tangent cones to tensor train varieties abstract as already done for the matrix case in 1 p 256 2 thm 6 1 p 1872 and 3 thm 3 2 we give a parametrization of the bouligand tangent cone of the variety of tensors of bounded tensor train tt rank we discuss how the proof generalizes to any binary hierarchical format the parametrization can be rewritten as an orthogonal sum of tt tensors its retraction onto the variety is particularly easy to compose we also give an implicit description of the tangent cone as the solution of a system of polynomial equations

superfast solution of linear convolutional volterra equations using qtt approximation we address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular toeplitz matrices and the recently proposed qtt format the inverses of such matrices can be computed by the divide and conquer and modified bini s algorithms for which we present the versions with the qtt approximation we also present an efficient formula for the shift of vectors given in qtt format which is used in the divide and conquer algorithm as a result we reduce the complexity of inversion from the fast fourier level o n log n to the speed of superfast fourier transform i e o log 2 n the results of the paper are illustrated by numerical examples

tensor train versus monte carlo for the multicomponent smoluchowski coagulation equation in this paper we present a novel numerical algorithm for the space homogeneous multicomponent multidimensional smoluchowski coagulation equation the number of components is considered as dimensionality the new methodology is based on the classical finite difference predictor corrector scheme in a straightforward implementation of this scheme however one would have to compute and store prohibitively many values of the grid function at the nodes of a multidimensional grid we propose to use special low parametric representations for the grid functions and as well for the coagulation kernel the corresponding multidimensional arrays are approximated by low rank tensor train decompositions reducing them to combinations of small low dimensional arrays eventually to matrices for which we can use fast algorithms of linear algebra instead of o n 2 d operations in the classical scheme we propose a new method that requires only o d 2 n log n operations where n is the number of nodes per axis in the space grid and d is the number of components in this work we accelerate the predictor corrector time scheme and use the trapezoidal rule for the computation of multidimensional integral operators thus the accuracy of the new method is o h 2 2 where h is the space grid step and is the time step

multigrid accelerated tensor approximation of function related multidimensional arrays in this paper we describe and analyze a novel tensor approximation method for discretized multidimensional functions and operators in mathbb r d based on the idea of multigrid acceleration the approach stands on successive reiterations of the orthogonal tucker tensor approximation on a sequence of nested refined grids on the one hand it provides a good initial guess for the nonlinear iterations to find the approximating subspaces on finer grids on the other hand it allows us to transfer from the coarse to fine grids the important data structure information on the location of the so called most important fibers in directional unfolding matrices the method indicates linear complexity with respect to the size of data representing the input tensor in particular if the target tensor is given by using the rank r canonical model then our approximation method is proved to have linear scaling in the univariate grid size n and in the input rank r the method is tested by three dimensional 3d electronic structure calculations for the multigrid accelerated low tucker rank approximation of the all electron densities having strong nuclear cusps we obtain high resolution of their 3d convolution product with the newton potential the accuracy of order 10 6 in max norm is achieved on large n times n times n grids up to n 1 6 cdot10 4 with the time scale in several minutes

similarity vs unitary similarity and perturbation analysis of sign characteristics complex and real indefinite inner products for selfadjoint and unitary matrices with respect to indefinite inner products on finite dimensional complex or real vector spaces the unitary similarity relation is studied in contrast to the similarity relation the number of unitary similarity classes in a similarity class of such matrices is identified it is proved that under sufficiently small perturbations similarity of selfadjoint or unitary with respect to an indefinite inner product matrices implies unitary similarity estimates are given for the sizes of perturbation that guarantee this property canonical forms and sign characteristics play a key role in the proofs a perturbation analysis is developed of the sign characteristics in the real case under sufficiently small structure preserving perturbations of both selfadjoint or unitary matrices and of the indefinite inner product

deterministic and probabilistic conditions for finite completability of low tucker rank tensor we investigate the fundamental conditions on the sampling pattern i e locations of the sampled entries for finite completability of a low rank tensor given some components of its tucker rank in order to find the deterministic necessary and sufficient conditions we propose an algebraic geometric analysis on the tucker manifold which allows us to incorporate multiple rank components in the proposed analysis in contrast with the conventional geometric approaches on the grassmannian manifold this analysis characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern which is closely related to finite completability of the sampled tensor where finite completability simply means that the number of possible completions of the sampled tensor is finite probabilistic conditions are then studied and a lower bound on the sampling probability is given which guarantees that the proposed deterministic conditions on the sampling patterns for finite completability hold with high probability furthermore using the proposed geometric approach for finite completability we propose a sufficient condition on the sampling pattern that ensures there exists exactly one completion of the sampled tensor

stable robust and super fast reconstruction of tensors using multi way projections in the framework of multidimensional compressed sensing cs we introduce an analytical reconstruction formula that allows one to recover an n th order data tensor underline bf x in bbr i 1 times i 2 times cdots times i n from a reduced set of multi way compressive measurements by exploiting its low multilinear rank structure moreover we show that an interesting property of multi way measurements allows us to build the reconstruction based on compressive linear measurements taken only in two selected modes independently of the tensor order n in addition it is proved that in the matrix case and in a particular case with 3rd order tensors where the same 2d sensor operator is applied to all mode 3 slices the proposed reconstruction underline bf x tau is stable in the sense that the approximation error is comparable to the one provided by the best low multilinear rank approximation where tau is a threshold parameter that controls the approximation error through the analysis of the upper bound of the approximation error we show that in the 2d case an optimal value for the threshold parameter tau tau 0 0 exists which is confirmed by our simulation results on the other hand our experiments on 3d datasets show that very good reconstructions are obtained using tau 0 which means that this parameter does not need to be tuned our extensive simulation results demonstrate the stability and robustness of the method when it is applied to real world 2d and 3d signals a comparison with state of the arts sparsity based cs methods specialized for multidimensional signals is also included a very attractive characteristic of the proposed method is that it provides a direct computation i e it is non iterative in contrast to all existing sparsity based cs algorithms thus providing super fast computations even for large datasets

