Sign rank versus VC dimension

This work studies the maximum possible sign rank of $N \times N$ sign matrices with a given VC dimension $d$. For $d=1$, this maximum is {three}. For $d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. For $d >2$, similar but slightly less accurate statements hold. {The lower bounds improve over previous ones by Ben-David et al., and the upper bounds are novel.} The lower bounds are obtained by probabilistic constructions, using a theorem of Warren in real algebraic topology. The upper bounds are obtained using a result of Welzl about spanning trees with low stabbing number, and using the moment curve. The upper bound technique is also used to: (i) provide estimates on the number of classes of a given VC dimension, and the number of maximum classes of a given VC dimension -- answering a question of Frankl from '89, and (ii) design an efficient algorithm that provides an $O(N/\log(N))$ multiplicative approximation for the sign rank. We also observe a general connection between sign rank and spectral gaps which is based on Forster's argument. Consider the $N \times N$ adjacency matrix of a $\Delta$ regular graph with a second eigenvalue of absolute value $\lambda$ and $\Delta \leq N/2$. We show that the sign rank of the signed version of this matrix is at least $\Delta/\lambda$. We use this connection to prove the existence of a maximum class $C\subseteq\{\pm 1\}^N$ with VC dimension $2$ and sign rank $\tilde{\Theta}(N^{1/2})$. This answers a question of Ben-David et al.~regarding the sign rank of large VC classes. We also describe limitations of this approach, in the spirit of the Alon-Boppana theorem. We further describe connections to communication complexity, geometry, learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC dimension". Additional results in this version: (i) Estimates on the number of maximum VC classes (answering a question of Frankl from '89). (ii) Estimates on the sign rank of large VC classes (answering a question of Ben-David et al. from '03). (iii) A discussion on the computational complexity of computing the sign-ran

Reaalisten varistojen vakausindeksi: Bröckerin ja Scheidererin teoria

Tässä työssä todistan Bröckerin ja Scheidererin teorian avoimille semi-algebrallisille perusjoukoille. Teoria osoittaa, että jokaiselle reaaliselle algebralliselle varistolle on olemassa yläraja niiden polynomien lukumäärässä, joiden avulla variston osajoukkona olevia avoimia semi-algebrallisia perusjoukkoja määritellään. Tätä lukuarvoa kutsutaan reaalisen variston vakausindeksiksi. Teoria pohjaa suljetuille reaalisille kunnille, jotka yleistävät reaalilukujen kuntaa. Reaalinen algebrallinen varisto on suljetun reaalisen kunnan osajoukko, joka on määritelty polynomiyhtälöiden ratkaisujoukkona. Jokainen semi-algebrallinen joukko on määritelty Boolen yhdistelmänä äärellisestä määrästä polynomien merkkiehtoja, jotka toteuttavat tietyt yhtäsuuruudet ja epäyhtälöt. Semi-algebralliset perusjoukot ovat ne semi-algebralliset joukot, jotka toteuttavat ainoastaan annetut yhtäsuuruudet ja epäyhtälöt polynomien merkkiehdoissa. Semi-algebrallisista perusjoukoista voidaan siis rakentaa kaikki semi-algebralliset joukot ottamalla perusjoukkojen äärelliset yhdisteet, leikkaukset ja joukkoerotukset. Tämä työ pyrkii esittämään riittävät esitiedot päätuloksen todistuksen syvällistä ja yksityiskohtaista ymmärtämistä varten. Ensimmäinen luku esittelee ja motivoi tulosta yleisellä tasolla. Toinen luku käsittelee tiettyjä edistyneitä algebrallisia rakenteita, joita vaaditaan päätuloksen todistuksessa. Näitä ovat muun muassa radikaalit, alkuideaalit, assosiatiiviset algebrat, renkaan ulottuvuuden käsite sekä tietyt tekijärakenteet. Kolmas luku määrittelee suljetut reaaliset kunnat ja semi-algebralliset joukot, jotka ovat tämän työn kulmakiviä. Kolmannessa luvussa myös kehitetään neliömuotojen teoriaa. Kolmannen luvun päätulos on Wittin teoria. Neljäs luku käsittelee Pfisterin muotoja, jotka ovat tietynlaisia neliömuotoja. Työssä määritellään yleiset Pfisterin muodot kuntien yli. Tämän jälkeen kehitään niiden teoriaa rationaalifunktioiden kunnan yli. Viidennessä luvussa esitetään kaksi konstruktiivista esimerkkiä Bröckerin ja Scheidererin teorian käytöstä yhdessä ja kahdessa ulottuvuudessa. Nämä esimerkit edelleen motivoivat tulosta ja sen mahdollisia algoritmisia ominaisuuksia. Työn päätteeksi todistetaan Bröckerin ja Scheidererin teoria, joka osoittaa, että reaalisen variston vakausindeksi on olemassa ja se on äärellinen kaikille reaalisille varistoille.In this work, I prove the theorem of Bröcker and Scheiderer for basic open semi-algebraic sets. The theorem provides an upper bound for a stability index of a real variety. The theory is based on real closed fields which generalize real numbers. A real variety is a subset of a real closed field that is defined by polynomial equalities. Every semi-algebraic set is defined by a boolean combination of polynomial equations and inequalities of the sign conditions involving a finite number of polynomials. The basic semi-algebraic sets are those semi-algebraic sets that are defined solely by the sign conditions. In other words, we can construct semi-algebraic sets from the basic semi-algebraic sets by taking the finite unions, intersections, and complements of the basic semi-algebraic sets. Then the stability index of a real variety indicates the upper bound of numbers of polynomials that are required to express an arbitrary semi-algebraic subset of the variety. The theorem of Bröcker and Scheiderer shows that such upper bound exists and is finite for basic open semi-algebraic subsets of a real variety. This work aims to be detailed in the proofs and represent sufficient prerequisites and references. The first chapter introduces the topic generally and motivates to study the theorem. The second chapter provides advanced prerequisites in algebra. One of such results is the factorial theorem of a total ring of fractions. Other advanced topics include radicals, prime ideals, associative algebras, a dimension of a ring, and various quotient structures. The third chapter defines real closed fields and semi-algebraic sets that are the fundamental building blocks of the theory. The third chapter also develops the theory of quadratic forms. The main result of this chapter is Witt’s cancellation theorem. We also shortly describe the Tsen-Lang theorem. The fourth chapter is about Pfister forms. Pfister forms are special kinds of quadratic forms that we extensively use in the proof of the main theorem. First, we define general Pfister forms over fields. Then we develop their theory over the fields of rational functions. Generally, Pfister forms share multiple similar properties as quadratic forms. The fifth chapter represents one- and two-dimensional examples of the main theorem. These examples are based on research that is done on constructive approaches to the theorem of Bröcker and Scheiderer. The examples clarify and motivate the result from an algorithmic perspective. Finally, we prove the main theorem of the work. The proof is heavily based on Pfister forms

