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Arithmetic Circuits: a survey of recent results and open questions
"... A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last fi ..."
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A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions. The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we
FROM SYLVESTERGALLAI CONFIGURATIONS TO RANK BOUNDS: IMPROVED BLACKBOX IDENTITY TEST FOR DEPTH3 CIRCUITS
"... Abstract. We study the problem of identity testing for depth3 circuits of top fanin k and degree d (called ΣΠΣ(k, d) identities). We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d kO(k) time blackbox identity test over ratio ..."
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Cited by 27 (6 self)
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Abstract. We study the problem of identity testing for depth3 circuits of top fanin k and degree d (called ΣΠΣ(k, d) identities). We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d kO(k) time blackbox identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes d O(k2)time. Our structure theorem essentially says that the number of independent variables in a real depth3 identity is very small. This theorem settles affirmatively the stronger rank conjectures posed by Dvir & Shpilka (STOC 2005) and Kayal & Saraf (FOCS 2009). Our techniques provide a unified framework that actually beats all known rank bounds and hence gives the best running time (for every field) for blackbox identity tests. Our main theorem (almost optimally) pins down the relation between higher dimensional SylvesterGallai theorems and the rank of depth3 identities in a very transparent manner. The existence of this was hinted at by Dvir & Shpilka (STOC 2005), but first proven, for reals, by Kayal & Saraf (FOCS 2009). We introduce the concept of SylvesterGallai rank bounds for any field, and show the intimate connection between this and depth3 identity rank bounds. We also prove the first ever theorem about high dimensional SylvesterGallai configurations over any field. Our proofs and techniques are very different from previous results and devise a very interesting ensemble of combinatorics and algebra. The latter concepts are ideal theoretic and involve a new Chinese remainder theorem. Our proof methods explain the structure of any depth3 identity C: there is a nucleus of C that forms a low rank identity, while the remainder is a high dimensional SylvesterGallai configuration. 1.
Elusive functions and lower bounds for arithmetic circuits
 Electronic Colloquium in Computational Complexity
, 2007
"... A basic fact in linear algebra is that the image of the curve f(x) = (x1, x2, x3,..., xm), say over C, is not contained in any m − 1 dimensional affine subspace of Cm. In other words, the image of f is not contained in the image of any polynomialmapping1 Γ: Cm−1 → Cm of degree 1 (that is, an affin ..."
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A basic fact in linear algebra is that the image of the curve f(x) = (x1, x2, x3,..., xm), say over C, is not contained in any m − 1 dimensional affine subspace of Cm. In other words, the image of f is not contained in the image of any polynomialmapping1 Γ: Cm−1 → Cm of degree 1 (that is, an affine mapping). Can one give an explicit example for a polynomial curve f: C → Cm, such that, the image of f is not contained in the image of any polynomialmapping Γ: Cm−1 → Cm of degree 2? In this paper, we show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. For example, any explicit f as above (with the right notion of explicitness2), of degree up to 2mo(1) , implies superpolynomial lower bounds for computing the permanent over C. More generally, we say that a polynomialmapping f: Fn → Fm is (s, r)elusive, if for every polynomialmapping Γ: Fs → Fm of degree r, Image(f) � ⊂ Image(Γ).
Blackbox identity testing for bounded top fanin depth3 circuits: the field doesn’t matter
 In Proceedings of the 43rd annual ACM Symposium on Theory of Computing (STOC
, 2011
"... Abstract. Let C be a depth3 circuit with n variables, degree d and top fanin k (called ΣΠΣ(k, d, n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically zero. Klivans & Spielman (STOC 2001) observed ..."
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Abstract. Let C be a depth3 circuit with n variables, degree d and top fanin k (called ΣΠΣ(k, d, n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically zero. Klivans & Spielman (STOC 2001) observed that the problem is open even when k is a constant. This case has been subjected to a serious study over the past few years, starting from the work of Dvir & Shpilka (STOC 2005). We give the first polynomial time blackbox algorithm for this problem. Our algorithm runs in time poly(n)dk, regardless of the base field. The only field for which polynomial time algorithms were previously known is F = Q (Kayal & Saraf, FOCS 2009, and Saxena & Seshadhri, FOCS 2010). This is the first blackbox algorithm for depth3 circuits that does not use the rank based approaches of Karnin & Shpilka (CCC 2008). We prove an important tool for the study of depth3 identities. We design a blackbox polynomial time transformation that reduces the number of variables in a ΣΠΣ(k, d, n) circuit to k variables, but preserves the identity structure. Key words. depth3 circuits; polynomial identity testing; derandomization; blackbox; Chinese remaindering; algebra homomorphism
Quasipolynomial hittingset for setdepth formulas
 In STOC
, 2013
"... Abstract. We call a depth4 formula C setdepth4 if there exists a (unknown) partition X1 unionsq · · · unionsq Xd of the variable indices [n] that the top product layer respects, i.e. C(x) = ∑k i=1 ..."
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Abstract. We call a depth4 formula C setdepth4 if there exists a (unknown) partition X1 unionsq · · · unionsq Xd of the variable indices [n] that the top product layer respects, i.e. C(x) = ∑k i=1
A CASE OF DEPTH3 IDENTITY TESTING, SPARSE FACTORIZATION AND DUALITY
"... Polynomial identity testing (PIT) problem is known to be challenging even for constant depth arithmetic circuits. In this work, we study the complexity of two special but natural cases of identity testing first is a case of depth3 PIT, the other of depth4 PIT. Our first problem is a vast generali ..."
