An enhanced line search scheme for complex-valued tensor decompositions. Application in DS-CDMA

Nion, Dimitri; Lathauwer, Lieven De

doi:10.1016/j.sigpro.2007.07.024

Cited by 110 publications

(87 citation statements)

References 9 publications

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“…This yields 3N (N − 1)/2 redundant rows. From definition (37), it can be verified that the null space of the matrix in (50) has at least dimensionality 3K, and contains the vectors…”

Section: Proof See Appendix Imentioning

confidence: 99%

CONFAC Decomposition Approach to Blind Identification of Underdetermined Mixtures Based on Generating Function Derivatives

Almeida

Luciani

Stegeman

et al. 2012

IEEE Trans. Signal Process.

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Abstract-This work proposes a new tensor-based approach to solve the problem of blind identification of underdetermined mixtures of complex sources exploiting the cumulant generating function (CGF) of the observations. We show that a collection of secondorder derivatives of the CGF of the observations can be stored in a third-order tensor following a constrained factor (CONFAC) decomposition with known constrained structure. In order to increase the diversity, we combine three derivative types into an extended third-order CONFAC decomposition. A detailed uniqueness study of this decomposition is provided, from which easy-to-check sufficient conditions ensuring the essential uniqueness of the mixing matrix are obtained.

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“…This yields 3N (N − 1)/2 redundant rows. From definition (37), it can be verified that the null space of the matrix in (50) has at least dimensionality 3K, and contains the vectors…”

Section: Proof See Appendix Imentioning

confidence: 99%

CONFAC Decomposition Approach to Blind Identification of Underdetermined Mixtures Based on Generating Function Derivatives

Almeida

Luciani

Stegeman

et al. 2012

IEEE Trans. Signal Process.

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show abstract

“…The discussion in [48] and [49] is limited to the real case. The complex case is addressed in [44] and [45].…”

Section: Computation: General Casementioning

confidence: 99%

Blind Identification of Underdetermined Mixtures by Simultaneous Matrix Diagonalization

Lathauwer

Castaing

2008

IEEE Trans. Signal Process.

188

134

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Abstract-In this paper, we study simultaneous matrix diagonalization-based techniques for the estimation of the mixing matrix in underdetermined independent component analysis (ICA). This includes a generalization to underdetermined mixtures of the well-known SOBI algorithm. The problem is reformulated in terms of the parallel factor decomposition (PARAFAC) of a higher-order tensor. We present conditions under which the mixing matrix is unique and discuss several algorithms for its computation.Index Terms-Canonical decomposition, higher order tensor, independent component analysis (ICA), parallel factor (PARAFAC) analysis, simultaneous diagonalization, underdetermined mixture.

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“…However, the ALS method is not guaranteed to converge to a global minimum or a stationary point, only to a solution where the objective function ceases to decrease. There are numerous extensions of the ALS method (e.g., incorporating a line-search procedure in the ALS procedure [46,55]). Along a related line, De Lathauwer et al [14] proposed higher-order power method (HOPM) on rank-one approximation of higher-order tensors, which can also be viewed as an extension of the ALS method.…”

mentioning

confidence: 99%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

161

177

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Abstract. In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we only accept a block update that achieves the maximum improvement, hence the name of our new search method: Maximum Block Improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proven. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem, thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.Key words. block coordinate descent, polynomial optimization problem, tensor form AMS subject classifications. 90C26, 90C30, 15A69, 49M271. Introduction. The optimization models whose objective and constraints are polynomial functions have recently attracted much research attention. This is in part due to an increased demand on the application side (cf. the sample applications in numerical linear algebra [50,26,28], material sciences [57], quantum physics [9,18], and signal processing [16,4,53]), and in part due to its own strong theoretical appeal. Indeed, polynomial optimization is a challenging task; at the same time it is rich enough to be fruitful. For instance, even the simplest instances of polynomial optimization, such as maximizing a cubic polynomial over a sphere, is NP-hard (Nesterov [42]). However, the problem is so elementary that it can be even attempted in an undergraduate calculus class. For readers interested in polynomial optimization with simple constraints, we refer to De Klerk [12] for a survey on the computational complexity of optimizing various classes of polynomial functions over a simplex, hypercube or sphere. In particular, De Klerk et al.[13] designed a polynomial-time approximation scheme (PTAS) for minimizing polynomials of fixed degree over the simplex.So far, a few results have been obtained for approximation algorithms with guaranteed worst-case performance ratios for higher degree polynomial optimization problems. Luo and Zhang [39] derived a polynomial-time approximation algorithm to optimize a multivariate quartic polynomial over a region defined by quadratic inequalities.

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An enhanced line search scheme for complex-valued tensor decompositions. Application in DS-CDMA

Cited by 110 publications

References 9 publications

CONFAC Decomposition Approach to Blind Identification of Underdetermined Mixtures Based on Generating Function Derivatives

CONFAC Decomposition Approach to Blind Identification of Underdetermined Mixtures Based on Generating Function Derivatives

Blind Identification of Underdetermined Mixtures by Simultaneous Matrix Diagonalization

Maximum Block Improvement and Polynomial Optimization

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