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43
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 ..."
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Cited by 101 (22 self)
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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
A survey of maxtype recursive distributional equations
 Annals of Applied Probability 15 (2005
, 2005
"... In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent cop ..."
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Cited by 89 (6 self)
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In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(·) is essentially a “maximum ” or “minimum” function. We draw attention to the theoretical question of endogeny: inthe associated recursive tree process X i,aretheX i measurable functions of the innovations process (ξ i)? 1. Introduction. Write
Parameter priors for directed acyclic graphical models and the characterization of several probability distributions
 MICROSOFT RESEARCH, ADVANCED TECHNOLOGY DIVISION
, 1999
"... We show that the only parameter prior for complete Gaussian DAG models that satisfies global parameter independence, complete model equivalence, and some weak regularity assumptions, is the normalWishart distribution. Our analysis is based on the following new characterization of the Wishart distri ..."
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Cited by 36 (1 self)
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We show that the only parameter prior for complete Gaussian DAG models that satisfies global parameter independence, complete model equivalence, and some weak regularity assumptions, is the normalWishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let W be an n × n, n ≥ 3, positivedefinite symmetric matrix of random variables and f(W) be a pdf of W. Then, f(W) is a Wishart distribution if and only if W11 − W12W −1 is independent 22 W ′ 12 of {W12, W22} for every block partitioning
2006) Some characterizations of the spherical harmonics coefficients for isotropic random fields
"... In this paper we provide some simple characterizations for the spherical harmonics coefficients of an isotropic random field on the sphere. The main result is a characterization of isotropic gaussian fields through independence of the coefficients of their development in spherical harmonics. ..."
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Cited by 28 (14 self)
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In this paper we provide some simple characterizations for the spherical harmonics coefficients of an isotropic random field on the sphere. The main result is a characterization of isotropic gaussian fields through independence of the coefficients of their development in spherical harmonics.
Efficient independent component analysis (I
, 2003
"... Independent component analysis (ICA) has been widely used for blind source separation in many fields such as brain imaging analysis, signal processing and telecommunication. Many statistical techniques based on Mestimates have been proposed for estimating the mixing matrix. Recently, several nonpar ..."
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Cited by 23 (4 self)
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Independent component analysis (ICA) has been widely used for blind source separation in many fields such as brain imaging analysis, signal processing and telecommunication. Many statistical techniques based on Mestimates have been proposed for estimating the mixing matrix. Recently, several nonparametric methods have been developed, but indepth analysis of asymptotic efficiency has not been available. We analyze ICA using semiparametric theories and propose a straightforward estimate based on the efficient score function by using Bspline approximations. The estimate is asymptotically efficient under moderate conditions and exhibits better performance than standard ICA methods in a variety of simulations.
An Algorithm For The Blind Identification Of Independent Signals With Sensors
 in: Sixth International Symposium on Signal Processing and its Applications (ISSPA’01
, 2001
"... This paper presents a novel procedure for the blind identification of a linear mixture of n sources with 2 sensors. It is shown that the second characteristic function obeys a partial differential equation (PDE) whose coefficients are directly related to the mixture coefficients. A uniqueness result ..."
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Cited by 22 (0 self)
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This paper presents a novel procedure for the blind identification of a linear mixture of n sources with 2 sensors. It is shown that the second characteristic function obeys a partial differential equation (PDE) whose coefficients are directly related to the mixture coefficients. A uniqueness result allows the design of an estimation procedure based on this PDE. An algorithm is therefore proposed and computer experiments illustrate its performances.
ON THE CHOICE OF m IN THE m OUT OF n BOOTSTRAP AND CONFIDENCE BOUNDS FOR EXTREMA
"... Abstract: For i.i.d. samples of size n, the ordinary bootstrap (Efron (1979)) is known to be consistent in many situations, but it may fail in important examples (Bickel, Götze and van Zwet (1997)). Using bootstrap samples of size m, where m → ∞ and m/n → 0, typically resolves the problem (Bickel e ..."
