bergman kernel from path integral we rederive the expansion of the bergman kernel on kahler manifolds developed by tian yau zelditch lu and catlin using path integral and perturbation theory and generalize it to supersymmetric quantum mechanics one physics interpretation of this result is as an expansion of the projector of wave functions on the lowest landau level in the special case that the magnetic field is proportional to the kahler form this is relevant for the quantum hall effect in curved space and for its higher dimensional generalizations other applications include the theory of coherent states the study of balanced metrics noncommutative field theory and a conjecture on metrics in black hole backgrounds discussed in 24 we give a short overview of these various topics from a conceptual point of view this expansion is noteworthy as it is a geometric expansion somewhat similar to the dewitt seeley gilkey et al short time expansion for the heat kernel but in this case describing the long time limit without depending on supersymmetry

adjacency spectra of random and complete hypergraphs abstract we present progress on the problem of asymptotically describing the adjacency eigenvalues of random and complete uniform hypergraphs there is a natural conjecture arising from analogy with random matrix theory that connects these spectra to that of the all ones hypermatrix several of the ingredients along a possible path to this conjecture are established and may be of independent interest in spectral hypergraph hypermatrix theory in particular we provide a bound on the spectral radius of the symmetric bernoulli hyperensemble and show that the spectrum of the complete k uniform hypergraph for k 2 3 is close to that of an appropriately scaled all ones hypermatrix

some bounds for the spectral radius of nonnegative tensors in this paper we extend the well known column sum bound of the spectral radius for nonnegative matrices to the tensor case an upper bound of the spectral radius for a nonnegative tensor is given via the largest eigenvalue of a symmetric tensor also we show some bounds of spectral radius of nonnegative tensors based on the sum of the entries in the other indices of tensors we demonstrate that our new results improve existing results the other main results of this paper is to provide a sharper ky fan type theorem and a comparison theorem for nonnegative tensors finally we make use of our bounds to give a perturbation bound for the spectral radius of symmetric nonnegative tensors this result is similar to the weyl theorem for the matrix case

singular values of a real rectangular tensor real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics in this paper we systematically study properties of singular values of a real rectangular tensor and give an algorithm to find the largest singular value of a nonnegative rectangular tensor numerical results show that the algorithm is efficient

determinants of laplacians the determinant of the laplacian on spinor fields on a riemann surface is evaluated in terms of the value of the selberg zeta function at the middle of the critical strip a key role in deriving this relation is played by the barnes double gamma function

hankel tensors vandermonde tensors and their positivities abstract an mth order n dimensional hankel tensor is defined as a tensor a satisfying a i 1 i m a i 1 i m m for some numbers a 0 a 1 a m n 1 a hankel tensor possesses a vandermonde decomposition vd a k 1 r k u k m where u k 1 w k w k 2 w k n 1 t is called a vandermonde vector v vector a is called a vandermonde tensor v tensor if a has a vd with each k 1 v tensors are the natural extension of vandermonde matrices it is easy to see that all even order v tensors are positive semidefinite psd and thus copositive an odd order real symmetric tensor is psd only if it is zero the problem when an odd order hankel tensor is copositive is open we present a necessary and sufficient condition for a rank 2 odd order symmetric real tensor to be copositive some necessary conditions for a general v tensor to be copositive are also presented the singularity of v tensors is also investigated and we show that a v tensor a is singular if its v rank is less than its dimension this condition becomes necessary if a is of odd order

some upper bounds on zt eigenvalues of tensors abstract in this paper we give upper bounds on zt spectral radius of a tensor a t 1 2 which extend the upper bounds of brauer to tensors moreover an upper bound on the z1 spectral radius is proposed via modulus sum of the entries of certain dimension of a which improves the upper bound given by li et al numerical experiments are given to illustrate the utility of the upper bound

comments on explicit criterion for the positive definiteness of a general quartic form the purpose of this note is to point out that the result in the above paper w h ku ieee trans autom control vol ac 10 no 3 pp 372 373 jul 1965 is incomplete and to give a complete and improved result

bergman kernels and local holomorphic morse inequalities let x be a hermitian manifold and let lk be a high power of a hermitian holomorphic line bundle over x local versions of demailly s holomorphic morse inequalities that give bounds on the dimension of the dolbeault cohomology groups associated to lk are presented after integration they give the usual holomorphic morse inequalities the local weak inequalities hold on any hermitian manifold x regardless of compactness and completeness the proofs which are elementary are based on a new approach to pointwise bergman kernel estimates where the kernels are estimated by a model kernel in open image in new window

