zeev dvir weizmann institute of science amir shpilka technion locally decodable codes with 2 queries...
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Zeev DvirZeev DvirWeizmann Institute of Weizmann Institute of
ScienceScience Amir ShpilkaAmir Shpilka
TechnionTechnion
Locally decodable codes with 2 Locally decodable codes with 2 queriesqueries
andand polynomial identity testing for depth polynomial identity testing for depth
3 circuits3 circuits
This talkThis talk• Explaining the title:Explaining the title:
– Locally Decodable codesLocally Decodable codes– Polynomial identity testingPolynomial identity testing– depth 3 circuitsdepth 3 circuits
• Results:Results:– Improved bounds for 2-queries LDC'sImproved bounds for 2-queries LDC's– Getting 2-LDC's from identically zero Getting 2-LDC's from identically zero
depth 3 circuits.depth 3 circuits.– Structural theorem for identically zero Structural theorem for identically zero
depth 3 circuitsdepth 3 circuits– PIT for depth 3 circuits PIT for depth 3 circuits
Locally decodable codesLocally decodable codes
xx11 ......
xxkk ...... xxnn yy11 yy22 ...... yyii ......
yyjj ...... yymm
xxkk
Def:Def: E: E: FFnn !! FFmm is is q-LDCq-LDC if x if xkk can be can be recovered from q entries of E(x). recovered from q entries of E(x).
Even if E(x) is corrupted in Even if E(x) is corrupted in m coordinates. m coordinates.
With high probability.With high probability.
AlgorithmAlgorithmk
w.h.p
Main questionsMain questions: constructing LDC's, : constructing LDC's, proving lower bounds on their length.proving lower bounds on their length.
Known constructions: Known constructions: q-LDC E: q-LDC E: FFnn !! FFmm with with m = exp(nm = exp(nloglog(q)/qloglog(q)/q¢¢log(q)log(q)) [) [BIKR02BIKR02].].
Lower boundsLower bounds::[[KT00KT00]: m = ]: m = (n(n1 + 1/q-11 + 1/q-1))[[GKST01GKST01]: In ]: In linear linear 2-LDC over 2-LDC over FF
m = exp(m = exp((n)- log|(n)- log|FF|)|)[[KdW03KdW03]: In 2-LDC over {]: In 2-LDC over {0,10,1} } m = m =
exp(exp((n)).(n)).
Our resultOur result: In : In linearlinear 2-LDC 2-LDC m = exp(m = exp((n)).(n)). Works for every field size, i.e. Works for every field size, i.e. FF==RR..
AssumptionAssumption: f has succinct representation.: f has succinct representation.
Motivation: Motivation: Natural problem, many Natural problem, many applications: primality testing, finding applications: primality testing, finding matching ...matching ...
Schwartz-ZippelSchwartz-Zippel: Evaluate f(x) at a random : Evaluate f(x) at a random point.point.
Long Term GoalsLong Term Goals: Deterministic algorithm.: Deterministic algorithm.
Short Term GoalsShort Term Goals: Restricted Models.: Restricted Models.
Polynomial identity testingPolynomial identity testing
f(x1,...,xn) 0 ?
General circuits: General circuits: Randomized algorithms Randomized algorithms [S80],[Z79],[CK97],[LV98],[AB03][S80],[Z79],[CK97],[LV98],[AB03]::
poly(poly(11//,size) time, n,size) time, n¢¢log(d/log(d/) random bits) random bits
Hardness vs. Randomness trade-off:Hardness vs. Randomness trade-off: [[KI03KI03] ]
PIT PIT 22 P P )) arithmetic lower bound for NEXP arithmetic lower bound for NEXP– NEXP NEXP ** P/poly P/poly oror – PERM PERM arithmetic P/poly arithmetic P/poly
Lower bounds for arithmetic circuits imply Lower bounds for arithmetic circuits imply sub-exponential time deterministic algs.sub-exponential time deterministic algs.
look for PIT where l.b. are known!look for PIT where l.b. are known!
Non-commutative formulasNon-commutative formulas: (vars do not commute): (vars do not commute)[N91][N91] exponential lower bound on formula size exponential lower bound on formula size[RS04] [RS04] PIT determ. poly-time in size of formulaPIT determ. poly-time in size of formula
““Depth 2” circuits: Depth 2” circuits: (sparse polynomials)(sparse polynomials)[BoT88],[GKS90],...,[KS01][BoT88],[GKS90],...,[KS01]: deterministic poly time.: deterministic poly time.
