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

We prove that a suitably separated family of n compact convex sets in R d can be met by k-flat 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 non-trivial 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 DMS93-22475, NSA grant MDA904-95-H-1012, and PSC-CUNY 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, CCR94-02640, and CCR94-24398. z Ohio Stat...

Let P be a d-dimensional polytope. The realization space of P is the space of all polytopes P # that are combinatorially equivalent to P , modulo affine transformations. We report

We prove generalization error bounds for predicting entries in a partially observed matrix by approximating the observed entries with a low-rank matrix. To do so, we bound the number of sign configurations of lowrank matrices using a result about realizable oriented matroids.

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 random-incremental 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...

In this thesis we present new algorithms to solve several very general problems of semi-algebraic geometry. These are currently the best algorithms for solving these problems. In addition, we have proved new bounds on the topological complexity of real semi-algebraic 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 semi-algebraic set. The second part of this thesis deals with connectivity questions of semialgebraic sets. We develop new techniques in order to give a...

Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension 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.

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.

In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n − 2)-spheres on 2n vertices as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that “cut across an ideal. ” Thus we arrive at a substantial generalization of Bier’s construction: the Bier posets Bier(P, I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular “generalized Bier spheres. ” In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P, I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields “many shellable spheres,” most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres.

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