generalized bergman kernels on symplectic manifolds we study the near diagonal asymptotic expansion of the generalized bergman kernel of the renormalized bochner laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold we show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them as a consequence we calculate the density of states function of the bochner laplacian and establish a symplectic version of the convergence of the induced fubini study metric we also discuss generalizations of the asymptotic expansion for non compact or singular manifolds as well as their applications our approach is inspired by the analytic localization techniques of bismut and lebeau

improving image classification quality via dissimilarity measure in non euclidean spaces this paper proposes an image classification scheme by learning the dissimilarity measure in non euclidean spaces specifically the dissimilarity representations of samples from a pseudo euclidean space are first constructed then the dissimilarity increment distribution information of each category is achieved with respect to the high order statistics of triplet neighbor points for each image finally a maximum a posteriori algorithm fused with the gaussian mixture model and triplet dissimilarity increments distribution is utilized to estimate the relevance of each image category with each input image experimental results conducted on a general image database demonstrate the effectiveness of the proposed scheme

on the asymptotic expansion of bergman kernel we study the asymptotic of the bergman kernel of the spin c dirac operator on high tensor powers of a line bundle

z eigenvalue methods for a global polynomial optimization problem as a global polynomial optimization problem the best rank one approximation to higher order tensors has extensive engineering and statistical applications different from traditional optimization solution methods in this paper we propose some z eigenvalue methods for solving this problem we first propose a direct z eigenvalue method for this problem when the dimension is two in multidimensional case by a conventional descent optimization method we may find a local minimizer of this problem then by using orthogonal transformations we convert the underlying supersymmetric tensor to a pseudo canonical form which has the same e eigenvalues and some zero entries based upon these we propose a direct orthogonal transformation z eigenvalue method for this problem in the case of order three and dimension three in the case of order three and higher dimension we propose a heuristic orthogonal transformation z eigenvalue method by improving the local minimum with the lower dimensional z eigenvalue methods and a heuristic cross hill z eigenvalue method by using the two dimensional z eigenvalue method to find more local minimizers numerical experiments show that our methods are efficient and promising

gradient skewness tensors and local illumination detection for images in this paper we propose the definition of d eigenvalue for an arbitrary order tensor related with a second order tensor d and introduce the gradient skewness tensor which involves a three order tensor derived from the skewness statistic of gradient images as we happen to find out that the skewness of oriented gradients can measure the directional characteristic of illumination in an image the local illumination detection problem for an image can be abstracted as solving the largest d eigenvalue of gradient skewness tensors we discuss the properties of d eigenvalues and especially for gradient skewness tensors we provide the calculation method of its d eigenvalues and d characteristic polynomial some numerical experiments show its effective application in illumination detection our method also presents excellent results in a class of image authenticity verification problems which is to distinguish artificial flat objects in a photograph

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

analyzing tensor power method dynamics in overcomplete regime we present a novel analysis of the dynamics of tensor power iterations in the overcomplete regime where the tensor cp rank is larger than the input dimension finding the cp decomposition of an overcomplete tensor is np hard in general we consider the case where the tensor components are randomly drawn and show that the simple power iteration recovers the components with bounded error under mild initialization conditions we apply our analysis to unsupervised learning of latent variable models such as multiview mixture models and spherical gaussian mixtures given the third order moment tensor we learn the parameters using tensor power iterations we prove it can correctly learn the model parameters when the number of hidden components k is much larger than the data dimension d up to k o d1 5 we initialize the power iterations with data samples and prove its success under mild conditions on the signal to noise ratio of the samples our analysis significantly expands the class of latent variable models where spectral methods are applicable our analysis also deals with noise in the input tensor leading to sample complexity result in the application to learning latent variable models

a procedure for the positive definiteness of forms of even order a positivity method for forms of even order is reported it is shown that testing for positivity of such forms can be reduced to a test for positive definiteness of parametric matrices special emphasis is placed on quartic and sixth order forms examples are given to illustrate the proposed criterion

markov chains with memory tensor formulation and the dynamics of power iteration a markov chain with memory is no different from the conventional markov chain on the product state space such a markovianization however increases the dimensionality exponentially instead markov chain with memory can naturally be represented as a tensor whence the transitions of the state distribution and the memory distribution can be characterized by specially defined tensor products in this context the progression of a markov chain can be interpreted as variants of power like iterations moving toward the limiting probability distributions what is not clear is the makeup of the second dominant eigenvalue that affects the convergence rate of the iteration if the method converges at all casting the power method as a fixed point iteration this paper examines the local behavior of the nonlinear map and identifies the cause of convergence or divergence as an application it is found that there exists an open set of irreducible and aperiodic transition probability tensors where the z eigenvector type power iteration fails to converge

