Hardness of Reconstructing Multivariate Polynomials.

1. Hardness of Reconstructing Multivariate Polynomials. Parikshit Gopalan U. Washington Subhash Khot NYU/Gatech Rishi Saket Gatech/NYU

2. Curve Fitting Problem:Given data points, find a low degree polynomial that fits best. Easy if there is a perfect fit. Well studied problem …

3. Curve Fitting through the ages

4. Curve Fitting through the ages

5. Curve Fitting through the ages

6. !

7. Statistics: Least Squares

8. Coding Theory Computational Learning Polynomial Reconstruction PCPs Cryptography Pseudorandom-ness

9. The Reconstruction Problem • Input: Degree d. • Output: A degree d polynomial that best fits the data. • In this talk: Finite fields, Hamming distance.

10. The Reconstruction Problem • Input: Degree d, set S, values f(x) for x 2 S. • Output: A degree d polynomial that best fits the data. • Parameters that matter: • Degree d, Field F. • Set S. • How good is the best fit? (error-rate )

11. Algorithms for Reconstruction • Univariate Case [Sudan, Guruswami-Sudan]: • Multivariate Case [Goldreich-Levin, Goldreich-Rubinfeld-Sudan, Arora-Sudan, Sudan-Trevisan-Vadhan]: • Can tolerate very high error rate . • Are these algorithms optimal?

12. Hardness Results: Univariate Case • Degree d polynomials, n points in F. • [Guruswami-Vardy]:NP-hard to tell if some degree d poly. has d +2 agreements. • [Guruswami-Sudan]: Can tell if some degree d poly. has (nd)0.5 agreement.

13. Hardness Results: Multivariate Case • Linear polynomialsoverF2 • [Hastad]:NP-hard to tell if • Some linear poly. satisfies 1-  fraction of points. • Every linear poly. satisfies less than 0.5 +  fraction of points. • Extends to any F and d =1. • Implies something for d < F. • d ¸ 2 over F2: Nothing known.

14. Our Results • Over F2 for any d, NP-hard to tell whether • Some linear polynomial satisfies 1-  fraction of points. • Every degree d polynomial satisfies at most 1 -2-d +  fraction of points. • SZ Lemma: For a degree d poly P  0 over F2, • Prx[ P(x)  0] ¸ 2-d.

15. Our Results • Over Fq for any d, NP-hard to tell whether • Some linear polynomial satisfies 1-  fraction of points. • Every degree d polynomial satisfies at most c(d,q)+  fraction of points. • c(d,q): Schwartz-Zippel for polynomials of total degree d over Fq.

16. Overview of Reduction • Reducing from Label-Cover. • Dictatorship Testing. • Consistency Testing. • Putting it all together.

17. Label Cover 1 Graph: G(V,E), |V| =n. Labels: [k] Edges:pe ½ [k] £ [k] Goal: Find a labeling satisfying all edges. 2 n 3 • Thm [PCP + Raz]: It is NP-hard to tell if • Some labeling satisfies all edges. • Every labeling satisfies · frac. of edges.

18. The Reduction Henceforth d =2, field = F2. X11 X12 … X1k Xn1 Xn2 … Xnk X21 X22 … X2k X31 X32 … X3k Constraints: Points in {0,1}nk+ values. Yes Case: Some L satisfies most constraints. No Case: No Q satisfies many constraints.

19. X11X12 … X1k Xn1 Xn2 … Xnk X21 X22 … X2k X31 X32 … X3k The Reduction • Ifl(v)is a good labelling, thenL = v Xvl(v)will satisfy most points.

20. The Reduction X11 X12 … X1k Xn1 Xn2 … Xnk X21X22 … X2k X31 X32 … X3k • Ifl(v)is a good labelling, thenL =v Xvl(v)will satisfy most points. • Any Q that does ¾ +  gives a labelling satisfying ’fraction of edges.

21. Overview of Reduction 3, 71, 99 • Dictatorship: • Q1 = Q(X11,…,X1k,0,..,0). • Q1looks like a Dictator X1j. • Will settle for small list.  17, 45 Constant independent of k. • Consistency: • Some pair of labels in the list satisfy .

22. Overview of Reduction 3, 71, 99 • Dictatorship: • Q1 = Q(X11,…,X1k,0,..,0). • Q1looks like a Dictator X1j. • Will settle for small list. • Can enforce this for  frac. of vertices.  17, 45 • Consistency: • Some pair of labels in the list satisfy . • Can enforce this for all edges.

23. Overview of Reduction 3, 71, 99  17, 45 • If Q does ¾ +  • Small list for  frac. of vertices. • Consistency for all edges. • Assign random labels from list. • Satisfies constant fraction of edges.

24. Overview of Reduction • Dictatorship Testing. • Consistency Testing. • Putting it all together.

25. Overview of Reduction • Dictatorship Testing. • Consistency Testing. • Putting it all together.

26. Dictatorship Testing for low-degree Polynomials. • Input:Q(X1,…,Xk) of degree 2. • Goal: Design a test s.t • Every dictatorship Xi passes w.p close to 1. • If Q does better than ¾, it is close to a dictatorship. • Test: Pick a random point x 2 {0,1}k. • Check if Q(x) = y. • Mini reconstruction problem! Small List

27. Dictatorship Testing for low-degree Polynomials. All polys. Quadratic polys. Dictatorships

28. Dictatorship Testing [Hastad, Bourgain, MOO] Hard to do with just 2 queries. All polys. Dictatorships

29. Dictatorship Testing for low-degree Polynomials. • Poly. is of low degree. • Allowed one query (!) Quadratic polys. Dictatorships

