harmonic measure versus hausdorff measures on repellers for holomorphic maps this paper is a continuation of a joint paper of the author with f przytycki and m urbanski we study a harmonic measure on a boundary of so called repelling boundary domain an important example is a basin of a sink for a rational map using the results of the above mentioned paper we prove that either the boundary of the domain is an analytically embedded circle or interval or else the harmonic measure is singular with respect to the hausdors measure corresponding to the function xc t t exp c log t logloglog 1 for some c 0 0 introduction and statement of results let q be a simply connected domain in the riemann sphere card q 2 let r d q be the riemann map from the unit disc d onto q since r has nontangential limits almost everywhere on s1 ad d the image r l of the length measure i on s1 can be considered this measure denoted usually by co and called a harmonic measure on aq has been studied a long time see mk2 and puz for the corresponding references with the most interesting results obtained recently in two remarkable papers mkl and mk2 for an increasing function 0 di di 0 0 o we define an outer measure for a c c a a lim inf diambj where the infimum is taken over all coverings of a with balls of a diameter smaller than a the diameter is computed with respect to the spherical metrics on in particular for 0 t ta we obtain the ol dimensional hausdorff measure ata will be denoted by aa let u be a probability measure on borel subsets of hausdorff dimension of u is defined as

a riemannian gradient sampling algorithm for nonsmooth optimization on manifolds in this paper an optimization method for nonsmooth locally lipschitz functions on complete riemannian manifolds is presented the method is based on approximating the subdifferential of the cost function at every iteration by the convex hull of transported gradients from tangent spaces at randomly generated nearby points to the tangent space of the current iterate and can hence be seen as a generalization of the well known gradient sampling algorithm to a riemannian setting a convergence result is obtained under the assumption that the cost function is bounded below and continuously differentiable on an open set of full measure and that the employed vector transport and retraction satisfy certain conditions which hold for instance for the exponential map and parallel transport then with probability one the algorithm produces iterates at which the cost function is differentiable and each cluster point of the iterates is a clarke stationary point modifications yielding only varepsilon stationary

collet eckmann and holder we prove that collet eckmann condition for rational functions which requires exponential expansion only along the critical orbits yields the holder regularity of fatou components this implies geometric regularity of julia sets with non hyperbolic and critically recurrent dynamics in particular polynomial collet eckmann julia sets are locally connected if connected and their hausdorff dimension is strictly less than 2 the same is true for rational collet eckmann julia sets with at least one non empty fully invariant fatou component

convergence of line search methods for unconstrained optimization line search methods are traditional and successful methods for solving unconstrained optimization problems its convergence has attracted more attention in recent years in this paper we analyze the general results on convergence of line search methods with seven line search rules it is clarified that the search direction plays a main role in these methods and that step size guarantees the global convergence in some cases it is also proved that many line search methods have same convergence property these convergence results can enable us to design powerful effective and stable algorithms in practice finally a class of special line search methods is investigated

the alternating linear scheme for tensor optimization in the tensor train format recent achievements in the field of tensor product approximation provide promising new formats for the representation of tensors in form of tree tensor networks in contrast to the canonical r term representation candecomp parafac these new formats provide stable representations while the amount of required data is only slightly larger the tensor train tt format siam j sci comput 33 2011 pp 2295 2317 a simple special case of the hierarchical tucker format j fourier anal appl 5 2009 p 706 is a useful prototype for practical low rank tensor representation in this article we show how optimization tasks can be treated in the tt format by a generalization of the well known alternating least squares als algorithm and by a modified approach mals that enables dynamical rank adaptation a formulation of the component equations in terms of so called retraction operators helps to show that many structural properties of the original problems transfer to the micro iterations giving what is to our knowledge the first stable generic algorithm for the treatment of optimization tasks in the tensor format for the examples of linear equations and eigenvalue equations we derive concrete working equations for the micro iteration steps numerical examples confirm the theoretical results concerning the stability of the tt decomposition and of als and mals but also show that in some cases high tt ranks are required during the iterative approximation of low rank tensors showing some potential of improvement