qtt rank one vectors with qtt rank one and full rank fourier images abstract quantics tensor train qtt a new data sparse format for one and multi dimensional vectors is based on a bit representation of mode indices followed by a separation of variables a radix 2 recursion that lays behind the famous fft algorithm can be efficiently applied to vectors in the qtt format if input and all intermediate vectors of the fft algorithm have moderate qtt ranks the resulted qtt fft algorithm outperforms the fft for large vectors and has asymptotically the same complexity as the superfast quantum fourier transform it is instructive to describe a class of such vectors explicitly we identify all vectors that have qtt ranks one on input intermediate steps and output of the fft algorithm we also give an example of qtt rank one vector that has the fourier image with full qtt ranks we show by numerical experiments that for certain rank one vectors with full rank fourier images the practical ranks remain moderate for large mode sizes

bivariate survival models induced by frailties abstract when two observed survival times depend via a proportional hazards model on the same unobserved variable called a frailty this common dependence induces an association between the observed times this article considers the class of bivariate survival distributions that can arise in this way extends it to allow negative association and shows that the observable bivariate distribution determines the unobserved frailty distribution up to a scale parameter a cross ratio function easily estimated even from censored data is the key to both the characterization results and inferential procedures and diagnostic plots that are introduced and illustrated by real examples the models considered are the natural counterpart for survival data of the latent variable models that have long been used in factor analysis for continuous data or in latent structure analysis for binary data the main distinguishing feature of survival data is the possibility of censoring which makes the standard methods difficu

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

solving the master equation without kinetic monte carlo in multiscale modeling of heterogeneous catalytic processes one crucial point is the solution of a markovian master equation describing the stochastic reaction kinetics usually this is too high dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic monte carlo method in this study we break the curse of dimensionality for the direct solution of the markovian master equation by exploiting the tensor train format for this purpose the performance of the approach is demonstrated on a first principles based reduced model for the co oxidation on the ruo2 110 surface we investigate the complexity for increasing system size and for various reaction conditions the advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness

semiparametric estimation of random effects using the cox model based on the em algorithm consider a survival experiment where individuals within a certain subset of the population share a common unobservable random frailty such a frailty could be an unobservable genetic or early environmental effect if individuals were in sibling groups or an environmental effect if individuals were grouped by households suppose that if the frailty omega is known the cox proportional hazards model for the observable covariates is valid with the consequence of the random effect being a multiplicative factor on the hazard rate assuming tht the random frailties follow a gamma distribution estimates of the fixed and random effects are obtained by using an em algorithm based on a profile likelihood construction the method developed is applied to the framingham heart study to examine the risks of smoking and cholesterol levels adjusting for potential random effects

the geometry of algorithms using hierarchical tensors in this paper the differential geometry of the novel hierarchical tucker format for tensors is derived the set ht k of tensors with fixed tree t and hierarchical rank k is shown to be a smooth quotient manifold namely the set of orbits of a lie group action corresponding to the non unique basis representation of these hierarchical tensors explicit characterizations of the quotient manifold its tangent space and the tangent space of ht k are derived suitable for high dimensional problems the usefulness of a complete geometric description is demonstrated by two typical applications first new convergence results for the nonlinear gauss seidel method on ht k are given notably and in contrast to earlier works on this subject the task of minimizing the rayleigh quotient is also addressed second evolution equations for dynamic tensor approximation are formulated in terms of an explicit projection operator onto the tangent space of ht k in addition a numerical comparison is made between this dynamical approach and the standard one based on truncated singular value decompositions 2013 elsevier inc

density matrix formulation for quantum renormalization groups a generalization of the numerical renormalization group procedure used first by wilson for the kondo problem is presented it is shown that this formulation is optimal in a certain sense as a demonstration of the effectiveness of this approach results from numerical real space renormalization group calculations for heisenberg chains are presented

convergence results for projected line search methods on varieties of low rank matrices via łojasiewicz inequality the aim of this paper is to derive convergence results for projected line search methods on the real algebraic variety mathcal m le k of real m times n matrices of rank at most k such methods extend riemannian optimization methods which are successfully used on the smooth manifold mathcal m k of rank k matrices to its closure by taking steps along gradient related directions in the tangent cone and afterwards projecting back to mathcal m le k considering such a method circumvents the difficulties which arise from the nonclosedness and the unbounded curvature of mathcal m k the pointwise convergence is obtained for real analytic functions on the basis of a ojasiewicz inequality for the projection of the antigradient to the tangent cone if the derived limit point lies on the smooth part of mathcal m le k i e in mathcal m k this boils down to more or less known results but with the benefit that asymptotic convergence rate estimates for specific step sizes

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

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

a low rank approach to the computation of path integrals we present a method for solving the reaction diffusion equation with general potential in free space it is based on the approximation of the feynman kac formula by a sequence of convolutions on sequentially diminishing grids for computation of the convolutions we propose a fast algorithm based on the low rank approximation of the hankel matrices the algorithm has complexity of o n r m log m n r 2 m flops and requires o m r floating point numbers in memory where n is the dimension of the integral r n and m is the mesh size in one dimension the presented technique can be generalized to the higher order diffusion processes

