Csr2011 june18 14_00_sudan

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  • 1. The Method of Multiplicities
    Madhu Sudan
    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
    1
  • 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
    2
    Mutliplicities @ CSR
  • 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
    3
    Mutliplicities @ CSR
  • 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
    4
    Mutliplicities @ CSR
  • 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.
    June 13-18, 2011
    5
    Mutliplicities @ CSR
  • 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
    6
    Mutliplicities @ CSR
  • 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
    7
    Mutliplicities @ CSR
  • 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
    8
    Mutliplicities @ CSR
  • 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
    9
    Mutliplicities @ CSR
  • 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
    10
    Mutliplicities @ CSR
  • 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
    11
    Mutliplicities @ CSR
  • 14. Part II: Kakeya Sets
    June 13-18, 2011
    12
    Mutliplicities @ CSR
  • 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
    13
    Mutliplicities @ CSR
  • 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!.
    June 13-18, 2011
    14
    Mutliplicities @ CSR
  • 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
    15
    Mutliplicities @ CSR
  • 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
    16
    Mutliplicities @ CSR
  • 19. Part III: Randomness Mergers & Extractors
    June 13-18, 2011
    17
    Mutliplicities @ CSR
  • 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?
    June 13-18, 2011
    18
    Mutliplicities @ CSR
  • 21. Randomness in Computation
    Readily available randomness
    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
    19
    Mutliplicities @ CSR
  • 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
    June 13-18, 2011
    20
    Mutliplicities @ CSR
    (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
    21
    Mutliplicities @ CSR
  • 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
    22
    Mutliplicities @ CSR
  • 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
    23
    Mutliplicities @ CSR
  • 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
    24
    Mutliplicities @ CSR
  • 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)
    June 13-18, 2011
    25
    Mutliplicities @ CSR
  • 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
    26
    Mutliplicities @ CSR
  • 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.
    June 13-18, 2011
    27
    Mutliplicities @ CSR
  • 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
    28
    Mutliplicities @ CSR
  • 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!
    June 13-18, 2011
    29
    Mutliplicities @ CSR
  • 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).
    June 13-18, 2011
    30
    Mutliplicities @ CSR
  • 33. Conclusions
    Combinatorics does have many “techniques” …
    Polynomial method + Multiplicity method adds to the body
    Supporting evidence:
    List decoding
    Kakeya sets
    Extractors/Mergers
    Others …
    June 13-18, 2011
    31
    Mutliplicities @ CSR
  • 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.
    June 13-18, 2011
    32
    Mutliplicities @ CSR
  • 35. Conclusions
    New (?) technique in combinatorics …
    Polynomial method + Multiplicity method
    Supporting evidence:
    List decoding
    Kakeya sets
    Extractors/Mergers
    Locally decodable codes …
    More?
    June 13-18, 2011
    33
    Mutliplicities @ CSR
  • 36. Thank You
    June 13-18, 2011
    Mutliplicities @ CSR
    34
  • 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).
    June 13-18, 2011
    35
    Mutliplicities @ CSR
  • 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. ><
    June 13-18, 2011
    36
    Mutliplicities @ CSR
  • 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.
    June 13-18, 2011
    37
    Mutliplicities @ CSR
  • 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.
    Add: small seed s (Additional randomness)
    Output: a uniform random variable Y.
    June 13-18, 2011
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    Mutliplicities @ CSR
    (Uniform, independent … nearly)
    (Biased, correlated)
    + small pure seed