harmonic gibbs and hausdorff measures on repellers for holomorphic maps i we prove that for a simply connected domain q c c whose boundary a q is self similar there is the following dichotomy concerning the harmonic measure x on a q viewed from 0 either a q is piecewise real analytic or else o is singular with respect to the hausdorff measure a notation o i a using makarov s function 4dc t t exp c log 1 t log log log 1 t for some c c x 0 and x is absolutely continuous with respect to a notation x c so if q has fractal boundary then the boundary compression and the radial growth of j logir l j for a riemann mapping r d are as strong respectively as fast as permitted by makarov s theory we prove that c x 2a 2 x for some asymptotic variance a 2 for a sequence of weakly dependent random variables and a lyapunov characteristic exponent x this includes the case where a q is a mixing repeller in ruelle s sense for a holomorphic map f defined on its neighbourhood the case a q is a quasi circle invariant under the action of a quasi fuchsian group for a pair of isomorphic compact surface fuchsian groups and the cases of the boundary of the snowflake and more generally of carleson s fractal jordan curves this dichotomy is partially deduced from the dichotomy concerning gibbs measures for holder continuous functions on an arbitrary mixing repeller x c c for a holomorphic map either i i ak where k is the hausdorff dimension of y

basis pursuit denoise with nonsmooth constraints level set optimization formulations with data driven constraints minimize a regularization functional subject to matching observations to a given error level these formulations are widely used particularly for matrix completion and sparsity promotion in data interpolation and denoising the misfit level is typically measured in the ell 2 norm or other smooth metrics in this paper we present a new flexible algorithmic framework that targets nonsmooth level set constraints including ell 1 ell infty and even ell 0 norms these constraints give greater flexibility for modeling deviations in observation and denoising and have significant impact on the solution measuring error in the ell 1 and ell 0 norms makes the result more robust to large outliers while matching many observations exactly we demonstrate the approach for basis pursuit denoise bpdn problems as well as for extensions of bpdn to matrix factorization with applications to interpolation and denoising of 4d seismic data the new methods are particularly promising for seismic applications where the amplitude in the data varies significantly and measurement noise in low amplitude regions can wreak havoc for standard gaussian error models

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

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

analyzing nonstationary spatial data using piecewise gaussian processes in many problems in geostatistics the response variable of interest is strongly related to the underlying geology of the spatial location in these situations there is often little correlation in the responses found in different rock strata so the underlying covariance structure shows sharp changes at the boundaries of the rock types conventional stationary and nonstationary spatial methods are inappropriate because they typically assume that the covariance between points is a smooth function of distance in this article we propose a generic method for the analysis of spatial data with sharp changes in the underlying covariance structure our method works by automatically decomposing the spatial domain into disjoint regions within which the process is assumed to be stationary but the data are assumed independent across regions uncertainty in the number of disjoint regions their shapes and the model within regions is dealt with in a fully bayesian fashion we illustrate our approach on a previously

matern cross covariance functions for multivariate random fields we introduce a flexible parametric family of matrix valued covariance functions for multivariate spatial random fields where each constituent component is a matern process the model parameters are interpretable in terms of process variance smoothness correlation length and colocated correlation coefficients which can be positive or negative both the marginal and the cross covariance functions are of the matern type in a data example on error fields for numerical predictions of surface pressure and temperature over the north american pacific northwest we compare the bivariate matern model to the traditional linear model of coregionalization

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

scaling of mandelbrot sets generated by critical point preperiodicity letz f z be a complex holomorphic function depending holomorphically on the complex parameter if for 0 a critical point off0 falls after a finite number of steps onto an unstable fixed point off0 then in the parameter space near 0 an infinity of more and more accurate copies of the mandelbrot set appears we compute their scaling properties

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

parallel tensor methods for high dimensional linear pdes abstract high dimensional partial differential equations pdes arise in a number of fields of science and engineering where they are used to describe the evolution of joint probability functions their examples include the boltzmann and fokker planck equations we develop new parallel algorithms to solve such high dimensional pdes the algorithms are based on canonical and hierarchical numerical tensor methods combined with alternating least squares and hierarchical singular value decomposition both implicit and explicit integration schemes are presented and discussed we demonstrate the accuracy and efficiency of the proposed new algorithms in computing the numerical solution to both an advection equation in six variables plus time and a linearized version of the boltzmann equation

self similarity of siegel disks and hausdorff dimension of julia sets let f z ez z where is an irrational number of bounded type according to siegel f is linearizable on a disk containing the origin in this paper we show the hausdorff dimension of the julia set j f is strictly less than two and if is a quadratic irrational such as the golden mean then the siegel disk for f is self similar about the critical point in the latter case we also show the rescaled first return maps converge exponentially fast to a system of commuting branched coverings of the complex plane