No sub-exp time deterministic algs. for depth > 2No sub-exp time deterministic algs. for depth > 2OpenOpen [ [KS01KS01]: depth 3 circuits w. top fan-in = 3.]: depth 3 circuits w. top fan-in = 3.
This paperThis paper: depth 3 circuits with small top fan-in:: depth 3 circuits with small top fan-in:deterministicdeterministic: quasi-polynomial time PIT alg. : quasi-polynomial time PIT alg.
(poly time for (poly time for multilinearmultilinear circuits). circuits).randomizedrandomized: polynomial time polylog random : polynomial time polylog random bits.bits.
New resultNew result: [: [KS06KS06] polynomial time algorithm.] polynomial time algorithm.
Depth 3 circuits - Depth 3 circuits - (k) circuits(k) circuits
+
XX XX XX XX
+++
top fan-in
x1 xn
ckc1
a1 an
1
a0
Li,j = t=1...n at¢xt + a0
Mi = j=1...diLi,j
...
M1
MkL1,1
C(x) = i=1...k ci¢Mi = ici
¢jLi,j
L1,d1
What's next:What's next:• Sketch of lower bound for 2-LDC.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 = 2.• PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3.• General depth 3 circuits (sketch)General depth 3 circuits (sketch)• Structural theorem for identically zero Structural theorem for identically zero
depth 3 circuits.depth 3 circuits.• PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits
Thm 1Thm 1 [ [GKST01GKST01]: For any linear 2-LDC over {0,1} of ]: For any linear 2-LDC over {0,1} of length m, m = exp(length m, m = exp((n)).(n)).
ProofProof:: Isoperimetric inequality.Isoperimetric inequality.
Thm 2Thm 2 [ [GKST01GKST01]: For any linear 2-LDC over ]: For any linear 2-LDC over FF of of length m, m = exp(length m, m = exp((n) – log|(n) – log|FF|).|).
Proof: Proof: combine next lemma with theorem 1.combine next lemma with theorem 1.
LemmaLemma [ [GKST01GKST01]: If ]: If 99 linear 2-LDC over linear 2-LDC over F F of length of length m then m then 99 linear 2-LDC over {0,1} of length | linear 2-LDC over {0,1} of length |FF||¢¢ m. m.
ProofProof: : randomly map all multiples of all coordinates to randomly map all multiples of all coordinates to {0,1}.{0,1}.
New LemmaNew Lemma: If : If 99 linear 2-LDC over linear 2-LDC over F F of length m of length m then then 99 linear 2-LDC over {0,1} of length m. linear 2-LDC over {0,1} of length m.
ProofProof: : randomly map randomly map well chosen multiplewell chosen multiple of each of each coordinate to {0,1}.coordinate to {0,1}.
What's next:What's next: Sketch of lower bound for 2-LDC.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 = 2.• PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3.• General depth 3 circuits (sketch)General depth 3 circuits (sketch)• Structural theorem for identically zero Structural theorem for identically zero
depth 3 circuits.depth 3 circuits.• PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits
Identically zero Identically zero (2) circuits(2) circuits
ReminderReminder: C(x) = c: C(x) = c11¢¢MM11 + c + c22¢¢MM22
LL11 LL22 ...... LLii ......
LLjj ...... LLdd
L'L'11 L'L'22 ...... L'L'ii ......
L'L'jj ...... L'L'dd
M1(x)=
M2(x)=FactFact: linear functions are irreducible : linear functions are irreducible polynomial.polynomial.
CorollaryCorollary: C: C0 then M0 then M11, M, M22 have the same have the same factors.factors.
CorollaryCorollary: : 99 matching i matching i j(i) s.t. L j(i) s.t. Lii ~ L' ~ L'j(i)j(i)
PIT algorithmPIT algorithm: look for such a matching.: look for such a matching.
What's next:What's next: Sketch of lower bound for 2-LDC.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 = 2.• PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3.• General depth 3 circuits (sketch)General depth 3 circuits (sketch)• Structural theorem for identically zero Structural theorem for identically zero
depth 3 circuits.depth 3 circuits.• PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits
PreliminariesPreliminariesClaimClaim: wlog linear functions are : wlog linear functions are
homogeneous (no constant term).homogeneous (no constant term).
