimizers for tetrahedral meshes in addition to material change edges we need to analyze each interior edge to create interior tetrahedra with the ambiguity free mesh ready we are able to apply quality improvement techniques to attain good mesh quality our main contribution lies in resolving topology ambiguity by analyzing a hybrid octree with both cubic and tetrahedral leaf cells and this method works for both homogeneous and multiple material domains we have applied our algorithm to three complicated material datasets and obtained good results

moving object detection using tensor based low rank and saliently fused sparse decomposition in this paper we propose a new low rank and sparse representation model for moving object detection the model preserves the natural space time structure of video sequences by representing them as three way tensors then it operates the low rank background and sparse foreground decomposition in the tensor framework on the one hand we use the tensor nuclear norm to exploit the spatio temporal redundancy of background based on the circulant algebra on the other we use the new designed saliently fused sparse regularizer sfs to adaptively constrain the foreground with spatio temporal smoothness to refine the existing foreground smooth regularizers the sfs incorporates the local spatio temporal geometric structure information into the tensor total variation by using the 3d locally adaptive regression kernel 3d lark what is more the sfs further uses the 3d lark to compute the space time motion saliency of foreground which is combined with the l 1 norm and improves the robustness of foreground extraction finally we solve the proposed model with globally optimal guarantee extensive experiments on challenging well known data sets demonstrate that our method significantly outperforms the state of the art approaches and works effectively on a wide range of complex scenarios

further results on moore penrose inverses of tensors with application to tensor nearness problems abstract the notion of the moore penrose inverse of a matrix has been extended to tensor case with einstein product recently in this paper we continue this work and propose the full rank decomposition of tensors by introducing a new kind of tensor rank which gives a novel representation of the moore penrose inverse moreover we also obtain several formulae related to the moore penrose inverse of a tensor and its i j k inverses in addition we consider the tensor nearness problem associated with a tensor equation a generalization of matrix nearness problems that arise in many areas of applied matrix computations it is shown that the unique solution to the tensor nearness problem can be represented by using the moore penrose inverses of the given tensors

the q racah krall type polynomials abstract in this paper the krall type polynomials obtained via the addition of two mass points to the weight function of the standard q racah polynomials are introduced several algebraic properties of these polynomials are obtained and some of their limit cases are discussed

remote sensing images destriping using unidirectional hybrid total variation and nonconvex low rank regularization abstract in this paper we propose a novel model for remote sensing images destriping which includes the schatten 1 2 norm and the unidirectional first order and high order total variation regularization the main idea is that the stripe layer is low rank and the desired image possesses smoothness across stripes therefore we use the schatten 1 2 norm regularization to depict the low rankness of stripes and use the unidirectional total variation and the unidirectional high order total variation to guarantee the smoothness of the underlying image we develop the alternating direction method of multipliers algorithm to solve the proposed model extensive experiments on synthetic and real data are reported to show the superiority of the proposed method over state of the art methods in terms of both quantitative and qualitative assessments

simultaneous tensor decomposition and completion using factor priors the success of research on matrix completion is evident in a variety of real world applications tensor completion which is a high order extension of matrix completion has also generated a great deal of research interest in recent years given a tensor with incomplete entries existing methods use either factorization or completion schemes to recover the missing parts however as the number of missing entries increases factorization schemes may overfit the model because of incorrectly predefined ranks while completion schemes may fail to interpret the model factors in this paper we introduce a novel concept complete the missing entries and simultaneously capture the underlying model structure to this end we propose a method called simultaneous tensor decomposition and completion stdc that combines a rank minimization technique with tucker model decomposition moreover as the model structure is implicitly included in the tucker model we use factor priors which are usually known a priori in real world tensor objects to characterize the underlying joint manifold drawn from the model factors by exploiting this auxiliary information our method leverages two classic schemes and accurately estimates the model factors and missing entries we conducted experiments to empirically verify the convergence of our algorithm on synthetic data and evaluate its effectiveness on various kinds of real world data the results demonstrate the efficacy of the proposed method and its potential usage in tensor based applications it also outperforms state of the art methods on multilinear model analysis and visual data completion tasks

