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Tensor-Rank and Lower Bounds for Arithmetic Formulas

Tensor-Rank and Lower Bounds for Arithmetic Formulas. Ran Raz Weizmann Institute. Arithmetic Formulas: Field: F Variables: X 1 ,...,X n Gates: Every gate in the formula computes a polynomial in F[X 1 ,...,X n ] Example: ( X 1 ¢ X 1 ) ¢ ( X 2 + 1 ). The Holy Grail:

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Tensor-Rank and Lower Bounds for Arithmetic Formulas

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  1. Tensor-Rank and Lower Bounds for Arithmetic Formulas Ran Raz Weizmann Institute

  2. Arithmetic Formulas: • Field: F • Variables: X1,...,Xn • Gates: • Every gate in the formula computes • a polynomial in F[X1,...,Xn] • Example:(X1¢ X1) ¢ (X2+ 1)

  3. The Holy Grail: • Super-polynomial lower bounds for • the size of arithmetic circuits and • formulas • (for explicit polynomials) • Our Result: • Connections between Tensor-Rank • and super-polynomial lower bounds • for arithmetic formulas

  4. Tensor-Rank: • A: [n]r! F is of rank 1 if • 9 a1,…,ar : [n] ! F s.t. • A = a1­ a2 ­… ­ ar, that is • A(i1,…,ir) = a1(i1) ¢¢¢ ar(ir) • Rank(A) = Min ks.t. A=A1+…+Ak • where A1,…,Ak are of rank 1 • 8A: [n]r! F Rank(A) · nr-1 • (generalization of matrix rank)

  5. Tensors and Polynomials: • Given A: [n]r! F and n¢r variables • x1,1,…,xr,n define

  6. Tensor-Rank and Arithmetic Circuits: • [Str73]: explicit A:[n]3!F of rank m • )explicit lower bound of (m) • for arithmetic circuits (for fA) • (may give lower bounds of up to(n2)) • (best known bound: (n)) • Our Result:8 r · logn/loglogn • explicit A:[n]r!F of rank nr(1-o(1)) • )explicit super-poly lower bound • for arithmetic formulas (for fA)

  7. Depth-3 vs. General Formulas: • Tensor-rank corresponds to depth-3 • set-multilinear formulas (for fA) • Corollary: strong enough lower bounds • for depth-3 formulas ) super-poly • lower bounds for general formulas • Folklore:strong enough bounds for depth-4 • circuits)exp bounds for general circuits • [AV08]:any exp bound for depth-4 • circuits)exp bound for general circuits

  8. The Tensor-Product Approach [Str]: • Given A1:[n1]r!F, A2:[n2]r!F • Define A = A1­A2 : [n1¢n2]r ! Fby • A((i1,j1),…,(ir,jr)) = A1(i1,…,ir) ¢ A2(j1,…jr) • For r=2, Rank(A) = Rank(A1)¢Rank(A2) • Is Rank(A) > Rank(A1)¢Rank(A2)/no(1) (8r) ? • YES ) super-poly lower bounds for • arithmetic formulas

  9. The Tensor-Product Approach [Str]: • Given A1:[n1]r!F, A2:[n2]r!F • Define A = A1­A2 : [n1¢n2]r ! Fby • A((i1,j1),…,(ir,jr)) = A1(i1,…,ir) ¢ A2(j1,…jr) • For r=2, Rank(A) = Rank(A1)¢Rank(A2) • Is Rank(A) > Rank(A1)¢Rank(A2)/no(1) (8r) ? • YES ) super-poly lower bounds for • arithmetic formulas • Proof: Let m=n1/r • Take A1,…,Ar:[m]r!Fof high rank • Let A = A1 ­ A2 ­ … ­ Ar : [n]r! F • How do we findA1,…,Arof high rank ? • We fix theirr¢n entries as inputs !

  10. Main Steps of the Proof: • 1) New homogenization and • multilinearization techniques • 2) Defining syntactic-rank of a • formula (bounds the tensor-rank) • 3)8s we find the formula of size s • with the largest syntactic-rank • 4) Compute the largest syntactic-rank of a poly-size formula

  11. Homogenous Polynomials: • P 2 F[X1,...,Xn] is homogenous if all • its monomials are of the same degree • Homogenous Formulas: • A formula is homogenous if each of • its nodes computes a homogenous • polynomial

  12. Homogenization: • Given a formula C of size s for a • homogenous polynomial f of deg r • give a homogenous formula D for f • [Str73]:D of size sO(log r) • (optimality conjectured in [NW95]) • Our Result:D of size • (where d = product depth ·O(log s)) • If s=poly(n), and r · logn • Size(D)=poly(n)

  13. Conclusions (of Step 1): • For r · logn/loglogn • 1) super-poly lower bounds for • homogenous formulas ) super-poly • lower bounds for general formulas • 2) super-poly lower bounds for • set-mult formulas ) super-poly • lower bounds for general formulas

  14. Thank You!

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