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Advancements in Locally Decodable Codes and Polynomial Identity Testing for Depth 3 Circuits

This talk presents significant advancements in the field of Locally Decodable Codes (LDCs) and Polynomial Identity Testing (PIT) focusing on depth 3 circuits. Notable improvements in bounds for 2-queries LDCs and structural theorems for identically zero depth 3 circuits are discussed. Key results include deriving 2-LDCs from circuits that evaluate to zero and establishing lower bounds. The implications suggest enhanced understanding for constructing LDCs and provide new insights for proving lower bounds on their lengths. The research integrates multiple facets of theoretical computer science, including circuit complexity.

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Advancements in Locally Decodable Codes and Polynomial Identity Testing for Depth 3 Circuits

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  1. Locally decodable codes with 2 queriesandpolynomial identity testing for depth 3 circuits Zeev Dvir Weizmann Institute of Science Amir Shpilka Technion

  2. This talk • Explaining the title: • Locally Decodable codes • Polynomial identity testing • depth 3 circuits • Results: • Improved bounds for 2-queries LDC's • Getting 2-LDC's from identically zero depth 3 circuits. • Structural theorem for identically zero depth 3 circuits • PIT for depth 3 circuits

  3. w.h.p Algorithm Locally decodable codes Def: E: Fn!Fm is q-LDC if xk can be recovered from q entries of E(x). Even if E(x) is corrupted in m coordinates. With high probability. k xk

  4. Main questions: constructing LDC's, proving lower bounds on their length. • Known constructions: q-LDC E: Fn!Fm with m = exp(nloglog(q)/q¢log(q)) [BIKR02]. • Lower bounds: • [KT00]: m = (n1 + 1/q-1) • [GKST01]: In linear 2-LDC over Fm = exp((n)- log|F|) • [KdW03]: In 2-LDC over {0,1} m = exp((n)). • Our result: In linear 2-LDC m = exp((n)). Works for every field size, i.e. F=R.

  5. f(x1,...,xn) Polynomial identity testing ?  0 Assumption: f has succinct representation. Motivation: Natural problem, many applications: primality testing, finding matching ... Schwartz-Zippel: Evaluate f(x) at a random point. Long Term Goals: Deterministic algorithm. Short Term Goals: Restricted Models.

  6. General circuits: Randomized algorithms [S80],[Z79],[CK97],[LV98],[AB03]: • poly(1/,size) time, n¢log(d/) random bits • Hardness vs. Randomness trade-off: [KI03] • PIT 2 P ) arithmetic lower bound for NEXP • NEXP * P/poly or • PERM  arithmetic P/poly • Lower bounds for arithmetic circuits imply sub-exponential time deterministic algs. •  look for PIT where l.b. are known!

  7. Non-commutative formulas: (vars do not commute) • [N91] exponential lower bound on formula size • [RS04] PIT determ. poly-time in size of formula • “Depth 2” circuits: (sparse polynomials) • [BoT88],[GKS90],...,[KS01]: deterministic poly time. • No sub-exp time deterministic algs. for depth > 2 • Open [KS01]: depth 3 circuits w. top fan-in = 3. • This paper: depth 3 circuits with small top fan-in:deterministic: quasi-polynomial time PIT alg. • (poly time for multilinear circuits).randomized: polynomial time polylog random bits. • New result: [KS06] polynomial time algorithm.

  8. + + + + M1 Mk L1,1 X X X X L1,d1 Depth 3 circuits - (k) circuits top fan-in ck c1 a1 an a0 Li,j = t=1...n at¢xt + a0 Mi = j=1...diLi,j ... x1 xn 1 C(x) = i=1...k ci¢Mi = ici¢jLi,j

  9. What's next: • Sketch of lower bound for 2-LDC. • PIT for depth 3 circuits with top fan-in = 2. • PIT for depth 3 circuits with top fan-in = 3. • General depth 3 circuits (sketch) • Structural theorem for identically zero depth 3 circuits. • PIT algorithms for depth 3 circuits

  10. Thm 1 [GKST01]: For any linear 2-LDC over {0,1} of length m, m = exp((n)). • Proof:Isoperimetric inequality. • Thm 2 [GKST01]: For any linear 2-LDC over F of length m, m = exp((n) – log|F|). • Proof: combine next lemma with theorem 1. • Lemma [GKST01]: If 9 linear 2-LDC over F of length m then 9 linear 2-LDC over {0,1} of length |F|¢ m. • Proof: randomly map all multiples of all coordinates to {0,1}. • New Lemma: If 9 linear 2-LDC over F of length m then 9 linear 2-LDC over {0,1} of length m. • Proof: randomly map well chosen multiple of each coordinate to {0,1}.

  11. What's next: • Sketch of lower bound for 2-LDC. • PIT for depth 3 circuits with top fan-in = 2. • PIT for depth 3 circuits with top fan-in = 3. • General depth 3 circuits (sketch) • Structural theorem for identically zero depth 3 circuits. • PIT algorithms for depth 3 circuits

  12. Identically zero (2) circuits • Reminder: C(x) = c1¢M1 + c2¢M2 M1(x)= M2(x)= Fact: linear functions are irreducible polynomial. Corollary: C0 then M1, M2 have the same factors. Corollary: 9 matching i  j(i) s.t. Li ~ L'j(i) PIT algorithm: look for such a matching.

