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12
Lower Bounds and Separations for Constant Depth Multilinear Circuits
"... We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a superpolynomial separation between ..."
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Cited by 15 (6 self)
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We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a superpolynomial separation between the size of productdepth 1 d and productdepth d + 1 multilinear circuits (where d is constant). That is, there exists a polynomial f such that • There exists a multilinear circuit of productdepth d + 1 and of polynomial size computing f. • Every multilinear circuit of productdepth d computing f has superpolynomial size. 1
Readonce Polynomial Identity Testing
"... An arithmetic readonce formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of readon ..."
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Cited by 11 (4 self)
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An arithmetic readonce formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of readonce formulas. the following are some of the results that we obtain. 1. Given k ROFs in n variables, over a field F, we give a deterministic (non blackbox) algorithm that checks whether they sum to zero or not. The running time of the algorithm is n O(k2). 2. We give an n O(d+k2) time deterministic algorithm for checking whether a black box holding the sum of k depth d ROFs in n variables computes the zero polynomial. In other words, we provide a hitting set of size n O(d+k2) for the sum of k depth d ROFs. If F  is too small then we make queries from a polynomial size extension field. This implies a deterministic algorithm that runs in time n O(d) for the reconstruction of depth d ROFs. 3. We give a hitting set of size exp ( Õ( √ n + k 2)) for the sum of k ROFs (without depth restrictions). In particular this implies a subexponential time deterministic algorithm for
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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Cited by 11 (3 self)
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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
Multilinear Formulas, MaximalPartition Discrepancy and MixedSources Extractors
, 2007
"... We study a new method for proving lower bounds for subclasses of arithmetic circuits. Roughly speaking, the lower bound is proved by bounding the correlation between the coefficients ’ vector of a polynomial and the coefficients ’ vector of any product of two polynomials with disjoint sets of variab ..."
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Cited by 8 (4 self)
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We study a new method for proving lower bounds for subclasses of arithmetic circuits. Roughly speaking, the lower bound is proved by bounding the correlation between the coefficients ’ vector of a polynomial and the coefficients ’ vector of any product of two polynomials with disjoint sets of variables. We prove lower bounds for several old and new subclasses of circuits. Monotone Circuits: We prove a tight 2 Ω(n) lower bound for the size of monotone arithmetic circuits. The highest previous lower bound was 2 Ω( √ n). Orthogonal Formulas: We prove a tight 2 Ω(n) lower bound for the size of orthogonal multilinear formulas (defined, motivated, and studied by Aaronson). Previously, nontrivial lower bounds were only known for subclasses of orthogonal multilinear formulas. NonCancelling Formulas: We define and study the new model of noncancelling multilinear formulas. Roughly speaking, in this model one is not allowed to sum two polynomials that almost cancel each other. The noncancelling multilinear model is a generalization of both the monotone model and the orthogonal model. We prove lower bounds of n Ω(1) for the depth of noncancelling multilinear formulas.
Balancing Syntactically Multilinear Arithmetic Circuits
, 2007
"... In their seminal paper, Valiant, Skyum, Berkowitz and Rackoff proved that arithmetic circuits can be balanced [VSBR]. That is, [VSBR] showed that for every arithmetic circuit Φ of size s and degree r, there exists an arithmetic circuit Ψ of size poly(r, s) and depth O(log(r) log(s)) computing the sa ..."
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Cited by 6 (4 self)
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In their seminal paper, Valiant, Skyum, Berkowitz and Rackoff proved that arithmetic circuits can be balanced [VSBR]. That is, [VSBR] showed that for every arithmetic circuit Φ of size s and degree r, there exists an arithmetic circuit Ψ of size poly(r, s) and depth O(log(r) log(s)) computing the same polynomial. In the first part of this paper, we follow the proof of [VSBR] and show that syntactically multilinear arithmetic circuits can be balanced. That is, we show that if Φ is syntactically multilinear, then so is Ψ. Recently, [R04b] proved a superpolynomial separation between multilinear arithmetic formula and circuit size. In the second part of this paper, we use the result of the first part to simplify the proof of this separation. That is, we construct a (simpler) polynomial f(x1,..., xn) such that • Every multilinear arithmetic formula computing f is of size n Ω(log(n)). • There exists a syntactically multilinear arithmetic circuit of size poly(n) and depth O(log 2 (n)) computing f. 1
TensorRank and Lower Bounds for Arithmetic Formulas
"... We show that any explicit example for a tensor A: [n] r → F with tensorrank ≥ nr·(1−o(1)) , (where r = r(n) ≤ log n / log log n), implies an explicit superpolynomial lower bound for the size of general arithmetic formulas over F. This shows that strong enough lower bounds for the size of arithmet ..."
