Results 1  10
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19
The Influence of Variables on Boolean Functions (Extended Abstract)
, 1988
"... Introduction This paper applies methods from harmonic analysis to prove some general theorems on boolean functions. The result that is easiest to describe says that "Boolean functions always have small dominant sets of variables." The exact definitions will be given shortly, but let us be more spec ..."
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Cited by 226 (20 self)
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Introduction This paper applies methods from harmonic analysis to prove some general theorems on boolean functions. The result that is easiest to describe says that "Boolean functions always have small dominant sets of variables." The exact definitions will be given shortly, but let us be more specific: Let f be an n\Gammavariable boolean function taking the value zero for half of the 2 n variable assignments. Then there is a set of o(n) variables such that almost surely the value of f is undetermined as long as these variables are not assigned values. This proves some of the conjectures made in [BL]. These new connections with harmonic analysis are very promising. Besides the results on boolean functions they enable us to prove new theorems on the rapid mixing of the random walk on the cube, as well as new theorems in the extremal theory of finite sets. We begin by reviewing some definitions from [BL].
Sphere Packing Numbers for Subsets of the Boolean nCube with Bounded VapnikChervonenkis Dimension
, 1992
"... : Let V ` f0; 1g n have VapnikChervonenkis dimension d. Let M(k=n;V ) denote the cardinality of the largest W ` V such that any two distinct vectors in W differ on at least k indices. We show that M(k=n;V ) (cn=(k + d)) d for some constant c. This improves on the previous best result of ((cn ..."
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Cited by 94 (4 self)
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: Let V ` f0; 1g n have VapnikChervonenkis dimension d. Let M(k=n;V ) denote the cardinality of the largest W ` V such that any two distinct vectors in W differ on at least k indices. We show that M(k=n;V ) (cn=(k + d)) d for some constant c. This improves on the previous best result of ((cn=k) log(n=k)) d . This new result has applications in the theory of empirical processes. 1 The author gratefully acknowledges the support of the Mathematical Sciences Research Institute at UC Berkeley and ONR grant N0001491J1162. 1 1 Statement of Results Let n be natural number greater than zero. Let V ` f0; 1g n . For a sequence of indices I = (i 1 ; . . . ; i k ), with 1 i j n, let V j I denote the projection of V onto I, i.e. V j I = f(v i 1 ; . . . ; v i k ) : (v 1 ; . . . ; v n ) 2 V g: If V j I = f0; 1g k then we say that V shatters the index sequence I. The VapnikChervonenkis dimension of V is the size of the longest index sequence I that is shattered by V [VC71] (t...
Introduction to Statistical Learning Theory
 In , O. Bousquet, U.v. Luxburg, and G. Rsch (Editors
, 2004
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Are there Hard Examples for Frege Systems?
"... It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. S ..."
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Cited by 20 (2 self)
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It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surprisingly, we conclude that no particularly good or convincing examples are known. The examples of combinatorial tautologies that we consider seem to give at most a quasipolynomial speedup of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl. It is
Pattern classification and learning theory
"... 1.1 A binary classification problem Pattern recognition (or classification or discrimination) is about guessing or predicting the unknown class of an observation. An observation is a collection of numerical measurements, represented by a ddimensional vector x. The unknown nature of the observation ..."
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Cited by 17 (7 self)
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1.1 A binary classification problem Pattern recognition (or classification or discrimination) is about guessing or predicting the unknown class of an observation. An observation is a collection of numerical measurements, represented by a ddimensional vector x. The unknown nature of the observation is called a class. It is denoted by y and takes values in the set f0; 1g. (For simplicity, we restrict our attention to binary classification.) In pattern recognition, one creates a function g(x) : R d! f0; 1g which represents one's guess of y given x. The mapping g is called a classifier. A classifier errs on x if g(x) 6 = y. To model the learning problem, we introduce a probabilistic setting, and let (X; Y) be an R d \Theta f0; 1gvalued random pair. The random pair (X; Y) may be described in a variety of ways: for example, it is defined by the pair (_; j), where _ is the probability measure for X and j is the regression of Y on X. More precisely, for a Borelmeasurable set A ` R d
Quantifying the Amount of Verboseness
, 1995
"... We study the fine structure of the classification of sets of natural numbers A according to the number of queries which are needed to compute the nfold characteristic function of A. A complete characterization is obtained, relating the question to finite combinatorics. In order to obtain an explic ..."