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

the atiyah singer theorems a probabilistic approach ii the lefschetz fixed point formulas abstract in this paper the lefschetz fixed point formulas of m f atiyah and i m singer ann of math 87 1968 484 530 546 604 for classical elliptic complexes are proved by using probabilistic methods

a new cluster isolation criterion based on dissimilarity increments this paper addresses the problem of cluster defining criteria by proposing a model based characterization of interpattern relationships taking a dissimilarity matrix between patterns as the basic measure for extracting group structure dissimilarity increments between neighboring patterns within a cluster are analyzed empirical evidence suggests modeling the statistical distribution of these increments by an exponential density we propose to use this statistical model which characterizes context to derive a new cluster isolation criterion the integration of this criterion in a hierarchical agglomerative clustering framework produces a partitioning of the data while exhibiting data interrelationships in terms of a dendrogram type graph the analysis of the criterion is undertaken through a set of examples showing the versatility of the method in identifying clusters with arbitrary shape and size the number of clusters is intrinsically found without requiring ad hoc specification of design parameters nor engaging in a computationally demanding optimization procedure

a new central limit theorem and decomposition for gaussian polynomials with an application to deterministic approximate counting one of the main results of this paper is a new multidimensional central limit theorem clt for multivariate polynomials under gaussian inputs roughly speaking the new clt shows that any collection of gaussian polynomials with small eigenvalues suitably defined must have a joint distribution which is close to a multidimensional gaussian distribution the new clt is proved using tools from malliavin calculus and stein s method a second main result of the paper which complements the new clt is a new decomposition theorem for low degree multilinear polynomials over gaussian inputs roughly speaking this result shows that up to some small error any such polynomial is very close to a polynomial which can be decomposed into a bounded number of multilinear polynomials all of which have extremely small eigenvalues an important feature of this decomposition theorem is the delicate control obtained between the number of polynomials in the decomposition versus their eigenvalues as the main application of these results we give a deterministic algorithm for approximately counting satisfying assignments of a degree d polynomial threshold function ptf over the domain 1 1 n this is a well studied problem from theoretical computer science more precisely given as input a degree d polynomial p x 1 dots x n over mathbb r n and a parameter epsilon 0 the algorithm approximates begin aligned mathop mathbf pr x sim 1 1 n p x ge 0 end aligned to within an additive pm epsilon in time o d epsilon 1 cdot mathrm poly n d since it is np hard to determine whether the above probability is nonzero any sort of efficient multiplicative approximation is almost certainly impossible even for randomized algorithms note that the running time of the algorithm as a function of n d the number of coefficients of a degree d ptf is a fixed polynomial the fastest previous algorithm for this problem kane corr arxiv 1210 1280 2012 based on constructions of unconditional pseudorandom generators for degree d ptfs runs in time n o d c 1 cdot epsilon c for all c 0

a seminorm regularized alternating least squares algorithm for canonical tensor decomposition abstract the regularization method could deal with the swamp effect of alternating least squares als algorithms for tensor decomposition usually the regularization term is a norm of the difference between the solution and the current iterate in this paper we show that the norm could be weakened to a seminorm so the selection of the regularization term could be more flexible to overcome the swamp effect and avoid the drawback that the hessian of the subproblem may get close to singular in the iterative process we propose a seminorm regularized als algorithm for solving the canonical tensor decomposition moreover in the new algorithm we introduce a novel extrapolation in the update of each mode factor which makes an immediate impression on the update of subsequent ones by assuming the boundness of the infinite sequence of iterates generated by the new algorithm we establish the global convergence and the weakly linear convergence rate of the sequence of iterates numerical experiments on synthetic and real world problems illustrate that the new method is efficient and promising

szego kernel asymptotics and morse inequalities on cr manifolds let x be an abstract compact orientable cr manifold of dimension 2n 1 n 2 and let l k be the k th tensor power of a cr complex line bundle l over x we assume that condition y q holds at each point of x in this paper we obtain a scaling upper bound for the szego kernel on 0 q forms with values in l k for large k after integration this gives weak morse inequalities analogues of the holomorphic morse inequalities of demailly by a refined spectral analysis we obtain also strong morse inequalities we apply the strong morse inequalities to the embedding of some convex concave manifolds

the infinitesimal lefschetz formulas a heat equation proof abstract in this paper the lefschetz formulas of atiyah and bott ann of math 86 1967 374 407 88 1968 451 491 and atiyah and singer ann of math 87 1968 546 604 93 1971 119 138 are proved by heat equation methods in infinitesimal form i e without using the usual localization on the fixed point set the proof is obtained by a homotopic deformation of the dirac operator and uses probabilistic methods the main applications are potentially in noncompact situations where fixed point formulas do not always hold while their infinitesimal analogues are still valid

