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54
Learning over Sets using Kernel Principal Angles
 Journal of Machine Learning Research
, 2003
"... We consider the problem of learning with instances defined over a space of sets of vectors. We derive a new positive definite kernel f (A,B) defined over pairs of matrices A,B based on the concept of principal angles between two linear subspaces. We show that the principal angles can be recovered ..."
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Cited by 79 (2 self)
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We consider the problem of learning with instances defined over a space of sets of vectors. We derive a new positive definite kernel f (A,B) defined over pairs of matrices A,B based on the concept of principal angles between two linear subspaces. We show that the principal angles can be recovered using only innerproducts between pairs of column vectors of the input matrices thereby allowing the original column vectors of A,B to be mapped onto arbitrarily highdimensional feature spaces.
Multichannel Blind Deconvolution: Fir Matrix Algebra And Separation Of Multipath Mixtures
, 1996
"... A general tool for multichannel and multipath problems is given in FIR matrix algebra. With Finite Impulse Response (FIR) filters (or polynomials) assuming the role played by complex scalars in traditional matrix algebra, we adapt standard eigenvalue routines, factorizations, decompositions, and mat ..."
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Cited by 74 (0 self)
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A general tool for multichannel and multipath problems is given in FIR matrix algebra. With Finite Impulse Response (FIR) filters (or polynomials) assuming the role played by complex scalars in traditional matrix algebra, we adapt standard eigenvalue routines, factorizations, decompositions, and matrix algorithms for use in multichannel /multipath problems. Using abstract algebra/group theoretic concepts, information theoretic principles, and the Bussgang property, methods of single channel filtering and source separation of multipath mixtures are merged into a general FIR matrix framework. Techniques developed for equalization may be applied to source separation and vice versa. Potential applications of these results lie in neural networks with feedforward memory connections, wideband array processing, and in problems with a multiinput, multioutput network having channels between each source and sensor, such as source separation. Particular applications of FIR polynomial matrix alg...
Discriminative Learning and Recognition of Image Set Classes Using Canonical Correlations
 IEEE Trans. Pattern Analysis and Machine Intelligence
, 2007
"... Abstract—We address the problem of comparing sets of images for object recognition, where the sets may represent variations in an object’s appearance due to changing camera pose and lighting conditions. Canonical Correlations (also known as principal or canonical angles), which can be thought of as ..."
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Cited by 49 (10 self)
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Abstract—We address the problem of comparing sets of images for object recognition, where the sets may represent variations in an object’s appearance due to changing camera pose and lighting conditions. Canonical Correlations (also known as principal or canonical angles), which can be thought of as the angles between two ddimensional subspaces, have recently attracted attention for image set matching. Canonical correlations offer many benefits in accuracy, efficiency, and robustness compared to the two main classical methods: parametric distributionbased and nonparametric samplebased matching of sets. Here, this is first demonstrated experimentally for reasonably sized data sets using existing methods exploiting canonical correlations. Motivated by their proven effectiveness, a novel discriminative learning method over sets is proposed for set classification. Specifically, inspired by classical Linear Discriminant Analysis (LDA), we develop a linear discriminant function that maximizes the canonical correlations of withinclass sets and minimizes the canonical correlations of betweenclass sets. Image sets transformed by the discriminant function are then compared by the canonical correlations. Classical orthogonal subspace method (OSM) is also investigated for the similar purpose and compared with the proposed method. The proposed method is evaluated on various object recognition problems using face image sets with arbitrary motion captured under different illuminations and image sets of 500 general objects taken at different views. The method is also applied to object category recognition using ETH80 database. The proposed method is shown to outperform the stateoftheart methods in terms of accuracy and efficiency. Index Terms—Object recognition, face recognition, image sets, canonical correlation, principal angles, canonical correlation analysis, linear discriminant analysis, orthogonal subspace method. Ç 1
Kernel Principal Angles for Classification Machines with Applications to Image Sequence Interpretation
, 2002
"... We consider the problem of learning with instances defined over a space of sets of vectors. We derive a new positive definite kernel f(A# B) defined over pairs of matrices A# B based on the concept of principal angles between two linear subspaces. We show that the principal angles can be recovered ..."
Abstract

Cited by 37 (6 self)
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We consider the problem of learning with instances defined over a space of sets of vectors. We derive a new positive definite kernel f(A# B) defined over pairs of matrices A# B based on the concept of principal angles between two linear subspaces. We show that the principal angles can be recovered using only innerproducts between pairs of column vectors of the input matrices thereby allowing the original column vectors of A# B to be mapped onto arbitrarily highdimensional feature spaces.