polynomial chaos expansions for dependent random variables abstract polynomial chaos expansions pce are well suited to quantifying uncertainty in models parameterized by independent random variables the assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate pce coefficients in contrast the application of pce to models of dependent variables is much more challenging three approaches can be used to construct pce of models of dependent variables the first approach uses mapping methods where measure transformations such as the nataf and rosenblatt transformation can be used to map dependent random variables to independent ones however we show that this can significantly degrade performance since the jacobian of the map must be approximated a second strategy is the class of dominating support methods in these approaches a pce is built using independent random variables whose distributional support dominates the support of the true dependent joint density we provide evidence that this approach appears to produce approximations with suboptimal accuracy a third approach the novel method proposed here uses gram schmidt orthogonalization gso to numerically compute orthonormal polynomials for the dependent random variables this approach has been used successfully when solving differential equations using the intrusive stochastic galerkin method and in this paper we use gso to build pce using a non intrusive stochastic collocation method the stochastic collocation method treats the model as a black box and builds approximations of the input output map from a set of samples building pce from samples can introduce ill conditioning which does not plague stochastic galerkin methods to mitigate this ill conditioning we generate weighted leja sequences which are nested sample sets to build accurate polynomial interpolants we show that our proposed approach gso with weighted leja sequences produces pce which are orders of magnitude more accurate than pce constructed using mapping or dominating support methods

cyclic dimensions kernel multiplicities and gohberg kaashoek numbers abstract two geometric characteristics namely cyclic dimensions and kernel multiplicities are introduced for a square matrix the connection between these characteristics and gohberg kaashoek numbers is studied on this basis two simple geometric proofs are given for the theorem about the change of the jordan structure of a given matrix under small perturbation

solution of linear systems and matrix inversion in the tt format tensors arise naturally in high dimensional problems in chemistry financial mathematics and many other areas the numerical treatment of such problems is difficult due to the curse of dimensionality the number of unknowns and the computational complexity grow exponentially with the dimension of the problem to break the curse of dimensionality low parametric representations or formats have to be used in this paper we make use of the tt format tensor train format which is one of the most effective stable representations of high dimensional tensors basic linear algebra operations in the tt format are now well developed our goal is to provide a black box type of solver for linear systems where both the matrix and the right hand side are in the tt format an efficient dmrg density matrix renormalization group method is proposed and several tricks are employed to make it work the numerical experiments confirm the effectiveness of our approach

efficient tensor completion for color image and video recovery low rank tensor train this paper proposes a novel approach to tensor completion which recovers missing entries of data represented by tensors the approach is based on the tensor train tt rank which is able to capture hidden information from tensors thanks to its definition from a well balanced matricization scheme accordingly new optimization formulations for tensor completion are proposed as well as two new algorithms for their solution the first one called simple low rank tensor completion via tt silrtc tt is intimately related to minimizing a nuclear norm based on tt rank the second one is from a multilinear matrix factorization model to approximate the tt rank of a tensor and is called tensor completion by parallel matrix factorization via tt tmac tt a tensor augmentation scheme of transforming a low order tensor to higher orders is also proposed to enhance the effectiveness of silrtc tt and tmac tt simulation results for color image and video recovery show the clear advantage of our method over all other methods

the change of the jordan structure of a mtrix under small perturbations abstract in this paper a complete description including multiplicity is given for the jordan structure of a matrix which is a small perturbation of a matrix of known jordan structure the problem solved here was solved independently as noted and the other solution has been published in english however the solutions are different the problem is important and it seems desirable to have this solution widely available

tt cross approximation for multidimensional arrays abstract as is well known a rank r matrix can be recovered from a cross of r linearly independent columns and rows and an arbitrary matrix can be interpolated on the cross entries other entries by this cross or pseudo skeleton approximation are given with errors depending on the closeness of the matrix to a rank r matrix and as well on the choice of cross in this paper we extend this construction to d dimensional arrays tensors and suggest a new interpolation formula in which a d dimensional array is interpolated on the entries of some tt cross tensor train cross the total number of entries and the complexity of our interpolation algorithm depend on d linearly so the approach does not suffer from the curse of dimensionality we also propose a tt cross method for computation of d dimensional integrals and apply it to some examples with dimensionality in the range from d 100 up to d 4000 and the relative accuracy of order 10 10 in all constructions we capitalize on the new tensor decomposition in the form of tensor trains tt decomposition

majorizing measures without measures we give a reformulation of majorizing measures that does not involve measures but rather special sequences of partitions this formulation is more convenient to perform chaining with different distances

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

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

a new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs the theta bodies of a polynomial ideal are a series of semidefinite programming relaxations of the convex hull of the real variety of the ideal in this paper we construct the theta bodies of the vanishing ideal of cycles in a binary matroid applied to cuts in graphs this yields a new hierarchy of semidefinite programming relaxations of the cut polytope of the graph if the binary matroid avoids certain minors we can characterize when the first theta body in the hierarchy equals the cycle polytope of the matroid specialized to cuts in graphs this result solves a problem posed by lovasz

approximation algorithms for model based compressive sensing compressive sensing cs states that a sparse signal can be recovered from a small number of linear measurements and that this recovery can be performed efficiently in polynomial time the framework of model based cs model cs leverages additional structure in the signal and provides new recovery schemes that can reduce the number of measurements even further this idea has led to measurement efficient recovery schemes for a variety of signal models however for any given model model cs requires an algorithm that solves the model projection problem given a query signal report the signal in the model that is closest to the query signal often this optimization can be computationally very expensive moreover an approximation algorithm is not sufficient for this optimization to provably succeed as a result the model projection problem poses a fundamental obstacle for extending model cs to many interesting classes of models in this paper we introduce a new framework that we call approximation tolerant model cs this framework includes a range of algorithms for sparse recovery that require only approximate solutions for the model projection problem in essence our work removes the aforementioned obstacle to model cs thereby extending model cs to a much wider class of signal models interestingly all our algorithms involve both the minimization and the maximization variants of the model projection problem we instantiate this new framework for a new signal model that we call the constrained earth mover distance cemd model this model is particularly useful for signal ensembles where the positions of the nonzero coefficients do not change significantly as a function of spatial or temporal location we develop novel approximation algorithms for both the maximization and the minimization versions of the model projection problem via graph optimization techniques leveraging these algorithms and our framework results in a nearly sample optimal sparse recovery scheme for the cemd model

