### Citations

3797 |
Density Estimation for Statistics and Data Analysis
- Silverman
- 1986
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Citation Context ... the �-th component of the corresponding random vector: �� � � � . Let now be a random vector, from which the samples � � ����� � have been observed. Then the kernel estimation of the PDF of is [12], =-=[13]-=-: And from there, the kernel estimation of the �-th component of JSF will be: Remark 1. �� � � � � � � � � (23) � The bandwidth parameter in the above equations determines the degree of smoothness of ... |

3777 |
Introduction to Statistical Pattern Recognition
- Fukunaga
- 1972
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Citation Context ...humb is given in the Remark 3 below. Remark 2. The bandwidth of the kernel can be different in each direction, which depends on the spread of data in that direction. However, as suggested by Fukunaga =-=[14]-=-, after a whitening process on data, we can use isotropic kernels (kernels with equal bandwidths in all directions). To state this idea more clearly, let denote the covariance matrix of , and be � � �... |

1848 |
Independent component analysis, a new concept? Signal processing
- Comon
- 1994
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Citation Context ...inear and memoryless (i.e. �� and �� , where � and � are regular matrices), then we have the well-known instantaneous linear mixtures, for which the equivalency of ICA and BSS has been already proved =-=[3]-=-: if the components of are independent, and if there is at most one Gaussian source, then the outputs will be equal to the source signals up to a scale and a permutation indeterminacy. Consequently, f... |

449 | Equivariant Adaptive Source Separation,”
- Cardoso, Laheld
- 1996
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Citation Context ...radient approach For these mixtures, the gradient of with respect to � has been obtained in equation (42). However, in linear instantaneous mixtures, it is preferable to use the equivariant algorithm =-=[24]-=-, [25], which results in a separation quality independent of the mixing matrix. In equivariant algorithm, instead of , the natural [24] (or relative [25]) gradient is used: � � � � � (49) �14 Initial... |

445 | Natural gradient works efficiently in learning
- Amari
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Citation Context ...t approach For these mixtures, the gradient of with respect to � has been obtained in equation (42). However, in linear instantaneous mixtures, it is preferable to use the equivariant algorithm [24], =-=[25]-=-, which results in a separation quality independent of the mixing matrix. In equivariant algorithm, instead of , the natural [24] (or relative [25]) gradient is used: � � � � � (49) �14 Initializatio... |

160 |
Smoothing Techniques with Implementation in S,
- Hardle
- 1991
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Citation Context ...nce of the �-th component of the corresponding random vector: �� � � � . Let now be a random vector, from which the samples � � ����� � have been observed. Then the kernel estimation of the PDF of is =-=[12]-=-, [13]: And from there, the kernel estimation of the �-th component of JSF will be: Remark 1. �� � � � � � � � � (23) � The bandwidth parameter in the above equations determines the degree of smoothne... |

147 | Source separation in post-nonlinear mixtures
- Taleb, Jutten
- 1999
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Citation Context ...ector is nothing but the score functions of its components, and hence it can be estimated by any estimation method of a score hal-00379405, version 1 - 28 Apr 2009 function (see for example [9], [8], =-=[11]-=-). In this section, the estimation of JSF and SFD is considered. A. Estimating JSF The methods presented here for estimating JSF are, in fact, the generalizations of the corresponding methods for esti... |

91 | An unsupervised ensemble learning method for nonlinear dynamic state-space models. Neural Computation
- Valpola, Karhunen
- 2002
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Citation Context ... by discussing their properties. 1Other regularization approaches have also been proposed, based on smooth mappings, i.e. multilayer perceptrons [4] or based on Bayesian models with ensemble learning =-=[5]-=-.3 A. Definitions Recall the definition of the score function of a random variable: Definition 1 (Score Function): The score function of a scalar random variable density, that is, � � � is the opposi... |

