Hardness of Reconstructing Multivariate Polynomials.
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Hardness of Reconstructing Hardness of Reconstructing Multivariate Polynomials.Multivariate Polynomials.
Parikshit GopalanParikshit Gopalan U. Washington U. Washington
Subhash Khot Subhash Khot NYU/Gatech NYU/Gatech
Rishi Saket Rishi Saket Gatech/NYUGatech/NYU
Curve FittingCurve 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 Curve Fitting through the agesages
Curve Fitting through the Curve Fitting through the agesages
Curve Fitting through the Curve Fitting through the agesages
!!
Statistics: Least SquaresStatistics: Least Squares
Polynomial Reconstruction
Coding Theory
Computational Learning
Cryptography
PCPs
Pseudorandom-ness
The Reconstruction The Reconstruction ProblemProblem
Input:Input: Degree d. Degree d. PointsPoints ValuesValues
xx11 f(xf(x11))
xxii f(xf(xii))
xxmm f(xf(xmm))
Output:Output: A degree d polynomial that A degree d polynomial that best best fitsfits the data. the data.
In this talk:In this talk: Finite fields, Hamming Finite fields, Hamming distance.distance.
The Reconstruction The Reconstruction ProblemProblem
Input:Input: Degree Degree dd, set , set SS, values , values f(x)f(x) for for x x 22 S S..
Output:Output: A degree d polynomial that A degree d polynomial that best fitsbest fits the data.the data.
Parameters that matter:Parameters that matter:
1.1. Degree Degree dd, Field , Field FF..
2.2. Set Set SS..
3.3. How good is the best fit? (error-rate How good is the best fit? (error-rate ))
Algorithms for Algorithms for ReconstructionReconstruction
Univariate Case Univariate Case [Sudan, Guruswami-Sudan]:[Sudan, Guruswami-Sudan]:
Multivariate Case Multivariate Case [Goldreich-Levin, Goldreich-[Goldreich-Levin, Goldreich-Rubinfeld-Sudan, Arora-Sudan, Sudan-Trevisan-Rubinfeld-Sudan, Arora-Sudan, Sudan-Trevisan-Vadhan]:Vadhan]:
Can tolerate very high error rate Can tolerate very high error rate ..Are these algorithms optimal?Are these algorithms optimal?
Hardness Results: Univariate Hardness Results: Univariate CaseCase
Degree Degree dd polynomials, polynomials, nn points in points in FF..
[Guruswami-Vardy]:[Guruswami-Vardy]: NPNP-hard to tell if some -hard to tell if some degree degree dd poly. has poly. has d +2d +2 agreements. agreements.
[Guruswami-Sudan]:[Guruswami-Sudan]: Can tell if some degree Can tell if some degree dd poly. has poly. has (nd)(nd)0.50.5 agreement. agreement.
Hardness Results: Multivariate Hardness Results: Multivariate CaseCase
Linear polynomialsLinear polynomials overover FF22
[Hastad][Hastad]:: NPNP-hard to tell if-hard to tell if Some linear poly. satisfies Some linear poly. satisfies 1- 1- fraction of fraction of points.points. Every linear poly. satisfies less than Every linear poly. satisfies less than 0.5 + 0.5 + fraction of points.fraction of points.
Extends to any Extends to any FF and and d =1d =1. . Implies something for Implies something for d < Fd < F..
d d ¸̧ 2 2 overover F F22:: Nothing known. Nothing known.
Our ResultsOur Results
Over Over FF22 for any for any dd, NP-hard to tell whether, NP-hard to tell whether Some Some linearlinear polynomial satisfies polynomial satisfies 1- 1- fraction of points.fraction of points. Every Every degree ddegree d polynomial satisfies at polynomial satisfies at most most 1 -2 1 -2-d -d + + fraction of points. fraction of points.
SZ Lemma:SZ Lemma: For a degree d poly P For a degree d poly P 0 over 0 over FF22, ,
PrPrxx[ P(x) [ P(x) 0] 0] ¸̧ 2 2-d-d..
d=1d=1 ½ + ½ +
d=2d=2 ¾ + ¾ +
Our ResultsOur Results
Over Over FFqq for any for any dd, NP-hard to tell whether, NP-hard to tell whether Some Some linearlinear polynomial satisfies polynomial satisfies 1- 1- fraction of points.fraction of points. Every Every degree ddegree d polynomial satisfies at polynomial satisfies at most most c(d,q)c(d,q) + + fraction of points. fraction of points.
c(d,q)c(d,q): Schwartz-Zippel for polynomials of : Schwartz-Zippel for polynomials of total degree total degree dd over over FFqq..
Overview of ReductionOverview of Reduction
Reducing from Label-Cover.Reducing from Label-Cover. Dictatorship Testing.Dictatorship Testing. Consistency Testing.Consistency Testing. Putting it all together.Putting it all together.
Label CoverLabel Cover
Graph: G(V,E), |V| =n.
Labels: [k]
Edges: e 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.
1
2n
3
X11 X1
2 … X1k
The ReductionThe Reduction
X21 X2
2 … X2k
X31 X3
2 … X3k
Xn1 Xn
2 … Xnk
Henceforth d =2, field = F2.
Constraints: Points in {0,1}nk + values.
Yes Case: Some L satisfies most constraints.
No Case: No Q satisfies many constraints.
The ReductionThe ReductionX1
1 X12 … X1
k
X21 X2
2 … X2k
X31 X3
2 … X3k
Xn1 Xn
2 … Xn
k
• If l(v) is a good labelling, then L = v Xv
l(v) will satisfy most points.
The ReductionThe ReductionX1
1 X12 … X1
k
X21 X2
2 … X2k
X31 X3
2 … X3k
Xn1 Xn
2 … Xn
k
• If l(v) is a good labelling, then L = v Xv
l(v) will satisfy most points.
• Any Q that does ¾ + gives a labelling satisfying ’fraction of edges.
Dictatorship:Dictatorship:
QQ11 = Q(X = Q(X1111,…,X,…,X11
kk,0,..,0).,0,..,0).
QQ11 looks like a Dictatorlooks like a Dictator X X11jj..
Will settle for small list.Will settle for small list.
Consistency:Consistency:
Some pair of labels in the list satisfy Some pair of labels in the list satisfy ..
3, 71, 99
17, 45
Overview of ReductionOverview of Reduction
Constant independe
nt of k.
Dictatorship:Dictatorship:
QQ11 = Q(X = Q(X1111,…,X,…,X11
kk,0,..,0).,0,..,0).
QQ11 looks like a Dictatorlooks like a Dictator X X11jj..
Will settle for small list.Will settle for small list.Can enforce this for Can enforce this for frac. of vertices. frac. of vertices.
Consistency:Consistency:
Some pair of labels in the list satisfy Some pair of labels in the list satisfy ..
Can enforce this for all edges.Can enforce this for all edges.
3, 71, 99
17, 45
Overview of ReductionOverview of Reduction
3, 71, 99
17, 45
Overview of ReductionOverview of Reduction
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 ReductionOverview of Reduction
Dictatorship Testing.Dictatorship Testing. Consistency Testing.Consistency Testing. Putting it all together.Putting it all together.
Overview of ReductionOverview of Reduction
Dictatorship Testing.Dictatorship Testing. Consistency Testing.Consistency Testing. Putting it all together.Putting it all together.
Dictatorship Testing for low-Dictatorship Testing for low-degree Polynomials.degree Polynomials.
Input:Input: Q(XQ(X11,…,X,…,Xkk)) of degree 2. of degree 2.
Goal:Goal: Design a test s.t Design a test s.t Every dictatorship Every dictatorship XXii passes w.p close to 1. passes w.p close to 1. If If QQ does better than ¾, it is close to a does better than ¾, it is close to a
dictatorship.dictatorship.
Test:Test: Pick a random point Pick a random point x x 22 {0,1} {0,1}kk..
Check if Check if Q(x) = yQ(x) = y..
Mini reconstruction problem!Mini reconstruction problem!
Small List
Dictatorship Testing for low-Dictatorship Testing for low-degree Polynomials.degree Polynomials.
Dictatorships
Quadratic polys.
All polys.
Dictatorship Testing Dictatorship Testing [Hastad, Bourgain, MOO][Hastad, Bourgain, MOO]
Dictatorships
All polys.
Hard to do with just 2 queries.
Dictatorship Testing for low-Dictatorship Testing for low-degree Polynomials.degree Polynomials.
Dictatorships
Quadratic polys.
Poly. is of low degree.
Allowed one query (!)
Dictatorship TestDictatorship Test
(0,…,0)
(1,…,1)
Dictatorship Test: Pick 2 {0,1}k from the -biased distribution. Check if Q() = 0.
• Uniform dist: Quadratic polys. are 3:1 balanced.
• -biased: Dictatorships are highly skewed.
• Is there a converse?
Each i =1 independentl
y. w.p
Dictatorship TestDictatorship 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 + … + X) + X2(X + …) passes w.p 1 - 2
Dictatorship TestDictatorship 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.
Q = X1X2 + X2X3, G(Q) =
Thm: If Q passes w.p ¾ + , then G(Q) has no large matchings.
1
2
3
Dictatorship TestDictatorship TestDictatorship 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
Dictatorship TestDictatorship TestDictatorship 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.
Q Q’Xi = 0 w.p 1- 2 Xi 2R {0,1}
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 TestDictatorship TestDictatorship 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 TestDictatorship 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.
Q = X1X2 + X2X3, G(Q) =
Thm: If Q passes w.p ¾ + , then G(Q) has no large matchings.
1
2
3
Small List: Vertex set of a maximal matching.
Dictatorship:Dictatorship: Assign a small list to a Assign a small list to a vertex.vertex.
Consistency:Consistency:
Some pair of labels in the list satisfy Some pair of labels in the list satisfy ..
3, 71, 99
17, 45
Overview of ReductionOverview of Reduction
Overview of ReductionOverview of Reduction
Dictatorship Testing.Dictatorship Testing. Consistency Testing.Consistency Testing. Putting it all together.Putting it all together.
Consistency TestingConsistency Testing
l(x) = l(y)
Consistency TestingConsistency Testing
X1 X2 … Xk Y1 Y2 … Yk
l(x) = l(y)
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 FoldingConsistency via Folding
X1 X2 … Xk Y1 Y2 … Yk
l(x) = l(y)
•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 the No case.
H
Consistency via FoldingConsistency 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.
H
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 FoldingTemplate for Folding
Want Q folded over a subspace H.
Compute nice basis (i, j).
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.
H
{0,1}n/H
Consistency via FoldingConsistency 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 ReductionSummary of Reduction
Each constraint gives H ½ {0,1}nk.
Fold over the span of all H.
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 ReductionOverview of Reduction
Dictatorship Testing.Dictatorship Testing. Consistency Testing.Consistency Testing. Putting it all together.Putting it all together.
Projections …Projections …X1
1 X12 … X1
k
X21 X2
2 … X2k
X31 X3
2 … X3k
Xn1 Xn
2 … Xn
k
Can handle equality, permutations. Need perfect completeness: no UGC. Have to deal with $#@%! projections.
Projections …Projections …
Decoding is a vertex cover for G(Qi).
Need to show that every two vertex covers intersect.
Projections …Projections …Do every two vertex covers of G intersect?
No:
Projections …Projections …Do every two vertex covers of G intersect?
… but in any three VCs, some pair intersects.
No:
Main TheoremMain Theorem
Over Over FF22 for any for any dd, NP-hard to tell whether, NP-hard to tell whether Some Some linearlinear polynomial satisfies polynomial satisfies 1- 1- fraction of points.fraction of points. Every Every degree ddegree d polynomial satisfies at polynomial satisfies at most most 1 -2 1 -2-d -d + + fraction of points. fraction of points.
Better Hardness?Better Hardness?Problem: Problem: Can we improve soundness to Can we improve soundness to 0.5 0.5
+ + ??
Bottleneck:Bottleneck: Dictatorship test. Dictatorship test.
Present analysis is optimal in general:Present analysis is optimal in general:
Q = (XQ = (X11 + ..+ X + ..+ Xkk)(X)(Xk+1k+1 + … +X + … +X2k2k) ) passes passes w.p ¾.w.p ¾.
Can assume that Q is balanced.Can assume that Q is balanced.
Thank You!Thank You!