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103
The geometry of the Newton method on noncompact Lie groups
 J. GLOBAL OPTIMIZ
, 2002
"... An important class of optimization problems involve minimizing a cost function on a Lie group. In the case where the Lie group is noncompact there is no natural choice of a Riemannian metric and it is not possible to apply recent results on the optimization of functions on Riemannian manifolds. I ..."
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Cited by 31 (4 self)
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An important class of optimization problems involve minimizing a cost function on a Lie group. In the case where the Lie group is noncompact there is no natural choice of a Riemannian metric and it is not possible to apply recent results on the optimization of functions on Riemannian manifolds. In this paper the invariant structure of a Lie group is exploited to provide a strong interpretation of a Newton iteration on a general Lie group. The paper unifies several previous algorithms proposed in the literature in a single theoretical framework. Local asymptotic quadratic convergence is proved for the algorithms considered.
A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups
 In Eighth International Conference on Control, Automation, Robotics and Vision
, 2004
"... Motivated by applications in fuzzy control, robotics and vision, this paper considers the problem of computing the centre of mass (precisely, the Karcher mean) of a set of points defined on a compact Lie group, such as the special orthogonal group consisting of all orthogonal matrices with unit d ..."
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Cited by 22 (5 self)
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Motivated by applications in fuzzy control, robotics and vision, this paper considers the problem of computing the centre of mass (precisely, the Karcher mean) of a set of points defined on a compact Lie group, such as the special orthogonal group consisting of all orthogonal matrices with unit determinant. An iterative algorithm, whose derivation is based on the geometry of the problem, is proposed. It is proved to be globally convergent. Interestingly, the proof starts by showing the algorithm is actually a Riemannian gradient descent algorithm with fixed step size. 1.
Interpolation based unitary precoding for spatial multiplexing MIMOOFDM with limited feedback,” in
 Proc. IEEE Globecom’04,
, 2004
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Joint Diagonalization of Correlation Matrices by Using Gradient Methods with Application to Blind Signal Separation
 in Proc. SAM, 2002
, 2002
"... Joint diagonalization of several correlation matrices is a powerful tool for blind signal separation. This paper addresses the blind signal separation problem for the case where the source signals are nonstationary and / or nonwhite, and the sensors are possibly noisy. We present cost functions fo ..."
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Cited by 16 (0 self)
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Joint diagonalization of several correlation matrices is a powerful tool for blind signal separation. This paper addresses the blind signal separation problem for the case where the source signals are nonstationary and / or nonwhite, and the sensors are possibly noisy. We present cost functions for jointly diagonalizing several correlation matrices. The corresponding gradients are derived and used in a gradientbased jointdiagonalization algorithms. Several variations are given, depending on desired properties of the separation matrix, e.g., unitary separation matrix. These constraints are either imposed by adding a penalty term to the cost function or by projecting the gradient onto the desired manifold. The performance of the proposed jointdiagonalization algorithm is verified by simulating a blind signal separation application.
Geometrical methods for nonnegative ICA: Manifolds, Lie groups and toral subalgebras
, 2004
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Penalty functionbased joint diagonalization approach for convolutive blind separation of nonstationary sources
 IEEE Transactions on Signal Processing
, 2005
"... Abstract—A new approach for convolutive blind source separation (BSS) by explicitly exploiting the secondorder nonstationarity of signals and operating in the frequency domain is proposed. The algorithm accommodates a penalty function within the crosspower spectrumbased cost function and thereby ..."
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Cited by 13 (1 self)
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Abstract—A new approach for convolutive blind source separation (BSS) by explicitly exploiting the secondorder nonstationarity of signals and operating in the frequency domain is proposed. The algorithm accommodates a penalty function within the crosspower spectrumbased cost function and thereby converts the separation problem into a joint diagonalization problem with unconstrained optimization. This leads to a new member of the family of joint diagonalization criteria and a modification of the search direction of the gradientbased descent algorithm. Using this approach, not only can the degenerate solution induced by a null unmixing matrix and the effect of large errors within the elements of covariance matrices at lowfrequency bins be automatically removed, but in addition, a unifying view to joint diagonalization with unitary or nonunitary constraint is provided. Numerical experiments are presented to verify the performance of the new method, which show that a suitable penalty function may lead the algorithm to a faster convergence and a better performance for the separation of convolved speech signals, in particular, in terms of shape preservation and amplitude ambiguity reduction, as compared with the conventional secondorder based algorithms for convolutive mixtures that exploit signal nonstationarity. Index Terms—Blind source separation, convolutive mixtures, frequency domain, orthogonal/nonorthogonal constraints, penalty function, speech signals. I.
A.M.: Analysis of head gesture and prosody patterns for prosodydriven headgesture animation
 IEEE Trans. Patt. Anal. and
, 2008
"... Abstract—We propose a new twostage framework for joint analysis of head gesture and speech prosody patterns of a speaker toward automatic realistic synthesis of head gestures from speech prosody. In the first stage analysis, we perform Hidden Markov Model (HMM)based unsupervised temporal segmentat ..."
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Cited by 12 (2 self)
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Abstract—We propose a new twostage framework for joint analysis of head gesture and speech prosody patterns of a speaker toward automatic realistic synthesis of head gestures from speech prosody. In the first stage analysis, we perform Hidden Markov Model (HMM)based unsupervised temporal segmentation of head gesture and speech prosody features separately to determine elementary head gesture and speech prosody patterns, respectively, for a particular speaker. In the second stage, joint analysis of correlations between these elementary head gesture and prosody patterns is performed using Multistream HMMs to determine an audiovisual mapping model. The resulting audiovisual mapping model is then employed to synthesize natural head gestures from arbitrary input test speech given a head model for the speaker. In the synthesis stage, the audiovisual mapping model is used to predict a sequence of gesture patterns from the prosody pattern sequence computed for the input test speech. The Euler angles associated with each gesture pattern are then applied to animate the speaker head model. Objective and subjective evaluations indicate that the proposed synthesis by analysis scheme provides natural looking head gestures for the speaker with any input test speech, as well as in “prosody transplant ” and “gesture transplant ” scenarios. Index Terms—Multimedia information systems, speech analysis, face and gesture recognition, pattern analysis and recognition, animation. Ç 1
Coordinateindependent sparse sufficient dimension reduction and variable selection
 The Annals of Statistics
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P.S.Krishnaprasad “Some Gradient Based Joint Diagonalization Methods for ICA
 in Proceedings of the 5th International Conference on Independent Component Analysis and Blind Source Separation
"... Abstract. We present a set of gradient based orthogonal and nonorthogonal matrix joint diagonalization algorithms. Our approach is to use the geometry of matrix Lie groups to develop continuoustime flows for joint diagonalization and derive their discretized versions. We employ the developed method ..."
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Cited by 11 (4 self)
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Abstract. We present a set of gradient based orthogonal and nonorthogonal matrix joint diagonalization algorithms. Our approach is to use the geometry of matrix Lie groups to develop continuoustime flows for joint diagonalization and derive their discretized versions. We employ the developed methods to construct a class of Independent Component Analysis (ICA) algorithms based on nonorthogonal joint diagonalization. These algorithms prewhiten or sphere the data but do not restrict the subsequent search for the (reduced) unmixing matrix to orthogonal matrices, hence they make effective use of both second and higher order statistics. 1