56 |
Criteria for multichannel signal separation,”
- Yellin, Weinstein
- 1994
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Citation Context ...tures, the mixing matrix composed of finite order Linear Time-Invariant (LTI) filters instead of scalars. For these mixtures, the separation system is: ��� ℄ � � �� �� � (47) As it has been proved in =-=[19]-=-, convolutive mixtures are separable, too: if � is determined to produce statistically independent outputs, then the sources are separated. In convolutive mixtures, the scale indeterminacy of instanta... |

44 | Mutual information approach to blind separation of stationary sources
- Pham
- 1999
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Citation Context ... vector with the same dimension, then: �� (19) where is the SFD of , and denotes higher order terms in . Remark. Equation (19) may be stated in the following form (which is similar to what is done in =-=[10]-=-): � �� � � (20) hal-00379405, version 1 - 28 Apr 2009 where and are bounded random vectors, � is a matrix with small entries, and � stands for a term that converges to zero faster than ���. This equa... |

24 |
A geometric approach for separating post nonlinear mixtures
- Babaie-Zadeh, Jutten, et al.
- 2002
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Citation Context ...r sensors. For this model, it is shown that the independence of outputs insures that �� � �� and � is a separating matrix [11] up to scale and permutation indeterminacies like in linear mixtures (see =-=[22]-=- for a simple geometric proof for bounded sources). In other words, PNL mixtures are separable: independence is sufficient for estimating the separating structure and restoring the sources. 4) Convolu... |

23 | Fast algorithm for estimating mutual information, entropies and score functions,”
- Pham
- 2003
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Citation Context ...ins with smallest ), for computing � from (38), we need the value of � which is not defined. In our simulations, we have used for these values, too. 4) Pham’s method: Recently D. T. Pham has proposed =-=[17]-=- a method for estimating the “conditional score function”, which appears in separating temporary correlated sources. The conditional score function of the random vector � ����� is defined by: � � � ��... |

16 |
Separating convolutive mixtures by mutual information minimization
- BABAIE-ZADEH, JUTTEN, et al.
- 2001
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Citation Context ...ches for minimizing are quiet general, and may be used in other mutual information optimization problems. II. MULTI-VARIATE SCORE FUNCTIONS Multi-variate score functions have been first introduced in =-=[7]-=-, by extending the concept of score function of a random variable to random vectors. The “gradient” of mutual information can be expressed in terms of these score functions (see Theorem 1). This secti... |

15 |
A penalty method for nonparametric estimation of the logarithmic derivative of a density function
- Cox
- 1985
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Citation Context ...re independent. For the - dimensional case too, � � if “ � and the other components of are independent”, that is, if ����� � � � ����� � � � ����� � � � ����� ,or � ����� � � � ����� � , �� � ����� . =-=[9]-=-. The next property is, in fact, the generalization of a similar property of the score function of a scalar random variable [8], Property 3: Let be a random vector with density and JSF (whose �-th com... |

13 |
Three easy ways for separating nonlinear mixtures
- Jutten, Babaie-Zadeh, et al.
- 2004
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Citation Context ...are statistically independent. This method, based on statistical independence, constitutes a generic approach called Independent Component Analysis (ICA). In general (nonlinear) case, it can be shown =-=[2]-=- that ICA does not lead to BSS. However, if some structural constraints are imposed Electrical engineering departement, Sharif university of technology, Tehran, Iran. Laboratoire des Images et des Sig... |

12 | Entropy optimization, application to blind source separation - TALEB, JUTTEN - 1997 |

11 |
Differential of mutual information function
- Babaie-Zadeh, Jutten, et al.
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Citation Context ...ng the “differential” of the mutual information, that is, its variation resulting from a small deviation in its argument, is very useful. Such a non-parametric differential has been recently proposed =-=[6]-=- (see also Section III). The aim of this paper is to consider two general gradient based approaches (which are called “gradient” and “MinimizationProjection” approaches) for minimizing mutual informat... |