ClaimClaim: A:: A:FFnnFFnn invertible linear map, then invertible linear map, then C(x)C(x)0 0 , , C(AC(A¢¢x)x)0.0.
DefinitionDefinition:: rr ,, rank(C) rank(C) ,, rank(linear functions in C). rank(linear functions in C).
Corollary 1Corollary 1: wlog L: wlog Lii's depend only on 's depend only on xx11,...,x,...,xrr..
Corollary 2Corollary 2: wlog x: wlog x11,...,x,...,xrr appear as linear appear as linear functions in C.functions in C.
InputInput: C(x) = c: C(x) = c11¢¢MM11 + c + c22¢¢MM2 2 + c+ c33¢¢MM33
xx11,...,x,...,xrr are linear functions in C are linear functions in C
wlog assumewlog assume g.c.d.(M g.c.d.(M11,M,M22,M,M33) = 1) = 1
M2(x)=
LL2d+12d+1 LL2d+22d+2 ...... LL2d+i2d+i ...... xxtt ...... LL3d3dM3(x)=
M1(x)=
Idea: Idea: reduction to reduction to (2): C(2): C0 0 ) ) C|C|xxss=0 =0 0 0
) ) if xif xss22MM11 then c then c22¢¢MM22||xxss=0=0+c+c33¢¢MM33||xxss=0=0=0.=0.
LemmaLemma: : 88xxss 99d pairs (d pairs (i,j(i)i,j(i)) s.t. L) s.t. Lii||xxss=0 =0 ~ L~ Lj(i)j(i)||xxss=0=0
0
LLd+1d+1 LLd+2d+2 ...... LLd+id+i ...... LLd+jd+j ...... LL2d2d
LL11 LL22 ...... LLii ...... xxss ...... LLdd
0
LemmaLemma: i=: i=1,21,2 L Lii22MMii: L: L11||xxss=0 =0 ~ L~ L22||xxss=0=0 ) ) xxss 22
span(Lspan(L11,L,L22))
ProofProof: Otherwise L: Otherwise L1 1 ~ L~ L22 )) L L11 | M | M11,M,M22
)) if C if C0 then L0 then L11 | M | M33 ) ) LL11 22 g.c.d(M g.c.d(M11,M,M22,M,M33) ) ??
DefineDefine E(x) = L E(x) = L11(x),...,L(x),...,L3d3d(x)(x)
ClaimClaim: : 88s s 99d pairs (i,j(i)) s.t. xd pairs (i,j(i)) s.t. xss 22 span(E(x) span(E(x)ii,E(x),E(x)j(i)j(i)).).
CorollaryCorollary: E is a 2-LDC of length 3d.: E is a 2-LDC of length 3d.
CorollaryCorollary: 3d=exp(: 3d=exp((r)) (r)) ) ) r=O(log(d))r=O(log(d))..
ThmThm: : If CIf C0 is 0 is (3) then rank(C) = O(log(d)).(3) then rank(C) = O(log(d)).
PIT AlgorithmPIT Algorithm: brute force. time = exp(log(d): brute force. time = exp(log(d)22).).
InputInput: C(x) = c: C(x) = c11¢¢MM11 + c + c22¢¢MM2 2 + c+ c33¢¢MM33
MM11 = = i=1...di=1...dLLii(x), M(x), M22==i=1...di=1...dLLd+id+i(x), M(x), M33==i=1...di=1...dLL2d+i2d+i(x) (x)
xx11,...,x,...,xrr are linear functions in C are linear functions in C
LemmaLemma: : 88xxss 99d pairs (d pairs (i,j(i)i,j(i)) s.t. L) s.t. Lii||xxss=0 =0 ~ L~ Lj(i)j(i)||xxss=0=0
What's next:What's next: Sketch of lower bound for 2-LDC.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 = 2. PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3.• General depth 3 circuits (sketch)General depth 3 circuits (sketch)• Structural theorem for identically zero Structural theorem for identically zero
depth 3 circuits.depth 3 circuits.• PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits
InputInput: C(x) = c: C(x) = c11¢¢MM11 + c + c22¢¢MM2 2 + c+ c33¢¢MM33 + ... + c + ... + ckk¢¢MMkk
DefDef: C is simple if g.c.d.(M: C is simple if g.c.d.(M11,...,M,...,Mkk)=1)=1
DefDef: : sim(C)sim(C) = C/g.c.d.(C) = C/g.c.d.(C)
DefDef: C is minimal if no sub-circuit is zero. : C is minimal if no sub-circuit is zero.
