For such a tensor, we give a description of its singular vector. Greedy approaches to symmetric orthogonal tensor decomposition. Generalized schur functions and generalized decomposable. We study their spectral properties and give a formula for all of their eigenvectors. V a vector space with dimv \\le 3, then every homogeneous element in \\lambdav is decomposable. Orthogonal tensor decomposition department of mathematics.
E, are called simple or decomposable symmetric ntensors. In this paper, we study various properties of symmetric tensors in. Dias da silva institute politknico viik real, portugal dedicated to alston s. My problem is, that i dont know the definition of homogeneous and decomposable. Read generalized schur functions and generalized decomposable symmetric tensors, linear algebra and its applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips.
Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. A spinj state corresponds to a boson if jis a positive integer, and corresponds to a fermion if j. Iterative methods for symmetric outer product tensor decomposition na liy, carmeliza navascaz, and christina glennz abstract. Symmetric eigenvalue decompositions for symmetric tensors lekheng lim university of california, berkeley january 29, 2009 contains joint work with pierre. Decomposable symmetric mappings between in nite dimensional. Facts about symmetric tensor decomposition i the generic rank of tensors in sdcn, denoted by r sd. A symmetric tensor is a higher order generalization of a symmetric matrix. Successive rankone approximations of nearly orthogonally. Symmetric tensors and symmetric tensor rank gipsalab. The answer to the corresponding question for the exterior power and symmetric power are also well known. However, unlike symmetric matrices, the above procedure for higher order tensors p. A symmetric tensor is orthogonally decomposable, in short, odeco, if it possesses an orthogonal decomposition.
Conditions for equality of decomposable symmetric tensors j. We not only show that successive rankone approximation algorithm exactly recovers the unitary decomposition of the unitarily. Such tensors are interesting because their decomposition can be found efficiently. Successive partialsymmetric rankone algorithms for almost. An equivalent way of representing a symmetric tensor t. Orthogonal decomposition of symmetric tensors article pdf available in siam journal on matrix analysis and applications 371 september 2014 with 128 reads how we measure reads. To characterize tensor products with small cartan components, we study the representation hlvi generated. In section 4 and 5 we characterise the decomposable mappings between symmetric tensor spaces of the same and di erent degrees, respectively. Any symmetric tensor can be decomposed into a linear combination of rank1. This reduces the proof of the main theorem for symmetrically odecoudeco tensors to the case of ordinary tensors. Many material properties and fields used in physics and engineering can be represented as symmetric tensor fields. Pdf a symmetric tensor is a higher order generalization of a symmetric matrix. Symmetric tensor decomposition in higher tensor powers.
As we mentioned in the introduction, a symmetric tensor t2smrn can equivalently be. Chapter 22 tensor algebras, symmetric algebras and. A symmetric tensor is orthogonally decomposable if it can be written as a linear combination of tensor powers of n orthonormal vectors. In section3we give a clear algebraic characterzation of all of the. For a general tensor u with components and a pair of indices i and j, u has symmetric and antisymmetric parts defined as. Motivated by the spectral theorem for real symmetric matrices. Also, in diffusion mri one often uses symmetric tensors to describe diffusion in the brain or other parts of the body. Orthogonal decomposition of symmetric tensors elina robeva university of california, berkeley abstract a real symmetric tensor is orthogonally decomposable or odeco if it can be written as a linear combination of symmetric powers of n vectors which form an orthonormal basis of rn. It is free math help boards we are an online community that gives free mathematics help any time of the day about any problem, no. The problem we shall be concerned with the problem of finding conditions for.
Condition for a tensor to be decomposable stack exchange. Successive rankone approximations for nearly orthogonally. Singular vectors of orthogonally decomposable tensors elina robeva and anna seigal abstract orthogonal decomposition of tensors is a generalization of the singular value decomposition of matrices. Covariant decomposition of symmetric tensors and the. One of our main conceptual contributions is to demonstrate that the success of such power iterations need not be viewed as a consequence of a linear or multilinear algebraic structure but instead relies on an orthogonal decomposition of the function f com. While the decomposability rank of a covariance rank 2 tensor a a is the same as the rank of the corresponding matrix, for higher covariance rank tensors we do not have general algorithms how to determine the decomposability rank.
Traditional courses on applied mathematics have emphasized problem solving techniques rather than the systematic development of concepts. In this paper, we introduce the almost unitarily decomposable conjugate partial symmetric tensors, which are different from the commonly studied orthogonally decomposable tensors by involving the conjugate terms in the decomposition and the perturbation term. The set of symmetric tensors of rank 1 is theveronese variety. In this paper, we study the spectral theory of orthogonally decomposable tensors. A tensor a that is antisymmetric on indices i and j has the property that the contraction with a tensor b that is symmetric on indices i and j is identically 0. In section2we are going to give motivation from machine learning for studying orthogonal tensor decomposition. Udayanandan associate professor department of physics nehru arts and science college, kanhangad 1. Orthogonal decomposition of symmetric tensors arxiv. Conditions for equallty of decomposable symmetric tensors. This paper is about linear transformations of the fcfold symmetric tensor product of an ndimensional vector space v which carry nonzero decomposable tensors to nonzero decomposable tensors. The set of orthogonally decomposable tensors can also be sn. Singular vectors of orthogonally decomposable tensors.
This book has been presented in such a clear and easy way that the students will have no difficulty. Orthogonally decomposable tensors auburn university. T hasrank 1if it is the dfold outer product of a vector v. Householder on the occasion of his seventyfifth birthday. We prove that any decomposable linear operator t from a symmetric kfold tensor product over a banach space. Conditions for equality of decomposable symmetric tensors. Journal of the operations research society of china 7. Symmetric eigenvalue decompositions for symmetric tensors. In this case, most powerful geometric tools developed for complex tensors can not be applied directly due to the fact that the euclidean closure of tensors with rank no greater than a. An orthogonal decomposition of a symmetric tensor t 2sdrn is a decomposition t xr i1 iv d i with corresponding f t xr i1 iv i xd such that the vectors v 1v r 2rn are orthonormal. As a result, it is possible for such courses to become terminal mathematics courses rather than. F for tensors is defined later, but is completely analogous to the matrix case. Greedy approaches to symmetric orthogonal tensor decomposition cun mu y, daniel hsuz, and donald goldfarb abstract.
This booklet contains an explanation about tensor calculus for students of physics. Postal 202 vila real, portugal submitted by marvin marcus abstract let xl. This process continues until a certain stopping criterion is met. Symmetric tensors an n n ntensor t t i1i2 i d issymmetricif it is unchanged under permuting indices. Symmetric eigenvalue decompositions for symmetric tensors lekheng lim university of california, berkeley january 29, 2009 contains joint work with pierre comon, jason morton, bernard mourrain, berkant savas. A real symmetric tensor is orthogonally decomposable or odeco if it can be written as a linear combination of symmetric powers of n vectors. A ktensor of this form is called a decomposable ktensor. Successive rankone approximations of nearly orthogonally decomposable symmetric tensors cun mu 1, daniel hsu2, and donald goldfarb 1department of industrial engineering and operations research, columbia university 2department of computer science, columbia university march 1, 2015 abstract. Similarly, if f is alternating, then we can define a skewsymmetric tensor power. Cartan components and decomposable tensors 311 on the other hand it is clear that the closure of the slnorbit of the highest weight vector consists of the nilpotent matrices of rank one. Finding the symmetric and orthogonal decomposition of a tensor is a recurring problem in signal processing, machine learning, and statistics.
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