The Method of Multiplicities Madhu Sudan Microsoft New England/MIT TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A Based on joint works with: V. Guruswami 98 S. Saraf 08 Z. Dvir, S. Kopparty, S. Saraf 09 September 23, Stanford1

of 33 Agenda A technique for combinatorics, via algebra: A technique for combinatorics, via algebra: Polynomial (Interpolation) Method + Multiplicity method Polynomial (Interpolation) Method + Multiplicity method List-decoding of Reed-Solomon Codes List-decoding of Reed-Solomon Codes Bounding size of Kakeya Sets Bounding size of Kakeya Sets Extractor constructions Extractor constructions September 23, Stanford

of 33 Part I: Decoding Reed-Solomon Codes Reed-Solomon Codes: Reed-Solomon Codes: Commonly used codes to store information (on CDs, DVDs etc.) Commonly used codes to store information (on CDs, DVDs etc.) Message: C 0, C 1, …, C d є F (finite field) Message: C 0, C 1, …, C d є F (finite field) Encoding: Encoding: View message as polynomial: M(x) = i=0 d C i x i View message as polynomial: M(x) = i=0 d C i x i Encoding = evaluations: { M( ® ) }_{ ® є F } Encoding = evaluations: { M( ® ) }_{ ® є F } Decoding Problem: Decoding Problem: Given: (x 1,y 1 ) … (x n,y n ) є F x F ; integers t,d; Given: (x 1,y 1 ) … (x n,y n ) є F x F ; integers t,d; Find: deg. d poly through t of the n points. Find: deg. d poly through t of the n points. September 23, Stanford

of 33 List-decoding? If #errors (n-t) very large, then several polynomials may agree with t of n points. If #errors (n-t) very large, then several polynomials may agree with t of n points. List-decoding problem: List-decoding problem: Report all such polynomials. Report all such polynomials. Combinatorial obstacle: Combinatorial obstacle: There may be too many such polynomials. There may be too many such polynomials. Hope – cant happen. Hope – cant happen. To analyze: Focus on polynomials P 1,…, P L and set of agreements S 1 … S L. To analyze: Focus on polynomials P 1,…, P L and set of agreements S 1 … S L. Combinatorial question: Can S 1, … S L be large, while n = | [ j S j | is small? Combinatorial question: Can S 1, … S L be large, while n = | [ j S j | is small? September 23, Stanford

of 33 List-decoding of Reed-Solomon codes Given L polynomials P 1,…,P L of degree d; and sets S 1,…,S L ½ F £ F s.t. Given L polynomials P 1,…,P L of degree d; and sets S 1,…,S L ½ F £ F s.t. |S i | = t |S i | = t S i ½ {(x,P i (x)) | x 2 F} S i ½ {(x,P i (x)) | x 2 F} How small can n = |S| be, where S = [ i S i ? How small can n = |S| be, where S = [ i S i ? Algebraic analysis from [S. 96, GuruswamiS 98] basis of decoding algorithms. Algebraic analysis from [S. 96, GuruswamiS 98] basis of decoding algorithms. September 23, Stanford

of 33 List-decoding analysis [S 96] Construct Q(x,y) 0 s.t. Construct Q(x,y) 0 s.t. Deg y (Q) < L Deg y (Q) < L Deg x (Q) < n/L Deg x (Q) < n/L Q(x,y) = 0 for every (x,y) 2 S = [ i S i Q(x,y) = 0 for every (x,y) 2 S = [ i S i Can Show: Can Show: Such a Q exists (interpolation/counting). Such a Q exists (interpolation/counting). Implies: t > n/L + dL ) (y – P i (x)) | Q Implies: t > n/L + dL ) (y – P i (x)) | Q Conclude: n ¸ L ¢ (t – dL). Conclude: n ¸ L ¢ (t – dL). (Can be proved combinatorially also; (Can be proved combinatorially also; using inclusion-exclusion) using inclusion-exclusion) If L > t/(2d), yield n ¸ t 2 /(4d) If L > t/(2d), yield n ¸ t 2 /(4d) September 23, Stanford

of 33 Focus: The Polynomial Method To analyze size of algebraically nice set S: To analyze size of algebraically nice set S: Find polynomial Q vanishing on S; Find polynomial Q vanishing on S; (Can prove existence of Q by counting coefficients … degree Q grows with |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. Use algebraic niceness of S to prove Q vanishes at other places as well. (In our case whenever y = P i (x) ). (In our case whenever y = P i (x) ). Conclude Q zero too often (unless S large). Conclude Q zero too often (unless S large). … (abstraction based on [Dvir]s work) September 23, Stanford