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Cited by 4 (2 self)
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We show that any explicit example for a tensor A: [n] r → F with tensorrank ≥ nr·(1−o(1)) , (where r = r(n) ≤ log n / log log n), implies an explicit superpolynomial lower bound for the size of general arithmetic formulas over F. This shows that strong enough lower bounds for the size of arithmetic formulas of depth 3 imply superpolynomial lower bounds for the size of general arithmetic formulas. One component of our proof is a new approach for homogenization and multilinearization of arithmetic formulas, that gives the following results: We show that for any nvariate homogenous polynomial f of degree r, if there exists a (fanin2) ( formula of size s and depth d for f then there exists a homogenous (d+r+1)) formula of size O r · s for f. In particular, for any r ≤ log n, if there exists a polynomial size formula for f then there exists a polynomial size homogenous formula for f. This refutes a conjecture of Nisan and Wigderson [NW95] and shows that superpolynomial lower bounds for homogenous formulas for polynomials of small degree imply superpolynomial lower bounds for general formulas. We show that for any nvariate setmultilinear polynomial f of degree r, if there exists a (fanin2) formula of size s and depth d for f then there exists a setmultilinear formula of size O ((d + 2) r · s) for f. In particular, for any r ≤ log n / log log n, if there exists a polynomial size formula for f then there exists a polynomial size setmultilinear formula for f. This shows that superpolynomial lower bounds for setmultilinear formulas for polynomials of small degree imply superpolynomial lower bounds for general formulas.
Deterministic identity testing of depth 4 multilinear circuits with bounded top fanin
, 2009
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NonCommutative Circuits and the SumofSquares Problem [Extended Abstract] ∗
"... We initiate a direction for proving lower bounds on the size of noncommutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of noncommutative arithmetic circuits and a problem about commutative degree four polynomials, the classical sumofsquares pr ..."
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Cited by 2 (2 self)
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We initiate a direction for proving lower bounds on the size of noncommutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of noncommutative arithmetic circuits and a problem about commutative degree four polynomials, the classical sumofsquares problem: find the smallest n such that there exists an identity (x 2 1+x 2 2+ · · ·+x 2 k)·(y 2 1+y 2 2+ · · ·+y 2 k) = f 2 1 +f 2 2 + · · ·+f 2 n, (1) where each fi = fi(X, Y) is bilinear in X = {x1,..., xk} and Y = {y1,..., yk}. Over the complex numbers, we show that a sufficiently strong superlinear lower bound on n in (1), namely, n ≥ k 1+ɛ with ε> 0, implies an exponential lower bound on the size of arithmetic circuits computing the noncommutative permanent. More generally, we consider such sumofsquares identities for any biquadratic polynomial h(X, Y), namely h(X, Y) = f 2 1 + f 2 2 + · · · + f 2 n. (2) Again, proving n ≥ k 1+ɛ in (2) for any explicit h over the complex numbers gives an exponential lower bound for the noncommutative permanent. Our proofs relies on several new structure theorems for noncommutative circuits, as well as a noncommutative analog of Valiant’s completeness of the permanent. We proceed to prove such superlinear bounds in some restricted cases. We prove that n ≥ Ω(k 6/5) in (1), if f1,..., fn are required to have integer coefficients. Over the real numbers, we construct an explicit biquadratic polynomial h such that n in (2) must be at least Ω(k 2). Unfortunately, these results do not imply circuit lower bounds.
Noncommutative circuits and the sumofsquares problem
 J. Amer. Math. Soc
"... 1.1. Noncommutative computation. Arithmetic complexity theory studies the computation of formal polynomials over some field or ring. Most of this theory is concerned with the computation of commutative polynomials. The basic model of computation is that of an arithmetic circuit. Despite decades of ..."
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Cited by 2 (2 self)
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1.1. Noncommutative computation. Arithmetic complexity theory studies the computation of formal polynomials over some field or ring. Most of this theory is concerned with the computation of commutative polynomials. The basic model of computation is that of an arithmetic circuit. Despite decades of work, the best
A Superpolynomial Lower Bound on the Size of Uniform Nonconstantdepth Threshold Circuits for the Permanent
, 902
"... Abstract. We show that the permanent cannot be computed by DLOGTIMEuniform threshold or arithmetic circuits of depth o(log log n) and polynomial size. ..."
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Abstract. We show that the permanent cannot be computed by DLOGTIMEuniform threshold or arithmetic circuits of depth o(log log n) and polynomial size.