geodesic gaussian processes for the parametric reconstruction of a free form surface reconstructing a free form surface from 3 dimensional 3d noisy measurements is a central problem in inspection statistical quality control and reverse engineering we present a new method for the statistical reconstruction of a free form surface patch based on 3d point cloud data the surface is represented parametrically with each of the three cartesian coordinates x y z a function of surface coordinates u v a model form compatible with computer aided design cad models this model form also avoids having to choose one euclidean coordinate say z as a response function of the other two coordinate locations say x and y as commonly used in previous euclidean kriging models of manufacturing data the u v surface coordinates are computed using parameterization algorithms from the manifold learning and computer graphics literature these are then used as locations in a spatial gaussian process model that considers correlations between two points on the surface a function of their ge

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

stability analysis of hierarchical tensor methods for time dependent pdes abstract we address the question of whether it is possible to stably integrate time dependent high dimensional linear pdes with hierarchical tensor methods and explicit time stepping schemes to this end we develop sufficient conditions for stability and convergence of tensor solutions evolving on tensor manifolds with constant rank we also argue that the applicability of pde solvers with explicit time stepping may be limited by time step restriction dependent on the dimension of the problem numerical applications are presented and discussed for variable coefficients linear hyperbolic and parabolic pdes

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

similarity between the mandelbrot set and julia sets the mandelbrot setm is self similar about any misiurewicz pointc in the sense that if we examine a neighborhood ofc inm with a very powerful microscope and then increase the magnification by a carefully chosen factor the picture will be unchanged except for a rotation the corresponding julia setj c is also self similar in the same sense with the same magnification factor moreover the two setsm andj c are similar in the sense that if we use a very powerful microscope to look atm andj c both focused atc the structures we see look like very much the same

nonparametric estimation of nonstationary spatial covariance structure abstract estimation of the covariance structure of spatial processes is a fundamental prerequisite for problems of spatial interpolation and the design of monitoring networks we introduce a nonparametric approach to global estimation of the spatial covariance structure of a random function z x t observed repeatedly at times ti i 1 t at a finite number of sampling stations xi i 1 2 n in the plane our analyses assume temporal stationarity but do not assume spatial stationarity or isotropy we analyze the spatial dispersions var z xi t z xj t as a natural metric for the spatial covariance structure and model these as a general smooth function of the geographic coordinates of station pairs xi xj the model is constructed in two steps first using nonmetric multidimensional scaling mds we compute a two dimensional representation of the sampling stations for which a monotone function of interpoint distances ij approximates the spatial dispersions mds transforms the problem

tree adaptive approximation in the hierarchical tensor format the hierarchical tensor format allows for the low parametric representation of tensors even in high dimensions d the efficiency of this representation strongly relies on an appropriate hierarchical splitting of the different directions 1 ldots d such that the associated ranks remain sufficiently small this splitting can be represented by a binary tree which is usually assumed to be given in this paper we address the question of finding an appropriate tree from a subset of tensor entries without any a priori knowledge on the tree structure we propose an agglomerative strategy that can be combined with rank adaptive cross approximation techniques such that tensors can be approximated in the hierarchical format in an entirely black box way numerical examples illustrate the potential and the limitations of our approach

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

fixed points of feigenbaum s type for the equationf p λx λf x existence and hyperbolicity of fixed points for the mapn p f x 1f p x withf p p fold iteration and f p 0 are given forp large these fixed points come close to being quadratic functions and our proof consists in controlling perturbation theory about quadratic functions

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

bayesian inference for non stationary spatial covariance structure via spatial deformations summary in geostatistics it is common practice to assume that the underlying spatial process is stationary and isotropic i e the spatial distribution is unchanged when the origin of the index set is translated and under rotation about the origin however in environmental problems such assumptions are not realistic since local influences in the correlation structure of the spatial process may be found in the data the paper proposes a bayesian model to address the anisot ropy problem following sampson and guttorp we define the correlation function of the spatial process by reference to a latent space denoted by d where stationarity and isotropy hold the space where the gauged monitoring sites lie is denoted by g we adopt a bayesian approach in which the mapping between g and d is represented by an unknown function d a gaussian process prior distribution is defined for d unlike the sampson guttorp approach the mapping of both gauged and ungauged sites is handled in a single framework and predictive inferences take explicit account of uncertainty in the mapping markov chain monte carlo methods are used to obtain samples from the posterior distributions two examples are discussed a simulated data set and the solar radiation data set that also was analysed by sampson and guttorp