of 33 Improved L-D. Analysis [G.+S. 98] Can we improve on the inclusion-exclusion bound? One that works when n > t 2 /(4d)? Can we improve on the inclusion-exclusion bound? One that works when n > t 2 /(4d)? Idea: Try fitting a polynomial Q that passes through each point with multiplicity 2. Idea: Try fitting a polynomial Q that passes through each point with multiplicity 2. Can find with Deg y < L, Deg x < 3n/L. Can find with Deg y < L, Deg x < 3n/L. If 2t > 3n/L + dL then (y-P i (x)) | Q. If 2t > 3n/L + dL then (y-P i (x)) | Q. Yields n ¸ (L/3).(2t – dL) Yields n ¸ (L/3).(2t – dL) If L>t/d, then n ¸ t 2 /(3d). If L>t/d, then n ¸ t 2 /(3d). Optimizing Q; letting mult. 1, get n ¸ t 2 /d Optimizing Q; letting mult. 1, get n ¸ t 2 /d September 23, Stanford

of 33 Aside: Is the factor of 2 important? Results in some improvement in [GS] (allowed us to improve list-decoding for codes of high rate) … Results in some improvement in [GS] (allowed us to improve list-decoding for codes of high rate) … But crucial to subsequent work But crucial to subsequent work [Guruswami-Rudra] construction of rate- optimal codes: Couldnt afford to lose this factor of 2 (or any constant > 1). [Guruswami-Rudra] construction of rate- optimal codes: Couldnt afford to lose this factor of 2 (or any constant > 1). September 23, Stanford

of 33 Focus: The Multiplicity Method To analyze size of algebraically nice set S: To analyze size of algebraically nice set S: Find poly Q zero on S (w. high multiplicity); Find poly Q zero on S (w. high multiplicity); (Can prove existence of Q by counting coefficients … degree Q grows with |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. Use algebraic niceness of S to prove Q vanishes at other places as well. (In our case whenever y = P i (x) ). (In our case whenever y = P i (x) ). Conclude Q zero too often (unless S large). Conclude Q zero too often (unless S large). September 23, Stanford

of 33 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 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? Over finite fields? Derivatives dont work; but Hasse derivatives do. What are these? Later… Derivatives dont work; but Hasse derivatives do. What are these? Later… There are {m+n choose n} such derivatives, for n-variate polynomials; There are {m+n choose n} such derivatives, for n-variate polynomials; Each is a linear function of coefficients of f. Each is a linear function of coefficients of f. September 23, Stanford

of 33 Part II: Kakeya Sets September 23, Stanford

of 33 Kakeya Sets K ½ F n is a Kakeya set if it has a line in every direction. K ½ F n is a Kakeya set if it has a line in every direction. I.e., 8 y 2 F n 9 x 2 F n s.t. {x + t.y | t 2 F} ½ K I.e., 8 y 2 F n 9 x 2 F n s.t. {x + t.y | t 2 F} ½ K F is a field (could be Reals, Rationals, Finite). F is a field (could be Reals, Rationals, Finite). Our Interest: Our Interest: F = F q (finite field of cardinality q). F = F q (finite field of cardinality q). Lower bounds. Lower bounds. Simple/Obvious: q n/2 · K · q n Simple/Obvious: q n/2 · K · q n Do better? Mostly open till [Dvir 2008]. Do better? Mostly open till [Dvir 2008]. September 23, Stanford

of 33 Kakeya Set analysis [Dvir 08] Find Q(x 1,…,x n ) 0 s.t. Find Q(x 1,…,x n ) 0 s.t. Total deg. of Q < q (let deg. = d) Total deg. of Q < q (let deg. = d) Q(x) = 0 for every x 2 K. ( exists if |K| < q n /n!) Q(x) = 0 for every x 2 K. ( exists if |K| < q n /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. 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. Line L ½ K ) Q| L = 0. Highest degree coefficient of Q| L is homogenous part of Q evaluated at y. Highest degree coefficient of Q| L is homogenous part of Q evaluated at y. Conclude: homogenous part of Q = 0. > <. Yields |K| ¸ q n /n!. Yields |K| ¸ q n /n!. September 23, Stanford

of 33 Multiplicities in Kakeya [Saraf, S 08] Fit Q that vanishes often? Fit Q that vanishes often? Good choice: #multiplicity m = n 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| 4 n < q n Can find Q 0 of individual degree < q, that vanishes at each point in K with multiplicity n, provided |K| 4 n < q n Q| L is of degree < qn. Q| L is of degree < qn. But it vanishes with multiplicity n at q points! 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 Conclude: |K| ¸ (q/4) n September 23, Stanford