ThmThm: C: C0 is simple and minimal, r = rank(C), 0 is simple and minimal, r = rank(C), d = deg(C). Then d = deg(C). Then 99 2-LDC E: 2-LDC E: FFaa FFbb s.t. s.t.
aa = r/ = r/22kk22log(d)log(d)k-3k-3 b b = kd = kd
CorollaryCorollary: rank(C) : rank(C) ·· O O((log(d)log(d)k-2k-2))ProofProof: induction on k. Assume x: induction on k. Assume x11,...,x,...,xrr 22 C. C.
Consider C|Consider C|xxii = 0 = 0 0. Top fan-in is k-1. Done? 0. Top fan-in is k-1. Done?
simple? minimal? rank?simple? minimal? rank?
M2(x)=
LL3,13,1 LL3,23,2 ...... LL3,i3,i ...... xxtt ...... LL3,d3,dM3(x)=
M1(x)=
0
LL2,12,1 LL2,22,2 ...... LL2,i2,i ...... LL2,j2,j ...... LL2,d2,d
LL1,11,1 LL1,21,2 ...... LL1,i1,i ...... xxss ...... LL1,d1,d
LLk,1k,1 LLk,2k,2 ...... LLk,ik,i ...... LLk,jk,j ...... LLk,dk,dMk(x)=
Is
...
Claim: 8xs 9Is s.t. (CIs)|xs=0 0 and minimal
Cor: 9I,r' ¸ r/2k s.t. (wlog) 8 1· s · r' (CI)|xs=0 0 and minimal.
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).
What's next:What's next: Sketch of lower bound for 2-LDC.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 = 2. PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3. General depth 3 circuits (sketch)General depth 3 circuits (sketch)• Structural theorem for identically zero Structural theorem for identically zero
depth 3 circuits.depth 3 circuits.• PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits
Structural theoremStructural theorem: C : C 0 is 0 is (k) then:(k) then:
99 partition I partition I11 t t II22 t t ... ... tt I Imm = [k] s.t. = [k] s.t.
CCIjIj 0 minimal 0 minimal (C = C(C = CII11 + C + CII22
+ ... + C + ... + CIImm))
rank(sim(Crank(sim(CIjIj)) )) ·· O(log(d) O(log(d)|I|Ijj|-2|-2))
PIT algorithmPIT algorithm: For each I : For each I ½½ [k] check [k] check whether rank(sim(Cwhether rank(sim(CII)) )) ·· O(log(d) O(log(d)|I|-2|I|-2) )
if yes then brute force check if Cif yes then brute force check if CII 0 0
if if 99 partition as in theorem then C partition as in theorem then C 0 0
Running time: Running time: exp(log(d)exp(log(d)k-1k-1))..
The Multilinear CaseThe Multilinear Case
If C is multilinear then If C is multilinear then rank(C)=d.rank(C)=d.
But we proved that if C=0 is simple But we proved that if C=0 is simple and minimal then and minimal then rank(C) rank(C) ·· polylog(d) polylog(d)
We get that We get that d d · · polylog(d) polylog(d)
Can only hold for finitely many values !Can only hold for finitely many values !
Conclusion: Conclusion: d d ·· O(1) O(1)
rank(C) rank(C) ·· dk dk ·· O(1) O(1)
Polynomial time algorithmPolynomial time algorithm
Open problemsOpen problems
• PIT algorithms for stronger models:PIT algorithms for stronger models:– Depth 3 circuitsDepth 3 circuits– Bounded depth Bounded depth
• Tightness of our results:Tightness of our results:– ConjectureConjecture: If C : If C 0 is 0 is (k) simple, (k) simple,
minimal then rank(C) = poly(k).minimal then rank(C) = poly(k).– [[KS06KS06] ] Not true for finite fields! Not true for finite fields! Example in of a circuit with top Example in of a circuit with top
fanin=3 and rank ~ log(d)fanin=3 and rank ~ log(d)
((MultilinearMultilinear))(Multilinear)(Multilinear)
top related