on low rank approximability of solutions to high dimensional operator equations and eigenvalue problems low rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high dimensional applications in this work we study the low rank approximability of solutions to linear systems and eigenvalue problems on hilbert spaces although this question is central to the success of all existing solvers based on low rank tensor techniques very few of the results available so far allow to draw meaningful conclusions for higher dimensions in this work we develop a constructive framework to study low rank approximability one major assumption is that the involved linear operator admits a low rank representation with respect to the chosen tensor format a property that is known to hold in a number of applications simple conditions which are shown to hold for a fairly general problem class guarantee that our derived low rank truncation error estimates do not deteriorate as the dimensionality increases c 2015 elsevier inc all rights reserved

asymptotic theory for the frailty model the frailty model is a generalization of cox s proportional hazards model which includes a random effect nielsen gill andersen and sorensen 1992 proposed an em algorithm to estimate the cumulative baseline hazard and the variance of the random effect here the asymptotic distribution of the estimators is given along with a consistent estimator of the asymptotic variance

sparse and low rank tensor estimation via cubic sketchings in this paper we propose a general framework for sparse and low rank tensor estimation from cubic sketchings a two stage non convex implementation is developed based on sparse tensor decomposition and thresholded gradient descent which ensures exact recovery in the noiseless case and stable recovery in the noisy case with high probability the non asymptotic analysis sheds light on an interplay between optimization error and statistical error the proposed procedure is shown to be rate optimal under certain conditions as a technical by product novel high order concentration inequalities are derived for studying high moment sub gaussian tensors an interesting tensor formulation illustrates the potential application to high order interaction pursuit in high dimensional linear regression

lipschitz properties of structure preserving matrix perturbations abstract it is proved that under small perturbations that preserve the jordan structure canonical transformation matrices have the lipschitz property analogous result is proved for matrices under additional structure such as hamiltonian skew hamiltonian and symplectic both real and complex cases are considered

on the interconnection between the higher order singular values of real tensors a higher order tensor allows several possible matricizations reshapes into matrices the simultaneous decay of singular values of such matricizations has crucial implications on the low rank approximability of the tensor via higher order singular value decomposition it is therefore an interesting question which simultaneous properties the singular values of different tensor matricizations actually can have but it has not received the deserved attention so far in this paper preliminary investigations in this direction are conducted while it is clear that the singular values in different matricizations cannot be prescribed completely independent from each other numerical experiments suggest that sufficiently small but otherwise arbitrary perturbations preserve feasibility an alternating projection heuristic is proposed for constructing tensors with prescribed singular values assuming their feasibility regarding the related problem of characterising sets of tensors having the same singular values in specified matricizations it is noted that orthogonal equivalence under multilinear matrix multiplication is a sufficient condition for two tensors to have the same singular values in all principal tucker type matricizations but in contrast to the matrix case not necessary an explicit example of this phenomenon is given

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

breaking the curse of dimensionality or how to use svd in many dimensions for d dimensional tensors with possibly large d 3 an hierarchical data structure called the tree tucker format is presented as an alternative to the canonical decomposition it has asymptotically the same and often even smaller number of representation parameters and viable stability properties the approach involves a recursive construction described by a tree with the leafs corresponding to the tucker decompositions of three dimensional tensors and is based on a sequence of svds for the recursively obtained unfolding matrices and on the auxiliary dimensions added to the initial spatial dimensions it is shown how this format can be applied to the problem of multidimensional convolution convincing numerical examples are given

tensor structured factorized calculation of two electron integrals in a general basis in this paper the problem of efficient grid based computation of the two electron integrals tei in a general basis is considered we introduce the novel multiple tensor factorizations of the tei unfolding matrix which decrease the computational demands for the evaluation of tei in several aspects using the reduced higher order svd the redundancy free product basis set is constructed that diminishes dramatically the initial number o n b 2 of three dimensional 3 d convolutions defined over cross products of n b basis functions to o n b scaling the tensor structured numerical integration with the 3 d newton convolving kernel is performed in one dimensional 1 d complexity thus enabling high resolution over fine 3 d cartesian grids furthermore using the quantized approximation of long vectors ensures the logarithmic storage complexity in the grid size finally we present and analyze two approaches to compute the cholesky decomposition of the tei matrix based on two types of precomput

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

greedy low rank approximation in tucker format of solutions of tensor linear systems abstract we propose a method for the approximation of tensors given either explicitly or implicitly as the solution of tensor linear systems in the tucker tensor format it is an iterative method that greedily constructs a suitable tensor product subspace in which to approximate the tensor by means of successive rank one approximations an approximation to the target tensor is then obtained by solving a suitable projection of the linear system into the constructed tensor subspace the proposed method has applications in tensor structured discretization methods for partial differential equations such as the finite difference method and isogeometric analysis in several numerical experiments we compare the method to the greedy rank one update or proper generalized decomposition approach as well as to quasi optimal approximations obtained by a truncated higher order singular value decomposition the proposed method outperforms the rank one update method significantly both in terms of error per iteration and error per unit of computation time

time dependent association measures for bivariate survival distributions abstract we propose time dependent association measures for application to bivariate survival analysis such association measures provide informative summaries for data on twins ophthalmic and auditory studies and for other matched pair designs we develop several desirable properties of time dependent association measures and study three measures motivated by these properties we examine the measures from a general bivariate survival perspective and for the proportional hazards frailty model we use monozygotic mz and dizygotic dz twin data from the danish twin registry to illustrate how these measures depend on the specification of the proportional hazards frailty model this model consists of two components a baseline hazard function and a frailty distribution we produce gamma and nonparametric maximum likelihood estimates of the frailty distribution and estimate a gompertz baseline hazard function for two of the measures a nonparametric estimate provides a comparison to the model based estim

