Property Testing and Affine Invariance Part II Madhu Sudan Harvard University December 29-30, 2015 IITB: Property Testing & Affine Invariance 1 of 29
Review of last lecture Property testing: Test global property with local inspection. E.g., test if mm-variate function over FF qq is deg. dd poly. #coefficients = exp min(mm, dd) query complexity (when dd qq ): pppppppp(dd) 2 Affine-invariant Property: PP ff: FF QQ mm FF qq ; QQ = qq nn PP is FF qq -linear PP is invariant under affine maps FF QQ mm FF QQ mm Generalize low-degree property; Testing results extend, strengthen; captures new properties. December 29-30, 2015 IITB: Property Testing & Affine Invariance 2 of 29
Today Positive aspect of New properties Lifted codes Testability of Lifted Codes Result statements Some proof ideas December 29-30, 2015 IITB: Property Testing & Affine Invariance 3 of 29
(Recall) Lifted Codes Base Code BB bb: FF QQ tt FF qq affine-invariant mm-dim Lift LL mm BB ff: FF QQ mm FF qq Last lecture: Used lifts to construct non-testable codes. Today: Use them to construct new, better, testable codes. December 29-30, 2015 IITB: Property Testing & Affine Invariance 4 of 29
Coding Theoretic Objective Want codes of (Properties with) High rate (many members) High Distance (pairwise far) Low Locality for Testing/Correcting (2-transitive and testable) Best known code (pre-2010) w. sublinear locality Bivariate polynomials, w. deg. kk < qq. PP ff: FF qq 2 FF qq ; length =dim. of ambient space = qq 2 Rate = dim P = kk2 < 1 ; locality l = qq = length length 2qq 2 2 December 29-30, 2015 IITB: Property Testing & Affine Invariance 5 of 29
Locality w. Rate > 11 22? 2010 [Kopparty,Saraf,Yekhanin] Multiplicity Codes : Locality = length εε ; Rate 1 Not known to be testable! 2011 [Viderman] Tensor Product Codes : Locality = length εε ; Rate 1 Testable, but not symmetric 2014 [Guo,Kopparty,S] Lifted Codes: Locality = length εε ; Rate 1 Testable + Symmetric! 2015 [Kopparty,Meir,RonZewi,Saraf]: Locality = length oo(1) ; Rate 1 December 29-30, 2015 IITB: Property Testing & Affine Invariance 6 of 29
Lifted Reed-Solomon Codes Base Code BB bb: FF qq FF qq deg bb kk = 1 δδ qq Lifted Code LL mm BB ff: FF mm qq FF qq ; Rate =? Distance =? Locality =? qq (obvious) mm-var. deg. kk poly LL mm BB Rate 1 mmmm Simple analysis: Dist δδ 1 qq Rate: εε > 0, mm, δδ > 0 s.t. Rate 1 εε mm! ; Dist δδ; December 29-30, 2015 IITB: Property Testing & Affine Invariance 7 of 29
Rate of bivariate Lifted RS codes BB = ff FF qq xx deg ff kk = 1 δδ qq} ; qq = 2 ss Will set δδ = 2 cc and let cc. Note: mm kk one of its cc MSBs is 0. LL 2 (BB) = ff: FF qq xx, yy ff yy=aaaa+bb BB, aa, bb} When is xx ii yy jj CC? (Will need to look at binary rep n of ii, jj.) December 29-30, 2015 IITB: Property Testing & Affine Invariance 8 of 29
Lucas s theorem & Rate Recall: rr 2 jj, if rr = rr ii 2 ii ii and jj = jj ii 2 ii ii ( rr ii, jj ii 0,1 ) and rr ii jj ii for all ii. Lucas s Theorem: xx rr supp aaaa + bb jj iff rr 2 jj. supp xx ii aaaa + bb jj xx ii+rr iff rr 2 jj So given ii, jj; rr 2 jj s. t. ii + rr mod qq > kk? ii 0 0 cc Pr ii,jj 15 16 cc 2 jj 0 0 December 29-30, 2015 IITB: Property Testing & Affine Invariance 9 of 29
Aside: Nikodym Sets NN FF qq mm is a Nikodym set if it almost contains a line through every point: aa FF qq mm, bb FF qq mm s.t. aa + tttt tt FF qq } NN {aa} Similar to Kakeya Set (which contain line in every direction). bb FF qq mm, aa FF qq mm s.t. aa + tttt tt FF qq } KK [Dvir], [DKSS]: KK, NN qq 2 mm December 29-30, 2015 IITB: Property Testing & Affine Invariance 10 of 29
Proof ( Polynomial Method ) Find low-degree poly FF 0 s.t. FF bb = 0, bb NN. deg FF < qq 1 provided NN < mm+qq 2 mm. But now FF LLaa = 0, Nikodym lines LL aa FF aa = 0 aa, Conclude NN mm+qq 2 mm contradicting FF 0. qqmm mm!. Multiplicities, more work, yields NN qq 2 But what do we really need from FF? FF comes from a large dimensional vector space. FF LL is low-degree! Using FF from lifted code yields NN 1 oo 1 qq mm December 29-30, 2015 IITB: Property Testing & Affine Invariance 11 of 29 mm. (provided qq of small characteristic).