multi view intact space learning it is practical to assume that an individual view is unlikely to be sufficient for effective multi view learning therefore integration of multi view information is both valuable and necessary in this paper we propose the multi view intact space learning misl algorithm which integrates the encoded complementary information in multiple views to discover a latent intact representation of the data even though each view on its own is insufficient we show theoretically that by combing multiple views we can obtain abundant information for latent intact space learning employing the cauchy loss a technique used in statistical learning as the error measurement strengthens robustness to outliers we propose a new definition of multi view stability and then derive the generalization error bound based on multi view stability and rademacher complexity and show that the complementarity between multiple views is beneficial for the stability and generalization misl is efficiently optimized using a novel iteratively reweight residuals irr technique whose convergence is theoretically analyzed experiments on synthetic data and real world datasets demonstrate that misl is an effective and promising algorithm for practical applications

an order p tensor factorization with applications in imaging operations with tensors or multiway arrays are increasingly prevalent in many applications involving multiway data analysis this paper extends a third order factorization strategy and tensor operations defined in a recent paper m e kilmer and c d martin linear algebra appl 435 2011 pp 641 658 to order p tensors the extension to order p tensors is explained in a recursive way but for computational speed is implemented directly using the fast fourier transform a major motivation for considering factorization strategies for order p tensors is to devise new types of algorithms for general order p tensors which can be used in applications we conclude with two applications in imaging the first application is image deblurring and the second application is video facial recognition both applications involve order 4 tensors

on unifying multi view self representations for clustering by tensor multi rank minimization in this paper we address the multi view subspace clustering problem our method utilizes the circulant algebra for tensor which is constructed by stacking the subspace representation matrices of different views and then rotating to capture the low rank tensor subspace so that the refinement of the view specific subspaces can be achieved as well as the high order correlations underlying multi view data can be explored by introducing a recently proposed tensor factorization namely tensor singular value decomposition t svd kilmer et al in siam j matrix anal appl 34 1 148 172 2013 we can impose a new type of low rank tensor constraint on the rotated tensor to ensure the consensus among multiple views different from traditional unfolding based tensor norm this low rank tensor constraint has optimality properties similar to that of matrix rank derived from svd so the complementary information can be explored and propagated among all the views more thoroughly and effectively the established model called t svd based multi view subspace clustering t svd msc falls into the applicable scope of augmented lagrangian method and its minimization problem can be efficiently solved with theoretical convergence guarantee and relatively low computational complexity extensive experimental testing on eight challenging image datasets shows that the proposed method has achieved highly competent objective performance compared to several state of the art multi view clustering methods

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

minimum weighted norm interpolation of seismic records in seismic data processing we often need to interpolate and extrapolate data at missing spatial locations the reconstruction problem can be posed as an inverse problem where from inadequate and incomplete data we attempt to reconstruct the seismic wavefield at locations where measurements were not acquired we propose a wavefield reconstruction scheme for spatially band limited signals the method entails solving an inverse problem where a wavenumber domain regularization term is included the regularization term constrains the solution to be spatially band limited and imposes a prior spectral shape the numerical algorithm is quite efficient since the method of conjugate gradients in conjunction with fast matrix vector multiplications implemented via the fast fourier transform fft is adopted the algorithm can be used to perform multidimensional reconstruction in any spatial domain