unsupervised learning of finite mixture models this paper proposes an unsupervised algorithm for learning a finite mixture model from multivariate data the adjective unsupervised is justified by two properties of the algorithm 1 it is capable of selecting the number of components and 2 unlike the standard expectation maximization em algorithm it does not require careful initialization the proposed method also avoids another drawback of em for mixture fitting the possibility of convergence toward a singular estimate at the boundary of the parameter space the novelty of our approach is that we do not use a model selection criterion to choose one among a set of preestimated candidate models instead we seamlessly integrate estimation and model selection in a single algorithm our technique can be applied to any type of parametric mixture model for which it is possible to write an em algorithm in this paper we illustrate it with experiments involving gaussian mixtures these experiments testify for the good performance of our approach

comments on a procedure for the positive definiteness of forms of even order hasan and hasan ieee trans automat control vol 41 p 615 17 1996 give a necessary and sufficient condition for the positive definiteness of a quadratic programming problem the purpose of this paper is to point out that the main result in the above mentioned paper by is erroneous

the tensor splitting with application to solve multi linear systems abstract in this paper firstly we introduce the variant tensor splittings and present some equivalent conditions for a strong m tensor based on the tensor splitting secondly the existence and uniqueness conditions of the solution for multi linear systems are given thirdly we propose some tensor splitting algorithms for solving multi linear systems with coefficient tensor being a strong m tensor as an application a tensor splitting algorithm for solving the multi linear model of higher order markov chains is proposed numerical examples are given to demonstrate the efficiency of the proposed algorithms

on the z eigenvalues of the adjacency tensors for uniform hypergraphs abstract the adjacency matrices for graphs are generalized to the adjacency tensors for uniform hypergraphs and some fundamental properties for the adjacency tensor and its z eigenvalues of a uniform hypergraph are obtained in particular some bounds on the smallest and the largest z eigenvalues of the adjacency tensors for uniform hypergraphs are presented

block tensors and symmetric embeddings abstract well known connections exist between the singular value decomposition of a matrix a and the schur decomposition of its symmetric embedding sym a 0 a a t 0 in particular if is a singular value of a then and are eigenvalues of the symmetric embedding the top and bottom halves of sym a s eigenvectors are singular vectors for a power methods applied to a can be related to power methods applied to sym a the rank of sym a is twice the rank of a in this paper we develop similar connections for tensors by building on l h lim s variational approach to tensor singular values and vectors we show how to embed a general order d tensor a into an order d symmetric tensor sym a through the embedding we relate power methods for a s singular values to power methods for sym a s eigenvalues finally we connect the multilinear and outer product rank of a to the multilinear and outer product rank of sym a

positivity and nonnegativity conditions of a quartic equation and related problems in this paper explicit conditions for positivity no real roots nonnegativity on positive real axis no positive real roots with odd multiplicity and stability aperiodicity all roots are real and negative and simple of a quartic or biquadratic equation are given the derived conditions from the known solution of the quartic equation are not only complete but simpler than those derived from sturm extended hurwitz inners and hankel methods because of abel s theorem no explicit solution in terms of the roots of an equation higher than quartic exists similar simplification for higher degree polynomial equations may not be possible furthermore explicit conditions for positivity and nonnegativity of equations of higher degree than four are extremely difficult to obtain and may not be possible the results of the paper will hopefully shed some light on a century old problem and thus enhance the engineering application of the derived condition to higher order systems

minimal gersgorin tensor eigenvalue inclusion set and its approximation for a tensor its minimal gersgorin tensor eigenvalue inclusion set is presented by establishing a sufficient and necessary condition for the elements belonging to this set we give a sequence of approximation sets and prove that the limit of this sequence is the minimal gersgorin tensor eigenvalue inclusion set for an irreducible tensor

dynamics of regular polynomial endomorphisms of c k a regular polynomial self mapping of complex euclidean space has a continuous extension to the hyperplane at infinity the induced mapping at infinity is used to describe the dynamics of the self mapping in a neighborhood of infinity a theory of external rays is developed to provide the connection between the behavior at infinity and the behavior on the finite plane

on the optimization landscape of tensor decompositions non convex optimization with local search heuristics has been widely used in machine learning achieving many state of art results it becomes increasingly important to understand why they can work for these np hard problems on typical data the landscape of many objective functions in learning has been conjectured to have the geometric property that all local optima are approximately global optima and thus they can be solved efficiently by local search algorithms however establishing such property can be very difficult in this paper we analyze the optimization landscape of the random over complete tensor decomposition problem which has many applications in unsupervised learning especially in learning latent variable models in practice it can be efficiently solved by gradient ascent on a non convex objective we show that for any small constant varepsilon 0 among the set of points with function values 1 varepsilon factor larger than the expectation of the function all the local maxima are approximate global maxima previously the best known result only characterizes the geometry in small neighborhoods around the true components our result implies that even with an initialization that is barely better than the random guess the gradient ascent algorithm is guaranteed to solve this problem however achieving such a initialization with random guess would still require super polynomial number of attempts our main technique uses kac rice formula and random matrix theory to our best knowledge this is the first time when kac rice formula is successfully applied to counting the number of local optima of a highly structured random polynomial with dependent coefficients

