Results 1 - 10
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101
Algebraic Algorithms for Sampling from Conditional Distributions
- Annals of Statistics
, 1995
"... We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so a ..."
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Cited by 156 (12 self)
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We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so an excursion into computational algebraic geometry.
Generalized principal component analysis (GPCA)
- IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2003
"... This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a ..."
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Cited by 75 (27 self)
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This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point. When the number of subspaces is known, we show that these polynomials can be estimated linearly from data; hence, subspace segmentation is reduced to classifying one point per subspace. We select these points optimally from the data set by minimizing certain distance function, thus dealing automatically with moderate noise in the data. A basis for the complement of each subspace is then recovered by applying standard PCA to the collection of derivatives (normal vectors). Extensions of GPCA that deal with data in a highdimensional space and with an unknown number of subspaces are also presented. Our experiments on low-dimensional data show that GPCA outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as K-subspaces and Expectation Maximization. We also present applications of GPCA to computer vision problems such as face clustering, temporal video segmentation, and 3D motion segmentation from point correspondences in multiple affine views.
Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry
, 2001
"... Suppose that 2d − 2 tangent lines to the rational normal curve z ↦ → (1: z:...: z d)ind-dimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the d-th Catalan n ..."
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Cited by 49 (14 self)
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Suppose that 2d − 2 tangent lines to the rational normal curve z ↦ → (1: z:...: z d)ind-dimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the d-th Catalan number. We prove that for real tangent lines, all these codimension 2 subspaces are also real, thus confirming a special case of a general conjecture of B. and M. Shapiro. This is equivalent to the following result: If all critical points of a rational function lie on a circle in the Riemann sphere (for example, on the real line), then the function maps this circle into a circle.
Algebraic Geometry of Bayesian Networks
- Journal of Symbolic Computation
, 2005
"... We study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. 1 ..."
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Cited by 44 (4 self)
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We study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. 1
Two-View Multibody Structure from Motion
, 2006
"... We present an algebraic geometric approach to 3-D motion estimation and segmentation of multiple rigid-body motions from noise-free point correspondences in two perspective views. Our approach exploits the algebraic and geometric properties of the so-called multibody epipolar constraint and its asso ..."
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Cited by 35 (15 self)
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We present an algebraic geometric approach to 3-D motion estimation and segmentation of multiple rigid-body motions from noise-free point correspondences in two perspective views. Our approach exploits the algebraic and geometric properties of the so-called multibody epipolar constraint and its associated multibody fundamental matrix, which are natural generalizations of the epipolar constraint and of the fundamental matrix to multiple motions. We derive a rank constraint on a polynomial embedding of the correspondences, from which one can estimate the number of independent motions as well as linearly solve for the multibody fundamental matrix. We then show how to compute the epipolar lines from the first-order derivatives of the multibody epipolar constraint and the epipoles by solving a plane clustering problem using Generalized PCA (GPCA). Given the epipoles and epipolar lines, the estimation of individual fundamental matrices becomes a linear problem. The clustering of the feature points is then automatically obtained from either the epipoles and epipolar lines or from the individual fundamental matrices. Although our approach is mostly designed for noise-free correspondences, we also test its performance on synthetic and real data with moderate levels of noise.
On Geometric and Algebraic Aspects of 3D Affine and Projective Structures from Perspective 2D Views
- In Proceedings of the 2nd European Workshop on Invariants, Ponta Delagada, Azores
, 1993
"... Part I of this paper investigates the differences --- conceptually and algorithmically --- between affine and projective frameworks for the tasks of visual recognition and reconstruction from perspective views. It is shown that an affine invariant exists between any view and a fixed view chosen as a ..."
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Cited by 22 (7 self)
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Part I of this paper investigates the differences --- conceptually and algorithmically --- between affine and projective frameworks for the tasks of visual recognition and reconstruction from perspective views. It is shown that an affine invariant exists between any view and a fixed view chosen as a reference view. This implies that for tasks for which a reference view can be chosen, such as in alignment schemes for visual recognition, projective invariants are not really necessary. The projective extension is then derived, showing that it is necessary only for tasks for which a reference view is not available --- such as happens when updating scene structure from a moving stereo rig. The geometric difference between the two proposed invariants are that the affine invariant measures the relative deviation from a single reference plane, whereas the projective invariant measures the relative deviation from two reference planes. The affine invariant can be computed from three correspondin...
Derandomization in Computational Geometry
, 1996
"... We survey techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. 1 Randomized algorithms and derandomization A rapid growth of knowledge about randomized algorithms stimulates research in derandomization, that is, repla ..."
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Cited by 17 (1 self)
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We survey techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. 1 Randomized algorithms and derandomization A rapid growth of knowledge about randomized algorithms stimulates research in derandomization, that is, replacing randomized algorithms by deterministic ones with as small decrease of efficiency as possible. Related to the problem of derandomization is the question of reducing the amount of random bits needed by a randomized algorithm while retaining its efficiency; the derandomization can be viewed as an ultimate case. Randomized algorithms are also related to probabilistic proofs and constructions in combinatorics (which came first historically), whose development has similarly been accompanied by the effort to replace them by explicit, non-random constructions whenever possible. Derandomization of algorithms can be seen as a part of an effort to map the power of randomness and explain its role. ...
Quadratic reductions of quadrilateral lattices
- J. Geom. Phys
, 1999
"... It is shown that quadratic constraints are compatible with the geometric integrability scheme of the multidimensional quadrilateral lattice equation. The corresponding Ribaucour-type reduction of the fundamental transformation of quadrilateral lattices is found as well, and superposition of the Riba ..."
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Cited by 14 (10 self)
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It is shown that quadratic constraints are compatible with the geometric integrability scheme of the multidimensional quadrilateral lattice equation. The corresponding Ribaucour-type reduction of the fundamental transformation of quadrilateral lattices is found as well, and superposition of the Ribaucour transformations is presented in the vectorial framework. Finally, the quadratic reduction approach is illustrated on the example of multidimensional circular lattices.
Analytical blind channel identification
- IEEE Trans. Signal Processing
, 2002
"... Abstract—In this paper, a novel analytical blind single-input single-output (SISO) identification algorithm is presented, based on the noncircular second-order statistics of the output. It is shown that statistics of order higher than two are not mandatory to restore identifiability. Our approach is ..."
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Cited by 12 (6 self)
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Abstract—In this paper, a novel analytical blind single-input single-output (SISO) identification algorithm is presented, based on the noncircular second-order statistics of the output. It is shown that statistics of order higher than two are not mandatory to restore identifiability. Our approach is valid, for instance, when the channel is excited by phase shift keying (PSK) inputs. It is shown that the channel taps need to satisfy a polynomial system of degree 2 and that identification amounts to solving the system. We describe the algorithm that is able to solve this particular system entirely analytically, thus avoiding local minima. Computer results eventually show the robustness with respect to noise and to channel length overdetermination. Identifiability issues are also addressed. Index Terms—Blind channel estimation, minimum shift keying, multipath channels, noncircularity, second-order statistics, timevarying channels. I.

