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# Testing Affine-Invariant Properties

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Testing Affine-Invariant Properties. Madhu Sudan Microsoft. Surveys: works with/of Eli Ben- Sasson , Elena Grigorescu , Tali Kaufman, Shachar Lovett, Ghid Maatouk , Amir Shpilka . TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A. - PowerPoint PPT Presentation
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Simple PCPs

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties1Testing Affine-Invariant PropertiesMadhu SudanMicrosoftTexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAASurveys: works with/of Eli Ben-Sasson, Elena Grigorescu, Tali Kaufman, Shachar Lovett, Ghid Maatouk, Amir Shpilka.of 401May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties2Property Testing of functions from D to R:Property P {D R}Distance(f,g) = Prx 2 D [f(x) g(x)](f,P) = ming 2 P [(f,g)]f is -close to g (f g) iff (f,g) .Local testability:P is (k, , )-locally testable if 9 k-query test Tf 2 P ) Tf accepts w.p. 1-.(f,P) > ) Tf accepts w.p. . Notes: want k(, ) = O(1) for ,= (1). of 40Classical Property Test: Linearity [BLR]Does f(x+y) = f(x) + f(y), for all x, y?Variation (Affineness): Is f(x+y) + f(0) = f(x) + f(y) , for all x, y?(roughly f(x) = a0 + i=1n ai xi )Test: Pick random x,y and verify above.Obvious: f affine ) passes test w.p. 1.BLR Theorem: If f is -far from every affine function, then it fails test w.p. ().

Ultimate goal of talk: To understand such testing results.May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties3of 40Affine-Invariant PropertiesDomain = K = GF(qn) (field with qn elements)Range = GF(q); q = power of prime p.P forms F-vector space.P invariant under affine transformations of domain. Affine transforms? x a.x + b, a K*, b K.Invariance? f P ) ga,b(x) = f(ax+b) P.affine permutation of domain leaves P unchanged.Quest: What makes affine-invariant property testable?May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties4of 40(My) GoalsWhy?BLR test has been very useful (in PCPs, LTCs).Other derivatives equally so (low-degree test).Proof magical! Why did 3 (4) queries suffice?Can we find other useful properties?

Program:Understand the proof better (using invariance).Get structural understanding of affine-invariant properties, visavis local testability.Get better codes/proofs?

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties5of 40Whys?Why InvarianceNatural way to abstract/unify common themes (in property testing).Graph properties, Boolean, Statistical etc.?Why affine-invariance:Abstracts linearity (affine-ness) testing.Low-degree testingBCH testing Why F-vector space?Easier to study (gives nice structure).Common feature (in above + in codes).May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties6of 40May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties7Contrast w. Combinatorial P.T. Algebraic Property = Code! (usually)Universe:{f:D R}PDont careMust rejectMust acceptPR is a field F; P is linear!of 40Basic Implications of Linearity [BHR]If P is linear, then:Tester can be made non-adaptive.Tester makes one-sided error (f 2 P ) tester always accepts).Motivates:Constraints: k-query test => constraint of size k:value of f at 1, k constrained to lie in subspace.Characterizations:If non-members of P rejected with positive probability, then P characterized by local constraints. functions satisfying all constraints are members of P.May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties8of 40f = assgmt to left

Right = constraints

Characterization of P: P = {f sat. all constraints}PictoriallyMay 23-28, 2011Bertinoro: Testing Affine-Invariant Properties9123D{0000,1100, 0011,1111}of 40Back to affine-invariance: More NotesWhy K F?Very few permutations (|K|2) !!Still 2-transitiveIncludes all properties from Fn to F that are affine-invariant over Fn.(Hope: Maybe find a new range of parameters?)Contrast with linear-invariance [Bhattacharyya et al.]Linear vs. Affine.Arbitrary P vs. F-vector space PLinear over Fn vs. Affine over K = GF(qn).