of 33 Comparing the bounds Simple: |K| ¸ q n/2 Simple: |K| ¸ q n/2 [Dvir]: |K| ¸ q n /n! [Dvir]: |K| ¸ q n /n! [SS]: |K| ¸ q n /4 n [SS]: |K| ¸ q n /4 n [SS] improves Simple even when q (large) constant and n 1 (in particular, allows q < n) [SS] improves Simple even when q (large) constant and n 1 (in particular, allows q < n) [MockenhauptTao, Dvir]: [MockenhauptTao, Dvir]: 9 K s.t. |K| · q n /2 n-1 + O(q n-1 ) 9 K s.t. |K| · q n /2 n-1 + O(q n-1 ) Can we do even better? Can we do even better? September 23, Stanford

of 33 Part III: Randomness Mergers & Extractors September 23, Stanford

of 33 Context One of the motivations for Dvirs work: One of the motivations for Dvirs work: Build better randomness extractors Build better randomness extractors Approach proposed in [Dvir-Shpilka] Approach proposed in [Dvir-Shpilka] Following [Dvir], new randomness merger and analysis given by [Dvir-Wigderson] Following [Dvir], new randomness merger and analysis given by [Dvir-Wigderson] Led to extractors matching known constructions, but not improving them … Led to extractors matching known constructions, but not improving them … What are Extractors? Mergers? … can we improve them? What are Extractors? Mergers? … can we improve them? September 23, Stanford

of 33 Random Uniform Randomness in Computation Alg X F(X) Randomness Processors Distribution A Distribution B Support industry: Readily available randomness Prgs, (seeded) extractors, limited independence generators, epsilon-biased generators, Condensers, mergers, September 23, Stanford

of 33 Randomness Extractors and Mergers Extractors: Extractors: Dirty randomness Pure randomness Dirty randomness Pure randomness Mergers: General primitive useful in the context of manipulating randomness. Mergers: General primitive useful in the context of manipulating randomness. k random variables 1 random variable k random variables 1 random variable September 23, Stanford (Biased, correlated) (Uniform, independent … nearly) + small pure seed (One of them uniform) (high entropy) (Dont know which, others potentially correlated) + small pure seed

of 33 Merger Analysis Problem Merger(X 1,…,X k ; s) = f(s), Merger(X 1,…,X k ; s) = f(s), where X 1, …, X k 2 F q n ; s 2 F q where X 1, …, X k 2 F q n ; s 2 F q and f is deg. k-1 function mapping F F n and f is deg. k-1 function mapping F F n s.t. f(i) = X i. s.t. f(i) = X i. (f is the curve through X 1,…,X k ) (f is the curve through X 1,…,X k ) Question: For what choices of q, n, k is Mergers output close to uniform? Question: For what choices of q, n, k is Mergers output close to uniform? Arises from [DvirShpilka05, DvirWigderson08]. Arises from [DvirShpilka05, DvirWigderson08]. Statistical high-deg. version of Kakeya problem. Statistical high-deg. version of Kakeya problem. September 23, Stanford

of 33 Concerns from Merger Analysis [DW] Analysis: Worked only if q > n. [DW] Analysis: Worked only if q > n. So seed length = log 2 q > log 2 n So seed length = log 2 q > log 2 n Not good enough for setting where k = O(1), and n 1. Not good enough for setting where k = O(1), and n 1. (Would like seed length to be O(log k)). (Would like seed length to be O(log k)). Multiplicity technique: Multiplicity technique: seems bottlenecked at mult = n. seems bottlenecked at mult = n. September 23, Stanford

of 33 General obstacle in multiplicity method Cant force polynomial Q to vanish with too high a multiplicity. Gives no benefit. Cant force polynomial Q to vanish with too high a multiplicity. Gives no benefit. E.g. Kakeya problem: Why stop at mult = n? E.g. Kakeya problem: Why stop at mult = n? Most we can hope from Q is that it vanishes on all of q n ; Most we can hope from Q is that it vanishes on all of q n ; Once this happens, Q = 0, if its degree is < q in each variable. Once this happens, Q = 0, if its degree is < q in each variable. So Q| L is of degree at most qn, so mult n suffices. Using larger multiplicity cant help! So Q| L is of degree at most qn, so mult n suffices. Using larger multiplicity cant help! Or can it? Or can it? September 23, Stanford

of 33 Extended method of multiplicities (In Kakeya context): (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? 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.) (Needed: Condition on multiplicity of zeroes of multivariate polynomials.) Perhaps Q can be shown to vanish with high multiplicity at each point in F n. Perhaps Q can be shown to vanish with high multiplicity at each point in F n. (Technical question: How?) (Technical question: How?) September 23, Stanford

