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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 23 (4 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.
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 21 (6 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
On Identity Testing of Tensors, Lowrank Recovery and Compressed Sensing
, 2011
"... We study the problem of obtaining efficient, deterministic, blackbox polynomial identity testing algorithms for depth3 setmultilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, whitebox polynomial identity testing algorithm (due to Raz and Shpilka [ ..."
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Cited by 16 (5 self)
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We study the problem of obtaining efficient, deterministic, blackbox polynomial identity testing algorithms for depth3 setmultilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, whitebox polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), but has no known such blackbox algorithm. We recast this problem as a question of finding a lowdimensional subspace H, spanned by rank 1 tensors, such that any nonzero tensor in the dual space ker(H) has high rank. We obtain explicit constructions of essentially optimalsize hitting sets for tensors of degree 2 (matrices), and obtain quasipolynomial sized hitting sets for arbitrary tensors (but this second hitting set is less explicit). We also show connections to the task of performing lowrank recovery of matrices, which is studied in the field of compressed sensing. Lowrank recovery asks (say, over R) to recover a matrix M from few measurements, under the promise that M is rank ≤ r. In this work, we restrict our attention to recovering matrices that are exactly rank ≤ r using deterministic, nonadaptive, linear measurements, that are free from noise. Over R, we provide a set (of size 4nr) of such measurements, from which M can be recovered in O(rn 2 + r 3 n) field operations,
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
Quasipolynomialtime Identity Testing of NonCommutative and ReadOnce Oblivious Algebraic Branching Programs
, 2012
"... We study the problem of obtaining efficient, deterministic, blackbox polynomial identity testing algorithms (PIT) for readonce oblivious algebraic branching programs (ABPs). This class has an efficient, deterministic, whitebox polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), ..."
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Cited by 13 (4 self)
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We study the problem of obtaining efficient, deterministic, blackbox polynomial identity testing algorithms (PIT) for readonce oblivious algebraic branching programs (ABPs). This class has an efficient, deterministic, whitebox polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), but prior to this work had no known such blackbox algorithm. Here we obtain the first quasipolynomial sized hitting sets for this class, when the order of the variables is known. This work can be seen as an algebraic analogue of the results of Nisan [Nis92] and ImpagliazzoNisanWigderson [INW94] for spacebounded pseudorandom generators. We also show that several other circuit classes can be blackbox reduced to readonce oblivious ABPs, including setmultilinear ABPs (a generalization of depth 3 setmultilinear formulas), noncommutative ABPs (generalizing noncommutative formulas), and (semi)diagonal depth4 circuits (as introduced by Saxena [Sax08], and recently shown by Mulmuley [Mul12] to have implications for derandomizing Noether’s Normalization Lemma). For setmultilinear ABPs and noncommutative ABPs, we give quasipolynomialtime blackbox PIT algorithms, where the latter case involves evaluations over the algebra of (D + 1) × (D + 1) matrices, where D is the depth of the ABP. For (semi)diagonal depth4 circuits, we obtain a blackbox PIT algorithm (over any characteristic) whose runtime is quasipolynomial in the runtime of Saxena’s whitebox algorithm, matching the concurrent work of Agrawal, Saha, and Saxena [ASS12]. Finally, by combining our results with the reconstruction algorithm of Klivans and Shpilka [KS06], we obtain deterministic reconstruction algorithms for the above circuit classes.
Derandomizing polynomial identity testing for multilinear constantread formulae
 Electronic Colloquium on Computational Complexity, Tech. Rep
"... Abstract—We present a polynomialtime deterministic algorithm for testing whether constantread 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. Before our work no subex ..."
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Abstract—We present a polynomialtime deterministic algorithm for testing whether constantread 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. Before our work no subexponentialtime deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in quasipolynomial time in general, and polynomial time for constant depth. 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 readonce formulae, and for multilinear depthfour circuits. Keywordsarithmetic circuit; boundeddepth circuit; derandomization; polynomial identity testing; I.
Blackbox identity testing of depth4 multilinear circuits
 In Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC
, 2011
"... We study the problem of identity testing for multilinear ΣΠΣΠ(k) circuits, i.e. multilinear depth4 circuits with fanin k at the top + gate. We give the first polynomialtime deterministic identity testing algorithm for such circuits. Our results also hold in the blackbox setting. The running time ..."
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Cited by 12 (3 self)
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We study the problem of identity testing for multilinear ΣΠΣΠ(k) circuits, i.e. multilinear depth4 circuits with fanin k at the top + gate. We give the first polynomialtime deterministic identity testing algorithm for such circuits. Our results also hold in the blackbox setting. The running time of our algorithm is (ns)O(k 3), where n is the number of variables, s is the size of the circuit and k is the fanin of the top gate. The importance of this model arises from [AV08], where it was shown that derandomizing blackbox polynomial identity testing for general depth4 circuits implies a derandomization of polynomial identity testing (PIT) for general arithmetic circuits. Prior to our work, the best PIT algorithm for multilinear ΣΠΣΠ(k) circuits [KMSV10] ran in quasipolynomialtime, with the running time being nO(k 6 log(k) log2 s). We obtain our results by showing a strong structural result for multilinear ΣΠΣΠ(k) circuits that compute the zero polynomial. We show that under some mild technical conditions, any gate of such a circuit must compute a sparse polynomial. We then show how to combine the structure theorem with a result by Klivans and Spielman [KS01], on the identity testing for sparse polynomials, to yield the full result.
Jacobian hits circuits: Hittingsets, lower bounds for depthD occurk formulas & depth3 transcendence degreek circuits
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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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Cited by 7 (3 self)
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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).
Algebraic Independence and Blackbox Identity Testing
 ICALP
, 2011
"... Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. The transcendence degree (trdeg) of a set {f1,..., fm} ⊂ F[x1,..., xn] of polynomials is the maximal size r of an algebraically independent subset. In this paper ..."
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Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. The transcendence degree (trdeg) of a set {f1,..., fm} ⊂ F[x1,..., xn] of polynomials is the maximal size r of an algebraically independent subset. In this paper we design blackbox and efficient linear maps ϕ that reduce the number of variables from n to r but maintain trdeg{ϕ(fi)}i = r, assuming fi’s sparse and small r. We apply these fundamental maps to solve several cases of blackbox identity testing: 1. Given a circuit C and sparse subcircuits f1,..., fm of trdeg r such that D: = C(f1,..., fm) has polynomial degree, we can test blackbox D for zeroness in poly(size(D))r time. 2. Define a ΣΠΣΠδ(k, s, n) circuit C to be of the form ∑k i=1