The Method of Multiplicities<br />Madhu Sudan<br />Microsoft New England/MIT<br />Based on joint works with:<br /><ul><li>...
 S. Saraf ‘08
 Z. Dvir, S. Kopparty, S. Saraf ‘09</li></ul>June 13-18, 2011<br />Mutliplicities @ CSR<br />TexPoint fonts used in EMF. <...
Agenda<br />A technique for combinatorics, via algebra:<br />Polynomial (Interpolation) Method + Multiplicity method<br />...
Part I: Decoding Reed-Solomon Codes<br />Reed-Solomon Codes:<br />Commonly used codes to store information (on CDs, DVDs e...
List-decoding?<br />If #errors (n-t) very large, then several polynomials may agree with t of n points.<br />List-decoding...
List-decoding of Reed-Solomon codes<br />Given L polynomials P1,…,PLof degree d;and sets S1,…,SL½ F £ F s.t.<br />|Si| = t...
List-decoding analysis [S ‘96]<br />Construct Q(x,y) ≠ 0 s.t.<br />Degy(Q) < L<br />Degx(Q) < n/L   <br />Q(x,y) = 0 for e...
Focus: The Polynomial Method<br />To analyze size of “algebraically nice” set S:<br />Find polynomial Q vanishing on S;<br...
Improved L-D. Analysis [G.+S. ‘98]<br />Can we improve on the inclusion-exclusion bound? One that works when n > t2/(4d)?<...
Aside: Is the factor of 2 important?<br />Results in some improvement in [GS] (allowed us to improve list-decoding for cod...
Focus: The Multiplicity Method<br />To analyze size of “algebraically nice” set S:<br />Find poly Q zero on S (w. high mul...
Multiplicity = ?<br />Over reals: Q(x,y) has root of multiplicity m+1 at (a,b) if every partial derivative of order up to ...
Part II: Kakeya Sets<br />June 13-18, 2011<br />12<br />Mutliplicities @ CSR<br />
Kakeya Sets<br />K ½Fn is a Kakeya set if it has a line in every direction.<br />I.e., 8 y 2Fn9 x 2Fns.t. {x + t.y | t 2 F...
Kakeya Set analysis [Dvir ‘08]<br />Find Q(x1,…,xn) ≠ 0 s.t.<br />Total deg. of Q < q (let deg. = d)<br />Q(x) = 0 for eve...
Multiplicities in Kakeya [Saraf, S ’08]<br />Fit Q that vanishes often?<br />Good choice: #multiplicity m = n<br />Can fin...
Comparing the bounds<br />Simple: |K| ¸qn/2<br />[Dvir]: |K| ¸qn/n!<br />[SS]: |K| ¸qn/4n<br />[SS] improves Simple even w...
Part III: Randomness Mergers & Extractors<br />June 13-18, 2011<br />17<br />Mutliplicities @ CSR<br />
Context<br />One of the motivations for Dvir’s work:<br />Build better “randomness extractors”<br />Approach proposed in [...
Randomness in Computation<br />Readily available randomness<br />Support industry:<br />F(X)<br />X<br />Alg<br />Random<b...
Randomness Extractors and Mergers<br />Extractors: <br />Dirty randomness  Pure randomness<br />Mergers: General primitiv...
Merger Analysis Problem <br />Merger(X1,…,Xk; s) = f(s),<br />where X1, …, Xk2Fqn; s 2Fq<br />        andfis deg. k-1funct...
Concerns from Merger Analysis<br />[DW] Analysis: Worked only if q > n.<br />So seed length = log2 q > log2 n<br />Not goo...
General obstacle in multiplicity method<br />Can’t force polynomial Q to vanish with too high a multiplicity. Gives no ben...
Extended method of multiplicities<br />(In Kakeya context):<br />Perhaps vanishing of Q with high multiplicity at each poi...
Vanishing of high-degree polynomials<br />Mult(Q,a) = multiplicity of zeroes of Qata.<br />I(Q,a) = 1 if mult(Q,a) > 0and0...
Multiplicities?<br />Q(X1,…,Xn) has zero of mult. m at a = (a1,…,an)if all (Hasse) derivatives of order < m vanish.<br />H...
Key Properties<br />Each derivative is a linear function of coefficients of Q. [Used in [GS’98], [SS’09] .] (Q+R)i = Qi + ...
Propagating multiplicities (in Kakeya)<br />FindQthat vanishes with multmonK<br />For everyiof order m/2, Q_ivanishes with...
Back to Kakeya<br />Find Q of degree d vanishing on K with multm. (can do if (m/n)n |K| < (d/n)n,dn > mn |K| )<br />Conclu...
Consequences for Mergers<br />Can analyze [DW] merger when q > k very small, n growing; <br />Analysis similar, more calcu...
Conclusions<br />Combinatorics does have many “techniques” …<br />Polynomial method + Multiplicity method adds to the body...
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Csr2011 june18 14_00_sudan

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

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