a full scale approximation of covariance functions for large spatial data sets summary gaussian process models have been widely used in spatial statistics but face tremendous computational challenges for very large data sets the model fitting and spatial prediction of such models typically require o n3 operations for a data set of size n various approximations of the covariance functions have been introduced to reduce the computational cost however most existing approximations cannot simultaneously capture both the large and the small scale spatial dependence a new approximation scheme is developed to provide a high quality approximation to the covariance function at both the large and the small spatial scales the new approximation is the summation of two parts a reduced rank covariance and a compactly supported covariance obtained by tapering the covariance of the residual of the reduced rank approximation whereas the former part mainly captures the large scale spatial variation the latter part captures the small scale local variation that is unexplained by the former part by combining the reduced rank representation and sparse matrix techniques our approach allows for efficient computation for maximum likelihood estimation spatial prediction and bayesian inference we illustrate the new approach with simulated and real data sets

fine structure of connectedness loci we study distributions of the harmonic and lebesgue measures on the boundary of the connectedness locus mathcal m d for uni critical polynomials z d c and dynamical properties which are typical with respect to the harmonic measure one result is a conformal similarity between mathcal m d and the corresponding julia set for almost every point of mathcal m d with respect to the harmonic measure it is also shown that the harmonic measure is supported on lebesgue density points of the complement of mathcal m d which are not accessible from outside within john angles and at which the boundary of mathcal m d spirals infinitely often in both directions in the case of quadratic polynomials we prove that every parameter c from the boundary of the mandelbrot set mathcal m that is recurrent under iterates of z 2 c but not infinitely tunable is a lebesgue weak density point of the complement of mathcal m a direct consequence of our theory is the yoccoz local connectivity theorem a new technical ingredient in the paper is the method of amplification which is based on promoting the small scale geometry of the parameter space at a typical point in the sense of harmonic measure to uniform geometry of the corresponding julia set the pointwise conformality of the similarity map is derived through inducing techniques and a classical twb theory

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

the hausdorff dimension of the boundary of the mandelbrot set and julia sets it is shown that the boundary of the mandelbrot set m has hausdorff dimension two and that for a generic c e am the julia set of z i z2 c also has hausdorff dimension two the proof is based on the study of the bifurcation of parabolic periodic points

quadratic julia sets with positive area we recently proved the existence of quadratic polynomials having a julia set with positive lebesgue measure we present the ideas of the proof and the techniques involved

low rank optimization with trace norm penalty the paper addresses the problem of low rank trace norm minimization we propose an algorithm that alternates between fixed rank optimization and rank one updates the fixed rank optimization is characterized by an efficient factorization that makes the trace norm differentiable in the search space and the computation of duality gap numerically tractable the search space is nonlinear but is equipped with a riemannian structure that leads to efficient computations we present a second order trust region algorithm with a guaranteed quadratic rate of convergence overall the proposed optimization scheme converges superlinearly to the global solution while maintaining complexity that is linear in the number of rows and columns of the matrix to compute a set of solutions efficiently for a grid of regularization parameters we propose a predictor corrector approach that outperforms the naive warm restart approach on the fixed rank quotient manifold the performance of the proposed algorithm is illustrated on p

stable als approximation in the tt format for rank adaptive tensor completion low rank tensor completion is a highly ill posed inverse problem particularly when the data model is not accurate and some sort of regularization is required in order to solve it in this article we focus on the calibration of the data model for alternating optimization we observe that existing rank adaption methods do not enable a continuous transition between manifolds of different ranks we denote this characteristic as instability under truncation as a consequence of this property arbitrarily small changes in the iterate can have arbitrarily large influence on the further reconstruction we therefore introduce a singular value based regularization to the standard alternating least squares als which is motivated by averaging in microsteps we prove its stability and derive a natural semi implicit rank adaption strategy we further prove that the standard als microsteps for completion problems are only stable on manifolds of fixed ranks and only around points that have what we define as internal tensor restricted isometry property itrip in conclusion numerical experiments are provided that show improvements of the reconstruction quality up to orders of magnitude in the new stable als approximation salsa compared to standard als and the well known riemannian optimization rttc

