Tensor Rank: Some Lower and Upper Bounds

Alexeev, Boris V.; Forbes, Michael A.; Tsimerman, Jacob

doi:10.1109/ccc.2011.28

Cited by 46 publications

(68 citation statements)

References 28 publications

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“…Lower bounds of 2n r/2 + n − O(r log n) for the tensor-rank of explicit tensors A : [n] r → F (for odd r) were recently proved in [AFT11]. We note also that it was proved by Håstad that computing the tensorrank is an NP-complete problem [H89].…”

Section: Given a Tensor A : [N]mentioning

confidence: 96%

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Tensor-rank and lower bounds for arithmetic formulas

Raz

2010

Proceedings of the Forty-Second ACM Symposium on Theory of Computing

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We show that any explicit example for a tensor A : [n] r → F with tensor-rank ≥ n r·(1−o (1)) , where r = r(n) ≤ log n/ log log n is super-constant, 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 super-polynomial 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 homogeneous polynomial f of degree r, if there exists a (fanin-2) formula of size s and depth d for f then there exists a homogeneous formula of size O d+r+1 r · s for f . In particular, for any r ≤ O(log n), if there exists a polynomial size formula for f then there exists a polynomial size homogeneous formula for f . This refutes a conjecture of Nisan and Wigderson [NW95] and shows that super-polynomial lower bounds for homogeneous 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 ≤ O(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.

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Section: Given a Tensor A : [N]mentioning

confidence: 96%

“…We note that the tensor-rank of most tensors A : [n] 3 → F is Θ(n 2 ). However, to date, no lower bound better than Ω(n) is known for the tensor-rank of any explicit tensor A : [n] 3 → F. Lower bounds of 3n − O(log n) were recently proved in [AFT11].…”

Section: Given a Tensor A : [N]mentioning

confidence: 99%

Tensor-rank and lower bounds for arithmetic formulas

Raz

2010

Proceedings of the Forty-Second ACM Symposium on Theory of Computing

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“…Improving the lower bound will require new techniques for explicit construction of linear-rank tensors with important consequences to circuit lower bounds; see for example Raz [7] and the paper by Alexeev, Forbes and Tsimerman [8] for state-of-theart tensor constructions. In general, we are still unable to upper-bound NQ NIH k ( f ) in terms of log nrank.…”

Section: Contributionsmentioning

confidence: 99%

Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication

Villagra

Nakanishi

Yamashita

et al. 2013

IEICE Trans. Inf. & Syst.

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SUMMARYIn this paper we study quantum nondeterminism in multiparty communication. There are three (possibly) different types of nondeterminism in quantum computation: i) strong, ii) weak with classical proofs, and iii) weak with quantum proofs. Here we focus on the first one. A strong quantum nondeterministic protocol accepts a correct input with positive probability and rejects an incorrect input with probability 1. In this work we relate strong quantum nondeterministic multiparty communication complexity to the rank of the communication tensor in the NumberOn-Forehead and Number-In-Hand models. In particular, by extending the definition proposed by de Wolf to nondeterministic tensor-rank (nrank), we show that for any boolean function f when there is no prior shared entanglement between the players, 1) in the Number-On-Forehead model the cost is upper-bounded by the logarithm of nrank( f ); 2) in the NumberIn-Hand model the cost is lower-bounded by the logarithm of nrank( f ). Furthermore, we show that when the number of players is o(log log n), we have NQP BQP for Number-On-Forehead communication. key words: multiparty communication complexity, quantum computation, quantum nondeterminism, tensor rank

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“…Let F be a sum of pairwise coprime monomials of degree 3 with rankr Σ-mon (n, 3). The only monomials that can appear are of the form x 3 , xy 2 , xyz, with ranks 1, 3, 4 respectively. We can replace each occurence in F of xyz with xy 2 + z 3 without changing the rank or number of variables.…”

mentioning

confidence: 99%

“…See for example [2], [20] for discussions of complexity-theoretic conclusions from tensors of high rank. See [18] for a more general introduction to connections between geometry and complexity.…”

mentioning

confidence: 99%