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How to Fool People to Work on Circuit Lower Bounds

This article explores the idea of fooling people to work on circuit lower bounds by presenting innocent-looking problems seemingly unrelated to proving circuit lower bounds. It discusses the concept of arithmetic circuits, elusive functions, tensor-rank, and lower bounds for arithmetic formulas. The article presents the Tensor-Product Approach as a method for finding super-polynomial lower bounds for arithmetic formulas.

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How to Fool People to Work on Circuit Lower Bounds

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  1. How to Fool People to Work on Circuit Lower Bounds Ran Raz Weizmann Institute & Microsoft Research

  2. The Only Barrier for Proving Super-Poly Lower Bounds…

  3. Why Super-Poly Lower Bounds Were Still not Proved ? • Maybe because not enough people are working on it…

  4. The Secret Plan: • Fooling people to work on circuit lower bounds…  • Coming up with innocent looking • clean and simple problems that • are seemingly unrelated to proving • circuit lower bounds, and whose • solution would imply strong circuit • lower bounds

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

  6. The Holy Grail: • Super-polynomial lower bounds • for circuit or formula size • I will present two innocent looking • problems that imply such bounds

  7. Elusive Functions and Lower Bounds for Arithmetic Circuits

  8. Polynomial Mappings: • f = (f1,...,fm): Cn! Cmis a • polynomial mapping of degreedif • f1,...,fmare polynomials of (total) • degreed • f is explicit if given a monomial M • and index i, the coefficient of M in • fi can be computed in poly time [Val]

  9. The Moments Curve: • f: C ! Cm • f(x) = (x,x2,x3,...,xm) • Fact: 8 affine subspace A ( Cm • 8 :Cm-1! Cm of (total) degree 1,

  10. The Exercise that Was Never Given: • Give an explicit f: C ! Cms.t.: • 8: Cm-1! Cm of degree2, • We require: f of degree · • [R08]:Any explicitf • )super-polynomial lower bounds • for the permanent

  11. Elusive Functions: • f: Cn! Cmis (s,r)-elusive if • 8: Cs! Cm of degreer, • [R08]: explicit constructions of • elusive functions imply lower bounds • for the size of arithmetic circuits

  12. Proof Idea: • Consider : Cs! Cm of degreer, that maps • a circuit to the polynomial computed by it • =polynomials that can be • computed by small circuits. • Proving lower bounds, • Finding points outside • Since • f hits a hard function • Add input variables of f as additional • input variables

  13. Lower Bounds for Depth-d Circuits: • [SS91], [R08]: • Lower bounds of n1+(1/d) • (using elusive functions)

  14. Tensor-Rank and Lower Bounds for Arithmetic Formulas

  15. 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)

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

  17. 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)) • [R09]: 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)

  18. 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

  19. 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

  20. 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 !

  21. 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

  22. 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

  23. 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]) • [R09]:D of size • (where d = product depth of C) • If s=poly(n), and r · logn/loglogn • Size(D)=poly(n)

  24. Thanks!

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