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Polynomial identity testing (PIT) problem is known to be challenging even for constant depth arithmetic circuits. In this work, we study the complexity of two special but natural cases of identity testing first is a case of depth3 PIT, the other of depth4 PIT. Our first problem is a vast generalization of: Verify whether a bounded top fanin depth3 circuit equals a sparse polynomial (given as a sum of monomial terms). Formally, given a depth3 circuit C, having constant many general product gates and arbitrarily many semidiagonal product gates, test if the output of C is identically zero. A semidiagonal product gate in C computes a product of the form m · ∏b, where m is a i=1 ℓei i monomial, ℓi is an affine linear polynomial and b is a constant. We give a deterministic polynomial time test, along with the computation of leading monomials of semidiagonal circuits over local rings. The second problem is on verifying a given sparse polynomial factorization, which is a classical question (von zur Gathen, FOCS 1983): Given multivariate sparse polynomials f, g1,..., gt explicitly, check if f = ∏ t i=1 gi. For the special case when every gi is a sum of univariate polynomials, we give a deterministic polynomial time test. We characterize the factors of such gi’s and even show how to test the divisibility of f by the powers of such polynomials. The common tools used are Chinese remaindering and dual representation. The dual representation of polynomials (Saxena, ICALP 2008) is a technique to express a productofsums of univariates as a sumofproducts of univariates. We generalize this technique by combining it with a generalized Chinese remaindering to solve these two problems (over any field).
Electronic Colloquium on Computational Complexity, 2006. 51 Ehud Friedgut. Necessary and sufficient conditions for sharp thresholds of graph properties and the kSAT problem
 J. of the AMS
, 1999
"... We initiate a study of inputoblivious proof systems, and present a few preliminary results regarding such systems. Our results offer a perspective on the intersection of the nonuniform complexity class P/poly with uniform complexity classes such as N P and IP. In particular, we provide a uniform c ..."
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Cited by 3 (0 self)
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We initiate a study of inputoblivious proof systems, and present a few preliminary results regarding such systems. Our results offer a perspective on the intersection of the nonuniform complexity class P/poly with uniform complexity classes such as N P and IP. In particular, we provide a uniform complexity formulation of the conjecture N P ⊂ P/poly and a uniform complexity characterization of the class IP ∩P/poly. These (and similar) results offer a perspective on the attempt to prove circuit lower bounds for complexity classes such as N P, PSPACE,
On the size of depththree boolean circuits for computing multilinear functions
 Electronic Coll. on Computational Complexity (ECCC
"... We propose that multilinear functions of relatively low degree over GF(2) may be good candidates for obtaining exponential1 lower bounds on the size of constantdepth Boolean circuits (computing explicit functions). Specifically, we propose to move gradually from linear functions to multilinear on ..."
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We propose that multilinear functions of relatively low degree over GF(2) may be good candidates for obtaining exponential1 lower bounds on the size of constantdepth Boolean circuits (computing explicit functions). Specifically, we propose to move gradually from linear functions to multilinear ones, and conjecture that, for any t ≥ 2, some explicit tlinear functions F: ({0, 1}n)t → {0, 1} require depththree circuits of size exp(Ω(tnt/(t+1))). Towards studying this conjecture, we suggest to study two frameworks for the design of depththree Boolean circuits computing multilinear functions, yielding restricted models for which lower bounds may be easier to prove. Both correspond to constructing a circuit by expressing the target polynomial as a composition of simpler polynomials. The first framework corresponds to a direct composition, whereas the second (and stronger) framework corresponds to nested composition and yields depththree Boolean circuits via a ”guessandverify ” paradigm. The corresponding restricted models of circuits are called Dcanonical and NDcanonical, respectively. Our main results are (1) a generic upper bound on the size of depththree Dcanonical
A Spectral Theory for Tensors
, 2011
"... In this paper we propose a general spectral theory for tensors. Our proposed factorization decomposes a tensor into a product of orthogonal and scaling tensors. At the same time, our factorization yields an expansion of a tensor as a summation of outer products of lower order tensors. Our proposed f ..."
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In this paper we propose a general spectral theory for tensors. Our proposed factorization decomposes a tensor into a product of orthogonal and scaling tensors. At the same time, our factorization yields an expansion of a tensor as a summation of outer products of lower order tensors. Our proposed factorization shows the relationship between the eigenobjects and the generalised characteristic polynomials. Our framework is based on a consistent multilinear algebra which explains how to generalise the notion of matrix hermicity, matrix transpose, and most importantly the notion of orthogonality. Our proposed factorization for a tensor in terms of lower order tensors can be recursively applied so as to naturally induces a spectral hierarchy for tensors. 1
ABELIAN TENSORS
"... Abstract. We analyze tensors in Cm⊗Cm⊗Cm satisfying Strassen’s equations for border rank m. Results include: two purely geometric characterizations of the CoppersmithWinograd tensor, a reduction to the study of symmetric tensors under a mild genericity hypothesis, and numerous additional equations ..."
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Abstract. We analyze tensors in Cm⊗Cm⊗Cm satisfying Strassen’s equations for border rank m. Results include: two purely geometric characterizations of the CoppersmithWinograd tensor, a reduction to the study of symmetric tensors under a mild genericity hypothesis, and numerous additional equations and examples. This study is closely connected to the study of the variety of mdimensional abelian subspaces of End(Cm) and the subvariety consisting of the Zariski closure of the variety of maximal tori, called the variety of reductions. 1.