TY - GEN

T1 - Interpolation of depth-3 arithmetic circuits with two multiplication gates

AU - Shpilka, Amir

PY - 2007

Y1 - 2007

N2 - In this paper we consider the problem of constructing a small arithmetic circuit for a polynomial for which we have oracle access. Our focus is on n-variate polynomials, over a finite field F, that have depth-3 arithmetic circuits with two multiplication gates of degree d. We obtain the following results: 1. Multilinear case: When the circuit is multilinear (multiplication gates compute multilinear polynomials) we give an algorithm that outputs, with probability 1 - o(1), all the depth-3 circuits two multiplication gates computing the same polynomial. The running time of the algorithm is poly(n, |F|). 2. General case: When the circuit is not multilinear we give a quasi-polynomial (in n, |F|) time algorithm that outputs, with probability 1-o(1), a succinct representation of the polynomial. In particular, if the depth-3 circuit for the polynomial is not of small depth- 3 rank (namely, after removing the g.c.d. of the two multiplication gates, the remaining linear functions span a not small linear space) then we output the depth-3 circuit itself. In case that the rank is small we output a depth-3 circuit with a quasi-polynomial number of multiplication gates.

AB - In this paper we consider the problem of constructing a small arithmetic circuit for a polynomial for which we have oracle access. Our focus is on n-variate polynomials, over a finite field F, that have depth-3 arithmetic circuits with two multiplication gates of degree d. We obtain the following results: 1. Multilinear case: When the circuit is multilinear (multiplication gates compute multilinear polynomials) we give an algorithm that outputs, with probability 1 - o(1), all the depth-3 circuits two multiplication gates computing the same polynomial. The running time of the algorithm is poly(n, |F|). 2. General case: When the circuit is not multilinear we give a quasi-polynomial (in n, |F|) time algorithm that outputs, with probability 1-o(1), a succinct representation of the polynomial. In particular, if the depth-3 circuit for the polynomial is not of small depth- 3 rank (namely, after removing the g.c.d. of the two multiplication gates, the remaining linear functions span a not small linear space) then we output the depth-3 circuit itself. In case that the rank is small we output a depth-3 circuit with a quasi-polynomial number of multiplication gates.

KW - Arithmetic circuits

KW - Depth-3

KW - Exact learning

KW - Interpolation

UR - http://www.scopus.com/inward/record.url?scp=35448989350&partnerID=8YFLogxK

U2 - 10.1145/1250790.1250833

DO - 10.1145/1250790.1250833

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AN - SCOPUS:35448989350

SN - 1595936319

SN - 9781595936318

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 284

EP - 293

BT - STOC'07

Y2 - 11 June 2007 through 13 June 2007

ER -