low rank tensor recovery via iterative hard thresholding abstract we study extensions of compressive sensing and low rank matrix recovery matrix completion to the recovery of low rank tensors of higher order from a small number of linear measurements while the theoretical understanding of low rank matrix recovery is already well developed only few contributions on the low rank tensor recovery problem are available so far in this paper we introduce versions of the iterative hard thresholding algorithm for several tensor decompositions namely the higher order singular value decomposition hosvd the tensor train format tt and the general hierarchical tucker decomposition ht we provide a partial convergence result for these algorithms which is based on a variant of the restricted isometry property of the measurement operator adapted to the tensor decomposition at hand that induces a corresponding notion of tensor rank we show that subgaussian measurement ensembles satisfy the tensor restricted isometry property with high probability under a certain almost optimal bound on the number of measurements which depends on the corresponding tensor format these bounds are extended to partial fourier maps combined with random sign flips of the tensor entries finally we illustrate the performance of iterative hard thresholding methods for tensor recovery via numerical experiments where we consider recovery from gaussian random measurements tensor completion recovery of missing entries and fourier measurements for third order tensors

ranking hubs and authorities using matrix functions the notions of subgraph centrality and communicability based on the exponential of the adjacency matrix of the underlying graph have been effectively used in the analysis of undirected networks in this paper we propose an extension of these measures to directed networks and we apply them to the problem of ranking hubs and authorities the extension is achieved by bipartization i e the directed network is mapped onto a bipartite undirected network with twice as many nodes in order to obtain a network with a symmetric adjacency matrix we explicitly determine the exponential of this adjacency matrix in terms of the adjacency matrix of the original directed network and we give an interpretation of centrality and communicability in this new context leading to a technique for ranking hubs and authorities the matrix exponential method for computing hubs and authorities is compared to the well known hits algorithm both on small artificial examples and on more realistic real world networks a few other ranking algorithms are also discussed and compared with our technique the use of gaussian quadrature rules for calculating hub and authority scores is discussed

difference equation for modifications of meixner polynomials abstract we consider a modification of meixner moment functional by adding a mass point at x 0 we obtain the resulting orthogonal polynomials identify them as hypergeometric 3f2 functions and derive the second order difference equation which these polynomials satisfy in such a way we give the solution to a problem raised by r askey 1991 in orthogonal polynomials and their applications p 418 baltzer ag scientific basel

on the properties for modifications of classical orthogonal polynomials of discrete variables abstract we consider a modification of moment functionals for some classical polynomials of a discrete variable by adding a mass point at x 0 we obtain the resulting orthogonal polynomials identify them as hypergeometric functions and derive the second order difference equation which these polynomials satisfy the corresponding tridiagonal matrices and associated polynomials were also studied

large margin multi viewinformation bottleneck in this paper we extend the theory of the information bottleneck ib to learning from examples represented by multi view features we formulate the problem as one of encoding a communication system with multiple senders each of which represents one view of the data based on the precise components filtered out from multiple information sources through a bottleneck a margin maximization approach is then used to strengthen the discrimination of the encoder by improving the code distance within the frame of coding theory the resulting algorithm therefore inherits all the merits of the ib principle and coding theory it has two distinct advantages over existing algorithms namely that our method finds a tradeoff between the accuracy and complexity of the multi view model and that the encoded multi view data retains sufficient discrimination for classification we also derive the robustness and generalization error bound of the proposed algorithm and reveal the specific properties of multi view learning first the complementarity of multi view features guarantees the robustness of the algorithm second the consensus of multi view features reduces the empirical rademacher complexity of the objective function enhances the accuracy of the solution and improves the generalization error bound of the algorithm the resulting objective function is solved efficiently using the alternating direction method experimental results on annotation classification and recognition tasks demonstrate that the proposed algorithm is promising for practical applications

algebraic and spectral properties of some quasiorthogonal polynomials encountered in quantum radiation the nodal structure of the wavefunctions of a large class of quantum mechanical potentials is often governed by the distribution of zeros of real quasiorthogonal polynomials it is known that these polynomials i may be described by an arbitrary linear combination of two orthogonal polynomials pn x and ii have real and simple zeros here the three term recurrence relation the second order differential equation and the distribution of zeros of quasiorthogonal polynomials of the classical class i e when pn x is a jacobi laguerre or hermite polynomial are derived and analyzed specifically the exact values of the newton sum rules and the wkb density of zeros of these polynomials are found