the witten complex and the degenerate morse inequalities on utilise le complexe introduit par witten pour demontrer des inegalites de morse dans le cas ou les points critiques de la fonction de morse sont isoles et aussi les inegalites de morse degenerees de bott

on new classes of nonnegative symmetric tensors in this paper we introduce three new classes of nonnegative forms or equivalently symmetric tensors and their extensions the newly identified nonnegative symmetric tensors constitute distinctive convex cones in the space of general symmetric tensors order six or above for the special case of quartic forms they collapse into the set of convex quartic homogeneous polynomial functions we discuss the properties and applications of the new classes of nonnegative symmetric tensors in the context of polynomial and tensor optimization numerical experiments for solving certain polynomial optimization models based on the new classes of nonnegative symmetric tensors are presented

inhomogeneous polynomial optimization over a convex set an approximation approach in this paper we consider computational methods for optimizing a multivariate inhomogeneous polynomial function over a general convex set the focus is on the design and analysis of polynomial time approximation algorithms the methods are able to deal with optimization models with polynomial objective functions in any fixed degrees in particular we first study the problem of maximizing an inhomogeneous polynomial over the euclidean ball a polynomial time approximation algorithm is proposed for this problem with an assured relative worst case performance ratio which is dependent only on the dimensions of the model the method and approximation ratio are then generalized to optimize an inhomogeneous polynomial over the intersection of a finite number of co centered ellipsoids furthermore the constraint set is extended to a general convex compact set specifically we propose a polynomial time approximation algorithm with a relative worst case performance ratio for polynomial optimization over some convex compact sets e g a polytope finally numerical results are reported revealing good practical performance of the proposed algorithms for solving some randomly generated instances

an eigenvalue method for testing positive definiteness of a multivariate form in this paper we present an eigenvalue method for testing positive definiteness of a multivariate form this problem plays an important role in the stability study of nonlinear autonomous systems via lyapunov s direct method in automatic control at first we apply the d andrea dickenstein version of the classical macaulay formulas of the resultant to compute the symmetric hyperdeterminant of an even order supersymmetric tensor by using the supersymmetry property we give detailed computation procedures for the bezoutians and specified ordering of monomials in this approach we then use these formulas to calculate the characteristic polynomial of a fourth order three dimensional supersymmetric tensor and give an eigenvalue method for testing positive definiteness of a quartic form of three variables some numerical results of this method are reported

uniqueness of nonnegative tensor approximations we show that for a nonnegative tensor a best nonnegative rank r approximation is almost always unique its best rank one approximation may always be chosen to be a best nonnegative rank one approximation and the set of nonnegative tensors with nonunique best rank one approximations forms an algebraic hypersurface we show that the last part holds true more generally for real tensors and thereby determine a polynomial equation so that a real or nonnegative tensor that does not satisfy this equation is guaranteed to have a unique best rank one approximation we also establish an analogue for real or nonnegative symmetric tensors in addition we prove a singular vector variant of the perron frobenius theorem for positive tensors and apply it to show that a best nonnegative rank r approximation of a positive tensor can never be obtained by deflation as an aside we verify that the euclidean distance ed discriminants of the segre variety and the veronese variety are hypersurfaces and give defining equations of these ed discriminants

the product of the eigenvalues of a symmetric tensor abstract we study e eigenvalues of a symmetric tensor f of degree d on a finite dimensional euclidean vector space v and their relation with the e characteristic polynomial of f we show that the leading coefficient of the e characteristic polynomial of f when it has maximum degree is the d 2 th power respectively the d 2 2 th power when d is odd respectively when d is even of the q discriminant where q is the d th veronese embedding of the isotropic quadric q p v this fact together with a known formula for the constant term of the e characteristic polynomial of f leads to a closed formula for the product of the e eigenvalues of f which generalizes the fact that the determinant of a symmetric matrix is equal to the product of its eigenvalues

maximum block improvement and polynomial optimization in this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem the new approach has the following three main ingredients first we establish a block coordinate descent type search method for nonlinear optimization with the novelty being that we accept only a block update that achieves the maximum improvement hence the name of our new search method maximum block improvement mbi convergence of the sequence produced by the mbi method to a stationary point is proved second we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the mbi approach third we propose a scheme to reach a kkt point of the polynomial optimization provided that a stationary solution for the relaxed tensor problem is available numerical experiments have shown that our new method works very efficiently for a majority of the test instances that we have experimented with the method finds the global optimal solution at a low computational cost