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties10of 40k-local constraintk-characterizedAffine-invariance & testabilityMay 23-28, 2011Bertinoro: Testing Affine-Invariant Properties11k-locally testablek-S-O-C [KS08]of 40Goal of this talkDefinition: Single-orbit-characterization (S-O-C)Known testable affine-invariant properties (all S-O-C!).Structure of Affine-invariant properties.Non testability resultsOpen questionsMay 23-28, 2011Bertinoro: Testing Affine-Invariant Properties12of 40Single-orbit-characterization (S-O-C)Many common properties are given by (Affine-)invarianceSingle constraint.Example: Affineness over GF(2)n:Affineness is affine-invariant.f(000000) - f(100000) f(010000) f(110000)S-O-C: Abstracts this notion. Suffices for testability [Kaufman+S08]Unifies all known testability results!!Nice structural properties.

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties13of 40S-O-C: Formal DefinitionConstraint: C = (1,,k;V Fk ); i KC satisfied by f if (f(1),,f(k)) V.

Orbit of constraint = {C o }, affine.C o = ((1),,(k); V).

P has k-S-O-C, if orbit(C) characterizes P.May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties14of 40Known testable properties - 0Theorem [Kaufman-S.08]:If P has a k-S-O-C, then P is k-locally testable.May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties15of 40k-local constraintk-characterizedAffine-invariance & testabilityMay 23-28, 2011Bertinoro: Testing Affine-Invariant Properties16k-locally testablek-S-O-C [KS08]of 40Known testable properties - 0Theorem [Kaufman-S.08]:If P has a k-S-O-C, then P is k-locally testable.

But who has k-S-O-C?Affine functions: over affine transforms of FnDegree d polynomials:again, over affine transforms of Fn

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties17of 40Known testable properties - 1Reed-Muller Property:View domain as Fn (n-variate functions)Parameter d.RM(d) = n-var. polynomials of degree d.Known to be qO(d/q)-locally testable: Test: Test if f restricted to O(d/q)-dimensional subspace is of degree d.Analysis: [Kaufman-Ron] (see appendix 1).Single-Orbit?Yes naturally over affine transforms of Fn.Yes unnaturally over K (field of size Fn).May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties18of 40Known testable properties - 2Sparse properties:Parameter t|P| |K|t

Testability:Conditioned on high-distance [Kaufman-Litsyn, Kaufman-S.]. (no need for aff. inv.)Unconditionally [Grigorescu, Kaufman, S. ], [Kaufman-Lovett] (for prime q).Also S-O-C.

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties19of 40Known Testable Properties - 3Intersections:P1 P2 always locally testable, also S-O-C.Sums: P1 + P2 (= {f1 + f2 | fi Pi})S-O-C iff P1 and P2 are S-O-C [BGMSS11]Lifts [BMSS11]Suppose F L K.P {L F} has k-S-O-C, with constraint C.Then Lift_{L K}(P) = property characterized by K-orbit(C).By Definition: Lift(P) is k-S-O-C.May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties20of 40Known Testable Properties - 1Finite combination of Lifts, Intersections, Sums of Sparse and Reed-Muller properties.

Known: They are testable (for prime q).Open: Are they the only testable properties?If so, Testability Single-Orbit.First target: n = prime:no lifts/intersections; only need to show that every testable property is sum of sparse and Reed-Muller property.May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties21of 40May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties22Affine-Invariant Properties: Structureof 40PreliminariesEvery function from K K, including K F, is a polynomial in K[x]So every property P = {set of polynomials}.Is set arbitrary? Any structure?Alternate representation:Tr(x) = x + xq + xq2 + + xqn-1Tr(x+y) = Tr(x)+Tr(y); Tr(x) = Tr(x), F.Tr: K F.Every function from K F is Tr(f) for some polynomial f K[x]. Any structure to these polynomials?May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties23of 40ExampleF = GF(2), K = GF(2n).Suppose P contains Tr(x11 + x3 + 1).What other functions must P contain (to be affine-invariant)?Claims: Let D = {0,1,3,5,9,11}.Then P contains every function of the form Tr(f), where f is supported on monomials with degrees from D.So Tr(x5),Tr(x9+x5),Tr(x11+x5+x3+x)+1 P.How? Why?May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties24of 40Structure - 1Definitions:Deg(P) = {d | 9 f P, with xd supp(f)}Fam(D) = {f: K F | supp(f) D}

Proposition: For affine-invariant property P P = Fam(Deg(P)). May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties25of 40Structure - 2Definitions: Shift(d) = {d, q.d, q2.d, } mod (qn-1).D is shift-closed if Shift(D) = D.e d : e = e0 + e1 p + ; d = d0 + d1 p + ; e d if ei di for all i.