mr ntd manifold regularization nonnegative tucker decomposition for tensor data dimension reduction and representation with the advancement of data acquisition techniques tensor multidimensional data objects are increasingly accumulated and generated for example multichannel electroencephalographies multiview images and videos in these applications the tensor objects are usually nonnegative since the physical signals are recorded as the dimensionality of tensor objects is often very high a dimension reduction technique becomes an important research topic of tensor data from the perspective of geometry high dimensional objects often reside in a low dimensional submanifold of the ambient space in this paper we propose a new approach to perform the dimension reduction for nonnegative tensor objects our idea is to use nonnegative tucker decomposition ntd to obtain a set of core tensors of smaller sizes by finding a common set of projection matrices for tensor objects to preserve geometric information in tensor data we employ a manifold regularization term for the core tensors constructed in the tucker decomposition an algorithm called manifold regularization ntd mr ntd is developed to solve the common projection matrices and core tensors in an alternating least squares manner the convergence of the proposed algorithm is shown and the computational complexity of the proposed method scales linearly with respect to the number of tensor objects and the size of the tensor objects respectively these theoretical results show that the proposed algorithm can be efficient extensive experimental results have been provided to further demonstrate the effectiveness and efficiency of the proposed mr ntd algorithm

dimensionality reduction in higher order signal processing and rank r1 r2 rn reduction in multilinear algebra abstract in this paper we review a multilinear generalization of the singular value decomposition and the best rank r 1 r 2 r n approximation of higher order tensors we show that they are important tools for dimensionality reduction in higher order signal processing we discuss applications in independent component analysis simultaneous matrix diagonalization and subspace variants of algorithms based on higher order statistics

image denoising using the higher order singular value decomposition in this paper we propose a very simple and elegant patch based machine learning technique for image denoising using the higher order singular value decomposition hosvd the technique simply groups together similar patches from a noisy image with similarity defined by a statistically motivated criterion into a 3d stack computes the hosvd coefficients of this stack manipulates these coefficients by hard thresholding and inverts the hosvd transform to produce the final filtered image our technique chooses all required parameters in a principled way relating them to the noise model we also discuss our motivation for adopting the hosvd as an appropriate transform for image denoising we experimentally demonstrate the excellent performance of the technique on grayscale as well as color images on color images our method produces state of the art results outperforming other color image denoising algorithms at moderately high noise levels a criterion for optimal patch size selection and noise variance estimation from the residual images after denoising is also presented

partial sum minimization of singular values in robust pca algorithm and applications robust principal component analysis rpca via rank minimization is a powerful tool for recovering underlying low rank structure of clean data corrupted with sparse noise outliers in many low level vision problems not only it is known that the underlying structure of clean data is low rank but the exact rank of clean data is also known yet when applying conventional rank minimization for those problems the objective function is formulated in a way that does not fully utilize a priori target rank information about the problems this observation motivates us to investigate whether there is a better alternative solution when using rank minimization in this paper instead of minimizing the nuclear norm we propose to minimize the partial sum of singular values which implicitly encourages the target rank constraint our experimental analyses show that when the number of samples is deficient our approach leads to a higher success rate than conventional rank minimization while the solutions obtained by the two approaches are almost identical when the number of samples is more than sufficient we apply our approach to various low level vision problems e g high dynamic range imaging motion edge detection photometric stereo image alignment and recovery and show that our results outperform those obtained by the conventional nuclear norm rank minimization method

characterization of the subdifferential of some matrix norms abstract a characterization is given of the subdifferential of matrix norms from two classes orthogonally invariant norms and operator or subordinate norms specific results are derived for some special cases

