Decomposition of quantics in sums of powers of linear forms

Comon, Pierre; Mourrain, Bernard

doi:10.1016/0165-1684(96)00079-5

Cited by 167 publications

(208 citation statements)

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“…The (27) with G ij the partial derivative of y + Ey] with respect to E ij at E = 0. These coe cients form a n n matrix, denoted r y], called the relative (29) The notion of natural gradient was independently introduced by Amari 2].…”

Section: A Relative Gradientmentioning

confidence: 99%

Blind signal separation: statistical principles

1998

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Abstract| Blind signal separation (BSS) and independent component analysis (ICA) are emerging techniques of array processing and data analysis, aiming at recovering unobserved signals or`sources' from observed mixtures (typically, the output of an array of sensors), exploiting only the assumption of mutual independence between the signals. The weakness of the assumptions makes it a powerful approach but requires to venture beyond familiar second order statistics. The objective of this paper is to review some of the approaches that have been recently developed to address this exciting problem, to show how they stem from basic principles and how they relate to each other.

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Section: A Relative Gradientmentioning

confidence: 99%

Blind signal separation: statistical principles

1998

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“…6.2/6.8/9.7/11.2 6.2 5.1 4.0 2 10 6.3/6.8/9.5/10.8 6.3 5.3 4.0 2 11 6.4/6.8/9.0/10.4 6.2 5.5 4.1 Table 3.3: QTT 2 -matrix-ranks of N × N -matrices for large N = 2 p .…”

Section: Proposition 316mentioning

confidence: 99%

“…e − x 2 diag(e −x 2 ) ∆ −1 2 1, ε = 10 −6 , 10 −7 , 10 −8 2 9 5.0 9.4 7.8 3.8 3.6/3.6/3.6 2 10 5.1 9.4 7.7 3.9 3.6/3.6/3.6 2 11 5.2 9.3 7.5 3.9 3.7/3.7/3.7 Table 3.2: QTT 2 -ranks of functional N × N -arrays on large grids, N = 2 p .…”

Section: Numerics On Qtt Approximation Of N-vectors and N-d-tensorsmentioning

confidence: 99%

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Tensors-structured numerical methods in scientific computing: Survey on recent advances

Khoromskij¹

2012

Chemometrics and Intelligent Laboratory Systems

161

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In the present paper, we give a survey of the recent results and outline future prospects of the tensor-structured numerical methods in applications to multidimensional problems in scientific computing. The guiding principle of the tensor methods is an approximation of multivariate functions and operators relying on certain separation of variables. Along with the traditional canonical and Tucker models, we focus on the recent quantics-TT tensor approximation method that allows to represent N -d tensors with log-volume complexity, O(d log N ). We outline how these methods can be applied in the framework of tensor truncated iteration for the solution of the high-dimensional elliptic/parabolic equations and parametric PDEs. Numerical examples demonstrate that the tensor-structured methods have proved their value in application to various computational problems arising in quantum chemistry and in the multi-dimensional/parametric FEM/BEM modeling-the tool apparently works and gives the promise for future use in challenging high-dimensional applications.

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“…Invariants of these group actions typically arise to reduce a problem or to decide if two objects, geometric or abstract, are obtained from one another by the action of a group element. [8,9,10,11,13,15,17,39,40,42,43,45,46,52,59] are a few recent references of applications. Both algebraic and differential invariant theories have become in recent years the subject of computational mathematics [13,14,17,40,60].…”

Section: Introductionmentioning

confidence: 99%