compressive multiplexing of correlated signals we present a general architecture for the acquisition of ensembles of correlated signals the signals are multiplexed onto a single line by mixing each one against a different code and then adding them together and the resulting signal is sampled at a high rate we show that if the m signals each band limited to w 2 hz can be approximated by a superposition of r m underlying signals then the ensemble can be recovered by sampling at a rate within a logarithmic factor of rw as compared with the cumulative nyquist rate of mw this sampling theorem shows that the correlation structure of the signal ensemble can be exploited in the acquisition process even though it is unknown a priori the reconstruction of the ensemble is recast as a low rank matrix recovery problem from linear measurements the architectures we are considering impose a certain type of structure on the linear operators although our results depend on the mixing forms being random this imposed structure results in a very different type of random projection than those analyzed in the low rank recovery literature to date

preconditioned low rank riemannian optimization for linear systems with tensor product structure the numerical solution of partial differential equations on high dimensional domains gives rise to computationally challenging linear systems when using standard discretization techniques the size of the linear system grows exponentially with the number of dimensions making the use of classic iterative solvers infeasible during the last few years low rank tensor approaches have been developed that allow one to mitigate this curse of dimensionality by exploiting the underlying structure of the linear operator in this work we focus on tensors represented in the tucker and tensor train formats we propose two preconditioned gradient methods on the corresponding low rank tensor manifolds a riemannian version of the preconditioned richardson method as well as an approximate newton scheme based on the riemannian hessian for the latter considerable attention is given to the efficient solution of the resulting newton equation in numerical experiments we compare the efficiency of our riemannian algorit

perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations abstract for selfadjoint matrices in an indefinite inner product possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation genericity is understood in the sense of algebraic geometry special attention is paid to the perturbation behavior of the sign characteristic typically under such a perturbation for every given eigenvalue the largest jordan block of the eigenvalue is destroyed and in case the eigenvalue is real all other jordan blocks keep their sign characteristic the new eigenvalues i e those eigenvalues of the perturbed matrix that are not eigenvalues of the original matrix are typically simple and in some cases information is provided about their sign characteristic if the new eigenvalue is real the main results are proved by using the well known canonical forms of selfadjoint matrices in an indefinite inner product a version of the brunovsky canonical form and on general results concerning rank one perturbations obtained

low rank tensor train coefficient array estimation for tensor on tensor regression the tensor on tensor regression can predict a tensor from a tensor which generalizes most previous multilinear regression approaches including methods to predict a scalar from a tensor and a tensor from a scalar however the coefficient array could be much higher dimensional due to both high order predictors and responses in this generalized way compared with the current low candecomp parafac cp rank approximation based method the low tensor train tt approximation can further improve the stability and efficiency of the high or even ultrahigh dimensional coefficient array estimation in the proposed low tt rank coefficient array estimation for tensor on tensor regression we adopt a tt rounding procedure to obtain adaptive ranks instead of selecting ranks by experience besides an ell 2 constraint is imposed to avoid overfitting the hierarchical alternating least square is used to solve the optimization problem numerical experiments on a synthetic data set and two real life data sets demonstrate that the proposed method outperforms the state of the art methods in terms of prediction accuracy with comparable computational complexity and the proposed method is more computationally efficient when the data are high dimensional with small size in each mode

computing functions of very large matrices with small tt qtt ranks by quadrature formulas abstract the computation of matrix functions using quadrature formulas and rational approximations of very large structured matrices using tensor trains tt and quantized tensor trains qtt is considered here the focus is on matrices with a small tt qtt rank some analysis of the error produced by the use of the tt qtt representation and the underlying approximation formula used is also provided promising experiments on exponential power mittag leffler and logarithm function of multilevel toeplitz matrices that are among those which generate a low tt qtt rank representation are also provided confirming that the proposed approach is feasible

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

optimization on the hierarchical tucker manifold applications to tensor completion in this work we develop an optimization framework for problems whose solutions are well approximated by hierarchical tucker ht tensors an efficient structured tensor format based on recursive subspace factorizations by exploiting the smooth manifold structure of these tensors we construct standard optimization algorithms such as steepest descent and conjugate gradient for completing tensors from missing entries our algorithmic framework is fast and scalable to large problem sizes as we do not require svds on the ambient tensor space as required by other methods moreover we exploit the structure of the gramian matrices associated with the ht format to regularize our problem reducing overfitting for high subsampling ratios we also find that the organization of the tensor can have a major impact on completion from realistic seismic acquisition geometries these samplings are far from idealized randomized samplings that are usually considered in the literature but are realizable in practical scenarios using these algorithms we successfully interpolate large scale seismic data sets and demonstrate the competitive computational scaling of our algorithms as the problem sizes grow

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

algorithms for numerical analysis in high dimensions nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension the separated representation introduced previously allows many operations to be performed with scaling that is formally linear in the dimension in this paper we further develop this representation by i discussing the variety of mechanisms that allow it to be surprisingly efficient ii addressing the issue of conditioning iii presenting algorithms for solving linear systems within this framework and iv demonstrating methods for dealing with antisymmetric functions as arise in the multiparticle schrodinger equation in quantum mechanics numerical examples are given

