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Hardness of Reconstructing Multivariate Polynomials. Parikshit Gopalan U. Washington Subhash Khot NYU/Gatech Rishi Saket Gatech/NYU Curve Fitting Problem: Given data points, find a low degree polynomial that fits best. Easy if there is a perfect fit. Well studied problem … Curve Fitting through the ages Curve Fitting through the ages Curve Fitting through the ages ! Statistics: Least Squares Computational Coding Learning Theory Polynomial Reconstruction PCPs Cryptography Pseudorandom -ness The Reconstruction Problem Input: Degree d. Points Values x1 f(x1) xi f(xi) xm f(xm) Output: A degree d polynomial that best fits the data. In this talk: Finite fields, Hamming distance. 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: 1. Degree d, Field F. 2. Set S. 3. How good is the best fit? (error-rate ) 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? 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. Hardness Results: Multivariate Case Linear polynomials over F2 [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. 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. d=1 ½+ SZ Lemma: For a degree d poly P 0 over F2, d=2 ¾ + Prx[ P(x) 0] ¸ 2-d. 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. Overview of Reduction Reducing from Label-Cover. Dictatorship Testing. Consistency Testing. Putting it all together. Label Cover 1 Graph: G(V,E), |V| =n. n 2 Labels: [k] 3 Edges: pe ½ [k] £ [k] Goal: Find a labeling satisfying all edges. Thm [PCP + Raz]: It is NP-hard to tell if • Some labeling satisfies all edges. • Every labeling satisfies · frac. of edges. The Reduction Henceforth d =2, field = F2. X11 X12 … X1k Xn 1 Xn 2 … Xn k 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. The Reduction X11 X12 … X1k Xn1 Xn2 … Xnk X21 X22 … X2k X31 X32 … X3k • If l(v) is a good labelling, then L = v Xvl(v) will satisfy most points. The Reduction X11 X12 … X1k Xn1 Xn2 … Xnk X21 X22 … X2k X31 X32 … X3k • If l(v) is a good labelling, then L = v Xvl(v) will satisfy most points. • Any Q that does ¾ + gives a labelling satisfying ’ fraction of edges. Overview of Reduction 3, 71, 99 Dictatorship: p Q1 = Q(X11,…,X1k,0,..,0). 17, 45 Q1 looks like a Dictator X1j. Will settle for small list. Constant independent of k. Consistency: Some pair of labels in the list satisfy p. Overview of Reduction 3, 71, 99 Dictatorship: p Q1 = Q(X11,…,X1k,0,..,0). 17, 45 Q1 looks like a Dictator X1j. Will settle for small list. Can enforce this for frac. of vertices. Consistency: Some pair of labels in the list satisfy p. Can enforce this for all edges. Overview of Reduction 3, 71, 99 p 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. Overview of Reduction Dictatorship Testing. Consistency Testing. Putting it all together. Overview of Reduction Dictatorship Testing. Consistency Testing. Putting it all together. 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. Small List Test: Pick a random point x 2 {0,1}k. Check if Q(x) = y. Mini reconstruction problem! Dictatorship Testing for low- degree Polynomials. All polys. Quadratic polys. Dictatorships Dictatorship Testing [Hastad, Bourgain, MOO] Hard to do with just 2 queries. All polys. Dictatorships Dictatorship Testing for low- degree Polynomials. Poly. is of low degree. Allowed one query (!) Quadratic polys. Dictatorships Dictatorship Test Dictatorship Test: Pick 2 {0,1}k from the -biased distribution. Check if Q() = 0. Each i =1 independently. w.p (1,…,1) • Uniform dist: Quadratic polys. are 3:1 balanced. • -biased: Dictatorships are highly skewed. (0,…,0) • Is there a converse? Dictatorship Test Dictatorship Test: Pick 2 {0,1}k from the -biased distribution. Check if Q() = 0. Xi passes w.p 1- . XiXj passes w.p 1- 2. X1(X1 + … + Xk) + X2(X1 + …) passes w.p 1 - 2 Dictatorship Test Dictatorship Test: Pick 2 {0,1}k from the -biased distribution. Check if Q() = 0. Define G(Q) to be the graph of Q. 2 Q = X1X2 + X2X3, G(Q) = 1 3 Thm: If Q passes w.p ¾ + , then G(Q) has no large matchings. 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. L + X L X1 1 2 2 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 Xi 2R {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. 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 42. • Events for each matching edge are independent. Dictatorship Test Dictatorship Test: Pick 2 {0,1}k from the -biased distribution. Check if Q() = 0. Define G(Q) to be the graph of Q. 2 Q = X1X2 + X2X3, G(Q) = 1 3 Thm: If Q passes w.p ¾ + , then G(Q) has no large matchings. Small List: Vertex set of a maximal matching. Overview of Reduction 3, 71, 99 Dictatorship: p Assign a small list to a 17, 45 vertex. Consistency: Some pair of labels in the list satisfy p. Overview of Reduction Dictatorship Testing. Consistency Testing. Putting it all together. Consistency Testing l(x) = l(y) Consistency Testing l(x) = l(y) X1 X2 … X k 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! Consistency via Folding l(x) = l(y) X1 X2 … X k Y1 Y2 … Yk •Yes case: Q = Xi + Yi for some i. • All of them vanish over H = (r,r). H • Constant on each coset of H. • Enforce this on Q even in the No case. Consistency via Folding Def: Q is folded over subspace H µ {0,1}k if Q is constant on every coset of H. 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. Template for Folding Want Q folded over a subspace H. Compute nice basis (i, j). H Ask for R(1,…,t). To test if Q(x) = y o Let x = (, ); test R() = y. For analysis: Rewrite R() as Q(x). Now Q is folded. {0,1}n/H Consistency via Folding l(x) = l(y) Fold over H = (r,r) for r 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. Summary of Reduction Each constraint p gives Hp ½ {0,1}nk. Fold over the span of all Hp. 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. Overview of Reduction Dictatorship Testing. Consistency Testing. Putting it all together. Projections … X11 X12 … X1k Xn1 Xn2 … Xnk X21 X22 … X2k X31 X32 … X3k Can handle equality, permutations. Need perfect completeness: no UGC. Have to deal with $#@%! projections. Projections … Decoding is a vertex cover for G(Qi). Need to show that every two vertex covers intersect. Projections … Do every two vertex covers of G intersect? No: Projections … Do every two vertex covers of G intersect? No: … but in any three VCs, some pair intersects. Main Theorem 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. Better Hardness? Problem: Can we improve soundness to 0.5 + ? Bottleneck: Dictatorship test. Present analysis is optimal in general: Q = (X1 + ..+ Xk)(Xk+1 + … +X2k) passes w.p ¾. Can assume that Q is balanced. Thank You!

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UC Berkeley, University of California, Berkeley, Berkeley, California, University of California, City of Berkeley, Bay Area, Berkeley, Haas School of Business, Berkeley County School District, Summer sessions

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posted: | 1/13/2011 |

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