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12
Learning with Matrix Factorization
, 2004
"... Matrices that can be factored into a product of two simpler matrices can serve as a useful and often natural model in the analysis of tabulated or highdimensional data. Models based on matrix factorization (Factor Analysis, PCA) have been extensively used in statistical analysis and machine learning ..."
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Cited by 39 (4 self)
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Matrices that can be factored into a product of two simpler matrices can serve as a useful and often natural model in the analysis of tabulated or highdimensional data. Models based on matrix factorization (Factor Analysis, PCA) have been extensively used in statistical analysis and machine learning for over a century, with many new formulations and models suggested in recent
Generalization Error Bounds for Collaborative Prediction with LowRank Matrices
 In Advances In Neural Information Processing Systems 17
, 2005
"... We prove generalization error bounds for predicting entries in a partially observed matrix by approximating the observed entries with a lowrank matrix. To do so, we bound the number of sign configurations of lowrank matrices using a result about realizable oriented matroids. ..."
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Cited by 22 (2 self)
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We prove generalization error bounds for predicting entries in a partially observed matrix by approximating the observed entries with a lowrank matrix. To do so, we bound the number of sign configurations of lowrank matrices using a result about realizable oriented matroids.
Bounding the Number of Geometric Permutations Induced by kTransversals
, 1994
"... We prove that a suitably separated family of n compact convex sets in R d can be met by kflat transversals in at most O(k) d 2 " 2 k+1 \Gamma 2 k '` n k + 1 "k(d\Gammak) or, for fixed k and d, O(n k(k+1)(d\Gammak) ) different order types. This is the first nontrivial upper bound for ..."
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Cited by 17 (5 self)
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We prove that a suitably separated family of n compact convex sets in R d can be met by kflat transversals in at most O(k) d 2 " 2 k+1 \Gamma 2 k '` n k + 1 "k(d\Gammak) or, for fixed k and d, O(n k(k+1)(d\Gammak) ) different order types. This is the first nontrivial upper bound for 1 ! k ! d \Gamma 1, and generalizes (asymptotically) the best upper bounds known for line transversals in R d , d ? 2. Introduction Let A be a family of n compact convex sets in R d . A line transversal of the family A is a line that intersects every member of A. If the sets in A are City College, City University of New York, New York, NY 10031, U.S.A. (jegcc@cunyvm.cuny.edu). Supported in part by NSF grant DMS9322475, NSA grant MDA90495H1012, and PSCCUNY grant 665343. y Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U.S.A. (pollack@geometry.nyu.edu). Supported in part by NSF grants DMS9400293, CCR9402640, and CCR9424398. z Ohio Stat...
Realization Spaces of 4Polytopes are Universal
 BULL. AMER. MATH. SOC
, 1995
"... Let P be a ddimensional polytope. The realization space of P is the space of all polytopes P # that are combinatorially equivalent to P , modulo affine transformations. We report ..."
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Cited by 13 (4 self)
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Let P be a ddimensional polytope. The realization space of P is the space of all polytopes P # that are combinatorially equivalent to P , modulo affine transformations. We report
There are asymptotically far fewer polytopes than we thought
 Bull. Amer. Math. Soc
, 1986
"... The problem of enumerating convex polytopes with n vertices in R d has been the object of considerable study going back to ancient times (see [4, §13.6] for some remarks about the history of this problem since the nineteenth century). While significant progress has been made when the number of verti ..."
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Cited by 11 (3 self)
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The problem of enumerating convex polytopes with n vertices in R d has been the object of considerable study going back to ancient times (see [4, §13.6] for some remarks about the history of this problem since the nineteenth century). While significant progress has been made when the number of vertices was not too much larger than the dimension [4], little had been known above dimension 3 in the general case despite considerable efforts: if ƒ (n, d) is the logarithm of the number of combinatorially distinct simplicial polytopes with n vertices in R d, the sharpest asymptotic bounds previously known for /(n,d) were cinlogn < /(n, d) < C2n d ^ 2 logn, with the lower bound due to Shemer [7], and the upper bound following from the (asymptotic) Upper Bound Theorem of Klee [5], leaving a wide gap between the two bounds. (Here, and in the sequel, we take the view that d is fixed and n — • oo; thus all constants depend on d.) The purpose of the
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 11 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Computational Geometry
 in Directions in Computational Geometry
, 1994
"... Computational geometry, the study of algorithms involving relatively simple geometric objects, is an active, exciting field. This chapter samples current research in computational geometry. Three topics are discussed at some length: the theory of arrangements, a randomincremental convex hull alg ..."
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Cited by 10 (0 self)
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Computational geometry, the study of algorithms involving relatively simple geometric objects, is an active, exciting field. This chapter samples current research in computational geometry. Three topics are discussed at some length: the theory of arrangements, a randomincremental convex hull algorithm, and robustness of geometric algorithms. 1 Introduction As pointed out by Robin Forrest and others[39, 79], the term "computational geometry" could be used for a variety of fields, including geometric modeling using curves and surfaces, computer proofs of geometric theorems, geometric design software, and the theory of perceptrons[71]. The purpose of this chapter is to give an impressionistic view of recent research in computational geometry, defined as the study of algorithms involving relatively simple geometric objects such as points and lines. A typical topic in this field is the analysis of an algorithm involving n objects in d dimensions; examples are the computation of the c...
Algorithms in Semialgebraic Geometry
, 1996
"... In this thesis we present new algorithms to solve several very general problems of semialgebraic geometry. These are currently the best algorithms for solving these problems. In addition, we have proved new bounds on the topological complexity of real semialgebraic sets, in terms of the paramete ..."
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Cited by 9 (0 self)
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In this thesis we present new algorithms to solve several very general problems of semialgebraic geometry. These are currently the best algorithms for solving these problems. In addition, we have proved new bounds on the topological complexity of real semialgebraic sets, in terms of the parameters of the polynomial system defining them, which improve some old and widely used results in this field. In the first part of the thesis we describe new algorithms for solving the decision problem for the first order theory of real closed fields and the more general problem of quantifier elimination. Moreover, we prove some purely mathematical theorems on the number of connected components and on the existence of small rational points in a given semialgebraic set. The second part of this thesis deals with connectivity questions of semialgebraic sets. We develop new techniques in order to give a...
Betti Number Bounds, Applications and Algorithms
 Current Trends in Combinatorial and Computational Geometry: Papers from the Special Program at MSRI, MSRI Publications Volume 52
, 2005
"... Abstract. Topological complexity of semialgebraic sets in R k has been studied by many researchers over the past fifty years. An important measure of the topological complexity are the Betti numbers. Quantitative bounds on the Betti numbers of a semialgebraic set in terms of various parameters (such ..."
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Cited by 9 (6 self)
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Abstract. Topological complexity of semialgebraic sets in R k has been studied by many researchers over the past fifty years. An important measure of the topological complexity are the Betti numbers. Quantitative bounds on the Betti numbers of a semialgebraic set in terms of various parameters (such as the number and the degrees of the polynomials defining it, the dimension of the set etc.) have proved useful in several applications in theoretical computer science and discrete geometry. The main goal of this survey paper is to provide an up to date account of the known bounds on the Betti numbers of semialgebraic sets in terms of various parameters, sketch briefly some of the applications, and also survey what is known about the complexity of algorithms for computing them. 1.