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

a new method for reconstruction of cross sections using tucker decomposition the full representation of a d variate function requires exponentially storage size as a function of dimension d and high computational cost in order to reduce these complexities function approximation methods called reconstruction in our context are proposed such as interpolation approximation etc the traditional interpolation model like the multilinear one has this dimensional ity problem to deal with this problem we propose a new model based on the tucker format a low rank tensor approximation method called here the tucker decomposition the tucker decomposition is built as a tensor product of one dimensional spaces where their one variate basis functions are constructed by an extension of the karhunenlo eve decomposition into high dimensional space using this technique we can acquire direction by direction the most important information of the function and convert it into a small number of basis functions hence the approximation for a given function needs less data than that of the multilinear model results of a test case on the neutron cross section reconstruction demonstrate that the tucker decomposition achieves a better accuracy while using less data than the multilinear interpolation

assessing heterogeneity and correlation of paired failure times with the bivariate frailty model we consider bivariate survival times for heterogeneous populations where heterogeneity induces deviations in an individual s risk of an event as well as associations between survival times the heterogeneity is characterized by a bivariate frailty model we measure the heterogeneity effects through deviations associated with hazard functions and an association function defined through the conditional hazard functions the cross ratio function proposed by oakes we show how the deviation and association measures are determined by the frailty distribution a gibbs sampling method is developed for bayesian inferences on regression coefficients frailty parameters and the heterogeneity measures the method is applied to a mental health care data set

low rank solvers for unsteady stokes brinkman optimal control problem with random data abstract we consider the numerical simulation of an optimal control problem constrained by the unsteady stokes brinkman equation involving random data more precisely we treat the state the control the target or the desired state as well as the viscosity as analytic functions depending on uncertain parameters this allows for a simultaneous generalized polynomial chaos approximation of these random functions in the stochastic galerkin finite element method discretization of the model the discrete problem yields a prohibitively high dimensional saddle point system with kronecker product structure we develop a new alternating iterative tensor method for an efficient reduction of this system by the low rank tensor train representation besides we propose and analyze a robust schur complement based preconditioner for the solution of the saddle point system the performance of our approach is illustrated with extensive numerical experiments based on two and three dimensional examples where the full problem size exceeds one billion degrees of freedom the developed tensor train scheme reduces the solution storage by two three orders of magnitude depending on discretization parameters

the automatic solution of partial differential equations using a global spectral method a spectral method for solving linear partial differential equations pdes with variable coefficients and general boundary conditions defined on rectangular domains is described based on separable representations of partial differential operators and the one dimensional ultraspherical spectral method if a partial differential operator is of splitting rank 2 such as the operator associated with poisson or helmholtz the corresponding pde is solved via a generalized sylvester matrix equation and a bivariate polynomial approximation of the solution of degree n x n y is computed in o n x n y 3 2 operations partial differential operators of splitting rank 3 are solved via a linear system involving a block banded matrix in o min n x 3 n y n x n y 3 operations numerical examples demonstrate the applicability of our 2d spectral method to a broad class of pdes which includes elliptic and dispersive time evolution equations the resulting pde solver is written in matlab and is publicly available as part of chebfun it can resolve solutions requiring over a million degrees of freedom in under 60 seconds an experimental implementation in the julia language can currently perform the same solve in 10 seconds

handwritten digit classification using higher order singular value decomposition in this paper we present two algorithms for handwritten digit classification based on the higher order singular value decomposition hosvd the first algorithm uses hosvd for construction of the class models and achieves classification results with error rate lower than 6 the second algorithm uses the hosvd for tensor approximation simultaneously in two modes classification results for the second algorithm are almost down at 5 even though the approximation reduces the original training data with more than 98 before the construction of the class models the actual classification in the test phase for both algorithms is conducted by solving a series least squares problems considering computational amount for the test presented the second algorithm is twice as efficient as the first one

learning tensors from partial binary measurements we generalize the 1 bit matrix completion problem to higher order tensors consider a rank r order d tensor t in mathbb r n times cdots times mathbb r n with bounded entries we show that when r o 1 such a tensor can be estimated efficiently from only m o r nd binary measurements this shows that the sample complexity of recovering a low rank tensor from 1 bit measurements of a subset of its entries is roughly the same as recovering it from unquantized measurements a result that had been known only in the matrix case i e when d 2 by using a certain atomic m norm as a convex proxy for rank we allow for approximately low rank tensors and give error bounds based on the m norm of the tensor we prove that the theoretical bounds are optimal both in the m norm bound and the size n moreover we show the advantage of directly using the low rank tensor structure rather than matricization both theoretically and numerically specifically we show how the 1 bit measurement model can be used for context aware recommender systems

lipschitz stability of canonical jordan bases of h selfadjoint matrices under structure preserving perturbations in this paper we study jordan structure preserving perturbations of matrices selfadjoint in the indefinite inner product the main result of the paper is lipschitz stability of the corresponding affiliation matrices the result can be reformulated as lipschitz stability under small perturbations of canonical jordan bases i e eigenvectors and generalized eigenvectors enjoying a certain flipped orthonormality relation of matrices selfadjoint in the indefinite inner product the proof relies upon the analysis of small perturbations of invariant subspaces where the size of a permutation of an invariant subspace is measured using the concepts of a gap and of a semigap

on the distance between lattices of invariant subspaces of matrices abstract we study the behavior of the lattice inv x of all invariant subspaces of a matrix x when x belongs to the class of matrices with fixed jordan structure i e with isomorphic lattices of invariant subspaces a larger class of matrices with fixed jordan structure corresponding to the eigenvalues of geometric multiplicity greater than one is also studied our main concern is analysis of the distance between the lattices of invariant subspaces

low rank tensor methods in galerkin based isogeometric analysis abstract the global patch wise geometry map which describes the computational domain is a new feature in isogeometric analysis this map has a global tensor structure inherited from the parametric spline geometry representation the use of this global structure in the discretization of partial differential equations may be regarded as a drawback at first glance as opposed to the purely local nature of high order classical finite elements in this work we demonstrate that it is possible to exploit the regularity of this structure and to identify the great potential for the efficient implementation of isogeometric discretizations first we formulate tensor product b spline bases as well as the corresponding mass and stiffness matrices as tensors in order to reveal their intrinsic structure second we derive an algorithm for the separation of variables in the integrands arising in the discretization this is possible by means of low rank approximation of the integral kernels we arrive at a compact separated representation of the integrals the separated form implies an expression of galerkin matrices as kronecker products of matrix factors with small dimensions this representation is very appealing due to the reduction in both memory consumption and computation times our benchmarks performed using the c library g smo demonstrate that the use of tensor methods in isogeometric analysis possesses significant advantages