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Cited by 11 (0 self)
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Abstract: For i.i.d. samples of size n, the ordinary bootstrap (Efron (1979)) is known to be consistent in many situations, but it may fail in important examples (Bickel, Götze and van Zwet (1997)). Using bootstrap samples of size m, where m → ∞ and m/n → 0, typically resolves the problem (Bickel et al. (1997), Politis and Romano (1994)). The choice of m is a key issue. In this paper, we consider an adaptive rule, proposed by Bickel, Götze, and van Zwet (personal communication), to pick m. We give general sufficient conditions for first order validity of the rule, and consider its higher order behavior when the ordinary bootstrap fails, and when it works. We then examine the behavior of the rule in the context of setting confidence bounds on high percentiles, such as the asymptotic expected maximum. Key words and phrases: Adaptive choice, bootstrap, choice of m, datadependent rule, extrema, m out of n bootstrap.
Blind identification of underdetermined mixtures based on the hexacovariance and higherorder cyclostationarity
 Proc. SSP’09
, 2009
"... Abstract—Blind identification of underdetermined mixtures can be addressed efficiently by using the second ChAracteristic Function (CAF) of the observations. Our contribution is twofold. First, we propose the use of a LevenbergMarquardt algorithm, herein called LEMACAF, as an alternative to an Alte ..."
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Cited by 8 (3 self)
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Abstract—Blind identification of underdetermined mixtures can be addressed efficiently by using the second ChAracteristic Function (CAF) of the observations. Our contribution is twofold. First, we propose the use of a LevenbergMarquardt algorithm, herein called LEMACAF, as an alternative to an Alternating Least Squares algorithm known as ALESCAF, which has been used recently in the case of real mixtures of real sources. Second, we extend the CAF approach to the case of complex sources for which the previous algorithms are not suitable. We show that the complex case involves an appropriate tensor stowage, which is linked to a particular tensor decomposition. An extension of the LEMACAF algorithm, called LEMACAFC is then proposed to blindly estimate the mixing matrix by exploiting this tensor decomposition. In our simulation results, we first provide performance comparisons between third and fourth order versions of ALESCAF and LEMACAF in various situations involving BPSK sources. Then, a performance study of LEMACAFC is carried out considering 4QAM sources. These results show that the proposed algorithm provides satisfying estimations especially in the case of a large underdeterminacy level. Index Terms—Blind identification, blind source separation, characteristic function, complex sources, underdetermined mixtures, tensor decompositions I.
Some Bayesian perspectives on statistical modelling
, 1988
"... I would like to thank my supervisor, Professor A. F. M. Smith, for all his advice and encourage ..."
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Cited by 6 (2 self)
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I would like to thank my supervisor, Professor A. F. M. Smith, for all his advice and encourage
Canonical Piecewise Linear Network for Nonlinear Filtering and Its Application to Blind Equalization
 SIGNAL PROCESS
"... Canonical piecewise linear structures provide a desirable compromise between the approximation ability of nonlinear models and the efficiency and theoretical accessibility of the linear domain, and they reduce the parameter storage requirement of piecewise linear models considerably by employing a g ..."
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Cited by 5 (0 self)
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Canonical piecewise linear structures provide a desirable compromise between the approximation ability of nonlinear models and the efficiency and theoretical accessibility of the linear domain, and they reduce the parameter storage requirement of piecewise linear models considerably by employing a global linear representation. We first study the representation and approximation ability of CPL network and show that we can always construct a partition for which a CPL representation exists, and that we can approximate any given continuous nonlinear mapping with a CPL network. We then present application of CPL network to blind equalization, show the ability of CPL equalizer to achieve blind equalization of a nonlinear channel, specifically, we show that if the distribution of the equalizer output is the same as that of the input to the channel, then the channel and equalizer combination is identity except for a sign and a delay factor. We also present experimental results to show the succ...