of 33 Vanishing of high-degree polynomials Mult(Q,a) = multiplicity of zeroes of Q at a. Mult(Q,a) = multiplicity of zeroes of Q at a. I(Q,a) = 1 if mult(Q,a) > 0 and 0 o.w. I(Q,a) = 1 if mult(Q,a) > 0 and 0 o.w. = min{1, mult(Q,a)} = min{1, mult(Q,a)} Schwartz-Zippel: for any S ½ F Schwartz-Zippel: for any S ½ F I(Q,a) · d. |S| n-1 where sum is over a 2 S n I(Q,a) · d. |S| n-1 where sum is over a 2 S n Can we replace I with mult above? Would strengthen S-Z, and be useful in our case. Can we replace I with mult above? Would strengthen S-Z, and be useful in our case. [DKSS 09]: Yes … (simple inductive proof [DKSS 09]: Yes … (simple inductive proof … that I can never remember) … that I can never remember) September 23, Stanford

of 33 Multiplicities? Q(X 1,…,X n ) has zero of mult. m at a = (a 1,…,a n ) if all (Hasse) derivatives of order < m vanish. Q(X 1,…,X n ) has zero of mult. m at a = (a 1,…,a n ) if all (Hasse) derivatives of order < m vanish. Hasse derivative = ? Hasse derivative = ? Formally defined in terms of coefficients of Q, various multinomial coefficients and a. Formally defined in terms of coefficients of Q, various multinomial coefficients and a. But really … But really … The i = (i1,…, in)th derivative is the coefficient of z 1 i1 …z n in in Q(z + a). The i = (i1,…, in)th derivative is the coefficient of z 1 i1 …z n in in Q(z + a). Even better … coeff. of z i in Q(z+x) Even better … coeff. of z i in Q(z+x) (defines ith derivative Q i as a function of x; can evaluate at x = a). (defines ith derivative Q i as a function of x; can evaluate at x = a). September 23, Stanford

of 33 Key Properties Each derivative is a linear function of coefficients of Q. [Used in [GS98], [SS09].] (Q+R) i = Q i + R i Each derivative is a linear function of coefficients of Q. [Used in [GS98], [SS09].] (Q+R) i = Q i + R i Q has zero of mult m at a, and S is a curve that passes through a, then Q| S has zero of mult m at a. [Used for lines in prior work.] Q has zero of mult m at a, and S is a curve that passes through a, then Q| S has zero of mult m at a. [Used for lines in prior work.] Q i is a polynomial of degree deg(Q) - j i i (not used in prior works) Q i is a polynomial of degree deg(Q) - j i i (not used in prior works) (Q i ) j Q i+j, but Q i+j (a) = 0 ) (Q i ) j (a) = 0 (Q i ) j Q i+j, but Q i+j (a) = 0 ) (Q i ) j (a) = 0 Q vanishes with mult m at a Q vanishes with mult m at a ) Q i vanishes with mult m - j i i at a. ) Q i vanishes with mult m - j i i at a. September 23, Stanford

of 33 Propagating multiplicities (in Kakeya) Find Q that vanishes with mult m on K Find Q that vanishes with mult m on K For every i of order m/2, Q_i vanishes with mult m/2 on K. For every i of order m/2, Q_i vanishes with mult m/2 on K. Conclude: Q, as well as all derivatives of Q of order m/2 vanish on F n Conclude: Q, as well as all derivatives of Q of order m/2 vanish on F n ) Q vanishes with multiplicity m/2 on F n ) Q vanishes with multiplicity m/2 on F n Next Question: When is a polynomial (of deg > qn, or even q n ) that vanishes with high multiplicity on q n identically zero? Next Question: When is a polynomial (of deg > qn, or even q n ) that vanishes with high multiplicity on q n identically zero? September 23, Stanford

of 33 Back to Kakeya Find Q of degree d vanishing on K with mult m. (can do if (m/n) n |K| m n |K| ) Find Q of degree d vanishing on K with mult m. (can do if (m/n) n |K| m n |K| ) Conclude Q vanishes on F n with mult. m/2. Conclude Q vanishes on F n with mult. m/2. Apply Extended-Schwartz-Zippel to conclude Apply Extended-Schwartz-Zippel to conclude (m/2) q n < d q n-1 (m/2) q n < d q n-1, (m/2) q < d, (m/2) n q n < d n = m n |K| Conclude: |K| ¸ (q/2) n Conclude: |K| ¸ (q/2) n Tight to within 2+o(1) factor! Tight to within 2+o(1) factor! September 23, Stanford