patched tube unitary transform for robust tensor completion abstract the aim of the robust tensor completion problem for third order tensors is to recover a low rank tensor from incomplete and or corrupted observations in this paper we develop a patched tubes unitary transform method for robust tensor completion the proposed method is to extract similar patched tubes to form a third order sub tensor and then a transformed tensor singular value decomposition is employed to recover such low rank incomplete and or corrupted sub tensor here the unitary transform matrix for transformed tensor singular value decomposition is constructed by using left singular vectors of the unfolding matrix arising from such sub tensor moreover we establish the perturbation results of the transformed tensor singular value decomposition for patched tubes tensor completion extensive numerical experiments on hyperspectral video and face data sets are presented to demonstrate the superior performance of the proposed patched tubes unitary transform method over testing state of the art robust tensor completion methods

bayesian conditional tensor factorizations for high dimensional classification abstractin many application areas data are collected on a categorical response and high dimensional categorical predictors with the goals being to build a parsimonious model for classification while doing inferences on the important predictors in settings such as genomics there can be complex interactions among the predictors by using a carefully structured tucker factorization we define a model that can characterize any conditional probability while facilitating variable selection and modeling of higher order interactions following a bayesian approach we propose a markov chain monte carlo algorithm for posterior computation accommodating uncertainty in the predictors to be included under near low rank assumptions the posterior distribution for the conditional probability is shown to achieve close to the parametric rate of contraction even in ultra high dimensional settings the methods are illustrated using simulation examples and biomedical applications supplementary materials for this artic

quality improvement of non manifold hexahedral meshes for critical feature determination of microstructure materials this paper describes a novel approach to improve the quality of non manifold hexahedral meshes with feature preservation for microstructure materials in earlier works we developed an octree based isocontouring method to construct unstructured hexahedral meshes for domains with multiple materials by introducing the notion of material change edge to identify the interface between two or more materials however quality improvement of non manifold hexahedral meshes is still a challenge in the present algorithm all the vertices are categorized into seven groups and then a comprehensive method based on pillowing geometric flow and optimization techniques is developed for mesh quality improvement the shrink set in the modified pillowing technique is defined automatically as the boundary of each material region with the exception of local non manifolds in the relaxation based smoothing process non manifold points are identified and fixed planar boundary curves and interior spatial curves are distinguished and then regularized using b spline interpolation and resampling grain boundary surface patches and interior vertices are improved as well finally the local optimization method eliminates negative jacobians of all the vertices we have applied our algorithms to two beta titanium datasets and the constructed meshes are validated via a statistics study finite element analysis of the 92 grain titanium is carried out based on the improved mesh and compared with the direct voxel to element technique

a general product of tensors with applications abstract we study a general product of two n dimensional tensors a and b with orders m 2 and k 1 this product satisfies the associative law and is a generalization of the usual matrix product using this product many concepts and known results of tensors can be simply expressed and or proved and a number of applications of it will be given using the associative law of this tensor product and some properties on the resultant of a system of homogeneous equations on n variables we define the similarity and congruence of tensors which are also the generalizations of the corresponding relations for matrices and prove that similar tensors have the same characteristic polynomials thus the same spectra we study two special kinds of similarity permutational similarity and diagonal similarity and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors we also define the direct product of tensors in matrix case it is also called the kronecker product and give its applications in the study of the spectra of two kinds of the products of hypergraphs we also give applications of this general product in the study of nonnegative tensors including a characterization of primitive tensors the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors

the generalized inverses of tensors and an application to linear models abstract in this paper we recall and extend some tensor operations then the generalized inverse of tensors is established by using tensor equations moreover we investigate the least squares solutions of tensor equations an algorithm to compute the moore penrose inverse of an arbitrary tensor is constructed finally we apply the obtained results to higher order gauss markov theorem

