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33
On the complexity of numerical analysis
 IN PROC. 21ST ANN. IEEE CONF. ON COMPUTATIONAL COMPLEXITY (CCC ’06
, 2006
"... We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation ..."
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Cited by 45 (5 self)
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We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model ” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a divisionfree straightline program producing an integer N, decide whether N> 0. • In the BlumShubSmale model, polynomial time computation over the reals (on discrete inputs) is polynomialtime equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. • The Generic Task of Numerical Computation is also polynomialtime equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.
Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits
 SIAM J. COMPUT
, 2007
"... In this work we study two, seemingly unrelated, notions. Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given ..."
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Cited by 30 (9 self)
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In this work we study two, seemingly unrelated, notions. Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero. We improve known results on LDCs and on polynomial identity testing and show a relation between the two notions. In particular we obtain the following results: (1) We show that if E: F n ↦ → F m is a linear LDC with two queries, then m = exp(Ω(n)). Previously this was known only for fields of size ≪ 2 n [O. Goldreich et al., Comput. Complexity, 15 (2006), pp. 263–296]. (2) We show that from every depth 3 arithmetic circuit (ΣΠΣ circuit), C, with a bounded (constant) top fanin that computes the zero polynomial, one can construct an LDC. More formally, assume that C is minimal (no subset of the multiplication gates sums to zero) and simple (no linear function appears in all the multiplication gates). Denote by d the degree of the polynomial computed by C and by r the rank of the linear functions appearing in C. Then we can construct a linear LDC with two queries that encodes messages of length r/polylog(d) by codewords of length O(d). (3) We prove a structural theorem for ΣΠΣ circuits, with a bounded top fanin, that
Polynomial identity testing for depth 3 circuits
 in Proceedings of the twentyfirst Annual IEEE Conference on Computational Complexity (CCC
, 2006
"... Abstract — We study ΣΠΣ(k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the first deterministic polynomial time blackbox identity test for ΣΠΣ(k) circuits over the field Q of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main ..."
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Cited by 26 (5 self)
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Abstract — We study ΣΠΣ(k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the first deterministic polynomial time blackbox identity test for ΣΠΣ(k) circuits over the field Q of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main technical result is a structural theorem for ΣΠΣ(k) circuits that compute the zero polynomial. In particular we show that if a ΣΠΣ(k) circuit C = ∑ i∈[k] Ai
PRIMES is in P
 Ann. of Math
, 2002
"... We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1 ..."
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Cited by 26 (2 self)
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We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1
Diagonal Circuit Identity Testing and Lower Bounds
, 2007
"... In this paper we give the first deterministic polynomial time algorithm for testing whether a diagonal depth3 circuit C(x1,..., xn) (i.e. C is a sum of powers of linear functions) is zero. We also prove an exponential lower bound showing that such a circuit will compute determinant or permanent onl ..."
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Cited by 12 (3 self)
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In this paper we give the first deterministic polynomial time algorithm for testing whether a diagonal depth3 circuit C(x1,..., xn) (i.e. C is a sum of powers of linear functions) is zero. We also prove an exponential lower bound showing that such a circuit will compute determinant or permanent only if there are exponentially many linear functions. Our techniques generalize to the following new results: 1. Suppose we are given a depth3 circuit (over any field F) of the form: C(x1,..., xn):= k� i=1 ℓ ei,1 i,1 · · · ℓei,s i,s where, the ℓi,j’s are linear functions living in F[x1,..., xn]. We can test whether C is zero deterministically in poly (nk, max{(1 + ei,1) · · · (1 + ei,s)  1 � i � k}) field operations. This immediately gives a deterministic poly(nk2 d) time identity test for general depth3 circuits of degree d. 2. We prove that if the above circuit C(x1,..., xn) computes the determinant � (or permanent) of an m × m formal matrix with a “small ” s = o then � m log m k = 2 Ω(m). Our lower bounds work for all fields F. (Previous exponential lower bounds for depth3 only work for nonzero characteristic.) 3. We present applications of our ideas to depth4 circuits and an exponentially faster identity test for homogeneous diagonal circuits (deterministically in poly(n k log(d)) field operations over finite fields).
HardnessRandomness Tradeoffs for Bounded Depth Arithmetic Circuits
"... In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x1,..., xm) that cannot be computed by a depth d arithmetic circuit of sma ..."
