Interpolation of Sparse Multivariate Polynomials over Large Finite Fields with Applications

Huang, Ming-Deh A.; Rao, Ashwin J.

doi:10.1006/jagm.1999.1045

Cited by 21 publications

(20 citation statements)

References 17 publications

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“…14 In Ben-Or and Tiwari [1988], it is shown how to interpolate polynomials over infinite fields using only 2t membership queries. Algorithms for interpolating polynomials over finite fields are given in Bshouty and Mansour [1995] and Huang and Rao [1996] provided that the fields are "big" enough (in Huang and Rao [1996], the field must contain ⍀(t 2 k ϩ tkn 2 ) elements and, in Bshouty and Mansour [1995], the field must contain ⍀(kn/log kn) elements). We require that the number of elements in the field is at least kn ϩ 1.…”

Section: Positive Resultsmentioning

confidence: 99%

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Learning functions represented as multiplicity automata

Beimel

Bergadano

Bshouty

et al. 2000

J. ACM

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We study the learnability of multiplicity automata in Angluin's exact learning model, and we investigate its applications. Our starting point is a known theorem from automata theory relating the This paper contains results from three conference papers. It is mostly based on the FOCS '96 paper by the authors. The rest of the results are from the STOC '96 paper of Bergadano, Catalano, and Varricchio and the EuroCOLT '97 paper of Beimel and Kushilevitz.Part of this research was done while A. Beimel was a Ph.D student at the Technion.number of states in a minimal multiplicity automaton for a function to the rank of its Hankel matrix. With this theorem in hand, we present a new simple algorithm for learning multiplicity automata with improved time and query complexity, and we prove the learnability of various concept classes. These include (among others):-The class of disjoint DNF, and more generally satisfy-O(1) DNF.-The class of polynomials over finite fields.-The class of bounded-degree polynomials over infinite fields.-The class of XOR of terms.-Certain classes of boxes in high dimensions.In addition, we obtain the best query complexity for several classes known to be learnable by other methods such as decision trees and polynomials over GF(2).While multiplicity automata are shown to be useful to prove the learnability of some subclasses of DNF formulae and various other classes, we study the limitations of this method. We prove that this method cannot be used to resolve the learnability of some other open problems such as the learnability of general DNF formulas or even k-term DNF for k ϭ (log n) or satisfy-s DNF formulas for s ϭ (1). These results are proven by exhibiting functions in the above classes that require multiplicity automata with super-polynomial number of states.

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Section: Positive Resultsmentioning

confidence: 99%

“…See, for example,Ben-Or and Tiwari [1988],Grigoriev et al [1990],Zippel [1990],Clausen et al [1991],Roth and Benedek [1991],Bshouty and Mansour [1995], andHuang and Rao [1996]. For more background and references, seeZippel [1993].15 For example, the polynomial ( x 1 ϩ 1) ( x 2 ϩ 1) .…”

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confidence: 98%

Learning functions represented as multiplicity automata

Beimel

Bergadano

Bshouty

et al. 2000

J. ACM

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“…In [6], Huang and Rao describe how to make the BenOr/Tiwari approach work over finite fields GF(q) with at least 4t(t − 2)d 2 + 1 elements. Their idea is to replace the primes 2, 3, 5, .…”

Section: Introductionmentioning

confidence: 99%

Parallel sparse polynomial interpolation over finite fields

Javadi

Monagan

2010

Proceedings of the 4th International Workshop on Parallel and Symbolic Computation

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We present a probabilistic algorithm to interpolate a sparse multivariate polynomial over a finite field, represented with a black box. Our algorithm modifies the algorithm of BenOr and Tiwari from 1988 for interpolating polynomials over rings with characteristic zero to characteristic p by doing additional probes.To interpolate a polynomial in n variables with t non-zero terms, Zippel's (1990) algorithm interpolates one variable at a time using O(ndt) probes to the black box where d bounds the degree of the polynomial. Our new algorithm does O(nt) probes. It interpolates each variable independently using O(t) probes which allows us to parallelize the main loop giving an advantage over Zippel's algorithm.We have implemented both Zippel's algorithm and the new algorithm in C and we have done a parallel implementation of our algorithm using Cilk [2]. In the paper we provide benchmarks comparing the number of probes our algorithm does with both Zippel's algorithm and Kaltofen and Lee's hybrid of the Zippel and Ben-Or/Tiwari algorithms.

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“…That is to say, given that the polynomial p = ∑ t i=1 m i is the sum of t monomials, we may wish to output a list of these monomials (and their respective coefficients), hence using at least t time steps simply to write down the list of coefficients. Several researchers have developed powerful interpolation and learning algorithms for a variety of contexts which achieve time bounds polynomial in all the relevant parameters, including t (see for example [4,11,16,20,23,31]). …”

Section: Introductionmentioning

confidence: 99%