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

multiple material marching cubes algorithm the accurate reconstruction of three dimensional 3d boundary surfaces from two dimensional 2d medical images is a crucial procedure in most applications of computational biomedical engineering this paper addresses an innovative system that efficiently reconstructs accurate multiple material 3d surface meshes from 2d medical images it is based on an enhanced marching cubes algorithm the multi material marching cubes algorithm m3c which extracts boundary surfaces between different materials within one sweep of the image stack in an integrated manner the continuity and integrity of the surfaces are ensured with this robust algorithm surface adjustment algorithms were also revised to adapt to the multiple material nature of the system copyright 2003 john wiley sons ltd

exposed faces and duality for symmetric and unitarily invariant norms abstract let be a unitarily invariant norm on the space of real or complex n m matrices and g the associated symmetric gauge function thus a g s a where s a is the decreasing sequence of singular values of a denote by b and bg the closed unit balls of and g in a previous paper we showed a close relationship between the faces of b and those of bg in particular to each face of b we associated a standard face of bg and we used this association to completely describe the matrices that are members of an individual face of in the present paper we consider the duality operator that transforms each exposed face of b into an exposed face of the unit ball of s dual and the duality operator that does the same with exposed faces of bg we show that these two operators are very nicely related among other results we prove the following given an exposed face e of b the standard face associated with the dual of e is precisely the dual of the standard face of bg associated with e e is an exposed face of b if and only if its associated standard face is an exposed face of bg as a by product we completely determine the subdifferential of in terms of the subdifferential of g and we completely characterize the matrices that are dual to a given matrix a with respect to

generalized tensor function via the tensor singular value decomposition based on the t product abstract in this paper we present the definition of generalized tensor function according to the tensor singular value decomposition t svd based on the tensor t product also we introduce the compact singular value decomposition t csvd of tensors from which the projection operators and moore penrose inverse of tensors are obtained we establish the cauchy integral formula for tensors by using the partial isometry tensors and apply it into the solution of tensor equations then we establish the generalized tensor power and the taylor expansion of tensors explicit generalized tensor functions are listed we define the tensor bilinear and sesquilinear forms and propose theorems on structures preserved by generalized tensor functions for complex tensors we established an isomorphism between complex tensors and real tensors in the last part of our paper we find that the block circulant operator establishes an isomorphism between tensors and matrices this isomorphism is used to prove the f stochastic structure is invariant under generalized tensor functions the concept of invariant tensor cones is raised

diagonalizing properties of the discrete cosine transforms since its introduction in 1974 by ahmed et al the discrete cosine transform dct has become a significant tool in many areas of digital signal processing especially in signal compression there exist eight types of discrete cosine transforms dcts we obtain the eight types of dcts as the complete orthonormal set of eigenvectors generated by a general form of matrices in the same way as the discrete fourier transform dft can be obtained as the eigenvectors of an arbitrary circulant matrix these matrices can be decomposed as the sum of a symmetric toeplitz matrix plus a hankel or close to hankel matrix scaled by some constant factors we also show that all the previously proposed generating matrices for the dcts are simply particular cases of these general matrix forms using these matrices we obtain for each dct a class of stationary processes verifying certain conditions with respect to which the corresponding dct has a good asymptotic behavior in the sense that it approaches karhunen loeve transform performance as the block size n tends to infinity as a particular result we prove that the eight types of dcts are asymptotically optimal for all finite order markov processes we finally study the decorrelating power of the dcts obtaining expressions that show the decorrelating behavior of each dct with respect to any stationary processes

low tucker rank tensor recovery via admm based on exact and inexact iteratively reweighted algorithms abstract in this paper we establish a non convex l p norm relaxation model for low tucker rank tensor recovery problem and equivalently transform it to a non convex minimization problem with separable structure by introducing series of auxiliary variables in particular we propose two alternating direction method of multipliers admm based on exact and inexact iteratively reweighted algorithms to solve the obtained non convex relaxation problem respectively which are proved to be convergent we implement the proposed algorithms in numerical experiments for solving low tucker rank tensor recovery problem on simulation data and real data and compare them with other existing state of art algorithms numerical results show the effectiveness of the proposed algorithms for solving low rank tensor recovery problem and image recovery