Testing December 29-30, 2015 IITB: Property Testing & Affine Invariance 12 of 29
Parameters of interest Property being tested PP ff: FF QQ mm FF qq Locality of test l Rejection ratio: εε min ff Pr [RRRRRRRRRRRRRRRRRR ff] δδ ff,pp δδ(ff Robustness: αα min llllllllll,pp llllllllll ) ff δδ(ff,pp) Ideally should compare l εε (or l αα ) December 29-30, 2015 IITB: Property Testing & Affine Invariance 13 of 29
Important Definition l-single orbit Property: PP gg: FF mm QQ FF qq is l-single orbit if there exists an l-local constraint CC s.t. ff PP affine AA: FF mm QQ FF mm QQ, ff AA satisfies CC (Single constraint + its orbit characterize PP) Lifted Property is QQ tt -single orbit. Most known algebraic properties had single orbit tests. Exceptions, pre-2008: Sparse Properties Post-2008: Sparse properties are single-orbit. December 29-30, 2015 IITB: Property Testing & Affine Invariance 14 of 29
Testing Theorems Thm 1 [KS08]: l-single orbit property is l-locally testable with εε 1 l 2 Thm 2 [HRS 13]: QQ εε s.t. tt, BB bb: FF QQ tt FF qq, mm LL mm BB is QQ tt -locally testable with soundness εε (Works for lifted codes. Soundness independent of tt. But depends on QQ) Thm 3 [GHS 15]: δδ αα s.t. if BB bb: FF QQ tt FF qq is a code of distance δδ then LL mm BB is QQ 2tt -locally testable with robustness αα (Generalizes low-degree testing. Stronger. Works even when dd qq) December 29-30, 2015 IITB: Property Testing & Affine Invariance 15 of 29
l-single-orbit l-locally testable PP FF mm l QQ FF qq given by αα 1,, αα l ; VV FF qq PP = ff AA, ff AA αα 1,, ff AA αα l VV Auto-correction based-proof: Fix ff s.t ρρ Pr Rejecting ff small Define gg from ff locally Prove gg close to ff Prove gg satisfies constraint AA Only possible gg xx = argmax ββ Pr ββ, ff(aa αα 2),, ff(aa αα l ) VV AA:AA αα 1 =xx December 29-30, 2015 IITB: Property Testing & Affine Invariance 16 of 29
Analysis (contd.) Vote AA xx = ββ ss. tt. ββ, ff AA αα 2,, ff(aa αα l ) VV gg xx = majority AA:AA αα1 =xx Vote AA xx Key Lemma: xx, Pr Vote AA xx = Vote BB xx AA,BB:AA αα 1 =BB αα 1 =xx 1 2lρρ [BLR,GLRSW,RS,AKLLR,KR,JPRZ] Proofs: Build a miracle l l matrix M: Rows indexed by AA 1 = AA, AA 2,, AA l Columns by BB 1 = BB, BB 2,, BB l MM iiii = AA ii αα jj = BB jj αα ii ii, jj Typical row/column random Why does such a matrix exist? December 29-30, 2015 IITB: Property Testing & Affine Invariance 17 of 29
Matrix Magic explained Wlog CC αα 1,, CC(αα tt ) independent; rest determined when CC random (affine). Random xx AA αα 2 BB(αα 2 ) BB(αα tt ) AA(αα tt ) AA αα l tt Determined Overdetermined? BB(αα l ) tt No! Linear algebra! December 29-30, 2015 IITB: Property Testing & Affine Invariance 18 of 29
Theorem 2: Context & Ideas Thm 2 [HRS 13]: QQ εε s.t. tt, BB bb: FF QQ tt FF qq, mm LL mm BB is QQ tt -locally testable with soundness εε Test: Obvious one: Pick random tt-dim subspace AA. Accept iff ff AA BB Claim: Pr AA ff AA BB εε δδ(ff, LL mm BB ) [BKSSZ] Special case: QQ = 2, BB = bb bb aa = 0 aa ; LL mm BB = mm-var deg tt 1 poly December 29-30, 2015 IITB: Property Testing & Affine Invariance 19 of 29
Theorem 2 (QQ = qq = 22, contd.) Alternative view of test: ff aa xx ff xx + aa ff(aa) discrete derivative deg ff < tt deg ff aa < tt 1 deg ff aa1,,aaaa < 0 ff aa1,,aaaa = 0 Rejection Prob. ρρ ff = Pr ff aa1 aaaa 0 aa1 aaaa 1 (1 2ρρ ff ) 2 dd special case of Gowers norm Strong Inverse Conjecture ρρ ff 1 2 as δδ ff, LL mm BB 1 2. Falsified by [LovettMeshulamSamorodnitsky],[GreenTao]: ff = SSSSmm tt xx 1 xx nn ; tt = 2 ss ; δδ ff, LL mm (BB) = 1 2 oo nn(1); ρρ ff 1 2 2 7 December 29-30, 2015 IITB: Property Testing & Affine Invariance 20 of 29