30. Dictatorship Test • Dictatorship Test: • Pick 2 {0,1}k from the -biased distribution. • Check if Q() = 0. Each i =1 independently. w.p  • Uniform dist: Quadratic polys. are 3:1 balanced. • -biased: Dictatorships are highly skewed. • Is there a converse? (1,…,1) (0,…,0)

31. Dictatorship Test • Dictatorship Test: • Pick 2 {0,1}k from the -biased distribution. • Check if Q() = 0. • Xipasses w.p1- . • XiXjpasses w.p 1- 2. • X1(X1 + … + Xk) + X2(X1 + …)passes w.p1 - 2

32. Dictatorship Test • Dictatorship Test: • Pick 2 {0,1}k from the -biased distribution. • Check if Q() = 0. 2 Define G(Q) to be the graph of Q. Q = X1X2 + X2X3, G(Q) = 3 1 Thm:If Q passes w.p ¾ + , then G(Q) has no large matchings.

33. Dictatorship Test • Dictatorship Test: • Pick 2 {0,1}k from the -biased distribution. • Check if Q() = 0. • Thm:If Q passes w.p ¾ + , then G(Q) has no large matchings. 1. Large matching: Independent monomials. 2. Only small matchings: Small vertex cover. X1L1 + X2L2

34. Dictatorship Test • Dictatorship Test: • Pick 2 {0,1}k from the -biased distribution. • Check if Q() = 0. • Thm:If Q does better than ¾, then G(Q) has no large matchings. Xi = 0 w.p 1- 2 Xi2R {0,1} Q Q’ c =? 0 • If G(Q) has a large matching, then Q’  0 w.h.p. • If Q’  0, then c =1 w.p ¸ ¼ (SZ lemma). • If Q does well, G(Q) has no large matchings.

35. Dictatorship Test • Dictatorship Test: • Pick 2 {0,1}k from the -biased distribution. • Check if Q() = 0. • Thm:If Q does better than ¾, then G(Q) has no large matchings. If G(Q) has a large matching, then Q’  0 w.h.p. • Each edge survives w.p 42. • Events for each matching edge are independent.

36. Dictatorship Test • Dictatorship Test: • Pick 2 {0,1}k from the -biased distribution. • Check if Q() = 0. 2 Define G(Q) to be the graph of Q. Q = X1X2 + X2X3, G(Q) = 3 1 Thm:If Q passes w.p ¾ + , then G(Q) has no large matchings. Small List:Vertex set of a maximal matching.

37. Overview of Reduction 3, 71, 99 • Dictatorship: • Assign a small list to a vertex.  17, 45 • Consistency: • Some pair of labels in the list satisfy .

38. Overview of Reduction • Dictatorship Testing. • Consistency Testing. • Putting it all together.

39. Consistency Testing l(x) = l(y)

40. Consistency Testing l(x) = l(y) X1 X2 … Xk Y1 Y2 … Yk Given Q(X1,…,Xk,Y1,…,Yk) s.t Q(Xi) and Q(Yj) both pass the dict. Test. Want Q(X1,..,Xk,0,…,0) = Q(0,…,0,Y1,…,Yk). Test: Q(r,0) = Q(0,r) for r 2R {0,1}k. • Two queries!

41. H Consistency via Folding l(x) = l(y) X1 X2 … Xk Y1 Y2 … Yk • Yes case:Q = Xi + Yi for some i. • All of them vanish over H = (r,r). • Constant on each coset of H. • Enforce this on Q even in theNocase.

42. H Consistency via Folding Def: Q is folded over subspace H µ {0,1}k if Q is constant on every coset of H. Examples: Linear polys., juntas. Thm:Q is folded over H iff for some nice basis (1,…,t,1,...,k-t), Q = R(1,…,t) is a t-junta for t = k – dim(H) In the nice basis (1,…,t,1,...,k-t) is: coset of H, js: position in coset.

43. H Template for Folding • Want Q folded over a subspace H. • Compute nice basis(i, j). • Ask forR(1,…,t). • To test ifQ(x) = y • Letx = (,); testR() = y. • For analysis: RewriteR()asQ(x). • Now Q is folded. {0,1}n/H

44. Consistency via Folding l(x) = l(y) Fold overH = (r,r)forr 2 {0,1}k. Polys. folded over H can be written as: Q(X1,…,Xk,Y1,…,Yk) = R(X1 +Y1, …, Xk + Yk) Gives Q(X1,…,Xk) = Q(Y1,…,Yk). List of Xis: Vertex set of maximal matching. Every two maximal matchings intersect.

45. Summary of Reduction • Each constraint givesH½ {0,1}nk. • Fold over the span of allH. • Run Dict. test on every vertex. • No explicit consistency tests. • If Q passes w.p ¾ + , • fraction of vertices do well on Dict. test. • Consistency for all edges by folding.

46. Overview of Reduction • Dictatorship Testing. • Consistency Testing. • Putting it all together.

47. Projections … X11X12 … X1k Xn1 Xn2 … Xnk X21 X22 … X2k X31 X32 … X3k • Can handle equality, permutations. • Need perfect completeness: no UGC. • Have to deal with $#@%! projections.

48. Projections … Decoding is a vertex cover for G(Qi). Need to show that every two vertex covers intersect.

49. Projections … Do every two vertex covers of G intersect? No:

50. Projections … Do every two vertex covers of G intersect? No: … but in any three VCs, some pair intersects.