tensor completion using total variation and low rank matrix factorization in this paper we study the problem of recovering a tensor with missing data we propose a new model combining the total variation regularization and low rank matrix factorization a block coordinate decent bcd algorithm is developed to efficiently solve the proposed optimization model we theoretically show that under some mild conditions the algorithm converges to the coordinatewise minimizers experimental results are reported to demonstrate the effectiveness of the proposed model and the efficiency of the numerical scheme

twist tensor total variation regularized reweighted nuclear norm based tensor completion for video missing area recovery abstract this paper focuses on recovering multi dimensional signal called tensor which is corrupted by random missing areas the performance of the conventional tensor completion techniques deteriorate when the tensor multi rank is large and or large missing areas moreover these techniques are weak in preserving the edges in the signals like images videos this paper proposes an efficient method to overcome these problems by simultaneously combining novel twist tensor total variation norm to exploit spatio temporal correlation and tensor singular value decomposition t svd based reweighted nuclear norm to improve low multi rank tensor recovery the twist tensor total variation norm takes care of edges in the recovered data and aids the recovery of missing areas by utilising the similarities in the adjacent samples the reweighted nuclear norm handles corrupted large rank tensors by sparsity enhancement via reweighting its singular values the effectiveness of the proposed method is established by applying to video completion problem and experimental results reveal that the algorithm outperforms its counterparts

third order tensors as linear operators on a space of matrices abstract a recently proposed tensor tensor multiplication m e kilmer c d martin l perrone a third order generalization of the matrix svd as a product of third order tensors tech rep tr 2008 4 tufts university october 2008 opens up new avenues to understanding the action of n n n tensors on a space of n n matrices in particular it emphasizes the need to understand the space of objects upon which tensors act this paper defines a free module and shows that every linear transformation on that module can be represented by tensor multiplication in addition it presents a generalization of ideas of eigenvalue and eigenvector to the space of n n n tensors

tensor robust principal component analysis with a new tensor nuclear norm in this paper we consider the tensor robust principal component analysis trpca problem which aims to exactly recover the low rank and sparse components from their sum our model is based on the recently proposed tensor tensor product or t product 14 induced by the t product we first rigorously deduce the tensor spectral norm tensor nuclear norm and tensor average rank and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm these definitions their relationships and properties are consistent with matrix cases equipped with the new tensor nuclear norm we then solve the trpca problem by solving a convex program and provide the theoretical guarantee for the exact recovery our trpca model and recovery guarantee include matrix rpca as a special case numerical experiments verify our results and the applications to image recovery and background modeling problems demonstrate the effectiveness of our method

second order difference equations for certain families of discrete polynomials abstract in this paper we will consider two algorithms which allow us to obtain second order linear difference equations for certain families of polynomials the corresponding algorithms can be implemented in any computer algebra system in order to obtain explicit expressions of the coefficients of the difference equations

the discrete cosine transform each discrete cosine transform dct uses n real basis vectors whose components are cosines in the dct 4 for example the j th component of boldv k is cos j frac 1 2 k frac 1 2 frac pi n these basis vectors are orthogonal and the transform is extremely useful in image processing if the vector boldx gives the intensities along a row of pixels its cosine series sum c k boldv k has the coefficients c k boldx boldv k n they are quickly computed from a fast fourier transform but a direct proof of orthogonality by calculating inner products does not reveal how natural these cosine vectors are we prove orthogonality in a different way each dct basis contains the eigenvectors of a symmetric second difference matrix by varying the boundary conditions we get the established transforms dct 1 through dct 4 other combinations lead to four additional cosine transforms the type of boundary condition dirichlet or neumann centered at a meshpoint or a midpoint determines the applications that are appropriate for each transform the centering also determines the period n 1 or n in the established transforms n frac 1 2 or n frac 1 2 in the other four the key point is that all these eigenvectors of cosines come from simple and familiar matrices