distribution of the eigenvalues of a random system of homogeneous polynomials abstract let f f 1 f n be a system of n complex homogeneous polynomials in n variables of degree d we call c an eigenvalue of f if there exists v c n 0 with f v v generalizing the case of eigenvalues of matrices d 1 we derive the distribution of when the f i are independently chosen at random according to the unitary invariant weyl distribution and determine the limit distribution for n

d eigenvalues of diffusion kurtosis tensors diffusion kurtosis imaging dki is a new model in medical engineering where a diffusion kurtosis dk tensor is involved a dk tensor is a fourth order three dimensional fully symmetric tensor in this paper we introduce d eigenvalues for a dk tensor the largest the smallest and the average d eigenvalues of a dk tensor correspond with the largest the smallest and the average apparent kurtosis coefficients akc of a water molecule in the space respectively we present their computational methods and discuss related anisotropy value of akc

continuation methods for computing z h eigenpairs of nonnegative tensors abstract in this paper a homotopy continuation method for the computation of nonnegative z h eigenpairs of a nonnegative tensor is presented we show that the homotopy continuation method is guaranteed to compute a nonnegative eigenpair additionally using degree analysis we show that the number of positive z eigenpairs of an irreducible nonnegative tensor is odd a novel homotopy continuation method is proposed to compute an odd number of positive z eigenpairs and some numerical results are presented

holomorphic morse inequalities for orbifolds we prove that demailly s holomorphic morse inequalities hold true for complex orbifolds by using a heat kernel method then we introduce the class of moishezon orbifolds and as an application of our inequalties we give a geometric criterion for a compact connected orbifold to be a moishezon orbifolds thus generalizing siu s and demailly s answers to the grauert riemenschneider conjecture to the orbifold case

spectral hypergraph theory of the adjacency hypermatrix and matroids abstract we define a matroid from an m uniform hypergraph spectral properties of the adjacency hypermatrix which can be deduced from the matroid are investigated in particular relationships between the coordinates of eigenvectors are revealed in terms of circuits of the matroid it is expected that these relationships will lead to a better initial input vector on the interior of the positive cone for numerical calculations of z eigenvalues using the ss hopm method and for h eigenvalues when using the nqz method

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

monotonically convergent algorithms for symmetric tensor approximation abstract reduced rank approximations to symmetric tensors find use in data compaction and in multi user blind source separation we derive iterative algorithms which feature monotonic convergence to a minimum of a frobenius norm approximation criterion for a certain rank r tucker product version of the approximation problem the approach exploits the gradient inequality for convex functions to establish monotonic convergence while sparing the cumbersome step size analysis required from a manifold gradient approach it likewise overcomes some limitations of symmetric versions of alternating least squares the computational load per iteration amounts to computing an unfolded matrix and a qr decomposition

positive definiteness of paired symmetric tensors and elasticity tensors abstract in this paper we consider higher order paired symmetric tensors and strongly paired symmetric tensors elasticity tensors and higher order elasticity tensors in solid mechanics are strongly paired symmetric tensors a strongly paired symmetric tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite positive definiteness of elasticity and higher order elasticity tensors is strong ellipticity in solid mechanics which plays an important role in nonlinear elasticity theory we mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional strongly paired symmetric tensors we first show that the concerned strongly paired symmetric tensor is positive definite if and only if its smallest m eigenvalue is positive second we propose several necessary and sufficient conditions under which the concerned strongly paired symmetric tensor is positive definite third we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional strongly paired symmetric tensor can be written as a sum of squares of polynomials and further propose several necessary and or sufficient conditions to judge whether the concerned strongly paired symmetric tensors are positive definite or not fourth by using semidefinite relaxation we propose a sequential semidefinite programming method to compute the smallest m eigenvalue of a fourth order three dimensional strongly paired symmetric tensor by which we can check positive definiteness of the concerned tensor the preliminary numerical results confirm our theoretical findings

an iterative algorithm based on strong h tensors for identifying positive definiteness of irreducible homogeneous polynomial forms abstract identifying the positive definiteness of an even degree homogeneous polynomial form arises in various applications such as evolutionary game dynamics automatical control magnetic resonance imaging and spectral hypergraph theory however it is difficult to determine whether a given homogeneous polynomial is positive definite or not because the problem is np hard in this paper an iterative algorithm of identifying the positive definiteness of irreducible homogeneous polynomial forms is proposed by identifying strong h tensors with weakly irreducibility the validity of the iterative algorithm is proved theoretically experimental results on multilinear systems high order markov chains and symmetric multi player games are presented to illustrate the applications of the proposed methods