OPUC on one foot
 Bull. Amer. Math. Soc
, 2005
"... Abstract. We present an expository introduction to orthogonal polynomials on the unit circle (OPUC). 1. ..."
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Cited by 32 (10 self)
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Abstract. We present an expository introduction to orthogonal polynomials on the unit circle (OPUC). 1.
Formally Biorthogonal Polynomials and a LookAhead Levinson Algorithm for General Toeplitz Systems
 Linear Algebra Appl
, 1993
"... Systems of linear equations with Toeplitz coefficient matrices arise in many important applications. The classical Levinson algorithm computes solutions of Toeplitz systems with only O(n 2 ) arithmetic operations, as compared to O(n 3 ) operations that are needed for solving general linear syst ..."
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Cited by 25 (2 self)
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Systems of linear equations with Toeplitz coefficient matrices arise in many important applications. The classical Levinson algorithm computes solutions of Toeplitz systems with only O(n 2 ) arithmetic operations, as compared to O(n 3 ) operations that are needed for solving general linear systems. However, the Levinson algorithm in its original form requires that all leading principal submatrices are nonsingular. In this paper, an extension of the Levinson algorithm to general Toeplitz systems is presented. The algorithm uses lookahead to skip over exactly singular, as well as illconditioned leading submatrices, and, at the same time, it still fully exploits the Toeplitz structure. In our derivation of this algorithm, we make use of the intimate connection of Toeplitz matrices with formally biorthogonal polynomials. In particular, the occurrence of singular or illconditioned submatrices corresponds to The research of this author was performed at the Research Institute for A...
Enhancing Analog Image Transmission Systems Using Digital Side Information: A New WaveletBased Image Coding Paradigm
, 2001
"... In this work we address digital transmission for enhancing, in a backward compatible way, the quality of analog image transmission systems. We propose a practical algorithm that treats the problem as one of wavelet image compression with side information (available in the form of a noisy analog v ..."
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Cited by 24 (2 self)
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In this work we address digital transmission for enhancing, in a backward compatible way, the quality of analog image transmission systems. We propose a practical algorithm that treats the problem as one of wavelet image compression with side information (available in the form of a noisy analog version of the image) present at the decoder. We propose a rate allocation technique to efficiently allocate the rate among the wavelet coefficients of the image. In typical instances of the problem, we get gains up to 2.5 dB over conventional methods that ignore the side information. Surprisingly, this is typically achieved by modifying a very small fraction of the wavelet coefficients (typically around 1020 %) of the conventional source coder. Extensions of our proposed image transmission framework to that of video transmission finds application in the upgrade of current analog television broadcast systems to digital TV. 1 Introduction Recent advances in communication and signal p...
The Generalized Schur Algorithm for the Superfast Solution of Toeplitz Systems
 in Rational Approximation and its Applications in Mathematics and Physics
, 1987
"... We review the connections between fast, O(n ), Toeplitz solvers and the classical theory of Szego polynomials and Schur's algorithm. ..."
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Cited by 19 (4 self)
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We review the connections between fast, O(n ), Toeplitz solvers and the classical theory of Szego polynomials and Schur's algorithm.
On and offline identification of linear state space models
 International Journal of Control
, 1989
"... A geometrically inspired matrix algorithm is derived for the identification of state space models for multivariable linear timeinvariant systems using (possibly noisy) inputoutput measurements only. As opposed to othermostly stochasticidentification schemes, no variancecovariance information ..."
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Cited by 17 (13 self)
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A geometrically inspired matrix algorithm is derived for the identification of state space models for multivariable linear timeinvariant systems using (possibly noisy) inputoutput measurements only. As opposed to othermostly stochasticidentification schemes, no variancecovariance information whatever is involved, and only a limited number of I/Odata are required for the determination of the system matrices. Hence, the algorithm can be best described and understood in the matrix formalism, and consists in the following two steps: First a state vector sequence is realized as the intersection of the row spaces of two block Hankel matrices, constructed with I/Odata. Then the system matrices are obtained at once from the least squares solution of a set of linear equations. When dealing with noisy data, this algorithm draws its excellent performance from repeated use of the numerically stable and accurate singular value decomposition Also, the algorithm is easily applied to slowly timevarying systems using windowing or exponential weighting. These results are illustrated by examples, including the identification of an industrial plant.
Boosted manifold principal angles for image setbased recognition, Pattern Recognition 40
, 2007
"... www.elsevier.com/locate/pr ..."