Publications (11)3.97 Total impact

Article: On Lower Bounds for ReadkTimes
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ABSTRACT: A syntactic readktimes branching program has the restriction that no variable occurs more than k times on any path (whether or not consistent) of the branching program. We rst extend the result in [30], to show that the =2 clique only function", which is easily seen to be computable by deterministic polynomial size readtwice programs, cannot be computed by nondeterministic polynomial size readonce programs, although its complement can be so computed. We then exhibit an explicit Boolean function f such that every nondeterministic syntactic readktimes branching program for computing f has size exp .  [Show abstract] [Hide abstract]
ABSTRACT: We show that there is a set of pointsp 1,p 2,...,p n such that any arithmetic circuit of depthd for polynomial evaluation (or interpolation) at these points has size$$\Omega \left( {\frac{{n\log n}}{{\log (2 + d/\log n}}} \right).$$ Moreover, for circuits of sublogarithmic depthd, we obtain a lower bound of Ω(dn 1+1/d ) on its size. 
Conference Paper: On representations by lowdegree polynomials
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ABSTRACT: In the first part of the paper we show that a subset S of a boolean cube B<sub>n</sub> embedded in the projective space P<sup>n</sup> can be approximated by a subset of B<sub>n</sub> defined by nonzeroes of a lowdegree polynomial only if the values of the Hilbert function of S are sufficiently small relative to the size of S. The use of this property provides a simple and direct technique for proving lower bounds on the size of ACC[p<sup>r</sup>] circuits. In the second part we look at the problem of computing manyoutput function by ACC[p<sup>r</sup>] circuit and give an example when such a circuit can be correct only at exponentially small fraction of assignments  [Show abstract] [Hide abstract]
ABSTRACT: A syntactic readktimes branching program has the restriction that no variable occurs more than k times on any path (whether or not consistent) of the branching program. We first extend the result in [31], to show that the "n/2 clique only function", which is easily seen to be computable by deterministic polynomial size readtwice programs, cannot be computed by nondeterministic polynomial size readonce programs, although its complement can be so computed. We then exhibit an explicit Boolean function f such that every nondeterministic syntactic readktimes branching program for computing f has size exp {Mathematical expression}  [Show abstract] [Hide abstract]
ABSTRACT: We examine the class of polynomial threshold functions using harmonic analysis and applies the results to derive lower bounds related to AC 0 functions. A Boolean function is polynomial threshold if it can be represented as the sign of a sparse polynomial (one that consists of a polynomial number of terms). The main result is that the class of polynomial threshold functions can be characterized using their spectral representation. It is proved that an nvariable Boolean function whose L 1 spectral norm is bounded by a polynomial in n is a polynomial threshold function, while a Boolean function whose L ∞ 1 spectral norm is not bounded by a polynomial in n is not a polynomial threshold function J. Bruck [SIAM J. Discrete Math. 3, 168177 (1990; Zbl 0695.94022)]. The motivation is that the characterization of polynomial threshold functions can be applied to obtain upper and lower bounds on the complexity of computing with networks of linear threshold elements. Results related to the complexity of computing AC 0 functions are presented. More applications of the characterization theorem are presented in J. Bruck [loc. cit.] and K. Y. Siu and J. Bruck [SIAM J. Discrete Math. 4, 423435 (1991; Zbl 0737.68042)]. 
Conference Paper: Lower bounds for polynomial evaluation and interpolation problems
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ABSTRACT: It is shown that there is a set of points p <sub>1</sub>, p <sub>2</sub>,. . ., p <sub>n</sub> such that any algebraic program of depth d for polynomial evaluation (or interpolation) at these points has size Ω( n log n /log d ). Moreover, if d is a constant, then a lower bound of Ω( n <sup>1+1/d</sup>) is obtained 
Conference Paper: Harmonic Analysis And The Complexity Of Computing With Threshold (Neural) Elements
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ABSTRACT: The main purpose of this talk is to introduce a useful tool for the analysis of discrete neural networks in which every node is a Boolean threshold gate. The difficulty in the analysis of neural networks arises from the fact that the basic processing elements (linear threshold gates) are nonlinear. The key idea in harmonic analysis of threshold functions is to represent the functions as polynomials over the field of real numbers. Answering different questions regarding neural networks becomes equivalent to answering questions related to the coefficients of these polynomials. We have applied these techniques and obtained many interesting and surprising results [1, 2, 3, 4]. The focus of this talk will be on presenting a theorem that characterizesusing spetral normsthe complexity of computing a Boolean function with threshold circuits [2, 3]. This result establishes the first known link between harmonic analysis and the complexity of computing with neural networks. 
Conference Paper: Polynomial threshold functions, AC functions and spectrum norms
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ABSTRACT: The class of polynomialthreshold functions is studied using harmonic analysis, and the results are used to derive lower bounds related to AC<sup>0</sup> functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). The main result is that polynomialthreshold functions can be characterized by means of their spectral representation. In particular, it is proved that a Boolean function whose L <sub>1</sub> spectral norm is bounded by a polynomial in n is a polynomialthreshold function, and that a Boolean function whose L <sub>∞</sub><sup>1</sup> spectral norm is not bounded by a polynomial in n is not a polynomialthreshold function. Some results for AC<sup>0</sup> functions are derived 
Conference Paper: On interpolation by analytic functions with special properties andsome weak lower bounds on the size of circuits with symmetric gates
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ABSTRACT: The author investigates the question of whether or not a specific Boolean function in n variables can be interpolated by an analytic function in the same variables whose partial derivatives of all orders span a subspace of low dimension in the space of analytic functions. The upper and lower bounds for this dimension yield some weak circuit lower bounds. For a particular function, an Ω( n /log n )size lower bound is obtained for its computation by a circuit whose gates are symmetric. For the same function an Ω( n ) lower bound is obtained for the circuit with mod<sub>k</sub> gates 
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ABSTRACT: We use algebraic methods to get lower bounds for complexity of different functions based on constant depth unbounded fanin circuits with the given set of basic operations. In particular, we prove that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(&Ogr;(n1/2k)) gates to calculate MODr functions for any r ≠ pm. This statement contains as special cases Yao's PARITY result [ Ya 85 ] and Razborov's new MAJORITY result [Ra 86] (MODm gate is an oracle which outputs zero, if the number of ones is divisible by m).
Publication Stats
808  Citations  
3.97  Total Impact Points  
Top Journals
Institutions

1993

Hebrew University of Jerusalem
 Rachel and Selim Benin School of Computer Science and Engineering
Yerushalayim, Jerusalem District, Israel


19901993

University of Toronto
 Department of Computer Science
Toronto, Ontario, Canada 
San Jose State University
San José, California, United States


1987

University of California, Berkeley
 Department of Mathematics
Berkeley, CA, United States