principal invariants and inherent parameters of diffusion kurtosis tensors a diffusion kurtosis dk tensor is a fourth order three dimensional fully symmetric tensor which is used in diffusion kurtosis imaging dki a new model in medical engineering to understand the biological and clinical meaning of the dk tensor we have to measure and calculate some quantities and parameters which are independent from coordinate system choices in this paper we study such quantities and parameters they include the largest the smallest and the average apparent kurtosis coefficients akc values which are invariant from the coordinate system choices and some parameters measured in the inherent coordinate system which is formed by the eigenvector system of the second order diffusion tensor we study their computational formulas and relationships

demailly s asymptotic morse inequalities a heat equation proof on donne une demonstration qui s appuie sur l equation de la chaleur pour les inegalites asymptotiques de demailly pour le complexe

gpu based parallel solver via the kantorovich theorem for the nonlinear bernstein polynomial systems this paper proposes a parallel solver for the nonlinear systems in bernstein form based on subdivision and the newton raphson method where the kantorovich theorem is employed to identify the existence of a unique root and guarantee the convergence of the newton raphson iterations since the kantorovich theorem accommodates a singular jacobian at the root the proposed algorithm performs well in a multiple root case moreover the solver is designed and implemented in parallel on graphics processing unit gpu with simd architecture thus efficiency for solving a large number of systems is improved greatly an observation validated by our experimental results

eigenvalues and invariants of tensors abstract a tensor is represented by a supermatrix under a co ordinate system in this paper we define e eigenvalues and e eigenvectors for tensors and supermatrices by the resultant theory we define the e characteristic polynomial of a tensor an e eigenvalue of a tensor is a root of the e characteristic polynomial in the regular case a complex number is an e eigenvalue if and only if it is a root of the e characteristic polynomial we convert the e characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor i e they are invariant under co ordinate system changes we call them principal invariants of that tensor the maximum number of principal invariants of mth order n dimensional tensors is a function of m and n we denote it by d m n and show that d 1 n 1 d 2 n n d m 2 m for m 3 and d m n m n 1 m for m n 3 we also define the rank of a tensor all real eigenvectors associated with nonzero e eigenvalues are in a subspace with dimension equal to its rank

semiclassical determinants of schrodinger operators on line bundles heat kernel methods are used to compute asymptotic expressions for the zeta regularized determinant and local heat invariants of a schrodinger operator acting on sections of the kth tensor power of a line bundle as k the result is an infinite series representation involving todd polynomials of the curvature

a new eigenvalue inclusion set for tensors and its applications abstract a new tensor eigenvalue inclusion set is given and proved to be tighter than those in l q qi 2005 18 and c q li y t li x kong 2014 12 in addition we study the eigenvalues lying on the boundary of the eigenvalue inclusion set provided by qi 2005 for weakly irreducible tensors as applications we give new bounds for the spectral radius of nonnegative tensors some sufficient conditions for a tensor to be a strong m tensor and some inequalities to identify the positive definiteness for an even order real supersymmetric tensor

lp estimates and existence theorems for the operator on complete kahler manifolds abstract in this paper we establish some l p estimates and existence theorems for the operator on a holomorphic hermitian vector bundle over a complete kahler manifold as applications we prove some vanishing theorems of the l p dolbeault cohomology on complete kahler manifolds with suitable geometric conditions

e cospectral hypergraphs and some hypergraphs determined by their spectra two k uniform hypergraphs are said to be cospectral e cospectral if their adjacency tensors have the same characteristic polynomial e characteristic polynomial a k uniform hypergraph h is said to be determined by its spectrum if there is no non isomorphic k uniform hypergraph cospectral with h in this note we give a method for constructing e cospectral hypergraphs which is similar with godsil mckay switching some hypergraphs are shown to be determined by their spectra

criterions for the positive definiteness of real supersymmetric tensors the positive definiteness of an even degree homogeneous polynomial form f x plays an important role in the stability study of nonlinear autonomous systems via lyapunov s direct method in automatic control and the positive definiteness of f x is equivalent to that of an even order supersymmetric tensor which defines f x in this paper we provide some criterions for identifying the positive definiteness of an even order real supersymmetric tensor moreover an iterative algorithm for identifying the positive definiteness of an even order real supersymmetric tensor is obtained numerical examples are given to verify the corresponding results

the number of eigenvalues of a tensor abstract eigenvectors of tensors as studied recently in numerical multilinear algebra correspond to fixed points of self maps of a projective space we determine the number of eigenvectors and eigenvalues of a generic tensor and we show that the number of normalized eigenvalues of a symmetric tensor is always finite we also examine the characteristic polynomial and how its coefficients are related to discriminants and resultants

semiclassical determinants of schrodinger operators on bundles with curvature of low rank the semiclassical k limit of a schrodinger operator k acting on the kth tensor power of a line bundle is studied an inductive argument is given to show that a previously derived infinite series representation for the leading term in the large k expansion of ln det k reduces under certain circumstances to simple expressions involving the riemann zeta function these results hold when the curvature of the bundle is confined to a single plane at each point

