24,368 research outputs found
Symmetric tensor decomposition
We present an algorithm for decomposing a symmetric tensor, of dimension n
and order d as a sum of rank-1 symmetric tensors, extending the algorithm of
Sylvester devised in 1886 for binary forms. We recall the correspondence
between the decomposition of a homogeneous polynomial in n variables of total
degree d as a sum of powers of linear forms (Waring's problem), incidence
properties on secant varieties of the Veronese Variety and the representation
of linear forms as a linear combination of evaluations at distinct points. Then
we reformulate Sylvester's approach from the dual point of view. Exploiting
this duality, we propose necessary and sufficient conditions for the existence
of such a decomposition of a given rank, using the properties of Hankel (and
quasi-Hankel) matrices, derived from multivariate polynomials and normal form
computations. This leads to the resolution of polynomial equations of small
degree in non-generic cases. We propose a new algorithm for symmetric tensor
decomposition, based on this characterization and on linear algebra
computations with these Hankel matrices. The impact of this contribution is
two-fold. First it permits an efficient computation of the decomposition of any
tensor of sub-generic rank, as opposed to widely used iterative algorithms with
unproved global convergence (e.g. Alternate Least Squares or gradient
descents). Second, it gives tools for understanding uniqueness conditions, and
for detecting the rank
Iterative Methods for Symmetric Outer Product Tensor Decompositions
We study the symmetric outer product decomposition which decomposes a fully
(partially) symmetric tensor into a sum of rank-one fully (partially) symmetric
tensors. We present iterative algorithms for the third-order partially
symmetric tensor and fourth-order fully symmetric tensor. The numerical
examples indicate a faster convergence rate for the new algorithms than the
standard method of alternating least squares
Generating Polynomials and Symmetric Tensor Decompositions
This paper studies symmetric tensor decompositions. For symmetric tensors,
there exist linear relations of recursive patterns among their entries. Such a
relation can be represented by a polynomial, which is called a generating
polynomial. The homogenization of a generating polynomial belongs to the apolar
ideal of the tensor. A symmetric tensor decomposition can be determined by a
set of generating polynomials, which can be represented by a matrix. We call it
a generating matrix. Generally, a symmetric tensor decomposition can be
determined by a generating matrix satisfying certain conditions. We
characterize the sets of such generating matrices and investigate their
properties (e.g., the existence, dimensions, nondefectiveness). Using these
properties, we propose methods for computing symmetric tensor decompositions.
Extensive examples are shown to demonstrate the efficiency of proposed methods.Comment: 35 page
Best rank one approximation of real symmetric tensors can be chosen symmetric
We show that a best rank one approximation to a real symmetric tensor, which
in principle can be nonsymmetric, can be chosen symmetric.
Furthermore, a symmetric best rank one approximation to a symmetric tensor is
unique if the tensor does not lie on a certain real algebraic variety.Comment: 14 page
Symmetric Tensor Decomposition by an Iterative Eigendecomposition Algorithm
We present an iterative algorithm, called the symmetric tensor eigen-rank-one
iterative decomposition (STEROID), for decomposing a symmetric tensor into a
real linear combination of symmetric rank-1 unit-norm outer factors using only
eigendecompositions and least-squares fitting. Originally designed for a
symmetric tensor with an order being a power of two, STEROID is shown to be
applicable to any order through an innovative tensor embedding technique.
Numerical examples demonstrate the high efficiency and accuracy of the proposed
scheme even for large scale problems. Furthermore, we show how STEROID readily
solves a problem in nonlinear block-structured system identification and
nonlinear state-space identification
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