of 33 Consequences for Mergers Can analyze [DW] merger when q > k very small, n growing; Can analyze [DW] merger when q > k very small, n growing; Analysis similar, more calculations. Analysis similar, more calculations. Yields: Seed length log q (independent of n). Yields: Seed length log q (independent of n). By combining it with every other ingredient in extractor construction: 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). Extract all but vanishing entropy (k – o(k) bits of randomness from (n,k) sources) using O(log n) seed (for the first time). September 23, Stanford

of 33 Conclusions Combinatorics does have many techniques … Combinatorics does have many techniques … Polynomial method + Multiplicity method adds to the body Polynomial method + Multiplicity method adds to the body Supporting evidence: Supporting evidence: List decoding List decoding Kakeya sets Kakeya sets Extractors/Mergers Extractors/Mergers Others … Others … September 23, Stanford

of 33 Other applications [Woodruff-Yekhanin 05]: An elegant construction of novel LDCs (locally decodable codes). [Outclassed by more recent Yekhanin/Efremenko constructions.] [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-Lev-Saraf-S. 09]: Higher dimensional Kakeya problems. [Kopparty-Saraf-Yekhanin yesterday]: Locally decodable codes with Rate 1. [Kopparty-Saraf-Yekhanin yesterday]: Locally decodable codes with Rate 1. September 23, Stanford

of 33 Conclusions New (?) technique in combinatorics … New (?) technique in combinatorics … Polynomial method + Multiplicity method Polynomial method + Multiplicity method Supporting evidence: Supporting evidence: List decoding List decoding Kakeya sets Kakeya sets Extractors/Mergers Extractors/Mergers Locally decodable codes … Locally decodable codes … September 23, Stanford

Thank You September 23, Stanford34

of 33 Mutliplicity = ? Q vanishes with multiplicity 2 at (a,b): 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. Q(a,b) = 0; \del Q/del x(a,b) = 0; \del Q/\del y(a,b) = 0. \del Q/del x? \del Q/del x? Hasse derivative; Hasse derivative; Linear function of coefficients of Q; 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). 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). September 23, Stanford

of 33 Concerns from Merger Analysis Recall Merger(X 1,…,X k ; s) = f(s), Recall Merger(X 1,…,X k ; s) = f(s), where X 1, …, X k 2 F q n ; s 2 F q where X 1, …, X k 2 F q n ; s 2 F q and f is deg. k-1 curve s.t. f(i) = X i. and f is deg. k-1 curve s.t. f(i) = X i. [DW08] Say X 1 random; Let K be such that ² fraction of choices of X 1,…,X k lead to bad curves such that ² fraction of ss such that Merger outputs value in K with high probability. [DW08] Say X 1 random; Let K be such that ² fraction of choices of X 1,…,X k lead to bad curves such that ² fraction of ss such that Merger outputs value in K with high probability. Build low-deg. poly Q vanishing on K; Prove for bad curves, Q vanishes on curve; and so Q vanishes on ² -fraction of X 1 s (and so ² -fraction of domain). Build low-deg. poly Q vanishing on K; Prove for bad curves, Q vanishes on curve; and so Q vanishes on ² -fraction of X 1 s (and so ² -fraction of domain). Apply Schwartz-Zippel. > < September 23, Stanford

of 33 What is common? Given a set in F q n with nice algebraic properties, want to understand its size. Given a set in F q n with nice algebraic properties, want to understand its size. Kakeya Problem: Kakeya Problem: The Kakeya Set. The Kakeya Set. Merger Problem: Merger Problem: Any set T ½ F n that contains ² -fraction of points on ² -fraction of merger curves. Any set T ½ F n that contains ² -fraction of points on ² -fraction of merger curves. If T small, then output is non-uniform; else output is uniform. If T small, then output is non-uniform; else output is uniform. List-decoding problem: List-decoding problem: The union of the sets. The union of the sets. September 23, Stanford

of 33 Randomness Extractors and Mergers Extractors: Extractors: Dirty randomness Pure randomness Dirty randomness Pure randomness Mergers: General primitive useful in the context of manipulating randomness. Mergers: General primitive useful in the context of manipulating randomness. Given: k (dependent) random variables Given: k (dependent) random variables X 1 … X k, such that one is uniform. X 1 … X k, such that one is uniform. Add: small seed s (Additional randomness) Add: small seed s (Additional randomness) Output: a uniform random variable Y. Output: a uniform random variable Y. September 23, Stanford (Biased, correlated) (Uniform, independent … nearly) + small pure seed