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

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

on manifolds of tensors of fixed tt rank recently the format of tt tensors hackbusch and kuhn in j fourier anal appl 15 706 722 2009 oseledets in siam j sci comput 2009 submitted oseledets and tyrtyshnikov in siam j sci comput 31 5 2009 oseledets and tyrtyshnikov in linear algebra appl 2009 submitted has turned out to be a promising new format for the approximation of solutions of high dimensional problems in this paper we prove some new results for the tt representation of a tensor u in mathbb r n 1 times cdots times n d and for the manifold of tensors of tt rank underline r as a first result we prove that the tt or compression ranks r i of a tensor u are unique and equal to the respective separation ranks of u if the components of the tt decomposition are required to fulfil a certain maximal rank condition we then show that the set mathbb t of tt tensors of fixed rank underline r locally forms an embedded manifold in mathbb r n 1 times cdots times n d therefore preserving the essential theoretical properties of the tucker format but often showing an improved scaling behaviour extending a similar approach for matrices conte and lubich in m2an 44 759 2010 we introduce certain gauge conditions to obtain a unique representation of the tangent space mathcal t u mathbb t of mathbb t and deduce a local parametrization of the tt manifold the parametrisation of mathcal t u mathbb t is often crucial for an algorithmic treatment of high dimensional time dependent pdes and minimisation problems lubich in from quantum to classical molecular dynamics reduced methods and numerical analysis 2008 we conclude with remarks on those applications and present some numerical examples

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

design and analysis of experiments on nonconvex regions modeling a response over a nonconvex design region is a common problem in diverse areas such as engineering and geophysics the tools available to model and design for such responses are limited and have received little attention we propose a new method for selecting design points over nonconvex regions that is based on the application of multidimensional scaling to the geodesic distance optimal designs for prediction are described with special emphasis on gaussian process models followed by a simulation study and an application in glaciology supplementary materials for this article are available online

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

harmonic measure and polynomial julia sets there is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the same for simply connected and for non simply connected domains in the plane because of the close relation to conformal mapping theory the simply connected case is much better understood and proving the above statement would give new results concerning the properties of harmonic measure in the general case we establish the conjecture in the category of domains bounded by polynomial julia sets the idea is to consider the coefficients of the dynamical zeta function as subharmonic functions on a slice of teichmuller s space of the polynomial and then to apply the maximum principle 1 dimension spectrum of harmonic measure in this paper we discuss some properties of harmonic measure in the complex plane for a domain and a point a let a denote the harmonic measure of evaluated at a the measure a can be defined for instance as the hitting distribution of a brownian motion started at a if e then a e is the probability that a random brownian path first hits the boundary at a point of e much work has been devoted to describing dimensional properties of when the domain is as general as possible in particular jones and wolff 7 proved that no matter what the domain is harmonic measure is concentrated on a borel set of hausdorff dimension at most one in other words dim 1 for all plane domains 1 1 we are interested in finding similar but stronger universal estimates involving the dimension spectrum of 1 1 universal spectrum for every positive we denote f dim z where z is the lower pointwise dimension of z lim inf 0 log b z log b z is a general notation for the disc with center z and radius the universal dimension spectrum is the function sup f 1 2 where the supremum is taken over harmonic measures of all planar domains the first author is supported by n s f grant dms 9970283 the second author is supported by n s f grant dms 9800714

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

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

black box approximation of tensors in hierarchical tucker format we derive and analyse a scheme for the approximation of order d tensors a rn1 nd in the hierarchical h tucker format a dimension multilevel variant of the tucker format and strongly related to the tt tensor train format for a fixed rank parameter k the storage complexity of a tensor in h tucker format is odk3 k i 1dni and we present a heuristic algorithm that finds an approximation to a tensor in the h tucker format in odk4 log d k2 i 1dni by inspection of only odk3 log d k2 i 1dni entries under mild assumptions tensors in the h tucker format are reconstructed for general tensors we derive error bounds that are based on the approximability of matrices matricizations of the tensor by few outer products of its rows and columns the construction parallelizes with respect to the order d and we also propose an adaptive approach that aims at finding the rank parameter for a given target accuracy e automatically

harmonic measure and expansion on the boundary of the connectedness locus the paper develops a technique for proving properties that are typical in the boundary of the connectedness locus with respect to the harmonic measure a typical expansion condition along the critical orbit is proved this condition implies a number of properties including the collet eckmann condition hausdorff dimension less than 2 for the julia set and the radial continuity in the parameter space of the hausdorff dimensions of totally disconnected julia sets