Theorem 2 (contd.) So ρρ ff 1 as δδ ff 1 ; but is ρρ ff > 0? 2 2 Prior to [BKSSZ]: ρρ ff > 4 tt δδ ff [BKSSZ] Lemma: ρρ ff min{εε, 2 tt δδ ff } Key ingredient in proof: Suppose δδ ff > 2 tt On how many hyperplanes HH can deg ff HH < tt? December 29-30, 2015 IITB: Property Testing & Affine Invariance 21 of 29
Hyperplanes δδ ff > 2 tt #{HH s.t. deg ff HH < tt}? 1. HH s.t. deg ff HH tt: defn of lifting. 2. Pr HH deg ff HH tt 1 qq deg xxii ff < qq 1. 3. What we needed: #{HH s.t. deg ff HH < tt} O(2 t ) December 29-30, 2015 IITB: Property Testing & Affine Invariance 22 of 29
General Lifted Properties Lemma: QQ cc s.t. if δδ ff QQ tt then # HH s. t. ff HH LL mm 1 (BB) cc QQ tt Ingredients in proof: qq = 2: Simple symmetry of subspaces, linear algebra. qq = 3: Roth s theorem General qq: Density Hales-Jewett theorem December 29-30, 2015 IITB: Property Testing & Affine Invariance 23 of 29
Theorem 3 : Context Thm 3 [GHS 15]: δδ αα s.t. if BB bb: FF QQ tt FF qq is a code of distance δδ then LL mm BB is QQ 2tt -locally testable with robustness αα Test not most natural one! Most natural: Inspect ff AA for tt-dim AA Our test: Inspect ff AA for 2tt-dim AA Based on [Raz-Safra], [BenSassonS],, [Viderman] Need to show: ff EE AA δδ ff AA, BB δδ(ff, LL mm BB ) Not previously known even when tt = 1 and BB = bb deg bb dd with dd = 1 εε qq December 29-30, 2015 IITB: Property Testing & Affine Invariance 24 of 29
Robust Testing of Lifted Codes For simplicity BB bb: FF qq FF qq (tt = 1). General geometry + symmetry Robust analysis with mm = 4 All mm How to analyze robustness of the test for constant mm? December 29-30, 2015 IITB: Property Testing & Affine Invariance 25 of 29
Tensors: Key to understanding Lifts Given FF ff: SS FF qq and GG gg: TT FF qq, FF GG = h: SS TT FF qq xx, yy, h, yy FF & h xx, GG FF mm = FF FF FF LL mm BB BB mm ; LL mm BB = TT TT BB mm (affine map TT) (mm 1)-dim test for BB mm : Fix coordinate at random and test if ff, xx ii, BB mm 1 [Viderman 13]: Test is αα δδ BB,mm -robust. Hope: Use LL mm BB = TT TT BB mm to show that testing for random TT(BB mm ) suffices; δδ AA ff, δδ BB ff small δδ AA BB ff small December 29-30, 2015 IITB: Property Testing & Affine Invariance 26 of 29
Actual Analysis Say testing LL 4 (BB) by querying 2-d subspace. Let PP aa = {ff ff line BB for all coordinate parallel LL 4 (BB) = aa PP aa ; lines, and lines in direction aa} PP aa not a tensor code, but modification of tensor analysis works! aa PP aa BB 4 is still an error-correcting code. So δδ PPaa ff, δδ PPbb ff small δδ PPaa PP bb ff small! Putting things together Theorem 3. December 29-30, 2015 IITB: Property Testing & Affine Invariance 27 of 29
Wrapping up Affine-Invariance Fruitful abstraction of low-degree property Many open questions Characterize OO(1)-locally testable properties. Rich properties beyond lifting? Beat polynomials for l = polylog(length) Invariance in Property Testing Pursue in other contexts? Other unifying generalizations? December 29-30, 2015 IITB: Property Testing & Affine Invariance 28 of 29
Thank You December 29-30, 2015 IITB: Property Testing & Affine Invariance 29 of 29