life table methods for heterogeneous populations distributions describing the heterogeneity taking account of heterogeneity between the individuals in population based mortality studies is important a systematic way of describing heterogeneity is by an unobserved quantity called frailty entering the hazard multiplicatively until now most studies have used a gamma distributed frailty which is mathematically convenient for example the distribution among survivors is also gamma this paper shows that several other distributions have equally simple properties the main example being the inverse gaussian distribution consequences of the different distributions are examined the inverse gaussian makes the population homogeneous with time whereas for the gamma the relative heterogeneity is constant

a low rank solver for the stochastic unsteady navier stokes problem abstract we study a low rank iterative solver for the unsteady navier stokes equations for incompressible flows with a stochastic viscosity the equations are discretized using the stochastic galerkin method and we consider an all at once formulation where the algebraic systems at all the time steps are collected and solved simultaneously the problem is linearized with picard s method to efficiently solve the linear systems at each step we use low rank tensor representations within the krylov subspace method which leads to significant reductions in storage requirements and computational costs combined with effective mean based preconditioners and the idea of inexact solve we show that only a small number of linear iterations are needed at each picard step the proposed algorithm is tested with a model of flow in a two dimensional symmetric step domain with different settings to demonstrate the computational efficiency

higher order principal component analysis for the approximation of tensors in tree based low rank formats this paper is concerned with the approximation of tensors using tree based tensor formats which are tensor networks whose graphs are dimension partition trees we consider hilbert tensor spaces of multivariate functions defined on a product set equipped with a probability measure this includes the case of multidimensional arrays corresponding to finite product sets we propose and analyse an algorithm for the construction of an approximation using only point evaluations of a multivariate function or evaluations of some entries of a multidimensional array the algorithm is a variant of higher order singular value decomposition which constructs a hierarchy of subspaces associated with the different nodes of the tree and a corresponding hierarchy of interpolation operators optimal subspaces are estimated using empirical principal component analysis of interpolations of partial random evaluations of the function the algorithm is able to provide an approximation in any tree based format with either a prescribed rank or a prescribed relative error with a number of evaluations of the order of the storage complexity of the approximation format under some assumptions on the estimation of principal components we prove that the algorithm provides either a quasi optimal approximation with a given rank or an approximation satisfying the prescribed relative error up to constants depending on the tree and the properties of interpolation operators the analysis takes into account the discretization errors for the approximation of infinite dimensional tensors for a tensor with finite and known rank in a tree based format the algorithm is able to recover the tensor in a stable way using a number of evaluations equal to the storage complexity of the representation of the tensor in this format several numerical examples illustrate the main results and the behavior of the algorithm for the approximation of high dimensional functions using hierarchical tucker or tensor train tensor formats and the approximation of univariate functions using tensorization

randomized interpolative decomposition of separated representations we introduce an algorithm to compute tensor interpolative decomposition dubbed ctd id for the reduction of the separation rank of canonical tensor decompositions ctds tensor id selects for a user defined accuracy a near optimal subset of terms of a ctd to represent the remaining terms via a linear combination of the selected terms ctd id can be used as an alternative to or in combination with the alternating least squares als algorithm we present examples of its use within a convergent iteration to compute inverse operators in high dimensions we also briefly discuss the spectral norm as a computational alternative to the frobenius norm in estimating approximation errors of tensor id we reduce the problem of finding tensor ids to that of constructing interpolative decompositions of certain matrices these matrices are generated via randomized projection of the terms of the given tensor we provide cost estimates and several examples of the new approach to the reduction of separation rank

sparse canonical temporal alignment with deep tensor decomposition for action recognition in this paper we solve three problems in action recognition sub action multi subject and multi modality by reducing the diversity of intra class samples the main stage contains canonical temporal alignment and key frames selection as we know temporal alignment aims to reduce the diversity of intra class samples however dense frames may yield misalignment or overlapped alignment and decrease recognition performance to overcome this problem we propose a sparse canonical temporal alignment scta method which selects and aligns key frames from two sequences to reduce diversity to extract better features from the key frames we propose a deep non negative tensor factorization dntf method to find a tensor subspace integrated with scta scheme first we model an action sequence as a third order tensor with spatiotemporal structure then we design a dntf scheme to find a tensor subspace in both spatial and temporal directions particularly in the first layer the original tensor is decomposed into two low rank tensors by ntf and in the second layer each low rank tensor is further decomposed by tensor train for time efficiency finally our framework composed of scta and dntf could solve the three problems and extract effective features for action recognition experiments on synthetic data msrdailyactivity3d and msractionpairs data sets show that our method works better than competitive methods in terms of accuracy

low rank tensor recovery using sequentially optimal modal projections in iterative hard thresholding sempiht iterative hard thresholding iht is a simple and effective approach to parsimonious data recovery its multilinear rank mrank based application to low rank tensor recovery lrtr is especially valuable given the difficulties involved in this problem in this paper we propose a novel iht algorithm for lrtr choosing sequential per mode svd truncation as its thresholding operator this operator is less costly than those used in existing iht algorithms for lrtr and often leads to superior performance furthermore by exploiting the sequential optimality of the employed modal projections we derive recovery guarantees relying on restricted isometry constants though these guarantees are suboptimal our numerical studies indicate that a quasi optimal number of gaussian measurements suffices for perfect data reconstruction we also investigate a continuation technique which yields a sequence of progressively more complex estimated models until attaining a target mrank when recovering real world data this

remarks on lipschitz properties of matrix groups actions it is proved that a large class of matrix group actions including joint similarity and congruence like actions as well as actions of the type of matrix equivalence have local lipschitz property under additional hypotheses global lipschitz property is proved these results are specialized and applied to obtain local lipschitz property of canonical bases of matrices that are selfadjoint in an indefinite inner product real complex and quaternionic matrices are considered

