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62
FROM SYLVESTER-GALLAI CONFIGURATIONS TO RANK BOUNDS: IMPROVED BLACK-BOX IDENTITY TEST FOR DEPTH-3 CIRCUITS
"... Abstract. We study the problem of identity testing for depth-3 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 black-box 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 depth-3 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 black-box 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 depth-3 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 black-box identity tests. Our main theorem (almost optimally) pins down the relation between higher dimensional Sylvester-Gallai theorems and the rank of depth-3 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 Sylvester-Gallai rank bounds for any field, and show the intimate connection between this and depth-3 identity rank bounds. We also prove the first ever theorem about high dimensional Sylvester-Gallai 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 depth-3 identity C: there is a nucleus of C that forms a low rank identity, while the remainder is a high dimensional Sylvester-Gallai configuration. 1.
Read-once Polynomial Identity Testing
"... An arithmetic read-once 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 read-on ..."
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Cited by 21 (6 self)
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An arithmetic read-once 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 read-once 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 black-box) 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 sub-exponential time deterministic algorithm for
Tensor-Rank and Lower Bounds for Arithmetic Formulas
"... We show that any explicit example for a tensor A: [n] r → F with tensor-rank ≥ nr·(1−o(1)) , (where r = r(n) ≤ log n / log log n), implies an explicit super-polynomial 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 20 (1 self)
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We show that any explicit example for a tensor A: [n] r → F with tensor-rank ≥ nr·(1−o(1)) , (where r = r(n) ≤ log n / log log n), implies an explicit super-polynomial 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 n-variate homogenous polynomial f of degree r, if there exists a (fanin-2) ( 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 super-polynomial lower bounds for general formulas. We show that for any n-variate set-multilinear polynomial f of degree r, if there exists a (fanin-2) formula of size s and depth d for f then there exists a set-multilinear 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 set-multilinear formula for f. This shows that super-polynomial lower bounds for set-multilinear formulas for polynomials of small degree imply super-polynomial lower bounds for general formulas.
On Identity Testing of Tensors, Low-rank Recovery and Compressed Sensing
, 2011
"... We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms for depth-3 set-multilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, white-box 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, black-box polynomial identity testing algorithms for depth-3 set-multilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), but has no known such black-box algorithm. We recast this problem as a question of finding a low-dimensional 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 optimal-size hitting sets for tensors of degree 2 (matrices), and obtain quasi-polynomial sized hitting sets for arbitrary tensors (but this second hitting set is less explicit). We also show connections to the task of performing low-rank recovery of matrices, which is studied in the field of compressed sensing. Low-rank 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, non-adaptive, 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,
A super-polynomial lower bound for regular arithmetic formulas.
- In Proc. 46th Annual ACM Symposium on the Theory of Computing,
, 2014
"... Abstract We consider arithmetic formulas consisting of alternating layers of addition (+) and multiplication (×) gates such that the fanin of all the gates in any fixed layer is the same. Such a formula Φ which additionally has the property that its formal/syntactic degree is at most twice the (tot ..."
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Cited by 15 (6 self)
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Abstract We consider arithmetic formulas consisting of alternating layers of addition (+) and multiplication (×) gates such that the fanin of all the gates in any fixed layer is the same. Such a formula Φ which additionally has the property that its formal/syntactic degree is at most twice the (total) degree of its output polynomial, we refer to as a regular formula. As usual, we allow arbitrary constants from the underlying field F on the incoming edges to a + gate so that a + gate can in fact compute an arbitrary F-linear combination of its inputs. We show that there is an (n 2 + 1)-variate polynomial of degree 2n in VNP such that any regular formula computing it must be of size at least n Ω(log n) . Along the way, we examine depth four (ΣΠΣΠ) regular formulas wherein all multiplication gates in the layer adjacent to the inputs have fanin a and all multiplication gates in the layer adjacent to the output node have fanin b. We refer to such formulas as ΣΠ [b] ΣΠ [a] -formulas. We show that there exists an n 2 -variate polynomial of degree n in VNP such that any ΣΠ
Quasipolynomial-time Identity Testing of Non-Commutative and Read-Once Oblivious Algebraic Branching Programs
, 2012
"... We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms (PIT) for read-once oblivious algebraic branching programs (ABPs). This class has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), ..."
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Cited by 14 (4 self)
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We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms (PIT) for read-once oblivious algebraic branching programs (ABPs). This class has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), but prior to this work had no known such black-box algorithm. Here we obtain the first quasi-polynomial 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 Impagliazzo-Nisan-Wigderson [INW94] for space-bounded pseudorandom generators. We also show that several other circuit classes can be black-box reduced to read-once oblivious ABPs, including set-multilinear ABPs (a generalization of depth 3 set-multilinear formulas), non-commutative ABPs (generalizing non-commutative formulas), and (semi-)diagonal depth-4 circuits (as introduced by Saxena [Sax08], and recently shown by Mulmuley [Mul12] to have implications for derandomizing Noether’s Normalization Lemma). For set-multilinear ABPs and non-commutative ABPs, we give quasi-polynomial-time black-box 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 depth-4 circuits, we obtain a black-box PIT algorithm (over any characteristic) whose run-time is quasi-polynomial 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.
Verifiable Delegation of Computation on Outsourced Data
, 2013
"... We address the problem in which a client stores a large amount of data with an untrusted server in such a way that, at any moment, the client can ask the server to compute a function on some portion of its outsourced data. In this scenario, the client must be able to efficiently verify the correct ..."
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Cited by 14 (5 self)
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We address the problem in which a client stores a large amount of data with an untrusted server in such a way that, at any moment, the client can ask the server to compute a function on some portion of its outsourced data. In this scenario, the client must be able to efficiently verify the correctness of the result despite no longer knowing the inputs of the delegated computation, it must be able to keep adding elements to its remote storage, and it does not have to fix in advance (i.e., at data outsourcing time) the functions that it will delegate. Even more ambitiously, clients should be able to verify in time independent of the input-size – a very appealing property for computations over huge amounts of data. In this work we propose novel cryptographic techniques that solve the above problem for the class of computations of quadratic polynomials over a large number of variables. This class covers a wide range of significant arithmetic computations – notably, many important statistics. To confirm the efficiency of our solution, we show encouraging performance results, e.g., correctness proofs have size below 1 kB
Quasi-polynomial hitting-set for set-depth- formulas
- In STOC
, 2013
"... Abstract. We call a depth-4 formula C set-depth-4 if there exists a (unknown) par-tition 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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Cited by 13 (4 self)
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Abstract. We call a depth-4 formula C set-depth-4 if there exists a (unknown) par-tition X1 unionsq · · · unionsq Xd of the variable indices [n] that the top product layer respects, i.e. C(x) = ∑k i=1
Black-box identity testing of depth-4 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 depth-4 circuits with fan-in k at the top + gate. We give the first polynomial-time deterministic identity testing algorithm for such circuits. Our results also hold in the black-box 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 depth-4 circuits with fan-in k at the top + gate. We give the first polynomial-time deterministic identity testing algorithm for such circuits. Our results also hold in the black-box 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 fan-in of the top gate. The importance of this model arises from [AV08], where it was shown that derandomizing black-box polynomial identity testing for general depth-4 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 quasi-polynomial-time, 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.