A Probabilistic Analysis of the Power of Arithmetic Filters

The assumption of real-number arithmetic, which is at the basis of conventional geometric algorithms, has been seriously challenged in recent years, since digital computers do not exhibit such capability. A geometric predicate usually consists of evaluating the sign of some algebraic expression. In most cases, rounded computations yield a reliable result, but sometimes rounded arithmetic introduces errors which may invalidate the algorithms. The rounded arithmetic may produce an incorrect result only if the exact absolute value of the algebraic expression is smaller than some (small) varepsilon, which represents the largest error that may arise in the evaluation of the expression. The threshold varepsilon depends on the structure of the expression and on the adopted computer arithmetic, assuming that the input operands are error-free. A pair (arithmetic engine,threshold) is an "arithmetic filter". In this paper we develop a general technique for assessing the efficacy of an arithmetic filter. The analysis consists of evaluating both the threshold and the probability of failure of the filter. To exemplify the approach, under the assumption that the input points be chosen randomly in a unit ball or unit cube with uniform density, we analyze the two important predicates "which-side" and "insphere". We show that the probability that the absolute values of the corresponding determinants be no larger than some positive value V, with emphasis on small V, is Theta(V) for the which-side predicate, while for the insphere predicate it is Theta(V^(2/3)) in dimension 1, O(sqrt(V)) in dimension 2, and O(sqrt(V) ln(1/V)) in higher dimensions. Constants are small, and are given in the paper.Comment: 22 pages 7 figures Results for in sphere test inproved in cs.CG/990702

Limit Cycles for two families of cubic systems

In this paper we study the number of limit cycles of two families of cubic systems introduced in previous papers to model real phenomena. The first one is motivated by a model of star formation histories in giant spiral galaxies and the second one comes from a model of Volterra type. To prove our results we develop a new criterion on non-existence of periodic orbits and we extend a well-known criterion on uniqueness of limit cycles due to Kuang and Freedman. Both results allow to reduce the problem to the control of the sign of certain functions that are treated by algebraic tools. Moreover, in both cases, we prove that when the limit cycles exist they are non-algebraic

Efficient algorithms for computing the Euler-Poincar\'e characteristic of symmetric semi-algebraic sets

Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincar\'e characteristic of real algebraic as well as semi-algebraic subsets of $\mathrm{R}^k$, which are defined by symmetric polynomials with coefficients in $\mathrm{D}$. We give algorithms for computing the generalized Euler-Poincar\'e characteristic of such sets, whose complexities measured by the number the number of arithmetic operations in $\mathrm{D}$, are polynomially bounded in terms of $k$ and the number of polynomials in the input, assuming that the degrees of the input polynomials are bounded by a constant. This is in contrast to the best complexity of the known algorithms for the same problems in the non-symmetric situation, which are singly exponential. This singly exponential complexity for the latter problem is unlikely to be improved because of hardness result ($\#\mathbf{P}$-hardness) coming from discrete complexity theory.Comment: 29 pages, 1 Figure. arXiv admin note: substantial text overlap with arXiv:1312.658