Abstract. The Vapnik-Chervonenkis (V-C) dimension is an important combinatorial tool in the analysis of learning problems in the PAC framework. For polynomial learnability, we seek upper bounds on the V-C dimension that are polynomial in the syntactic complexity of concepts. Such upper bounds are automatic for discrete concept classes, but hitherto little has been known about what general conditions guarantee polynomial bounds on V-C dimension for classes in which concepts and examples are represented by tuples of real numbers. In this paper, we show that for two general kinds of concept class the V-C dimension is polynomially bounded in the number of real numbers used to define a problem instance. One is classes where the criterion for membership of an instance in a concept can be expressed as a formula (in the first-order theory of the reals) with fixed quantification depth and exponentially-bounded length, whose atomic predicates are polynomial inequalities of exponentially-bounded degree. The other is classes where containment of an instance in a concept is testable in polynomial time, assuming we may compute standard arithmetic operations on reals exactly in constant time. Our results show that in the continuous case, as in the discrete, the real barrier to efficient learning in the Occam sense is complexity-theoretic and not information-theoretic. We present examples to show how these results apply to concept classes defined by geometrical figures and neural nets, and derive polynomial bounds on the V-C dimension for these classes. Keywords: Concept learning, information theory, Vapnik-Chervonenkis dimension, Milnor’s theorem 1.

The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...

We consider a model of analog computation which can recognize various languages in real time. We encode an input word as a point in R d by composing iterated maps, and then apply inequalities to the resulting point to test for membership in the language. Each class of maps and inequalities, such as quadratic functions with rational coefficients, is capable of recognizing a particular class of languages; for instance, linear and quadratic maps can have both stack-like and queue-like memories. We use methods equivalent to the VapnikChervonenkis dimension to separate some of our classes from each other, e.g. linear maps are less powerful than quadratic or piecewise-linear ones, polynomials are less powerful than elementary (trigonometric and exponential) maps, and deterministic polynomials of each degree are less powerful than their non-deterministic counterparts. Comparing these dynamical classes with various discrete language classes helps illuminate how iterated maps can...

We introduce a new method for proving explicit upper bounds on the VC Dimension of general functional basis networks, and prove as an application, for the first time, that the VC Dimension of analog neural networks with the sigmoidal activation function oe(y) = 1=1+e \Gammay is bounded by a quadratic polynomial O((lm) 2 ) in both the number l of programmable parameters, and the number m of nodes. The proof method of this paper generalizes to much wider class of Pfaffian activation functions and formulas, and gives also for the first time polynomial bounds on their VC Dimension. We present also some other applications of our method.

We consider s polynomials P 1 ; : : : ; P s in k ! s variables with coefficients in an ordered domain A contained in a real closed field R, each of degree at most d. We present a new algorithm which computes a point in each connected component of each non-empty sign condition over P 1 ; : : : ; P s . The output is the set of points together with the sign condition at each point. The algorithm uses s(s=k) k d O(k) arithmetic operations in A. The algorithm is nearly optimal in the sense that the size of the output can be as large as s(O(sd=k)) k . Previous algorithms of Canny and Renegar used (sd) O(k) operations [5, 7, 8, 15]. We use either these algorithms in the case s = 1 as a subroutine in our algorithm. As a bonus, our algorithm yields an independent proof of the bound on the number of connected components in all non-empty sign conditions ([14]) and also yields an independent proof of a theorem of Warren 1 Courant Institute of Mathematical Sciences, New York University, N...

Leonard J. Schulman College of Computing Georgia Institute of Technology Atlanta GA 30332-0280 ABSTRACT We address the problem of partitioning a set of n points into clusters, so as to minimize the sum, over all intracluster pairs of points, of the cost associated with each pair. We obtain a randomized approximation algorithm for this problem, for the cost functions ` 2 2 ; `1 and `2 , as well as any cost function isometrically embeddable in ` 2 2 .

We introduce a new method for proving explicit upper bounds on the VC Dimension of general functional basis networks, and prove as an application, for the first time, the VC Dimension of analog neural networks with the sigmoid activation function oe(y) = 1=1 + e \Gammay to be bounded by a quadratic polynomial in the number of programmable parameters.

Several researchers, including Leonid Levin, Gerard 't Hooft, and Stephen Wolfram, have argued that quantum mechanics will break down before the factoring of large numbers becomes possible. If this is true, then there should be a natural "Sure/Shor separator"---that is, a set of quantum states that can account for all experiments performed to date, but not for Shor's factoring algorithm. We propose as a candidate the set of states expressible by a polynomial number of additions and tensor products. Using a recent lower bound on multilinear formula size due to Raz, we then show that states arising in quantum error-correction require n## additions and tensor products even to approximate, which incidentally yields the first superpolynomial gap between general and multilinear formula size of functions. More broadly, we introduce a complexity classification of pure quantum states, and prove many basic facts about this classification. Our goal is to refine vague ideas about a breakdown of quantum mechanics into specific hypotheses that might be experimentally testable in the near future.

We prove old and new results on the complexity of computing the dimension of algebraic varieties. In particular, we show that this problem is NP-complete in the Blum-Shub-Smale model of computation over C , that it admits a s O(1) D O(n) deterministic algorithm, and that for systems with integer coefficients it is in the Arthur-Merlin class under the Generalized Riemann Hypothesis. The first two results are based on a general derandomization argument. 1 Introduction We wish to compute the dimension of an algebraic variety V ` C n defined by a system of algebraic equations f 1 (x) = 0; : : : ; f s (x) = 0 (1) where f i 2 C [X 1 ; : : : ; Xn ]. This can be formalized as a decision problem DIMC . An instance of DIMC is a system of this form together with an integer d n. An instance is accepted if the variety defined by the system has dimension at least d. We also consider for each fixed value of d the restriction DIM d C of DIMC . For instance, DIM 0 C is the problem of dec...

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