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties26of 40Structure - 3Proposition: For every affine-invariant property P, Deg(P) is p-shadow-closed and q-shift-closed.(Shadowing comes from affine-transforms; Shifts come from range being F).

Proposition: For every p-shadow-closed, q-shift-closed family D, Fam(D) is affine-invariant and D = Deg(Fam(D)) May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties27of 40Example revisitedTr(x11 + x3) PDeg(P) 11, 3 (definition of Deg)Deg(P) 11, 9, 5, 3, 1, 0 (shadow-closure)Deg(P) Tr(x11), Tr(x9) etc. (shift-closure).Fam(Deg(P)) Tr(x11) etc. (definition of Fam).P Tr(x11) (P = Fam(Deg(P)))

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties28of 40What kind of properties have k-S-O-C?(Positive results interpreted structurally)

If propery has all degrees of q-weight at most k then it is RM and has (qk)-S-O-C:q-weight(d) = i di, where d = d0 + d1 q + Also, if P = Fam(D) & D = Shift(S) for small shadow-closed S, then P is k(|S|)-S-O-C.(Alternate definition of sparsity.)

Other examples from Intersection, Sum, Lift.

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties29of 40What affine-invariant properties are not locally testable.Very little known.

Specific examples:GKS08: Exists a-i property with k-local constraint which is not k-locally characterized.

BMSS11: Exists k-locally characterized a-i property that is not testable.

BSS10: If wt(d) k for some d in Deg(P), then P does not have a k-local constraint.

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties30of 40[BS10]k-local constraintk-characterizedAffine-invariance & testabilityMay 23-28, 2011Bertinoro: Testing Affine-Invariant Properties31k-locally testablek-S-O-C [KS08][GKS08][BMSS11]weight-k degreesof 40Quest in lower boundGiven degree set D (shadow-closed, shift-closed) prove it has no S-O-C.

Equivalently: Prove there are no 1 k F, 1 k K such that i=1k i id = 0 for every d D. i=1k i id 0 for every minimal d D.

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties32of 40PictoriallyMay 23-28, 2011Bertinoro: Testing Affine-Invariant Properties331d2dkdM(D) = Is there a vector (1,,k) in itsright kernel?Can try to prove NO by provingmatrix has full rank.Unfortunately, few techniques to prove non-square matrix has high rank.of 40Non-testable Property - 1AKKLR (Alon,Kaufman,Krivelevich,Litsyn,Ron) Conjecture:If a linear property is 2-transitive and has a k-local constraint then it is testable.[GKS08]: For every k, there exists affine-invariant property with 8-local constraint that is not k-locally testable.P = Fam(Shift({0,1} [ {1+2,1+22,,1+2k})).May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties34of 40Proof (based on [BMSS11])F = GF(2); K = GF(2n);Pk = Fam(Shift({0,1} [ {1 + 2i | i {1,,k}}))

Let Mi =

If Ker(Mi) = Ker(Mi+1), then Ker(Mi+2) = Ker(Mi)Ker(Mk+1) = would accept all functions in Pk+1So Ker(Mi) must go down at each step, implying Rank(M_{i+1}) > Rank(M_i).May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties3512i1222k2

22ik2iof 40Stronger CounterexampleGKS counterexample: Takes AKKLR question too literally; Of course, a non-locally-characterizable property can not be locally tested.

