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134
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 192 (16 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 algebrogeometric 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 117 (29 self)
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This paper presents an algebrogeometric 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 lowdimensional data show that GPCA outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as Ksubspaces 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.
Ranksparsity incoherence for matrix decomposition
, 2009
"... Abstract. Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown lowrank matrix. Our goal is to decompose the given matrix into its sparse and lowrank components. Such a problem arises in a number of applications in model and system identification, and is int ..."
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Cited by 81 (10 self)
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Abstract. Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown lowrank matrix. Our goal is to decompose the given matrix into its sparse and lowrank components. Such a problem arises in a number of applications in model and system identification, and is intractable to solve in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components, by minimizing a linear combination of the ℓ1 norm and the nuclear norm of the components. We develop a notion of ranksparsity incoherence, expressed as an uncertainty principle between the sparsity pattern of a matrix and its row and column spaces, and use it to characterize both fundamental identifiability as well as (deterministic) sufficient conditions for exact recovery. Our analysis is geometric in nature with the tangent spaces to the algebraic varieties of sparse and lowrank matrices playing a prominent role. When the sparse and lowrank matrices are drawn from certain natural random ensembles, we show that the sufficient conditions for exact recovery are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems.
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 57 (5 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
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)inddimensional 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 dth Catalan n ..."
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Cited by 56 (17 self)
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Suppose that 2d − 2 tangent lines to the rational normal curve z ↦ → (1: z:...: z d)inddimensional 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 dth 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.
TwoView Multibody Structure from Motion
, 2006
"... We present an algebraic geometric approach to 3D motion estimation and segmentation of multiple rigidbody motions from noisefree point correspondences in two perspective views. Our approach exploits the algebraic and geometric properties of the socalled multibody epipolar constraint and its asso ..."
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Cited by 42 (15 self)
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We present an algebraic geometric approach to 3D motion estimation and segmentation of multiple rigidbody motions from noisefree point correspondences in two perspective views. Our approach exploits the algebraic and geometric properties of the socalled 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 firstorder 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 noisefree correspondences, we also test its performance on synthetic and real data with moderate levels of noise.
A Unified Algebraic Approach to 2D and 3D Motion Segmentation
 IN EUROPEAN CONFERENCE ON COMPUTER VISION
, 2004
"... We present an analytic solution to the problem of estimating multiple 2D and 3D motion models from twoview correspondences or optical flow. The key to our approach is to view the estimation of multiple motion models as the estimation of a single multibody motion model. This is possible thanks ..."
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Cited by 40 (15 self)
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We present an analytic solution to the problem of estimating multiple 2D and 3D motion models from twoview correspondences or optical flow. The key to our approach is to view the estimation of multiple motion models as the estimation of a single multibody motion model. This is possible thanks to two important algebraic facts. First, we show that all the image measurements, regardless of their associated motion model, can be fit with a real or complex polynomial. Second, we show
The Convex Geometry of Linear Inverse Problems
, 2010
"... In applications throughout science and engineering one is often faced with the challenge of solving an illposed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constr ..."
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Cited by 38 (10 self)
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In applications throughout science and engineering one is often faced with the challenge of solving an illposed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered are those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include wellstudied cases such as sparse vectors (e.g., signal processing, statistics) and lowrank matrices (e.g., control, statistics), as well as several others including sums of a few permutations matrices (e.g., ranked elections, multiobject tracking), lowrank tensors (e.g., computer vision, neuroscience), orthogonal matrices (e.g., machine learning), and atomic measures (e.g., system identification). The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial
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 23 (8 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...