Blackbox Identity Testing for Bounded Top-Fanin Depth-3 Circuits: The Field Doesn't Matter

Saxena, Nitin; Seshadhri, C.

doi:10.1137/10848232

Cited by 41 publications

(31 citation statements)

References 33 publications

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“…The currently known blackbox PIT algorithms work only for further restricted depth-3 and depth-4 circuits. The case of bounded top fanin depth-3 circuits has received great attention and has blackbox PIT algorithms [15][16][17][18][19][20][21]. The analogous case for depth-4 circuits is open.…”

mentioning

confidence: 99%

Algebraic independence and blackbox identity testing

Beecken¹,

Mittmann²,

Saxena³

2013

Information and Computation

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mentioning

confidence: 99%

Algebraic independence and blackbox identity testing

Beecken¹,

Mittmann²,

Saxena³

2013

Information and Computation

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“…Technically, it is a path of a subcircuit. A path consists of at most k nodes, so that rank of all the forms in the path is at most k+ rk(radsp(I)) Hence, it is a low-rank certificate for the nonzeroness of C. We would like to stress the importance of this theorem, especially since it is central to later improvements for depth-3 PIT [Saxena and Seshadhri 2011b]. THEOREM 4.6 (CERTIFICATE FOR A NONIDENTITY).…”

Section: Applying Chinese Remaindering To Circuitsmentioning

confidence: 99%

From sylvester-gallai configurations to rank bounds

Saxena¹,

Seshadhri

2013

J. ACM

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We study the problem of identity testing for depth-3 circuits of top fanin k and degree d. We give a new structure theorem for such identities that improves the known deterministic d k O(k) -time blackbox identity test over rationals [Kayal and Saraf, 2009] to one that takes d O(k 2 ) -time. Our structure theorem essentially says that the number of independent variables in a real depth-3 identity is very small. This theorem affirmatively settles the strong rank conjecture posed by Dvir and Shpilka [2006].We devise various algebraic tools to study depth-3 identities, and use these tools to show that any depth-3 identity contains a much smaller nucleus identity that contains most of the "complexity" of the main identity. The special properties of this nucleus allow us to get near optimal rank bounds for depth-3 identities. The most important aspect of this work is relating a field-dependent quantity, the Sylvester-Gallai rank bound, to the rank of depth-3 identities. We also prove a high-dimensional Sylvester-Gallai theorem for all fields, and get a general depth-3 identity rank bound (slightly improving previous bounds). ACM Reference Format:Saxena, N. and Seshadhri, C. 2013. From sylvester-gallai configurations to rank bounds: Improved blackbox identity test for depth-3 circuits.

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“…For depth-two formulae, several deterministic polynomial-time blackbox algorithms are known (Agrawal 2003;Arvind & Mukhopadhyay 2010;Ben-Or & Tiwari 1988;Bläser et al 2009;Klivans & Spielman 2001). For depth three, the state of the art is a deterministic polynomial-time blackbox algorithm when the fanin of the top gate is fixed to any constant (Saxena & Seshadhri 2012). The same is known for depth four when the formulae are multilinear, i.e., when every gate in the formula computes a polynomial of degree at most one in each variable (Saraf & Volkovich 2011).…”

Section: Is There An Efficient Deterministic Identity Test For Arithmmentioning

confidence: 99%

Deterministic polynomial identity tests for multilinear bounded-read formulae

2015

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We present a polynomial-time deterministic algorithm for testing whether constant-read multilinear arithmetic formulae are identically zero. In such a formula, each variable occurs only a constant number of times, and each subformula computes a multilinear polynomial. Our algorithm runs in time s O(1) · n k O(k) , where s denotes the size of the formula, n denotes the number of variables, and k bounds the number of occurrences of each variable. Before our work, no subexponential-time deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in time n k O(k) +O(k log n) in general, and time n k O(k 2 ) +O(kD) for depth D. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of read-once formulae and for multilinear depth-four formulae.

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Blackbox Identity Testing for Bounded Top-Fanin Depth-3 Circuits: The Field Doesn't Matter

Abstract: Let C be a depth-3 circuit with n variables, degree d and top fanin k (called ΣΠΣ (k, d, n)

Cited by 41 publications

References 33 publications

Algebraic independence and blackbox identity testing

Algebraic independence and blackbox identity testing

From sylvester-gallai configurations to rank bounds

Deterministic polynomial identity tests for multilinear bounded-read formulae

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