Depth-3 arithmetic formulae over fields of characteristic zero

Amir Shpilka; Avi Wigderson

Depth-3 arithmetic formulae over fields of characteristic zero

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

In this paper we prove near quadratic lower bounds for depth-3 arithmetic formulae over fields of characteristic zero. Such bounds are obtained for the elementary symmetric functions, the (trace of) iterated matrix multiplication, and the determinant. As corollaries we get the first non-trivial lower bounds for computing polynomials of constant degree, and a gap between the power depth-3 arithmetic formulas and depth-4 arithmetic formulas. The main technical contribution relates the complexity of computing a polynomial in this model to the wealth of partial derivatives it has on every affine subspace of small co-dimension. Lower bounds for related models utilize an algebraic analog of Nechiporuk lower bound on Boolean formulae.

Original language	English
Title of host publication	Proceedings of the Annual IEEE Conference on Computational Complexity
Publisher	IEEE Comp Soc
Pages	87-96
Number of pages	10
ISBN (Print)	0769500757
State	Published - 1999
Externally published	Yes
Event	Proceedings of the 1999 14th Annual IEEE Conference on Computational Complexity - Atlanta, GA, USA Duration: 4 May 1999 → 6 May 1999

Publication series

Name	Proceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)	1093-0159

Conference

Conference	Proceedings of the 1999 14th Annual IEEE Conference on Computational Complexity
City	Atlanta, GA, USA
Period	4/05/99 → 6/05/99

Cite this

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title = "Depth-3 arithmetic formulae over fields of characteristic zero",

abstract = "In this paper we prove near quadratic lower bounds for depth-3 arithmetic formulae over fields of characteristic zero. Such bounds are obtained for the elementary symmetric functions, the (trace of) iterated matrix multiplication, and the determinant. As corollaries we get the first non-trivial lower bounds for computing polynomials of constant degree, and a gap between the power depth-3 arithmetic formulas and depth-4 arithmetic formulas. The main technical contribution relates the complexity of computing a polynomial in this model to the wealth of partial derivatives it has on every affine subspace of small co-dimension. Lower bounds for related models utilize an algebraic analog of Nechiporuk lower bound on Boolean formulae.",

author = "Amir Shpilka and Avi Wigderson",

year = "1999",

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series = "Proceedings of the Annual IEEE Conference on Computational Complexity",

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Shpilka, A & Wigderson, A 1999, Depth-3 arithmetic formulae over fields of characteristic zero. in Proceedings of the Annual IEEE Conference on Computational Complexity. Proceedings of the Annual IEEE Conference on Computational Complexity, IEEE Comp Soc, pp. 87-96, Proceedings of the 1999 14th Annual IEEE Conference on Computational Complexity, Atlanta, GA, USA, 4/05/99.

Depth-3 arithmetic formulae over fields of characteristic zero. / Shpilka, Amir; Wigderson, Avi.

Proceedings of the Annual IEEE Conference on Computational Complexity. IEEE Comp Soc, 1999. p. 87-96 (Proceedings of the Annual IEEE Conference on Computational Complexity).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Depth-3 arithmetic formulae over fields of characteristic zero

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