  13. What's next: • Sketch of lower bound for 2-LDC. • PIT for depth 3 circuits with top fan-in = 2. • PIT for depth 3 circuits with top fan-in = 3. • General depth 3 circuits (sketch) • Structural theorem for identically zero depth 3 circuits. • PIT algorithms for depth 3 circuits

  14. Preliminaries • Claim: wlog linear functions are homogeneous (no constant term). • Claim: A:FnFn invertible linear map, then C(x)0 , C(A¢x)0. • Definition:r, rank(C) , rank(linear functions in C). • Corollary 1: wlog Li's depend only on x1,...,xr. • Corollary 2: wlog x1,...,xr appear as linear functions in C.

  15. 0 • Input: C(x) = c1¢M1 + c2¢M2 + c3¢M3 • x1,...,xr are linear functions in C • wlog assume g.c.d.(M1,M2,M3) = 1 M1(x)= M2(x)= 0 M3(x)= • Idea: reduction to (2): C0 ) C|xs=0 0 • ) if xs2M1 then c2¢M2|xs=0+c3¢M3|xs=0=0. • Lemma: 8xs9d pairs (i,j(i)) s.t. Li|xs=0 ~ Lj(i)|xs=0

  16. Input: C(x) = c1¢M1 + c2¢M2 + c3¢M3 • M1 = i=1...dLi(x), M2=i=1...dLd+i(x), M3=i=1...dL2d+i(x) • x1,...,xr are linear functions in C • Lemma: 8xs9d pairs (i,j(i)) s.t. Li|xs=0 ~ Lj(i)|xs=0 • Lemma: i=1,2 Li2Mi: L1|xs=0 ~ L2|xs=0) xs2 span(L1,L2) • Proof: Otherwise L1~ L2) L1 | M1,M2) if C0 then L1 | M3) L12 g.c.d(M1,M2,M3) ? • Define E(x) = L1(x),...,L3d(x) • Claim: 8s 9d pairs (i,j(i)) s.t. xs2 span(E(x)i,E(x)j(i)). • Corollary: E is a 2-LDC of length 3d. • Corollary: 3d=exp((r)) ) r=O(log(d)). • Thm: If C0 is (3) then rank(C) = O(log(d)). • PIT Algorithm: brute force. time = exp(log(d)2).

  17. What's next: • Sketch of lower bound for 2-LDC. • PIT for depth 3 circuits with top fan-in = 2. • PIT for depth 3 circuits with top fan-in = 3. • General depth 3 circuits (sketch) • Structural theorem for identically zero depth 3 circuits. • PIT algorithms for depth 3 circuits

  18. Input: C(x) = c1¢M1 + c2¢M2 + c3¢M3 + ... + ck¢Mk • Def: C is simple if g.c.d.(M1,...,Mk)=1 • Def: sim(C) = C/g.c.d.(C) • Def: C is minimal if no sub-circuit is zero. • Thm: C0 is simple and minimal, r = rank(C), d = deg(C). Then 9 2-LDC E: FaFb s.t. a = r/2k2log(d)k-3 b = kd • Corollary: rank(C) · O(log(d)k-2) • Proof: induction on k. Assume x1,...,xr2 C. Consider C|xi = 0 0. Top fan-in is k-1. Done? • simple? minimal? rank?

  19. Claim: 8xs9Is s.t. (CIs)|xs=0 0 and minimal 0 M1(x)= M2(x)= Is M3(x)= ... Mk(x)= Cor: 9I,r' ¸ r/2k s.t. (wlog) 8 1· s · r' (CI)|xs=0 0 and minimal.

  20. Cor: 9I,r' ¸ r/2k s.t. 8 1 · s · r' (CI)|xs=0 0 and minimal. • Optimistic: done? • Problematic: what's the rank of (CI)|xs=0 ? • Optimistic: lemma: rank(CI) ¸ r' ¸ r/2k • Problematic: (CI)|xs=0 not simple • Optimistic: consider sim((CI)|xs=0 ) (removing g.c.d.) • Problematic: what happens to the rank? • Optimistic: eh ... • Lemma: 9 xi s.t. • rank(sim((CI)|xs=0)) ¸ rank(CI)/2klog(d) • Proof: … • End of proof: induction on (CI)|xi=0 (from Lemma).

  21. What's next: • Sketch of lower bound for 2-LDC. • PIT for depth 3 circuits with top fan-in = 2. • PIT for depth 3 circuits with top fan-in = 3. • General depth 3 circuits (sketch) • Structural theorem for identically zero depth 3 circuits. • PIT algorithms for depth 3 circuits

  22. Structural theorem: C  0 is (k) then: • 9 partition I1t I2t ... t Im = [k] s.t. • CIj 0 minimal (C = CI1 + CI2 + ... + CIm) • rank(sim(CIj)) · O(log(d)|Ij|-2) • PIT algorithm: For each I ½ [k] check whether rank(sim(CI)) · O(log(d)|I|-2) • if yes then brute force check if CI 0 • if 9 partition as in theorem then C  0 • Running time: exp(log(d)k-1).

  23. The Multilinear Case • If C is multilinear then rank(C)=d. • But we proved that if C=0 is simple and minimal then rank(C) · polylog(d) • We get that d · polylog(d) • Can only hold for finitely many values ! • Conclusion: d · O(1) • rank(C) · dk · O(1) • Polynomial time algorithm

  24. Open problems (Multilinear) • PIT algorithms for stronger models: • Depth 3 circuits • Bounded depth • Tightness of our results: • Conjecture: If C  0 is (k) simple, minimal then rank(C) = poly(k). • [KS06] Not true for finite fields! Example in of a circuit with top fanin=3 and rank ~ log(d) (Multilinear)

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