wedderburn rank reduction and krylov subspace method for tensor approximation part 1 tucker case new algorithms are proposed for the tucker approximation of a 3 tensor accessed only through a tensor by vector by vector multiplication subroutine in the matrix case the krylov methods are methods of choice to approximate the dominant column and row subspaces of a sparse or structured matrix given through a matrix by vector operation using the wedderburn rank reduction formula we propose a matrix approximation algorithm that computes the krylov subspaces and can be generalized to 3 tensors the numerical experiments show that on quantum chemistry data the proposed tensor methods outperform the minimal krylov recursion of savas and elden

dynamically orthogonal tensor methods for high dimensional nonlinear pdes abstract we develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high dimensional time dependent nonlinear partial differential equations pdes the key idea relies on a hierarchical decomposition of the approximation space obtained by splitting the independent variables of the problem into disjoint subsets this process which can be conveniently visualized in terms of binary trees yields series expansions analogous to the classical tensor train and hierarchical tucker tensor formats by enforcing dynamic orthogonality conditions at each level of the binary tree we obtain coupled evolution equations for the modes spanning each subspace within the hierarchical decomposition this allows us to effectively compute the solution to high dimensional time dependent nonlinear pdes on tensor manifolds of constant rank with no need for rank reduction methods we also propose new algorithms for dynamic addition and removal of modes within each subspace numerical examples are presented and discussed for high dimensional hyperbolic and parabolic pdes in bounded domains

quasioptimality of maximum volume cross interpolation of tensors we consider a cross interpolation of high dimensional arrays in the tensor train format we prove that the maximum volume choice of the interpolation sets provides the quasioptimal interpolation accuracy that differs from the best possible accuracy by the factor which does not grow exponentially with dimension for nested interpolation sets we prove the interpolation property and propose greedy cross interpolation algorithms we justify the theoretical results and measure speed and accuracy of the proposed algorithm with numerical experiments

on local convergence of alternating schemes for optimization of convex problems in the tensor train format alternating linear schemes als with the alternating least squares algorithm a notable special case provide one of the simplest and most popular choices for the treatment of optimization tasks by tensor methods an according adaptation of als for the recent tt tensor train format i v oseledets siam j sci comput 33 2011 pp 2295 2317 known in quantum computations as matrix product states has recently been investigated in s holtz t rohwedder and r schneider siam j sci comput 34 2012 pp a683 a713 with the present work the positive practical experience with tt als is backed up with an according local linear convergence theory for the optimization of convex functionals j the main assumption entering the proof is that the redundancy introduced by the tt parametrization tau matches the null space of the hessian of the induced functional j j circ tau and we give conditions under which this assumption can be expected to hold in particular this is the case if

tensor train decomposition a simple nonrecursive form of the tensor decomposition in d dimensions is presented it does not inherently suffer from the curse of dimensionality it has asymptotically the same number of parameters as the canonical decomposition but it is stable and its computation is based on low rank approximation of auxiliary unfolding matrices the new form gives a clear and convenient way to implement all basic operations efficiently a fast rounding procedure is presented as well as basic linear algebra operations examples showing the benefits of the decomposition are given and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19 dimensional operator

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

riemannian optimization for high dimensional tensor completion tensor completion aims to reconstruct a high dimensional data set where the vast majority of entries is missing the assumption of low rank structure in the underlying original data allows us to cast the completion problem into an optimization problem restricted to the manifold of fixed rank tensors elements of this smooth embedded submanifold can be efficiently represented in the tensor train or matrix product states format with storage complexity scaling linearly with the number of dimensions we present a nonlinear conjugate gradient scheme within the framework of riemannian optimization which exploits this favorable scaling numerical experiments and comparison to existing methods show the effectiveness of our approach for the approximation of multivariate functions finally we show that our algorithm can obtain competitive reconstructions from uniform random sampling of few entries compared to adaptive sampling techniques such as cross approximation

numerical solution of the hartree fock equation in multilevel tensor structured format in this paper we describe a novel method for a robust and accurate iterative solution of the self consistent hartree fock equation in mathbb r 3 based on the idea of tensor structured computation of the electron density and the nonlinear hartree and nonlocal exchange operators at all steps of the iterative process we apply the self consistent field scf iteration to the galerkin discretization in a set of low separation rank basis functions that are solely specified by the respective values on a three dimensional cartesian grid the approximation error is estimated by o h 3 where h o n 1 is the mesh size of an n times n times n tensor grid while the numerical complexity to compute the galerkin matrices scales linearly in n log n we propose the tensor truncated version of the scf iteration using the traditional direct inversion in the iterative subspace scheme enhanced by the multilevel acceleration with the grid dependent termination criteria at each discretization level this implies that the overall computational cost scales almost linearly in the univariate problem size n numerical illustrations are presented for the all electron case of h 2 o and the pseudopotential case of ch 4 and ch 3 oh molecules the proposed scheme is not restricted to a priori given rank 1 basis sets allowing analytically integrable convolution transform with the newton kernel that opens further perspectives for promotion of the tensor structured methods in computational quantum chemistry

discretized dynamical low rank approximation in the presence of small singular values low rank approximations to large time dependent matrices and tensors are the subject of this paper these matrices and tensors either are given explicitly or are the unknown solutions of matrix and tensor differential equations based on splitting the orthogonal projection onto the tangent space of the low rank manifold novel time integrators for obtaining approximations by low rank matrices and low rank tensor trains were recently proposed by standard theory the lie trotter and strang projector splitting methods are first and second order accurate respectively but the usual error bounds break down when the low rank approximation has small singular values this happens when the singular values of the solution decay without a distinct gap or when the effective rank of the solution is overestimated on the other hand the integrators are exact when given time dependent matrices or tensors are already of the prescribed rank we provide an error analysis which unifies these properties we show that in c

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea