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• 1. The Method of Multiplicities
Microsoft New England/MIT
Based on joint works with:
• V. Guruswami ‘98
• 2. S. Saraf ‘08
• 3. Z. Dvir, S. Kopparty, S. Saraf ‘09
June 13-18, 2011
Mutliplicities @ CSR
TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.: AAA
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• 4. Agenda
A technique for combinatorics, via algebra:
Polynomial (Interpolation) Method + Multiplicity method
List-decoding of Reed-Solomon Codes
Bounding size of Kakeya Sets
Extractor constructions
(won’t cover) Locally decodable codes
June 13-18, 2011
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• 5. Part I: Decoding Reed-Solomon Codes
Reed-Solomon Codes:
Commonly used codes to store information (on CDs, DVDs etc.)
Message: C0, C1, …, Cdє F (finite field)
Encoding:
View message as polynomial: M(x) = i=0dCixi
Encoding = evaluations: { M(®) }_{ ®є F }
Decoding Problem:
Given: (x1,y1) … (xn,yn) є F x F; integers t,d;
Find: deg. dpoly through t of the n points.
June 13-18, 2011
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• 6. List-decoding?
If #errors (n-t) very large, then several polynomials may agree with t of n points.
List-decoding problem:
Report all such polynomials.
Combinatorial obstacle:
There may be too many such polynomials.
Hope – can’t happen.
To analyze: Focus on polynomials P1,…, PLand set of agreements S1 … SL.
Combinatorial question: Can S1, … SLbe large, while n = | [jSj | is small?
June 13-18, 2011
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• 7. List-decoding of Reed-Solomon codes
Given L polynomials P1,…,PLof degree d;and sets S1,…,SL½ F £ F s.t.
|Si| = t
Si½ {(x,Pi(x)) | x 2 F}
How small can n = |S| be, where S = [iSi ?
Algebraic analysis from [S. ‘96, GuruswamiS ’98] basis of decoding algorithms.
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• 8. List-decoding analysis [S ‘96]
Construct Q(x,y) ≠ 0 s.t.
Degy(Q) < L
Degx(Q) < n/L
Q(x,y) = 0 for every (x,y) 2 S = [iSi
Can Show:
Such a Q exists (interpolation/counting).
Implies: t > n/L + dL) (y – Pi(x)) | Q
Conclude: n ¸ L¢ (t – dL).
(Can be proved combinatorially also;
using inclusion-exclusion)
If L > t/(2d), yield n ¸t2/(4d)
June 13-18, 2011
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• 9. Focus: The Polynomial Method
To analyze size of “algebraically nice” set S:
Find polynomial Q vanishing on S;
(Can prove existence of Q by counting coefficients … degree Q grows with |S|.)
Use “algebraic niceness” of S to prove Q vanishes at other places as well.
(In our case whenever y = Pi(x) ).
Conclude Q zero too often (unless S large).
… (abstraction based on [Dvir]’s work)
June 13-18, 2011
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• 10. Improved L-D. Analysis [G.+S. ‘98]
Can we improve on the inclusion-exclusion bound? One that works when n > t2/(4d)?
Idea: Try fitting a polynomial Q that passesthrough each point with “multiplicity” 2.
Can find with Degy < L, Degx < 3n/L.
If 2t > 3n/L + dLthen (y-Pi(x)) | Q.
Yields n ¸ (L/3).(2t – dL)
If L>t/d, then n ¸t2/(3d).
Optimizing Q; letting mult. 1, get n ¸t2/d
June 13-18, 2011
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• 11. Aside: Is the factor of 2 important?
Results in some improvement in [GS] (allowed us to improve list-decoding for codes of high rate) …
But crucial to subsequent work
[Guruswami-Rudra] construction of rate-optimal codes: Couldn’t afford to lose this factor of 2 (or any constant > 1).
June 13-18, 2011
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• 12. Focus: The Multiplicity Method
To analyze size of “algebraically nice” set S:
Find poly Q zero on S (w. high multiplicity);
(Can prove existence of Q by counting coefficients … degree Q grows with |S|.)
Use “algebraic niceness” of S to prove Q vanishes at other places as well.
(In our case whenever y = Pi(x) ).
Conclude Q zero too often (unless S large).
June 13-18, 2011
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• 13. Multiplicity = ?
Over reals: Q(x,y) has root of multiplicity m+1 at (a,b) if every partial derivative of order up to m vanishes at 0.
Over finite fields?
Derivatives don’t work; but “Hasse derivatives” do. What are these? Later…
There are {m+n choose n} such derivatives, for n-variate polynomials;
Each is a linear function of coefficients of f.
June 13-18, 2011
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• 14. Part II: Kakeya Sets
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• 15. Kakeya Sets
K ½Fn is a Kakeya set if it has a line in every direction.
I.e., 8 y 2Fn9 x 2Fns.t. {x + t.y | t 2 F} ½ K
F is a field (could be Reals, Rationals, Finite).
Our Interest:
F = Fq(finite field of cardinality q).
Lower bounds.
Simple/Obvious: qn/2· K ·qn
Do better? Mostly open till [Dvir 2008].
June 13-18, 2011
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• 16. Kakeya Set analysis [Dvir ‘08]
Find Q(x1,…,xn) ≠ 0 s.t.
Total deg. of Q < q (let deg. = d)
Q(x) = 0 for every x 2 K. (exists if |K| < qn/n!)
Prove that (homogenous deg. d part of) Q vanishes on y, if there exists a line in direction y that is contained in K.
Line L ½ K )Q|L = 0.
Highest degree coefficient of Q|L is homogenous part of Q evaluated at y.
Conclude: homogenous part of Q = 0. ><.
Yields |K| ¸qn/n!.
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• 17. Multiplicities in Kakeya [Saraf, S ’08]
Fit Q that vanishes often?
Good choice: #multiplicity m = n
Can find Q ≠ 0 of individual degree < q,that vanishes at each point in K with multiplicity n, provided |K| 4n < qn
Q|Lis of degree < qn.
But it vanishes with multiplicity n at q points!
So it is identically zero ) its highest degree coeff. is zero. ><
Conclude: |K| ¸ (q/4)n
June 13-18, 2011
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• 18. Comparing the bounds
Simple: |K| ¸qn/2
[Dvir]: |K| ¸qn/n!
[SS]: |K| ¸qn/4n
[SS] improves Simple even when q (large) constant and n 1 (in particular, allows q < n)
[MockenhauptTao, Dvir]:
9 K s.t. |K| ·qn/2n-1 + O(qn-1)
Can we do even better?
June 13-18, 2011
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• 19. Part III: Randomness Mergers & Extractors
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• 20. Context
One of the motivations for Dvir’s work:
Build better “randomness extractors”
Approach proposed in [Dvir-Shpilka]
Following [Dvir] , new “randomness merger” and analysis given by [Dvir-Wigderson]
Led to “extractors” matching known constructions, but not improving them …
What are Extractors? Mergers? … can we improve them?
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• 21. Randomness in Computation
Support industry:
F(X)
X
Alg
Random
Distribution A
Uniform
Randomness Processors
Prgs, (seeded) extractors, limited independence generators, epsilon-biased generators,
Condensers, mergers,
Distribution B
June 13-18, 2011
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• 22. Randomness Extractors and Mergers
Extractors:
Dirty randomness  Pure randomness
Mergers: General primitive useful in the context of manipulating randomness.
k random variables  1 random variable
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(Uniform, independent … nearly)
(Biased, correlated)
+ small pure seed
(One of them uniform)
(high entropy)
(Don’t know which, others
potentially correlated)
+ small pure seed
• 23. Merger Analysis Problem
Merger(X1,…,Xk; s) = f(s),
where X1, …, Xk2Fqn; s 2Fq
andfis deg. k-1function mapping F Fn
s.t.f(i) = Xi.
(fis the curve through X1,…,Xk)
Question: For what choices of q, n, k isMerger’soutput close to uniform?
Arises from [DvirShpilka’05, DvirWigderson’08].
“Statistical high-deg. version” of Kakeya problem.
June 13-18, 2011
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• 24. Concerns from Merger Analysis
[DW] Analysis: Worked only if q > n.
So seed length = log2 q > log2 n
Not good enough for setting where k = O(1), and n 1.
(Would like seed length to be O(log k)).
Multiplicity technique:
seems bottlenecked at mult= n.
June 13-18, 2011
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• 25. General obstacle in multiplicity method
Can’t force polynomial Q to vanish with too high a multiplicity. Gives no benefit.
E.g. Kakeya problem: Why stop at mult= n?
Most we can hope from Q is that it vanishes on all of qn;
Once this happens, Q = 0, if its degree is < q in each variable.
So Q|Lis of degree at most qn, so multn suffices. Using larger multiplicity can’t help!
Or can it?
June 13-18, 2011
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• 26. Extended method of multiplicities
(In Kakeya context):
Perhaps vanishing of Q with high multiplicity at each point shows higher degree polynomials (deg > q in each variable) are identically zero?
(Needed: Condition on multiplicity of zeroes of multivariate polynomials .)
Perhaps Q can be shown to vanish with high multiplicity at each point in Fn.
(Technical question: How?)
June 13-18, 2011
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• 27. Vanishing of high-degree polynomials
Mult(Q,a) = multiplicity of zeroes of Qata.
I(Q,a) = 1 if mult(Q,a) > 0and0 o.w.
=min{1, mult(Q,a)}
Schwartz-Zippel: for any S ½ F
 I(Q,a) · d. |S|n-1 where sum is overa 2Sn
Can we replace Iwithmult above? Would strengthen S-Z, and be useful in our case.