a c eigenvalue problem for tensors with applications to higher order multivariate markov chains abstract in this paper we study a new tensor eigenvalue problem which involves e and s eigenvalues as its special cases some theoretical results such as existence of an eigenvalue and the number of eigenvalues are given for an application of the proposed eigenvalue problem we establish a tensor model for a higher order multivariate markov chain the core issue of this problem is to study a stationary probability distribution of a higher order multivariate markov chain a sufficient condition of the unique stationary positive distribution is given an algorithm for computing stationary probability distribution is also developed numerical examples of applications in stock market modeling sales demand prediction and biological sequence analysis are given to illustrate the proposed tensor model and the computed stationary probability distribution can provide a better prediction in these markov chain applications

statistical modeling of dissimilarity increments for d dimensional data application in partitional clustering this paper addresses the use of high order dissimilarity models in data mining problems we explore dissimilarities between triplets of nearest neighbors called dissimilarity increments dis we derive a statistical model of dis for d dimensional data d did assuming that the objects follow a multivariate gaussian distribution empirical evidence shows that the d did is well approximated by the particular case d 2 we propose the application of this model in clustering with a partitional algorithm that uses a merge strategy on gaussian components experimental results in synthetic and real datasets show that clustering algorithms using did usually outperform well known clustering algorithms

a levenberg marquardt method for solving semi symmetric tensor equations abstract in this paper we propose a levenberg marquardt lm method for solving tensor equations with semi symmetric coefficient tensor and prove its global convergence and local quadratic convergence under the local error bound condition which is weaker than non singularity as application we solve h eigenvalue of real semi symmetric tensor by the lm method at last some numerical examples are provided to illustrate the efficiency and validity of these methods proposed

a modified newton s method for best rank one approximation to tensors in this paper a modified newton s method for the best rank one approximation problem to tensor is proposed we combine the iterative matrix of jacobi gauss newton jgn algorithm or alternating least squares als algorithm with the iterative matrix of grq newton method and present a modified version of grq newton algorithm a line search along the projective direction is employed to obtain the global convergence preliminary numerical experiments and numerical comparison show that our algorithm is efficient

on determinants of laplacians on riemann surfaces determinants of laplacians on tensors and spinors of arbitrary weights on compact hyperbolic riemann surfaces are computed in terms of values of selberg zeta functions at half integer points

some variational principles for z eigenvalues of nonnegative tensors many important spectral properties of nonnegative matrices have recently been successfully extended to higher order nonnegative tensors for example see chang et al 2008 2011 friedland et al in press lim 2005 liu et al 2010 ng et al 2010 qi et al 2007 yang and yang 2010 2 3 9 17 23 24 27 28 however most of these results focus on the h eigenvalues introduced by qi 2005 2007 25 26 the key results of this paper reveal some similarities as well as some crucial differences between z eigenvalues and h eigenvalues of a nonnegative tensor in particular neither the positive z eigenvalue nor the associated positive z eigenvector of an irreducible nonnegative tensor has to be unique in general as demonstrated by example 2 7 furthermore the collatz type min max characterizations of the largest positive z eigenvalue of an irreducible nonnegative tensor is only partially true in general as seen in theorem 4 7 and example 4 8

the degree of the e characteristic polynomial of an even order tensor the e characteristic polynomial of an even order supersymmetric tensor is a useful tool in determining the positive definiteness of an even degree multivariate form in this paper for an even order tensor we first establish the formula of its e characteristic polynomial by using the classical macaulay formula of resultants then give an upper bound for the degree of that e characteristic polynomial examples illustrate that this bound is attainable in some low order and dimensional cases

approximation algorithms for homogeneous polynomial optimization with quadratic constraints in this paper we consider approximation algorithms for optimizing a generic multi variate homogeneous polynomial function subject to homogeneous quadratic constraints such optimization models have wide applications e g in signal processing magnetic resonance imaging mri data training approximation theory and portfolio selection since polynomial functions are non convex the problems under consideration are all np hard in general in this paper we shall focus on polynomial time approximation algorithms in particular we first study optimization of a multi linear tensor function over the cartesian product of spheres we shall propose approximation algorithms for such problem and derive worst case performance ratios which are shown to be dependent only on the dimensions of the model the methods are then extended to optimize a generic multi variate homogeneous polynomial function with spherical constraint likewise approximation algorithms are proposed with provable approximation performance ratios furthermore the constraint set is relaxed to be an intersection of co centered ellipsoids namely we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin and propose polynomial time approximation algorithms with provable worst case performance ratios numerical results are reported illustrating the effectiveness of the approximation algorithms studied