Results 1  10
of
39
Symmetric tensors and symmetric tensor rank
 Scientific Computing and Computational Mathematics (SCCM
, 2006
"... Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. An ..."
Abstract

Cited by 46 (19 self)
 Add to MetaCart
Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We
Phylogenetic ideals and varieties for the general Markov model
 Math. Biosciences
"... Abstract. The general Markov model of the evolution of biological sequences along a tree leads to a parameterization of an algebraic variety. Understanding this variety and the polynomials, called phylogenetic invariants, which vanish on it, is a problem within the broader area of Algebraic Statisti ..."
Abstract

Cited by 42 (5 self)
 Add to MetaCart
Abstract. The general Markov model of the evolution of biological sequences along a tree leads to a parameterization of an algebraic variety. Understanding this variety and the polynomials, called phylogenetic invariants, which vanish on it, is a problem within the broader area of Algebraic Statistics. For an arbitrary trivalent tree, we determine the full ideal of invariants for the 2state model, establishing a conjecture of PachterSturmfels. For the κstate model, we reduce the problem of determining a defining set of polynomials to that of determining a defining set for a 3leaved tree. Along the way, we prove several new cases of a conjecture of GarciaStillmanSturmfels on certain statistical models on star trees, and reduce their conjecture to a family of subcases. 1.
Canonical Tensor Decompositions
 ARCC WORKSHOP ON TENSOR DECOMPOSITION
, 2004
"... The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short pap ..."
Abstract

Cited by 31 (14 self)
 Add to MetaCart
The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short paper, such as the maximal achievable rank of a tensor of given dimensions, or the computation of a CanD. Some questions are answered, and it turns out that the answers depend on the choice of the underlying field, and on tensor symmetry structure, which outlines a major difference compared to matrices.
Induction for secant varieties of Segre varieties
 Trans. Amer. Math. Soc
"... This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calc ..."
Abstract

Cited by 25 (4 self)
 Add to MetaCart
This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective tsecant varieties to Segre varieties for t ≤ 6. We generalize a theorem of CatalisanoGeramitaGimigliano on nondefectivity of tensor powers of P n. We determine the set of p for which unbalanced Segre varieties have defective psecant varieties. In addition, we completely describe the dimensions of the secant varieties to the deficient Segre varieties P 1 ×P 1 ×P n ×P n and P 2 × P 3 × P 3. In the final section we propose a series of conjectures about
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
Tensor Decompositions, Alternating Least Squares and Other Tales
 JOURNAL OF CHEMOMETRICS
, 2009
"... This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple “bottlenecks”, and on “swamps”. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple “bottlenecks”, and on “swamps”. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the ELS enhancement in these algorithms is discussed. Computer simulations feed this discussion.
The border rank of the multiplication of 2 × 2 matrices is seven
 J. Amer. Math. Soc
"... One of the leading problems of algebraic complexity theory is matrix multiplication. The naïve multiplication of two n × n matrices uses n 3 multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
One of the leading problems of algebraic complexity theory is matrix multiplication. The naïve multiplication of two n × n matrices uses n 3 multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, Hopcroft and Kerr [12] and
Geometric complexity theory and tensor rank, arXiv:1011.1350
, 2010
"... Mulmuley and Sohoni [25, 26] proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tens ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
Mulmuley and Sohoni [25, 26] proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tensors, in particular for matrix multiplication. We thus study specific orbit closure problems for the group G = GL(W1) × GL(W2) × GL(W3) acting on the tensor product W = W1 ⊗ W2 ⊗ W3 of complex finite dimensional vector spaces. Let Gs = SL(W1) × SL(W2) × SL(W3). A key idea from [26] is that the irreducible Gsrepresentations occurring in the coordinate ring of the Gorbit closure of astabletensorw∈Ware exactly those having a nonzero invariant with respect to the stabilizer group of w. However, we prove that by considering Gsrepresentations, only trivial lower bounds on border rank can be shown. It is thus necessary to study Grepresentations, which leads to geometric extension problems that are beyond the scope of the subgroup restriction problems emphasized in [25, 26] and its follow up papers. We prove a very modest lower bound on the border rank of matrix multiplication tensors using Grepresentations. This shows at least that the barrier for Gsrepresentations can be overcome. To advance, we suggest the coarser approach to replace the semigroup of representations of a tensor by its moment polytope. We prove first results towards determining the moment polytopes of matrix multiplication and unit tensors. A full version of this paper is available at arxiv.org/abs/1011.1350
Genericity and Rank Deficiency of High Order Symmetric Tensors
 Proc. IEEE Int. Conference on Acoustics, Speech, and Signal Processing (ICASSP
, 2006
"... Blind Identification of UnderDetermined Mixtures (UDM) is involved in numerous applications, including MultiWay factor Analysis (MWA) and Signal Processing. In the latter case, the use of HighOrder Statistics (HOS) like Cumulants leads to the decomposition of symmetric tensors. Yet, little has be ..."
Abstract

Cited by 8 (6 self)
 Add to MetaCart
Blind Identification of UnderDetermined Mixtures (UDM) is involved in numerous applications, including MultiWay factor Analysis (MWA) and Signal Processing. In the latter case, the use of HighOrder Statistics (HOS) like Cumulants leads to the decomposition of symmetric tensors. Yet, little has been published about rankrevealing decompositions of symmetric tensors. Definitions of rank are discussed, and useful results on Generic Rank are proved, with the help of tools borrowed from Algebraic Geometry. 1.
New lower bounds for the border rank of matrix multiplication
"... Abstract. The border rank of the matrix multiplication operator for n × n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n 2 − n. Our bounds are better than the previous lower bound (due to Li ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
Abstract. The border rank of the matrix multiplication operator for n × n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n 2 − n. Our bounds are better than the previous lower bound (due to Lickteig in 1985) of 3 2 n2 + n 2 − 1 for all n ≥ 3. The bounds are obtained by finding new equations that bilinear maps of small border rank must satisfy, i.e., new equations for secant varieties of triple Segre products, that matrix multiplication fails to satisfy.