Weaker conjecture:Every k-locally characterized affine-invariant (2-transitive) property is locally testable.Alas, not true: [BMSS]May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties36of 40[BMSS] CounterExampleRecall: Every known locally characterized property was locally testableEvery known locally testable property is S-O-C.Need a locally characterized property which is (provably) not S-O-C.Idea:Start with sparse family Pi.Lift it to get Qi (still S-O-C).Take intersection of superconstantly many such properties. Q = i QiMay 23-28, 2011Bertinoro: Testing Affine-Invariant Properties37of 40Example: Sums of S-O-C propertiesSuppose D1 = Deg(P1) and D2 = Deg(P2)Then Deg(P1 + P2) = D1 [ D2.Suppose S-O-C of P1 is C1: f(a1) + + f(ak) = 0; and S-O-C of P2 is C2: f(b1) + + f(bk) = 0.Then every g P1 + P2 satisfies: i,j g(ai bj) = 0Doesnt yield S-O-C, but applied to random constraints in orbit(C1), orbit(C2) does!Proof uses wt(Deg(P1)) k.May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties38of 40ConcludingAffine-invariance gives nice umbrella to capture algebraic property testing:Important (historically) for PCPs, LTCs, LDCs.Incorporates symmetry.Would be nice to have a complete characterization of testability of affine-invariant properties.Understanding (severely) lacking.Know: Cant be much better than Reed-Muller.Can they be slightly better? YES!May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties39of 40Thank You!May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties40of 40Appendix 0: Main ReferencesEarly results mentioning invariance:[Babai, Shpilka, Stefankovic] Cyclic codes[AKKLR] 2-transitivity[Goldreich-Sheffett] Lower bounds on randomness required.Affine-invariance[Kaufman-Sudan, STOC 08][Grigorescu, Kaufman, Sudan, CCC 08][Grigorescu, Kaufman, Sudan, Random 09] [Kaufman, Lovett, ECCC TR10-065][Ben-Sasson, Sudan, ECCC TR 10-108][Ben-Sasson, Maatouk, Shpilka, Sudan, CCC11][Ben-Sasson, Grigorescu, Maatouk, Shpilka, Sudan, ECCC TR 11-079]Other related themes:Fair amount of work on (non-linear, linear-invariance) refs omitted.Kaufman+Wigderson, ??? other algebraic invariances (1-transitive)Goldreich + Kaufman general relationships between invariance and testingMay 23-28, 2011Bertinor: Testing Affine-Invariant Properties41of 40Appendix 1Reed-Muller testing:Early works (PCP etc.): consider only d < q.[Rubinfeld, Sudan 92][Arora, Safra 92][ALMSS 92]Variations (multilinear, low indiv. degree) due to [BFL,BFLS,FGLSS].d > n: qq = 2: [Alon,Kaufman,Krivelevich,Litsyn,Ron 02]general q: [Kaufman Ron 04](prime q): [Jutla Patthak Rudra Zuckerman 04.]Tight results:q=2: O(2^d)-locally testable: [Bhattacharyya,Kopparty,Schoenebeck,Sudan,Zuckerman 09]general q: q^{d/(q-q/p} [Haramaty Shpilka Sudan 11]tight for prime q; open for general q.

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties42of 40Appendix 2: Analysis of k-S-O-C testProperty P (k-S-O-C) given by 1,,k; V 2 Fk

P = {f | f(A(1)) f(A(k)) 2 V, 8 affine A:KnKn}

Rej(f) = ProbA [ f(A(1)) f(A(k)) not in V ]

Wish to show: If Rej(f) < 1/k3, then (f,P) = O(Rej(f)).

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties43of 40Appendix 2: BLR AnalogRej(f) = Prx,y [ f(x) + f(y) f(x+y)] <

Define g(x) = majorityy {Votex(y)}, where Votex(y) = f(x+y) f(y).

Step 0: Show (f,g) small

Step 1: 8 x, Pry,z [Votex(y) Votex(z)] small.

Step 2: Use above to show g is well-defined and a homomorphism.

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties44of 40Appendix 2: BLR Analysis of Step 1Why is f(x+y) f(y) = f(x+z) f(z), usually?May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties45- f(x+z)f(y)- f(x+y)f(z)-f(y)f(x+y+z)-f(z)0?of 40Appendix 2: Generalizationg(x) = that maximizes, over A s.t. A(1) = x, PrA [,f(A(2),,f(A(k)) 2 V]

Step 0: (f,g) small.

Votex(A) = s.t. , f(A(2))f(A(k)) 2 V (if such exists)

Step 1 (key): 8 x, whp Votex(A) = Votex(B).Step 2: Use above to show g 2 P.

May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties46of 40Appendix 2: Matrix Magic?May 23-28, 2011Bertinoro: Testing Affine-Invariant Properties47A(2)B(k)B(2)A(k)xtSay A(1) A(t) independent; rest dependenttRandomNo ChoiceDoesnt Matter!of 40

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