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Cited by 12 (2 self)
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In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x1,..., xm) that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic algorithm to test whether a given depth d − 8 circuit is identically zero or not (assuming the individual degrees of the tested circuit are not too high). In particular, if we are guaranteed that the tested circuit computes a multilinear polynomial then we can perform the identity test efficiently. To the best of our knowledge this is the first hardnessrandomness tradeoff for bounded depth arithmetic circuits. The above results are obtained using the the arithmetic NisanWigderson generator of [KI04] together with a new theorem on bounded depth circuits, which is the main technical contribution of our work. This theorem deals with polynomial equations of the form P (x1,..., xn, y) ≡ 0 and shows that if P has a circuit of depth d and size s and if the polynomial f(x1,..., xn) satisfies P (x1,..., xn, f(x1,..., xn)) ≡ 0 then f has a circuit of depth d + 3 and size O(s · r + m r), where m is the total degree of f and r is the degree of y in P.
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 11 (2 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.
The ideal membership problem and polynomial identity testing
, 2007
"... Abstract. Given a monomial ideal I = 〈m1, m2, · · · , mk 〉 where mi are monomials and a polynomial f as an arithmetic circuit the Ideal Membership Problem is to test if f ∈ I. We study this problem and show the following results. (a) If the ideal I = 〈m1, m2, · · · , mk 〉 for a constant k then ..."
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Cited by 10 (1 self)
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Abstract. Given a monomial ideal I = 〈m1, m2, · · · , mk 〉 where mi are monomials and a polynomial f as an arithmetic circuit the Ideal Membership Problem is to test if f ∈ I. We study this problem and show the following results. (a) If the ideal I = 〈m1, m2, · · · , mk 〉 for a constant k then there is a randomized polynomialtime membership algorithm to test if f ∈ I. This result holds even for f given by a blackbox, when f is of small degree. (b) When I = 〈m1, m2, · · · , mk 〉 for a constant k and f is computed by a ΣΠΣ circuit with output gate of bounded fanin we can test whether f ∈ I in deterministic polynomial time. This generalizes the KayalSaxena result [KS07] of deterministic polynomialtime identity testing for ΣΠΣ circuits with bounded fanin output gate. (c) When k is not constant the problem is coNPhard. However, the problem is upper bounded by coAM PP over the field of rationals, and by coNP ModpP over finite fields. (d) Finally, we discuss identity testing for certain restricted depth 4 arithmetic circuits. For ideals I = 〈f1, · · · , fℓ 〉 where each fi ∈ F[x1, · · · , xk] is an arbitrary polynomial but k is a constant, we show similar results as (a) and (b) above. 1
Black box polynomial identity testing of generalized depth3 arithmetic circuits with bounded top fanin
 in IEEE Conference on Computational Complexity
"... In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the blackbox polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in ..."
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Cited by 9 (4 self)
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In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the blackbox polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in the form f(¯x) = ∑ k i=1 hi(¯x) · gi(¯x), where each hi is a polynomial that depends on only ρ linear functions, and each gi is a product of linear functions (when hi = 1, for each i, then we get the class of depth3 circuits with k multiplication gates, also known as ΣΠΣ(k) circuits, but the general case is much richer). When maxi(deg(hi · gi)) = d we say that f is computable by a ΣΠΣ(k, d, ρ) circuit. We obtain the following results. 1. A deterministic blackbox identity testing algorithm for ΣΠΣ(k, d, ρ) circuits that runs in quasipolynomial time (for ρ = polylog(n + d)). In particular this gives the first blackbox quasipolynomial time PIT algorithm for depth3 circuits with k multiplication gates. 2. A deterministic blackbox identity testing algorithm for readk ΣΠΣ circuits (depth3 circuits where each variable appears at most k times) that runs in time n 2O(k2). In particular
An Almost Optimal Rank Bound for Depth3 Identities
"... Abstract—We show that the rank of a depth3 circuit (over any field) that is simple, minimal and zero is at most O(k 3 log d). The previous best rank bound known was 2 O(k2) (log d) k−2 by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we als ..."
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Cited by 7 (2 self)
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Abstract—We show that the rank of a depth3 circuit (over any field) that is simple, minimal and zero is at most O(k 3 log d). The previous best rank bound known was 2 O(k2) (log d) k−2 by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank Ω(k log d)). Our rank bound significantly improves (dependence on k exponentially reduced) the best known deterministic blackbox identity tests for depth3 circuits by Karnin and Shpilka (CCC 2008). Our techniques also shed light on the factorization pattern of nonzero depth3 circuits, most strikingly: the rank of linear factors of a simple, minimal and nonzero depth3 circuit (over any field) is at most O(k 3 log d). The novel feature of this work is a new notion of maps between sets of linear forms, called ideal matchings, used to study depth3 circuits. We prove interesting structural results about depth3 identities using these techniques. We believe that these can lead to the goal of a deterministic polynomial time identity test for these circuits. I.