10 | C.: Batch algorithm for source separation in postnonlinear mixtures
- Taleb, Jutten
- 1999
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Citation Context ...ach parameter of the model. For example, neural networks and polynomial models have already been used for modeling � �’s [11], [27]. Another approach, which is used in this paper, is ‘non-parametric’ =-=[28]-=-, [26]. In this approach, no parametric model is assumed for ��’s, and the ‘gradient’ of the criterion with respect to these functions, which is itself a function, is directly calculated. To clarify t... |

9 | Blind separating convolutive post-nonlinearmixtures
- Babaie-Zadeh, Jutten, et al.
- 2001
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Citation Context ..., i.e. � and � (with scalar entries) are replaced by filter matrices � and � (where entries are filters) in Fig. 2. The separability of this model can be deduced from the separability of PNL mixtures =-=[23]-=-, for finite impulse response filters. VII. APPLICATION TO LINEAR INSTANTANEOUS MIXTURES Here, as an illustration of the “gradient” and “MP” approaches, we utilize them for separating linear instantan... |

7 | Minimization-projection (MP) approach for blind source separation in different mixing models
- Babaie-Zadeh, Jutten, et al.
- 2003
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Citation Context ...dence of does not imply the source separation (recall that without stractural constraints, output independence does not insure separation). Then, the idea of the Minimization-Projection (MP) approach =-=[18]-=- 4The scalar (Euclidean) product of two matrices and is defined by � � � ��� �� ��. Moreover, it can be easily seen that for vectors and and matrix � we have � � Å �� .12 for overcoming this problem ... |

5 | Jutten & A. Taleb “Parametric approach to blind deconvolution of nonlinear channels
- Solé, C
(Show Context)
Citation Context ...lculated. Finally the steepest descent gradient algorithm is applied on each parameter of the model. For example, neural networks and polynomial models have already been used for modeling � �’s [11], =-=[27]-=-. Another approach, which is used in this paper, is ‘non-parametric’ [28], [26]. In this approach, no parametric model is assumed for ��’s, and the ‘gradient’ of the criterion with respect to these fu... |

3 |
ICA of Linear and Nonlinear Mixtures Based on Mutual Information
- Almeida
(Show Context)
Citation Context ...tion starts with a review of definitions, and continues by discussing their properties. 1Other regularization approaches have also been proposed, based on smooth mappings, i.e. multilayer perceptrons =-=[4]-=- or based on Bayesian models with ensemble learning [5].3 A. Definitions Recall the definition of the score function of a random variable: Definition 1 (Score Function): The score function of a scala... |

3 |
Spline smoothing and nonparanetric regression
- Eubank
- 1988
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Citation Context ...eparation algorithm tries to minimize these variations. � � � � (35) Practically, following Property 4, �� � is a regression from � to � , which can be calculated for instance using smoothing splines =-=[16]-=-. The final estimation procedure is summarized in the following steps: 1) From the observed values � � ����� � estimate ��� � �� � � ����� (e.g. by kernel estimators). 2) Compute the smoothing spline ... |

3 |
Séparation aveugle des sources en mélange convolutif
- Simon
- 1999
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Citation Context ...stically independent outputs, then the sources are separated. In convolutive mixtures, the scale indeterminacy of instantaneous mixtures extends to a filtering indeterminacy. However, it can be shown =-=[20]-=- that the effect of each source on each sensor (that is, what a sensor would receive if all other sources were zero), can be found after the source separation. In convolutive mixtures, the independenc... |

2 |
Using multivariate score functions in source separation: Application to post non-linear mixtures
- Babaie-Zadeh, Jutten, et al.
- 2002
(Show Context)
Citation Context ...rmation minimization approaches of section V (gradient and MP approaches) in separating PNL mixtures. A. Gradient approach The gradient approach for separating PNL mixtures has been first reported in =-=[26]-=-. The parameters of the separating system (Fig. 2) are the matrix � and the functions � �’s, and for using the gradient approach, the gradients of must be calculated with respect to these parameters. ... |