[DKSS ‘09]: Yes … (simple inductive proof
… that I can never remember)
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• 28. Multiplicities?
Q(X1,…,Xn) has zero of mult. m at a = (a1,…,an)if all (Hasse) derivatives of order < m vanish.
Hasse derivative = ?
Formally defined in terms of coefficients of Q, various multinomial coefficients and a.
But really …
The i = (i1,…, in)thderivative is the coefficient of z1i1…znin in Q(z + a).
Even better … coeff. of zi in Q(z+x)
(defines ith derivative Qi as a function of x; can evaluate at x = a).
June 13-18, 2011
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• 29. Key Properties
Each derivative is a linear function of coefficients of Q. [Used in [GS’98], [SS’09] .] (Q+R)i = Qi + Ri
Q has zero of multm ata, and S is a curve that passes through a, then Q|Shas zero of multmat a. [Used for lines in prior work.]
Qiis a polynomial of degree deg(Q) - jii (not used in prior works)
(Qi)j ≠ Qi+j, but Qi+j(a) = 0 ) (Qi)j(a) = 0
Qvanishes with multmata
)Qivanishes with multm - j iiat a.
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• 30. Propagating multiplicities (in Kakeya)
FindQthat vanishes with multmonK
For everyiof order m/2, Q_ivanishes with multm/2on K.
Conclude: Q, as well as all derivatives of Q of order m/2 vanish on Fn
) Qvanishes with multiplicity m/2 onFn
Next Question: When is a polynomial (of deg >qn, or even qn) that vanishes with high multiplicity onqnidentically zero?
June 13-18, 2011
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• 31. Back to Kakeya
Find Q of degree d vanishing on K with multm. (can do if (m/n)n |K| < (d/n)n,dn > mn |K| )
Conclude Q vanishes on Fn with mult. m/2.
Apply Extended-Schwartz-Zippel to conclude
(m/2) qn < d qn-1
, (m/2) q < d
, (m/2)nqn < dn = mn |K|
Conclude: |K| ¸ (q/2)n
Tight to within 2+o(1) factor!
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• 32. Consequences for Mergers
Can analyze [DW] merger when q > k very small, n growing;
Analysis similar, more calculations.
Yields: Seed length log q (independent of n).
By combining it with every other ingredient in extractor construction:
Extract all but vanishing entropy (k – o(k) bits of randomness from (n,k) sources) using O(log n) seed (for the first time).
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• 33. Conclusions
Combinatorics does have many “techniques” …
Polynomial method + Multiplicity method adds to the body
Supporting evidence:
List decoding
Kakeya sets
Extractors/Mergers
Others …
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• 34. Other applications
[Woodruff-Yekhanin ‘05]: An elegant construction of novel “LDCs (locally decodable codes)”. [Outclassed by more recent Yekhanin/Efremenko constructions.]
[Kopparty-Lev-Saraf-S. ‘09]: Higher dimensional Kakeya problems.
[Kopparty-Saraf-Yekhanin‘2011]: Locally decodable codes with Rate  1.
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• 35. Conclusions
New (?) technique in combinatorics …
Polynomial method + Multiplicity method
Supporting evidence:
List decoding
Kakeya sets
Extractors/Mergers
Locally decodable codes …
More?
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• 36. Thank You
June 13-18, 2011
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• 37. Mutliplicity = ?
Q vanishes with multiplicity 2 at (a,b):
Q(a,b) = 0; del Q/del x(a,b) = 0; del Q/del y(a,b) = 0.
del Q/del x?
Hasse derivative;
Linear function of coefficients of Q;
Q zero with multiplicity m at (a,b) and C=(x(t),y(t)) is curve passing through (a,b), then Q|C has zero of multiplity m at (a,b).
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• 38. Concerns from Merger Analysis
Recall Merger(X1,…,Xk; s) = f(s),
whereX1, …, Xk2Fqn; s 2Fq
and f is deg. k-1 curves.t. f(i) = Xi.
[DW08] Say X1random; Let Kbe such that ²fraction of choices of X1,…,Xk lead to “bad” curves such that²fraction of s’s such that Merger outputs value in Kwith high probability.
Build low-deg. poly Qvanishing on K; Prove for “bad” curves,Q vanishes on curve; and so Q vanishes on ²-fraction of X1’s (and so ²-fraction of domain).
Apply Schwartz-Zippel. ><
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• 39. What is common?
Given a set in Fqn with nice algebraic properties, want to understand its size.
Kakeya Problem:
The Kakeya Set.
Merger Problem:
Any set T ½Fnthat contains ²-fraction of points on ²-fraction of merger curves.
If T small, then output is non-uniform; else output is uniform.
List-decoding problem:
The union of the sets.
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• 40. Randomness Extractors and Mergers
Extractors:
Dirty randomness  Pure randomness
Mergers: General primitive useful in the context of manipulating randomness.
Given: k (dependent) random variables
X1 … Xk, such that one is uniform.