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Cited by 16 (6 self)
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We study the fine structure of the classification of sets of natural numbers A according to the number of queries which are needed to compute the nfold characteristic function of A. A complete characterization is obtained, relating the question to finite combinatorics. In order to obtain an explicit description we consider several interesting combinatorial problems. 1 Introduction In the theory of bounded queries, we measure the complexity of a function by the number of queries to an oracle which are needed to compute it. The field has developed in various directions, both in complexity theory and in recursion theory; see Gasarch [21] for a recent survey. One of the original concerns is the classification of sets A of natural numbers by their "query complexity," i.e., according to the number of oracle queries that are needed to compute the nfold characteristic function F A n = x 1 ; : : : ; x n : (ØA (x 1 ); : : : ; ØA (x n )). In [3, 8] a set A is called verbose iff F A n is com...
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.
Small Forbidden Configurations
 II, Electr. J. Comb
, 1993
"... In the present paper we continue the work begun by Sauer, Perles, Shelah and Anstee on forbidden configurations of 01 matrices. We give asymptotically exact bounds for all possible 2 × l forbidden submatrices and almost all 3 × l ones. These bounds are improvements of the general bounds ..."
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Cited by 4 (2 self)
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In the present paper we continue the work begun by Sauer, Perles, Shelah and Anstee on forbidden configurations of 01 matrices. We give asymptotically exact bounds for all possible 2 × l forbidden submatrices and almost all 3 × l ones. These bounds are improvements of the general bounds, or else new constructions show that the general bound is best possible. It is interesting to note that up to the present state of our knowledge every forbidden configuration results in polynomial asymptotic.
A constrained version of Sauer's Lemma
, 2004
"... We generalize Sauer's Lemma to finite V Cdimension classes functions on [n] = . . . , n} which have a margin of at least N on every element in a sample S [n] of cardinality l, where the margin h (x) of h on a point [n] is defined as the largest nonnegative integer a such that h is c ..."
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Cited by 3 (3 self)
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We generalize Sauer's Lemma to finite V Cdimension classes functions on [n] = . . . , n} which have a margin of at least N on every element in a sample S [n] of cardinality l, where the margin h (x) of h on a point [n] is defined as the largest nonnegative integer a such that h is constant on the interval I a (x) = [x a, x + a].
Defect Sauer Results
"... : In this paper we present a unified account of various results concerning traces of set systems, including the original lemma proved independently by Sauer [14], Shelah [15], and Vapnik and Chervonenkis [16], and extend these results in various directions. Included are a new criterion for a set sys ..."
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Cited by 3 (0 self)
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: In this paper we present a unified account of various results concerning traces of set systems, including the original lemma proved independently by Sauer [14], Shelah [15], and Vapnik and Chervonenkis [16], and extend these results in various directions. Included are a new criterion for a set system to be extremal for the Sauer inequality and upper and lower bounds, obtained by random methods, for the trace of a set system of size n r guaranteed on some ffn sized subset of f1; 2; : : : ; ng. 1 Notation A set system is a subset of P(n), the power set of [n] = f1; 2; : : : ; ng. The complement of a set I ae [n] is written I c , all other set differences are written out explicitly, e.g., P(n) n A. The system consisting of all sets of size k is written [n] (k) , while A (k) = A " [n] (k) , for any set system A. We define [n] (k) , A (k) , etc. similarly. Given A ae P(n) and I ae [n] we write I +A for fA [ I : A 2 Ag. Introduction A striking extremal result concerning ...