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86
A multilinear singular value decomposition
 SIAM J. Matrix Anal. Appl
, 2000
"... Abstract. We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higherorder tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, firstorder perturbation effects, etc., are ..."
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Cited by 472 (22 self)
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Abstract. We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higherorder tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, firstorder perturbation effects, etc., are analyzed. We investigate how tensor symmetries affect the decomposition and propose a multilinear generalization of the symmetric eigenvalue decomposition for pairwise symmetric tensors.
On the best rank1 approximation of higherorder supersymmetric tensors
 SIAM J. Matrix Anal. Appl
, 2002
"... Abstract. Recently the problem of determining the best, in the leastsquares sense, rank1 approximation to a higherorder tensor was studied and an iterative method that extends the wellknown power method for matriceswasproposed for itssolution. Thishigherorder power method is also proposed for th ..."
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Cited by 76 (1 self)
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Abstract. Recently the problem of determining the best, in the leastsquares sense, rank1 approximation to a higherorder tensor was studied and an iterative method that extends the wellknown power method for matriceswasproposed for itssolution. Thishigherorder power method is also proposed for the special but important class of supersymmetric tensors, with no change. A simplified version, adapted to the special structure of the supersymmetric problem, is deemed unreliable, asitsconvergence isnot guaranteed. The aim of thispaper isto show that a symmetric version of the above method converges under assumptions of convexity (or concavity) for the functional induced by the tensor in question, assumptions that are very often satisfied in practical applications. The use of this version entails significant savings in computational complexity as compared to the unconstrained higherorder power method. Furthermore, a novel method for initializing the iterative processisdeveloped which hasbeen observed to yield an estimate that liescloser to the global optimum than the initialization suggested before. Moreover, its proximity to the global optimum is a priori quantifiable. In the course of the analysis, some important properties that the supersymmetry of a tensor implies for its square matrix unfolding are also studied.
Local Convergence of the Alternating Least Squares Algorithm For Canonical Tensor Approximation
, 2011
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Hyperbolic Polynomials and Convex Analysis
, 1998
"... Abstract. A homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t ↦ → p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, G˚arding proved in 1951 that the largest such root is a convex function of x, and s ..."
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Cited by 39 (4 self)
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Abstract. A homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t ↦ → p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, G˚arding proved in 1951 that the largest such root is a convex function of x, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize G˚arding’s result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convexanalytic tools for such symmetric functions, of interest in interiorpoint methods for optimization problems over related cones. 1
TENSORCUR DECOMPOSITIONS FOR TENSORBASED DATA
 SIAM J. MATRIX ANAL. APPL.
, 2008
"... Motivated by numerous applications in which the data may be modeled by a variable subscripted by three or more indices, we develop a tensorbased extension of the matrix CUR decomposition. The tensorCUR decomposition is most relevant as a data analysis tool when the data consist of one mode that i ..."
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Cited by 36 (10 self)
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Motivated by numerous applications in which the data may be modeled by a variable subscripted by three or more indices, we develop a tensorbased extension of the matrix CUR decomposition. The tensorCUR decomposition is most relevant as a data analysis tool when the data consist of one mode that is qualitatively different from the others. In this case, the tensorCUR decomposition approximately expresses the original data tensor in terms of a basis consisting of underlying subtensors that are actual data elements and thus that have a natural interpretation in terms of the processes generating the data. Assume the data may be modeled as a (2+1)tensor, i.e., an m×n×p tensor A in which the first two modes are similar and the third is qualitatively different. We refer to each of the p different m × n matrices as “slabs ” and each of the mn different pvectors as “fibers.” In this case, the tensorCUR algorithm computes an approximation to the data tensor A that is of the form CUR, where C is an m×n×c tensor consisting of a small number c of the slabs, R is an r × p matrix consisting of a small number r of the fibers, and U is an appropriately defined and easily computed c × r encoding matrix. Both C and R may be chosen by randomly sampling either slabs or fibers according to a judiciously chosen and datadependent probability distribution, and both c and r depend on a rank parameter k, an error parameter ɛ, and a failure probability δ. Under
A randomized algorithm for a tensorbased generalization of the singular value decomposition
, 2007
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BRST Cohomology and Phase Space Reduction in Deformation Quantization
 COMMUN. MATH. PHYS
, 2000
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The deformation quantization of certain superPoisson brackets and BRST cohomology, in Conférence Moshé Flato
, 1999
"... To the memory of Moshé Flato. Submitted to the proceedings of the conférence Moshé Flato. On every split supermanifold equipped with the Rothstein superPoisson bracket we construct a deformation quantization by means of a Fedosovtype procedure. In other words, the supercommutative algebra of all s ..."
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Cited by 26 (2 self)
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To the memory of Moshé Flato. Submitted to the proceedings of the conférence Moshé Flato. On every split supermanifold equipped with the Rothstein superPoisson bracket we construct a deformation quantization by means of a Fedosovtype procedure. In other words, the supercommutative algebra of all smooth sections of the dual Grassmann algebra bundle of an arbitrarily given vector bundle E (equipped with a fibre metric) over a symplectic manifold M will be deformed by a series of bidifferential operators having first order supercommutator proportional to the Rothstein superbracket. Moreover, we discuss two constructions related to the above result, namely the quantized BRSTcohomology for a locally free Hamiltonian Lie group action and the classical BRST cohomology in the general coistropic (or reducible) case without using a ‘ghosts of ghosts’ scheme.
M.E.Valcher Algebraic aspects of 2D convolutional codes, submitted to
 IEEE Trans.Inf.Th
, 1993
"... Twodimensional (2D) codes are introduced as linear shiftinvariant spaces of admissible signals on the discrete plane. Convolutional and, in particular, basic codes are characterized both in terms of their internal properties and by means of their inputoutput representations. The algebraic structu ..."
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Cited by 19 (11 self)
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Twodimensional (2D) codes are introduced as linear shiftinvariant spaces of admissible signals on the discrete plane. Convolutional and, in particular, basic codes are characterized both in terms of their internal properties and by means of their inputoutput representations. The algebraic structure of the class of all encoders that correspond to a given convolutional code is investigated and the possibility of obtaining 2D decoders, free from catastrophic errors, as well as efficient syndrome decoders is considered. Some aspects of the state space implementation of 2D encoders and decoders via (finite memory) 2D systems are finally discussed.
Numerical Taylor expansions of invariant manifolds in large dynamical systems
, 1996
"... In this paper we develop a numerical method for computing higher order local approximations of invariant manifolds, such as stable, unstable or center manifolds near steady states of a dynamical system. The underlying system is assumed to be large in the sense that a large sparse Jacobian at the equ ..."
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Cited by 16 (1 self)
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In this paper we develop a numerical method for computing higher order local approximations of invariant manifolds, such as stable, unstable or center manifolds near steady states of a dynamical system. The underlying system is assumed to be large in the sense that a large sparse Jacobian at the equilibrium occurs, for which only a linear (black box) solver and a low dimensional invariant subspace is available, but for which methods like the QRAlgorithm are considered to be too expensive. Our method is based on an analysis of the multilinear Sylvester equations for the higher derivatives which can be solved under certain nonresonance conditions. These conditions are weaker than the standard gap conditions on the spectrum which guarantee the existence of the invariant manifold. The final algorithm requires the solution of several large linear systems with a bordered Jacobian. To these systems we apply a block elimination method recently developed by Govaerts and Pryce (1991, 1993).